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# Optimization Models for Integrated Production, Capacity and Revenue Management

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OPTIMIZATION MODELS FOR INTEGRA TED PRODUCTION, CAPACITY AND REVENUE MANAGEMENT By YASEMIN MERZIFONLUOGLU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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To my family.

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ACKNOWLEDGMENTS I would like to thank a ll those people who have helped to make this thesis possible. Firstly, I would like to express my sincer e gratitude to Dr. Joseph Geunes for being the perfect supervisor for me. He has become more of a mentor and a friend to me than a professor. He has had always time to disc uss my research ideas, to listen my problems, to answer my e-mails, and to carefully edit my writing. Thanks to his relieving attitude and his understanding, I have lived through ma ny difficulties in the last four years. I also consider myself very lucky to wo rk with Dr. Edwin Romeijn. He has an amazing eye for detail and his interesting resear ch ideas helped to shape this thesis. I also want to thank Dr. Elif Akal and Dr. Seluk Ereng for their helpful comments and for participating in my dissertation committee. I would like to express my warmest gratitude to my parents, Nurhan and Nfer Merzifonluo lu for being courageous and patient enough when I was making life changing decisions and for teaching me that I would never be alone regardless of the place that I live. I am the person that I am t oday thanks to their endless faith and trust in me and my abilities. I w ould like to thank my brother, Abdurrahman Merzifonluo lu, for his unmatched friendship and guidance in life as a big brother. I could not have imagined reaching this goal without their unconditional love and support. Lastly, I would like to express my appreciation to Eray Uzgren fo r giving me the extra motivation and support to complete this thesis. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.............................................................................................................x ABSTRACT.......................................................................................................................xi CHAPTER 1 INTRODUCTION...................................................................................................1 1.1 Integrated Capacity, Demand and Production Planning Models with Subcontracting and Overtime Options.........................................................3 1.2 Capacitated Production Planning Mode ls with Price Sensitive Demand and General Concave Revenue Functions...................................................5 1.3 Uncapacitated Production Planning Models with Demand Fulfillment Flexibility.....................................................................................................7 1.4 Demand Assignment Models under Uncertainity........................................8 1.5 Research Scope and Thesis Outline.............................................................9 2 LITERATURE REVIEW......................................................................................11 2.1 Requirements Planning..............................................................................11 2.2 Pricing/Demand Management with Production Planning..........................12 2.3 Capacity Planning......................................................................................16 2.4 Subcontracting...........................................................................................17 2.5 Overtime Planning.....................................................................................18 3 INTEGRATED CAPACITY, DEMAND AND PRODUCTION PLANNING MODELS WITH SUBCONTRACTI NG AND OVERTIME OPTIONS.............20 Introduction............................................................................................................20 3.1 Model and Solution Approach with Fixed Procurement Capacity............25 3.1.1 Problem Definition and Mode l Formulation for Single Uncapacitated Subcontractor.........................................................25 3.1.2 Determining Candidate De mand Vectors for an RI.......................33 3.1.3 Optimal Cost Calculation for an RI...............................................36 v

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3.1.3.1 Regular capacity as integer multiple of overtime capacity..............................................................................37 3.1.3.2 Regular capacity as any positive multiple of overtime capacity..............................................................................39 3.1.4 Complexity of Solution Approach.................................................42 3.2 Capacity Planning......................................................................................43 3.3 Multiple Subcontractors and Subcontractor Capacities.............................47 3.3.1 Uncapacitated Subcontractors........................................................47 3.3.2 Capacitated Subcontractors............................................................49 3.3.2.1 Determining candidate demand levels for a regeneration interval...............................................................................50 3.3.2.2 Regular capacity as integer multiple of overtime capacity..............................................................................51 3.3.2.3 Regular capacity as any positive multiple of overtime capacity..............................................................................53 3.3.2.4 Complexity of solution approach.......................................56 3.3.3 Capacity Planning with Multiple Subcontractors..........................57 3.4 Conclusions................................................................................................59 4 CAPACITATED PRODUCTION PL ANNING MODELS WITH PRICE SENSITIVE DEMAND AND GENERAL CONCAVE REVENUE FUNCTIONS.........................................................................................................61 Introduction............................................................................................................61 4.1 Model and Solution Approach with Dynamic Prices................................63 4.1.1 Problem Definition and Model Formulation..................................63 4.1.2 Development of Solution Approach for DCRPP...........................66 4.1.2.1 Properties of optimal RI demand vectors...........................67 4.1.2.2 Characterizing optimal demand in RIs containing one fractional procurement period............................................70 4.1.2.3 Characterizing optimal demands in RIs containing no fractional procurement period............................................72 4.1.2.4 Complexity of overall solution approach...........................76 4.1.2.5 Refining the solution approach..........................................78 4.2 Model and Solution Approach with a Constant Price................................81 4.2.1 Model Description.........................................................................81 4.2.2 Linearity of Cost in Demand Effect...............................................84 4.2.3 Characterizing the Structure of the Optimal Cost Function...........86 4.2.4 Number of Breakpoints of ..........................................................88 4.2.5 Solution Approach for SCRPP.......................................................91 4.3 Conclusion.................................................................................................93 5 UNCAPACITATED PRODUCTIO N PLANNING MODELS WITH DEMAND FULFILLMENT FLEXIBILITY........................................................95 Introduction............................................................................................................95 5.1 Problem Definition and Model Formulation..............................................99 vi

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5.2 Polynomial Time Solution under Special Cost/Revenue Structures........103 5.3 Dual Based Heuristic Algorithm for General Revenue Parameter Values......................................................................................................107 5.3.1 Economic Interpretation of the Dual and Complementary Slackness Relationships...............................................................117 5.3.2 Creating a Feasible Primal Solution............................................118 5.4 Computational Testing and Results.........................................................120 5.4.1 Analysis of results........................................................................123 5.5 Conclusions..............................................................................................126 6 DEMAND ASSIGNMENT MODELS UNDER UNCERTAINTY....................128 Introduction..........................................................................................................128 6.1 Problem Definition and Model Formulation............................................133 6.2 Branch and Price Scheme........................................................................136 6.2.1 Column Generation Algorithm....................................................137 6.2.1.1 Column generation for [SSSPL].......................................137 6.2.1.2 Initial columns.................................................................138 6.2.2 Pricing Problem...........................................................................138 6.2.3 Branching Scheme.......................................................................139 6.2.4 Rounding Heuristic......................................................................141 6.3 Static Stochastic Knapsack Problem........................................................141 6.3.1 Linear Relaxation of Restricted Static Stochastic Knapsack Problem........................................................................................144 6.3.1.1 KKT conditions................................................................145 6.3.1.2 Analysis of KKT conditions............................................146 6.3.1.3 KKT based algorithm.......................................................149 6.3.2 Branch and Bound Scheme..........................................................155 6.4 Preference Order Greedy Heuristic..........................................................156 6.5 Numerical Study......................................................................................157 6.6 Conclusion...............................................................................................162 7 CONCLUSION....................................................................................................164 APPENDIX A NP HARD PROOF FOR THE CA PACITATED PRODUCTION PLANNING PROBLEM WITH PRICING AND CA PACITATED SUBCONTRACTORS..170 B NP HARD PROOF FOR THE UN CAPACITATED PR ODUCTION AND LOCATION PLANNING MODEL WITH DEMAND FULLFILLMENT FLEXIBILITY.....................................................................................................172 C NP HARD PROOF FOR THE STATIC STOCHASTIC ASSIGNMENT PROBLEM...........................................................................................................176 vii

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D CHARACTERISTICS OF THE OBJECTIVE FUNCTION OF THE DEMAND ASSIGNMENT PROBLE M WITH A SINGLE DECISION VARIABLE.........................................................................................................179 LIST OF REFERENCES.................................................................................................182 BIOGRAPHICAL SKETCH...........................................................................................189 viii

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LIST OF TABLES Table page 3-1 Complexity results with M subcontractors (integral )...........................................59 3-2 Complexity results with M subcontractors (general )............................................59 5-1 Demand time windows for the exam ple problem shown in Figure 5 .1.................101 5-2 Demand ( dj) and setup cost values (St) for example problem 1 .............................112 5-3 Problem 1-Iteration 0 .............................................................................................113 5-4 Problem 1-Iteration 1 .............................................................................................114 5-5 Problem 1-Iteration 2 .............................................................................................114 5-6 Demand and setup cost values for example problem 2..........................................115 5-7 Problem 2-Iteration 0 .............................................................................................115 5-8 Problem 2-Iteration 1 .............................................................................................116 5-9 Order rejection rates under different cost parameter value settings .......................124 6-1 Problem sizes in the numerical study.....................................................................158 6-2 Computation times for various problem settings...................................................159 6-3 Performance evaluation table for heuristic algorithm............................................161 ix

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LIST OF FIGURES Figure page 3-1 Piecewise linear concave revenue function ..............................................................30 3-2 Network representation of the [CPPPPL]..................................................................32 4-1 Candidate subgradient values and corresponding candida te demand values...........69 4-2 Illustration of Proposition 4 .4...................................................................................80 4-3 An arbitrarily selected n-period regeneration interval.............................................89 5-1 Fixed-charge network flow repr esentation of the [DFFP] problem .......................101 5-2 Structure of longest path graph ..............................................................................105 5-3 Impact of dimensions of flexibility on profit .........................................................123 5-4 Profit levels as a percentage of the maximum profitability (FLEX(D, T))............125 6-1 Supply chain network for 5 facil ities and 10 downstream demand points .............128 6-2 Description of problem parameters on an example distribution network ..............133 6-3 Comparison of computation times for various problem settings...........................160 6-4 Performance evaluation for heuristic algorithm.....................................................161 6-5 Performance comparison for high and low overflow costs....................................162 x

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Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION MODELS FOR INTEGRA TED PRODUCTION, CAPACITY AND REVENUE MANAGEMENT By Yasemin Merzifonluoglu August 2006 Chair: Joseph Geunes Major Department: Industrial and Systems Engineering This thesis provides new planning models for making synchronized decisions on capacity, demand management and production/i nventory planning in supply chains. These models focus on the tradeoffs between capacity costs, production costs, costs for assigning customer demands to different supp ly resources and revenues associated with satisfying customer demands. Within this cl ass of models, we study various degrees of flexibility on the part of a supplier of goods including flexibility in demand and capacity management. We consider integrated production, capacity, and pricing planning problems, where a goods price may change throughout a planning horizon, as well as contexts in which a constant price is require d for the entire horizon. We also consider production planning models in which a supplier may not have a great deal of price setting flexibility, but may wish to be selective in its choice of markets (or customers) and the timing of demand fulfillment, as a result of the unique fulfillment costs associated with different markets (or customers). We also investigate the role of capacity planning in xi

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these contexts, including capacity acquisition problems that require setting the suppliers best capacity level for an entire planni ng horizon. We examine subcontracting and overtime as mechanisms for short-term capacity flexibility. We also consider logistics supply network design problems that determine the best allocation of downstream demands to upstream facilities in uncertain demand environments. We used polyhedral properties and dynamic programming techniqu es to provide polynomial-time solution approaches for obtaining optimal solutions fo r some of the problems that are not NPHard. When the problem is NP-Hard, we proposed very efficient heuristic solution approaches which are developed considering part icular features of the problems. We also employed a Branch and Price method for the large scale nonlinear assignment problems. xii

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CHAPTER 1 INTRODUCTION Our research provides a new set of pl anning models for simultaneously making decisions on capacity, demand manageme nt (pricing, demand selection, demand assignment) and production/inve ntory planning. Although thes e decisions are extremely interrelated in practice, coordination among them has not been fully addressed in the literature. Therefore, the models we pr esent generally concentrate on the critical tradeoffs between capacity costs, production costs, costs associated with assigning customer demands to different supply f acilities (e.g., transportation cost) and the revenues associated with satisfying customer demands. Within this class of models, we consider various degrees of flexibility in demand management. In most production and invent ory planning models in the operations literature, demand is exogenously determined (possibly characterized by some probability distribution). On the other hand, some past studies exist that assume that demand is completely endogenous. In such models, a manufacturing department might produce according to a predefined production schedule, and marketing and sales departments then attempt to realize sales according to this out put plan. The practical reality, however, is typically neither of these two extremes, becau se demand is neither absolutely exogenous nor endogenous. To more closely reflect the complexities in operations practices, our models consider demand and capacity decisi ons together, which results in optimal demand levels that are influenced by capaci ty and production costs, and vice versa. 1

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2 We examine demand management decisions through pricing, demand selection and demand assignment models. We consider dynamic pricing problems where an items price may change throughout a production ho rizon, as well as contexts in which a constant price is required for the entire hor izon. The pricing mode ls we provide permit us to establish the best demand schedule base d on the suppliers available resources and cost structure in conjunction with customer responses to pr ices. Some of our pricing models may also be interpreted as demand selection problems, where a supplier must decide whether or not a part icular customer order is economically attractive enough to accept and produce. In these demand selecti on problems, we typically assume that customers offer a price and provide a required sh ipping date for their orders. In practice, however, customers may also be flexible in te rms of both shipping date and price, and in these cases, an items price may be a function of the actual delivery time. We therefore provide solution methods for cases in which th e price associated with a customers order may depend on the delivery time. In additi on to pricing and demand selection models, the decision maker often faces the problem of assigning potential customer markets to available resources in the most profitable way. Although assignment problems have been studied extensively in the ope rations literature, demand uncer tainty has not been fully addressed in such settings. We therefore pr ovide contributions in the area of assignment problems under demand uncertainty. In addition to flexibility on the demand side we also focus on capacity adjustments as a mechanism for matching supply with demand, including capacity acquisition problems that determine the best production ca pacity for an entire planning horizon. As mechanisms for short-term capacity flexib ility, we also examine subcontracting and

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3 overtime options as two distinct choices for managing capacity. With the recent increase in contract manufacturing, production planning models that include subcontracting and outsourcing decisions have receiv ed a significant amount of a ttention in the literature. On the other hand, overtime decisions in c onjunction with capacity and demand planning models have not been completely examined in the operations literature. Our models also provide an option for a producers output to be completely subcontract ed instead of using internal production, in cases where the s ubcontracting option is more economically advantageous. Therefore, in addition to in ternal capacity management decisions, our models can be used to deal with traditional make-or-buy questions. In the broadest sense, our research pr imarily considers production and inventory planning contexts, where various pricing and cap acity issues also play important roles in maximizing a firms profit. In addition to si ngle stage models, our work also considers two-stage supply chain problems, where demand assignment under uncertainty is the main focus. This introductory chap ter provides an overview of the thesis. 1.1 Integrated Capacity, Demand and Pro duction Planning Models with Subcontracting and Overtime Options In this study, we outline two basic models for capacity, demand, and production planning; the first case assumes a fixed capac ity level for the manufacturer, and assumes that this capacity is exogenously predet ermined. The resulting model and solution approach lay the groundwork for the case in which capacity is a decision variable. We consider a manufacturer producing a good to satisfy price-dependent demand over a finite number of time periods. Th e objective is to determine the production (regular and overtime) schedule, inventory qua ntities, subcontracting, and demand levels in order to maximize net profit.

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4 In this problem context, we allow inte rnal production to cons ist of regular and overtime production. Since, in practice, the available regular production hours are often a bottleneck, a manufacturer may benef it from extra working hours, or by adding temporary workers. The total production output in such cases therefore consists of regular production plus any overtime and the a ssociated cost functions are often concave in total output, reflecting economies of scale in production. We therefore assume that the cost of regular production is a nondecreasing concave functio n of total output from regular time production. Overtime cost is an incremental cost (over regular time production cost) and is also a concave a nd nondecreasing function of the overtime production level in a period. A producer either supplies demand using in ternal production (regular or overtime), or purchases finished products from a s ubcontractor (or simultaneously utilizes both internal and external resources). We first assume a single subcontractor without capacity limits, where cost function is concave for each subcontractor. We have also considered multiple (non-identical) subcontractors, bot h with and without subcontracting capacity limits in order to generalize our approach. The revenue functions associated with satisfying demands may vary among periods, and is characterized by a non-d ecreasing piecewise-linear function of the demand satisfied in a period. We also a llow linear inventory holding cost for the inventory remaining at the end of each period. This problem minimizes a concave cost function over a set of network flow constraints, and therefore an optimal extreme point solution exists. In any extreme point solution, the basic variables create a spanning tree in th e network. Our suggested

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5 dynamic programming solution primarily relies on the spanning tree structure of extreme point solutions in this network. We us e dynamic programming methods to provide polynomial-time solution algorithms for obtaining an optimal solution for this class of problems. Chapter 3 also focuses on capacity acquisition decisions. We consider the case where a manufacturer attempts to determine its optimal internal capacity level for the planning horizon. We consider a concave cap acity cost function for internal production capacity, which is a decision variable. We again have the minimization of a concave function over a polyhedron, which implies that an optimal extreme point solution exists. We characterize extreme point properties of the associated polyhedron, which permit considering a polynomial number of distinct ca pacity levels in order to determine an optimal capacity level. For each of the candida te capacity levels associated with extreme point solutions, we can then solve a fixed capacity problem. We also consider the capacity acquisition problem under various assumptions regarding the number and capacity levels of subcontractors. 1.2 Capacitated Production Planning Models with Price Sensitive Demand and General Concave Revenue Functions Chapter 4 continues to cons ider discrete-time, finite -horizon operations planning models with capacity limits, where demand for a good is price sensitive. In contrast to the problem discussed in Chapter 3, we do not consider the availability temporary capacity expansion mechanisms, such as employing overtime hours and subcontracting (we continue to assume time-invariant pr oduction capacities, however). Instead, we focus our attention on handling more general concave re venue functions.

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6 We account for economies of scale in pr oduction of a good using a fixed plus variable production cost structur e. We also consider linear holding costs in the amount of inventory held in a period. Within this class of problems we consider both the case in which price may vary dynamically by period, as well as the case in which a single price is chosen for the entire planning horizon. In the dynamic pricing problem class, we consider a finite horizon planning model for a single item with production capacity eq ual to some positive value in every period. Demand in a period is a decision variable, and we assume that any value of demand in a period implies a corresponding unit price. The total revenue in a period is determined by a nondecreasing concave function of dema nd, where the corresponding price as a function of demand equals tota l revenue divided by demand, i. e., revenue/demand. We assume that production in any period requires incurring an order cost plus an additional variable cost. A holding cost is incurred fo r each unit remaining in inventory at the end of period. The amount of inventory remaining at the end of a period must be nonnegative. Our goal is to maximize total re venue less production and holding costs. In Chapter 4, we show that we can solve this model in polynomial time using a dynamicprogramming-based approach. We also consider the problem setting wher e a manufacturer requires setting a single price for a good over the entire planning horiz on. For this case, we assume that the demand in a period is given by a nonincreasi ng function of price. We define total revenue in a period as the product of price and total demand. The suggested algorithm runs in polynomial time under some mild assumptions on the revenue functions. The solution method along with the complexity results is explained in Chapter 4.

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7 1.3 Uncapacitated Production Planning Models with Demand Fulfillment Flexibility In addition to pricing and demand selection decisions, Chapter 5 recognizes the flexibility manufacturers often have to adjust order shipment times for a given set of production orders. In many such settings the net profit of an order depends on the time at which the order is satisfied. Our model acc ounts for these demand-timing decisions as well as order acceptance decisions, along with their production and inventory planning implications. For this class of problem s, we do not account for production capacity limits. We consider a discrete planning horizon a nd a set of candidate demands (or orders) for a single good produced by a supplier. We co nsider a fixed plus linear production cost structure and linear holding costs. As in th e previous demand selection type models, each candidate demand represents a request for a fixed amount. Net revenue for satisfying a candidate demand depends on the delivery period. We assume without loss of generality that for each candidate demand source, a customer-specified delivery time window exists during which the customer will accept delivery. The producer wishes to maximize net profit during the planning horizon, defined as the total net revenue from order acceptance and delivery-timing decisions, less the total setup, variable production, and holding costs. Interestingly, this problem may also be considered as a faci lity location problem, where the supplier wishes to maximize his profit with full flexibility to choose customer demands. This problem is proved to be NP-Hard (A ppendix B), which implies that we cannot reasonably seek an efficient (polynomial-tim e) solution method for this problem in its most general form. Under certain mild assumptions on costs and revenues, we can,

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8 however, solve the problem in polynomial time. In particular, under specialized cost assumptions that imply that no speculative motives exist for holding inventory or backlogging (i.e., there is no reason to pr oduce earlier than ne cessary due to the anticipation of production cost increases, or to produce later than necessary to take advantage of lower penalty costs later), our problem can be solved in polynomial time by employing a dynamic programming methods. As mentioned earlier, the ge neral uncapacitated case of the problem, in which the revenue parameter values can take any arbitr ary values, is NP-Hard. However, the dual of the LP relaxation of the problem leads to a very efficient heuristic procedure. Although the procedure does not guarantee an optimal solution to ev ery instance of the problem, for certain special cases, optimality is achieved. In addition, this dual approach provides a range of managerial insights about the problem setting. We also tested the relative benefits of different dimensions of supplier flexibilit y, considering possible combinations of demand selection and timing flexibility. 1.4 Demand Assignment Models under Uncertainity In Chapter 6 we introduce demand uncertain ty to our planning models. Although stochasticity broadens the applicability, it brings additional complications to the models and solution methods. Chapter 6, therefore, solely considers the problem settings where the planner determines the best allocation of customer demands to available resources. We consider a two-echelon model with an upstream supply echelon and a downstream demand echelon. The upstream echelon might be manufacturing or warehouse facilities, while the downstream echelon might be retail si tes. Each resources supply capacity is assumed to be known and each retailer site implies a known probability distribution of demand. For such a network, our models deal with the best assignment of demands to

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9 available resources. In addition, we assume that the cost of assigning a demand to a resource is resource specific. Customer demands are assigned to supply facilities based on their expected assignment costs and the availa ble capacities of the resource s. After demand realization, the assigned demands may exceed the available capacity of one or more resources. In such cases an additional penalty cost is in curred, which is associated with the capacity shortage. We wish to minimize the total exp ected cost, which includes the expected cost of satisfying demands using available resour ces and the penalty costs for exceeding the capacity of each supply source. The assignment problem introduced in Chapter 6 is a nonlinear Integer Programming Problem which can be solved by enumeration, e.g., a Branch and Bound algorithm. Branch and Price is commonl y used when solving such large-scale assignment problems. Branch and Price is a generalization of th e linear programming (LP) based Branch and Bound scheme, specifically designed to handle integer programming (IP) formulations that contain a huge number of variable s. In Chapter 6, we described the Branch and Price scheme for our problem starting with a Column Generation scheme. The pricing problems en countered in the Bran ch and Price scheme are Static Stochastic Knapsack problems which are interesting theoretically and practically on their own. We also present an efficient and novel solution method for the linear relaxation of the associated St atic Stochastic Knapsack problem. 1.5 Research Scope and Thesis Outline In general, we focus on developing e fficient solution methods for operations management problems involving production/i nventory planning, as well as demand and capacity management decisions. We consider dynamic pricing and capacity adjustment

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10 problems under various subcontractor assump tions, production and inventory planning problems with general concave revenue func tions, uncapacitated production planning problems with flexibility in demand timi ng and two stage demand assignment models under uncertainty. A wealth of past res earch on production planning models seeks to meet prescribed demands at minimum cost. In many contexts, demand comes from independent sources, not all of which are ne cessarily profitable. We consider such planning problems from a different perspec tive, assuming certain demand characteristics are decision variables. This thesis, therefore, primarily aims to fill gaps in literature by providing optimization models involving integrated production, capacity and demand management in the supply chains. Chapter 2 provides an introducto ry literature review for Chapters 3, 4, and 5. In Chapter 3, we discuss the production planning problem where demand is price dependent and various ty pes of capacity adjustments are available, including capacity acquisition, overtime, and subcontracting. In Chapter 4, we present a similar pricing and capacitated production planning model which fills a gap in the literature by considering price-dependen t demand with general concave revenue functions and fixed production capacities. In Chapter 5, we introduce demand timing flexibility into our produc tion and demand management models, where an items price may also change according to the actual delivery date. Chapter 6 recognizes the importance of demand uncertainty and s upply chain network design by providing a planning model that assigns customer dema nds to available resources in the most profitable way. Finally, Chapter 7 conc ludes with a summary and results.

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CHAPTER 2 LITERATURE REVIEW The focus in Chapters 3, 4, and 5 is production and inventory planning with integrated demand and capacity decisions. In this chapter, we pr ovide an introductory literature review for these integrated decisi ons. Since the focus of Chapter 6 differs slightly from these chapters, the related litera ture will be discussed within that chapter. We classify the relevant literature for Chapter 3, 4, and 5 into five categories: requirements planning, capacity planning, s ubcontracting, overtime and pricing/demand management. These are the primary elements of the models we will present, and for the most part represent distinct research streams. 2.1 Requirements Planning Wagner and Whitin (1958) first modeled the classical uncapacitated economic lotsizing problem (ELSP), which addresses th e tradeoff between setup and holding costs under dynamic, deterministic demand. Sin ce their original work appeared, many generalizations of the basic problem have been studied (e.g., Zangwill 1969, Love 1972, Thomas 1970, Afentakis and Gavish 1986). The capacitated version of the dynamic requirements planning problem has also been well researched (see Florian and Klein 1971, Baker, Dixon, Magazine and Silver 1978) This past research on dynamic requirements planning problems assumes demand s and capacities are predetermined. In these models, demands must be filled as they occur, or in models allowing backlogging, demands can be met later periods. In either case, these models assume that all demand 11

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12 must be met during the time horizon, using th e (fixed) capacity levels available to the firm. 2.2 Pricing/Demand Management with Production Planning The first integrated dynamic lot sizi ng and pricing analysis was provided by Thomas (1970). He generalized the Wa gner-Whitin (1958) model by characterizing demand in each of a finite number of time pe riods as function of price, treating each periods price as a decision variable. Geune s, Romeijn, and Taaffe (2006) considered a more general form of this model with tim e-invariant production capacities and piecewiselinear, nondecreasing, and concave revenue functions for each period. Our dynamicpricing models in Chapter 3 generalize this wo rk to the case where capacity decisions are taken into consideration. Chapter 4 also generalizes this work by considering more general concave revenue func tions. To address contexts with demand selection and production economies of scale, Geunes, Shen, and Romeijn (2004) also considered integrated production planning and market se lection decisions in a continuous-time model with market-specific constant and dete rministic demand rates. Loparic, Pochet and Wolsey (2001) also considered a relate d problem where the manufacturer maximizes profit, and does not require satisfying all demand, but sets lower bounds on inventory to account for safety stock requirements. Th eir model, however, assumed that only one demand source exists in every period and revenue gained from this demand is proportional to the satisfied demand. Biller, Chan, Simchi-Levi, and Swann (2005) considered a related dynamic-pricing pr oblem in which revenue is concave and nondecreasing in the demand satisfied, procurem ent capacity limits vary with time, and procurement costs are linear in the procurem ent volume. They note that the addition of setup costs to the model would result in a dynamic programming approach with solution

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13 time that grows exponentially in the size of the problem under timevarying capacities, but do not consider the equal-capacity case. Recently, Deng and Yano (2006) studied an integrated pricing and production planning problem under time-varying capacities, leading to a solution approach with an expone ntial running time. They also considered the time-invariant capacity case, and showed that it is polynomially solvable. Our solution methodology in Chapter 4 differs esse ntially from theirs with an improved running time. In addition, in Chapter 4 we address interesting insights about the relationships between optimal prices in diffe rent periods, and the relationships between optimal price vectors and production plans. In addition to providing solution methods fo r the time-varying pr ice case, Chapter 4 generalizes the past methods for constant-priced goods to account for time-invariant capacities. Kunreuther and Schrage (1973) first considered the problem of setting a single price over an entire planning horizon wi th an uncapacitated lot-sizing-based cost structure (with fixed-charge procurement cost structures a nd linear holding costs), and provided a heuristic solution approach for this problem. Gilbert (1999) provided a polynomial-time solution method for this probl em under the assumpti on of stationary costs. Van den Heuvel and Wagelmans (2006) subsequently showed that the more general version of the problem with time-varying costs can be solved in polynomial time. Our constant-pricing model in Chapter 4 generalizes this work to account for timeinvariant procurement capacities. Our work requires a much more general characterization of the propert ies of optimal solutions, and provides the first solution method for combined pricing and capacitated procurement planning with constant-priced goods and economies of scale in procurement (note that Gilbert 2000, also considered a

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14 periodic multi-product planning problem with constant-priced goods that share procurement capacity, although procurement cost s were linear in the amount of an item procured). Moreover, the stru ctural properties of optimal so lutions that we provide (both in the time-varying and constant-priced goods cas es) can lead to insights for developing solution methods for more general classes of profit maximization problems with general concave revenue functions and fixed-charge co st structures. Bha ttacharjee and Ramesh (2000) also considered the pricing problem for perishable goods, assuming demand can be characterized as a function of price. They studied structural properties of the optimal profit function, and provided heuristic methods to solve the problem. In addition to demand selection and prici ng flexibility, firms may have flexibility in delivery timing of the selected demands. In a number of practic al contexts, customers may allow a grace period (also called a de mand time window) during which a particular demand or order can be satisfied. Lee, etinkaya, and Wagelmans (2001) modeled and solved general lot-sizing problems with demand time windows. Their model still requires, however, that each demand is ultimately satisfied during its predetermined time window. That is, they considered demand-timi ng flexibility without th e benefits of order (demand) selection and rejection decisions. The approach we present in Chapter 5 integrates demand selection and a more gene ral version of demand time windows for lotsizing problems, providing two dimensions of demand planning flexibility. Charnsirisakskul, Griffin, and Keskinocak (2004) considered a similar model that focuses on the economic benefits of lead time flex ibility and order sel ection decisions in production planning with finite production ca pacities. Their model assumes that each order has a preferred due date and a latest acceptable due da te, after which the customer

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15 will not accept delivery. A tardiness penalty is incurred if an order is completed after the preferred period. Our models in Chapter 5, on the other hand, allow a market (or customer) to provide any period-specific per unit revenue values for delivery within the acceptable range of delivery dates. This al lows customers to specify any subset of acceptable delivery periods, and this subset n eeds not consist of consecutive periods, as in the case with past models that consider demand time windows. In addition, Charnsirisakskul et al. (2004) did not provide any tailor ed solution procedures for exploiting the special structure of the mode l; rather, they primarily studied the model parameter settings under which lead time flexib ility is most beneficial, and relied on the CPLEX solver for model solution. Moodie (1999 ) also considered pr icing and lead time negotiation strategies as a mechanism for influencing demand with timeand pricesensitive customers under fixed cap acity using a simulation model. Recent operations management literature also discusses additional mechanisms for affecting demand in order to increase net profit after subtracting operations costs. Crandall and Markland (1996) classified se veral demand management approaches for service industries, including capacity ma nagement and general demand influencing strategies. Iyer et al. (2003) use pos tponement (with an associated customer reimbursement) as a mechanism for managing demand surges under limited capacity. Calosso, Cantamessa, Vu, and Villa (2003) modeled a business-to-business electronic negotiation process in a make-to-order environm ent where the firm determines the jobs it will bid on (or accept) using a goal progr amming approach. Recent literature on available to promise (ATP) functions (e .g., Pibernik 2005, Chen, Zhao, and Ball 2002) considers order acceptance in a rolling fashion, based on a production or supply chain

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16 systems constraints and resource availabil ity levels. Order acceptance decisions have also been addressed in sche duling contexts, where the m achine utilization level and expected lead times (based on the set of previously accepted jobs) drive acceptance decisions (e.g., see Ten Kate 1994, for a simu lation-based approach for this scheduling problem class). 2.3 Capacity Planning Production capacities specify the abilities and limitations of a firm in producing outputs. The capacity expansion literature is concerned with determining the size, timing, and location of additional cap acity installations. Luss (1982), Love (1973), and Li and Tirupati (1994) provide examples of work on dynamic capacity expansion and reduction. These studies do not, however, consider more detailed dynamic production decisions; moreover, demand values are predefined (possibly according to some timebased function) and ar e not price dependent. When demand is uncertain and capacity is expensive, capacity may be insufficient to meet demand. In these situations, firms em ploy strategies such as pricing, backlogging or advance inventory build-up to manage shortages (see van Mieghem 2003). When shortages are allowed, one should account for the impacts of backlogging or lost sales with related demand-shortage penalties in th e model. Manne (1961) considered settings with backlogging where only capacity expans ion is allowed. Van Mieghem and Rudi (2002) also considered capacity additions us ing a growing stochastic demand model with backlogging. In most operations management models, demand is primarily treated as exogenous and may contain some associated uncertaint y. Another extreme might be the case in which demand is completely treated as endogenous. For example, manufacturing

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17 produces according to a plan based on its pr oduction capabilities and marketing and sales do what is necessary to realize sales according to that output pl an. The practical reality is typically somewhere between these two extr eme cases. An example of work that considers such effects is C achon and Lariviere (1999), who c onsider situations in which demand is influenced by a scarcity of capacit y. Kouvelis and Milner (2002) also present a two stage model that addr esses the effects of demand and supply uncertainties on capacity expansion decisions. In our models we also consider demand and capacity decisions together. Because of this, optim al demand levels are influenced by capacity costs (and vice versa) and are neither completely endogenous nor exogenous. 2.4 Subcontracting Subcontracting and outsourci ng have been the subjects of a number of recent studies due to their increased use in pract ice. Gaimon (1994) presents a model that investigates subcontracting as an alternative to capacity expansion. She also examines the effects of using subcontracting on prici ng services. Lee et al. (1997), Logendran and Puvanunt (1997), and Logendran and Ramakrishna (1997) considered subcontracting models for cellular manufacturing and flexib le manufacturing systems. Atamtrk and Hochbaum (2001) consider the tradeoffs between capacity acqui sition, subcontracting, and production and inventory decisions (with production economies of scale) to satisfy non-stationary deterministic demand over a finite horizon. Our work in Chapter 3 generalizes their results to account for cap acitated overtime availability and pricedependent demand. Our model also permits characterizing the impacts of multiple, capacitated subcontractors, whereas Atamtrk and Hochbaum (2001) focused on a single uncapacitated subcontractor. C oordination issues related to subcontracting, capacity and investment decisions are discussed by van Mieghem (1999), Kamien and Li (1990), and

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18 Cachon and Harker (2002), who also pres ent game-theoretic models related to subcontracting. In addition, Bertrand and Sr idharan (2001) study heuristic decision rules for subcontracting in a make-to-order manufact uring system in an effort to maximize utilization while minimizing tardy deliveries. 2.5 Overtime Planning Models for overtime planning have been addressed in a number of contexts. Kunreuther and Morton (1974) developed a production planning model that considers overtime, lost sales, simple subcontrac ting, undertime and backlogging, when production costs are linear in volume. Dixon et al. (1983) considered an other model that deals with the size and timing of replenishments for an item with time-varying demand. In their model, regular time and overtime production opti ons (the latter at a cost) are available and production capacities can also vary with time, but are not decision variables. They provided a heuristic approach for minimizing cost. Adshead and Price (1989) described simulation experiments of an actual make-t o-stock shop to examine the impact of changes in the decision rules used to cont rol overtime on cost performance. Ozdamar and Birbil (1998) considered capacitated lot sizing and facility loading with overtime decisions and setup times, minimizing total ta rdiness on unrelated pa rallel processors. They developed hybrid heuristics involving search techniques such as simulated annealing, tabu search, and genetic algorithms. Dellaert a nd Melo (1998) addressed a stochastic single-item production system in a make-to-order environment to determine the optimal size of a producti on lot and minimize the sum of setup costs, holding costs for orders that are finished before their promised delivery dates, penalty costs for orders that are not satisfied on time (and are therefor e backordered), and overtime costs. Pinker

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19 and Larson (2003) developed a model for flexible workforce management in environments with uncertain ty in the demand for labor.

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CHAPTER 3 INTEGRATED CAPACITY, DEMAND AND PRODUCTION PLANNING MODELS WITH SUBCONTRACTING AND OVERTIME OPTIONS Introduction The primary function of a production planni ng system is to determine how to best meet demand utilizing a firms production capacity. Production planning systems found in practice typically take a set of prescribed demands and predefined capacity levels as input, and determine how to meet the prescr ibed demands at a minimum cost without violating capacity limits. These prescribed demand levels often result from a markets response to the price of a good (when the pr oducer has a degree of monopoly power), while the predefined capacity levels are a consequence of a producers capacity investments. Pricing and capacity invest ment decisions therefore impose a set of constraints within which the production planning system must work. As a result, overall production system performance is affected no t only by the production planning decisions themselves, but also by demandand capacity-re lated decisions that typically precede the production planning process. Achieving ma ximum performance from a given production system thus requires an ability to determin e the best match between supply capacity and demand, based on the systems operating and cap acity costs and the market response to price. To address this problem, this thesis provides modeling and so lution techniques for integrated capacity, demand, and production planning decisions. Current practice typically addresses cap acity, production/inventory, and pricing decisions separately as part of a hierarchy of decisions. Capacity decisions are often 20

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21 made by strategic operations managers, while sales and pricing d ecisions are made by marketing and sales departments, and production/inventory deci sions are made by production planners. Traditional approaches in the operations literature reflect this practice by sequentially considering thes e decisions according to their relative importance, or based on the length of the associated planning horizon for each decision type. Using such an approach, each successive optimization model imposes constraints on the model at the next level in the hierar chy (Graves 2002). Generally, the last link in this decision process is production/inventor y planning, and for these decision problems, demand and capacity are taken as fixed para meters that are exogenous to the model. Recent literature has begun to recognize the importance of considering demand and capacity level decisions in production pla nning. For example, Bradley and Arntzen (1999) recently discussed the benefits of simultaneous consideration of capacity and production decisions. Similarly, Geunes, Romeij n and Taaffe (2006) provided analytical models that consider capacita ted production and pricing deci sions together in order to maximize a producers profit. This chapte r takes a further step by providing planning models that address critical tradeoffs betw een capacity/production costs and increased revenues by simultaneously considering capaci ty, pricing (demand management), and production/inventory planning decisions under economies of scale in production costs over a finite planning horizon. The pricing co mponent of the models allows a supplier to selectively determine the demand levels it will satisfy. On the capacity side, in addition to setting an internal base capacity level (w hich is time invariant), our models allow two types of short-term capacity adjustment s, through subcontr acting and capacityconstrained overtime. When a firm can in fluence its demand levels through pricing

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22 strategies, these decisions imp act the optimal amount of mediumto long-term internal production capacity, as well as the capacity us age strategy when different forms of shortterm capacity adjustments are available. A great number of traditional production planning models take demand as predetermined and exogenous, whereas we c onsider pricing decisions that determine demand levels. Similarly, traditional models typically take some initial starting capacity as given, and consider capacity adjustments that have an associated variable cost, such as a hiring or layoff cost per worker. For exam ple, classical aggregate planning models use linear programming techniques to make long term aggregate production, inventory, and personnel planning decisions (e.g. Manne 1961, Holt, Modigliani, Muth, and Simon 1960). In contrast, our models determine an optimal level of fixed (time-invariant) internal capacity for a horizon before the st art of that horizon, where a capacity cost function exists that is concave in the am ount of capacity acquired. The producer may then draw on flexible short-term capacity adjustments through overtime and subcontracting options. These short-term capacity adjustments allow a firm to use a chase strategy for meeting demand fluctuat ions; alternatively the producer may use inventory as a mechanism for applying a lev el strategy that does not employ short-term flexible capacity sources. As mechanisms for short-term capacity fl exibility, we focus on subcontracting and overtime as two distinct choi ces. Although both of these alte rnatives can be used to provide extra capacity in the short run, in practice these are two completely distinct options and should therefore be modeled distin ctly. In particular, the cost and capacity structures typically dictate th at overtime is only used after regular internal capacity has

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23 been exhausted, while subcontracting can also be used instead of internal production. When output requirements cannot be met dur ing regular working hours, employees may be scheduled to work overtime hours. In these cases, the production cost is structured in such a way that, although economies of scale in production will apply to both regular and overtime production, regular intern al production is always uti lized before using overtime options. Moreover, the amount of available overtime capacity is t ypically proportional to the amount of regular internal capacity. Fo r instance, in telephone call centers, the amount of overtime is often limited to 50% of the total amount of regular production time (Gans and Zhou 2002). We provide a scheme fo r modeling overtime co st that leads to total internal production cost that is pi ecewise-concave in the production level in a period. This scheme permits modeling a very general class of to tal production cost functions while retaining analytical tractability. As an alternative to overtime, ex cess demand may be subcontracted. Subcontracting is not an unco mmon practice in a variety of contexts and may be utilized for two reasons (Day 1956). First, in-house production capacity (re gular plus overtime) may not be sufficient (as in the call center case), and second, in -house production (in particular overtime production) may actually be more expensive than subcontracting. While we may expect that typically the marg inal cost of subcontra cting is higher than (regular) marginal in-house production cost (s ince otherwise one w ould subcontract all demand, resulting in complete outsourcing), the models we consider in this chapter do permit complete internal production (make) or complete outsourcing (buy). Therefore, they can be used to address traditional make-or-buy questions. Ultimately, however, our models are most appropriate for combinations of make/buy decisions.

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24 The primary contribution of this chapter is the integration of various important planning decisions in single, tractable model under a fairly general set of assumptions on cost structures and production dynamics. To our knowledge, no model currently exists in the literature that simultaneously determines optimal capacity and demand levels with dynamic price-dependent demands, economies of scale in production costs, and subcontracting and capacity-constrained overt ime options. Our approach for modeling demand permits application of the model more broadly to contexts in which pricing does not apply, but a supplier can accept or reject production orders based on order-dependent net revenues (after subtracting any variable fulfillment costs). Beyond applications the model might have in short-term operations planning, it can also provide substantial valu e through its ability to determine an optimal fit between demand levels and supply capacities. That is the model can provide a benchmark for the ideal fit between demand and capacity le vels in a production system, given the production systems operations cost structures, capacity options, and constraints. Therefore, even in contexts that do not le nd themselves to dyna mic pricing or order acceptance/rejection decisions, strategic deci sion makers can use the model to gain a better understanding of how cu rrent capacity and demand conditions deviate from a best-case scenario. The extent of such devi ations can then be an alyzed to develop new strategies for demand and capacity management. As we later show, solving our model re quires minimizing a concave function over a polyhedron (as discussed in Section 3.1, th is concavity is ensured by employing piecewise-linear and concave revenue curves within each planning period). We characterize important extreme point propert ies for this polyhedron that permit using

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25 dynamic programming techniques to provide po lynomial-time solutions. Relative to most of the classical planning models, the worst-case performance is typically a relatively high order polynomial function of the pla nning horizon length. Given the number of simultaneous decisions addressed by the m odel, however, the resulting polynomial solvability is a reasonably powerful result th at permits solution of large-scale problems in reasonably fast computing time. The organization of this chapter is as follows. Section 3.1 describes the modeling and solution approach used for the capaci tated production planning and pricing problem when the capacity level is exogenously determined. In Section 3.2, the capacity planning problem is addressed, extending the models of Section 3.1 to the cas e in which capacity is a decision variable. Section 3.3 summari zes our results for various assumptions on subcontractor parameters, such as an uncap acitated versus capacitated subcontractor and single versus multiple subcontractors. Fina lly Section 3.4 concludes with a summary. 3.1 Model and Solution Approach with Fixed Procurement Capacity 3.1.1 Problem Definition and Model Form ulation for Single Uncapacitated Subcontractor In this section we consider our basic model with a fi xed capacity level that is exogenously predetermined. The resulting model and solution approach lay the groundwork for the case in which capacity is a decision variable, which we consider in Section 3.2. We consider a manufacturer producing a good to satisfy price-dependent demand over a finite number of time periods, T The manufacturer can affect demand through pricing (implying some degree of relative monopoly for the good) and can draw on overtime and subcontracting as mechanisms for short-term capacity flexibility. The objective is to determine the production (re gular and overtime) schedule, inventory

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26 quantities, subcontracting, and demand levels (through pricing), in order to maximize net profit. In practice, the available regular prod uction hours a producer has are often a capacity bottleneck. A manufacturer may often benefit from employing extra working hours, or by adding temporary workers. We therefore distinguish between regular and overtime production, the latter of which can be viewed as a temporary production capacity increase in a period. We assume that the amount of available overtime capacity is often proportional to the amount of regular internal capacity. We will denote the ratio between regular and overtime production capacity by a positive constant We expect that, in practice, will often be integral and small, say between 1 and 3. For example, if a firm regularly utilizes two shifts per da y, but has the option of adding a third, then regular-time capacity is twice overtime capacity and = 2. However, in general we will allow to take any positive value, and ther efore regular-time capacity can be any positive multiple of overtime capacity (thus there is no loss of generality here in defining this ratio between regular and overtime capacity). We next define K and K as the total amount of available regular and overt ime capacity, respectively, in any time period, where K is a positive constant. The total regul ar and overtime capacity in a period therefore equals ( + 1) K Let xt denote the total internal production quantity, consis ting of regular production rt plus any overtime production zt. Let pt( xt) denote a corresponding base internal production cost function in period t In addition to this base cost, output produced during overtime in period t incurs an incremental overtime cost given by the function ot( zt). That is, when no overtime is used in period t then zt = 0 and xt = rt and the production cost

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27 equals pt( rt); if overtime is used in period t then zt > 0 and xt = rt + zt and the production cost equals pt( rt + zt) + ot( zt). The producer may also purchase finished products from a subcontractor. Let yt denote this subcontracted quantity in period t and let gt( yt) be subcontracting cost function. We assume the subcontractors capacity is unlimited, although we discuss the impli cations of limited subcontract or capacity in Section 3.3. We assume that the cost functions pt, ot, and gt are all nonnegative a nd nondecreasing. In addition, we also assume that they are con cave, representing the pr esence of economies of scale. Note, however, that due to th e presence of production capacities and overtime costs, the total internal production cost function depends on the amount of regular capacity and is not necessarily concave in the total quantity produced. Finally, for convenience and without loss of generality, we assume that pt(0) = ot(0) = gt(0) = 0 for all t We assume that there is a one-to-one correspondence between price and demand in any period (except possibly when the price equals zero), where demand is a downwardsloping function of price (see Gilbert (1999) and Geunes et al. (2006)). That is, a quantity of demand satisfied in period t say Dt, implies a unique valu e of the price in period t We therefore work directly with de mand values as decision variables rather than prices. The revenue in period t is assumed to be a nondecreasing and concave function of the demand satisfied in period t and is denoted by Rt( Dt). The concavity of Rt( Dt) is consistent with standard economics models that assume decreasing marginal revenue in output (see Gilbert 1999). For c onvenience and without loss of generality, we assume that Rt(0) = 0 for all t Let it be the decision variable denoting the inventory level

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28 at the end of period t and let ht be the corresponding nonnega tive per unit inventory holding cost. Our Capacitated Production Planning Probl em with Pricing [CPPP] can now be formulated as follows: [CPPP] Minimize 11()TT tttttttttt tt p xozgyhiRD (3.1) Subject to: tttttDiyxi 1 t = 1, 2, T (3.2) Krt t = 1, 2, T (3.3) t = 1, 2, T (3.4) tzK tt t x rz t = 1, 2, T (3.5) t = 1, 2, T ,,,,0tttttyirzD (3.6) 00Tii The objective function (3.1) minimizes production, overtime, subcontracting and holding costs less revenue from satisfied demand (thus the negative of the objective function value provides the net profit). Constraint set (3.2) represents inventory balance constraints. Note that this balance cons traint takes demand as a decision variable. Constraint sets (3.3) and (3.4) ensure that regular producti on is limited by capacity level K and overtime production is limited by capacity level K Constraint set (3.5) implies that internal production consists of both regular and overtime production. Note that the [CPPP] model does not expl icitly require regular capacity to be exhausted before overtime capacity. Suppose, how ever, that we have a feasible solution to the [CPPP] for which, in some period t some overtime is utilized and regular capacity

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29 is not exhausted, i.e., zt > 0 and Krt Then we can define an alternative solution with tttzrzK and max,ttrKr tz t (which is also feasible, because ttttt x rzrzx ). This new solution is at least as profitable as the first since t t x x and and the incremental overtime cost function ot is nonnegative and nondecreasing. Thus we can assume without lo ss of optimality that the facility employs overtime production only if regul ar-time capacity is exhausted. tzz t The [CPPP] minimizes the difference between concave functions. This problem is a difficult global optimization problem in general (Horst and Tuy 1990, Geunes et al. 2006). However, if the revenue functions in each period are piecewise-linear we can exploit the concavity of the remainder of the objective function. We will discuss a setting in which such a revenue function arises natura lly. However, in general this choice of revenue function form may serve as a close approximation to the actual revenue function. In the remainder of this chapter, we ther efore assume that th e revenue function in period t consists of Jt consecutive linear segments with widths djt and positive slopes rjt (where, for convenience, we define max 1,...,max{}tTt J J ). The concavity of the revenue function dictates that the sl opes are decreasing; moreover, we add a final segment having slope zero, beginning at some upper bound on the total possible demand level (see Figure 3.1). This upper bound may occur, for example, at the demand value that results when the price is set to the variable cost (while further reducing the pr ice below such a value may increase total revenue, such solutions wi ll never be profitable, and need not be considered; similarly, since costs increase for demands satisfied beyond this upper bound on demand in our model, such solutions will never be optimal for our model).

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30 Rt( Dt) Dt rjt d jt Figure 3-1. Piecewise linear concave revenue function. We modify our model for such reve nue function structures by letting vjt denote the decision variable for the amount of demand satisfied within the jth segment of the revenue curve in period t (note that we only need su ch variables for the first Jt such segments, since the (Jt+1)st segment implies zero additional revenue at additional cost). We thus reformulate the problem with piecewise linear revenue functions, which we refer to as [CPPPPL], as follows: [CPPPPL] Minimize 11()tTT tttttttt jtjt tt 1J j p xozgyhirv jt Subject to : 1 1tJ tttt jixyiv t = 1, 2, T (3.7) 0 vjt djt, t = 1, 2, T j = 1, Jt, (3.8) (3.3) (3.6) and yt, it, rt, zt 0, t = 1, 2, T (3.9) The objective function of [CPPPPL] is concave in the decision variables. Observe that for demand segments within a period, the model will naturally select those segments with higher slopes first. We therefore need not explicitly impose constraints specifying that a segments vjt variable can only be positive if vj-1,t = dj-1,t. Note that this problem can

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31 also be interpreted as an equivalent order selection problem where a manufacturer produces goods to satisfy demands corresponding to different orders over a finite number of time periods. In this context we allow o ffering different prices for different orders, where each order provides (possibly) a unique price. In this interpretation, each ( j t ) segment corresponds to a customer order where djt is the quantity associated with order j in period t and rjt is the unit revenue. In addition, vjt is the decision variable for the amount delivered corresponding to order j in period t Observe that the special case in which = 1 and ot( zt) = for zt > 0 and for all t represents contexts in whic h no overtime options exist and regular-time capacity equals K If, in addition gt( yt) = for yt > 0 and for all t then no subcontracting will be utilized, and the resulting model generalizes the problem studied by Geunes, et al. (2006) to account for general concave pr oduction costs (they considered only fixed plus linear production costs). Thus the [CPPPPL] generalizes the capacitated production planning and pricing problem considered by Geunes, et al. (2006) in two important ways. First, we allow the additional options of subcontracting and capacitated overtime, whereas their model considered only a regular-time cap acity limit. Second, the production cost function takes a general concave form (a s do the overtime and subcontracting cost functions). Moreover, the solution procedur e we provide in this chapter improves upon the worst-case complexity of the algo rithm provided by Geunes et al. (2006). The [CPPPPL] can be modeled as a concave-cost network flow problem as depicted in Figure 3.2. The flow on the arcs from periods to demand segments (or customer orders) represents demand satisfaction, and each has capacity djt. The flow on arc (r t ) represents regul ar production, (o, t ) overtime production, (s t ) subcontracting, and the

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32 flow on arc ( t t +1) denotes the inventor y carried from period t to period t+ 1. There is no capacity for subcontracting while regular production arcs have capacity K and overtime arcs have capacity K D vjt r o s 1 2 3 T Demand Segments or Customer Orders Periods Subcontractor Regular Production Overtime Production xt rt yt zt i1 i2 iT -2 iT -1 1 2 3 T T -2 T -1 T -2 T -1 Dummy Source Figure 3-2. Network repr esentation of the [CPPPPL]. Since the [CPPPPL] minimizes a concave cost function over a set of network flow constraints, an optimal extreme point soluti on exists. In any extr eme point solution, the basic variables create a spanning tree in th e network (Ahuja, Magnant i, and Orlin 1993). Before proceeding, we provide an important definition of the concept of a Regeneration Interval (RI) as provided by Florian and Klein (1971). Regeneration Interval: Given a feasible production pl an, a Regeneration Interval (RI) ( t t' ) is a sequence of consecutive periods t t + 1, t' 1 with and for = t t + 1, t' 2 (where 1 t < t .T +1). 1'10ttii 0 i The following proposition states important ch aracteristics of those RIs that can be contained in an extr eme point solution. Proposition 3.1. In an extreme point solution for [CPPPPL], any RI can have at most one period t with 0 < rt < K (fractional regular production), at most one period t with 0 < zt < K (fractional overtime production), at most one period with yt

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33 > 0 (subcontracting), or at most one period with 0 < vjt < djt, but not any of these together. Proof. Proof of this result follows from th e spanning tree property by showing that any solution that violates the conditions of the proposition results in an undirected cycle in the network. Any solution to the [CPPPPL] can be decomposed into a sequence of RIs. More importantly, any extreme point solution to the [CPPPPL] can be decomposed into a sequence of RIs that satisfy the characteriza tion in Proposition 3.1. Therefore, if we can find the minimum cost associated with each possible RI while observing the structure of Proposition 3.1, then the overall problem can be solved using an acyclic shortest path graph containing a path corresponding to every sequence of possible RIs. Using Proposition 3.1 we can classify the RIs that may be associated with extreme points into two types: Fractional supply: There is exactly one positive s upply quantity that is not at capacity (either 0 < xt < K or K < xt < (+1) K or yt > 0), and all (period, segment) pairs that are used are filled to capacity ( vjt {0,djt}). Fractional demand: Any positive regular and overtime production is at full capacity and no subcontracting takes place ( xt {0, K (+1) K } and yt = 0), and there exists at most one (period, segmen t)-pair in which the segment-capacity is used partially (0 < vjt < djt). In order to efficiently determine the mini mal cost of these types of RIs, we will next discuss how to identify candidate demand levels for a given RI in polynomial time. 3.1.2 Determining Candidate Demand Vectors for an RI In the previous section we have seen that we need to consider at most two types of RIs associated with extreme point solutions For each RI type, we will characterize a candidate set of demand levels such that at least one of these demand levels provides an optimal solution for the given RI type.

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34 We first reformulate our problem in a mo re compact form, which allows us to characterize a key property of optimal demand levels for any given RI. We can define the inventory level in period t as the cumulative production amount less satisfied demand, i.e., 1111tttJ t jixy jv 1J j which allows us to reformulate the problem as follows: Minimize 11()tTT tttttt jtjt ttcxsyoz v Subject to : 11110ttttJ jt jxyv t = 1, T (3.3) (3.6) (3.8) and t = 1, T j = 1, Jt, ,,0tttyrz where we have used the following set of redefined cost functions and revenue coefficients: ()(), ()(), .T ttttt t T ttttt t T jtjt tcxpxxh s ygyyh rh Note that the functions ct and st inherit the properties of pt and gt, i.e., they are nonnegative, nondecreasing, and concave, and ct(0) = st(0) = 0 for all t The jt values will play an important part in our solution approach, as the following propositions illustrate. Proposition 3.2. For each RI, we only need to consider solutions for which vk > 0 (where period is in the RI) implies that vjt = djt for all ( j t ) such that jt >k and period t is in the RI. Proof. Consider an RI containing periods t and and suppose that we have a corresponding solution for which vjt < djt and vk > 0 while jt >k. Then a small but positive amount of demand can be shifted from period s kth segment to period t s jth segment without changing any supply le vels. The change in the objective

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35 function per unit of shif ted demand is equal to kjt < 0, which implies that the original solution cannot be optimal for the RI. This implies the desired result. Proposition 3.3. For each RI of the first type [fractional supply], we only need to consider solutions for which vk > 0 (where period is in the RI) implies that vjt = djt for all ( j t ) such that jt k and period t is in the RI. Proof. The desired result follows imme diately from Proposition 3.2 for all ( j t ) such that period t is in the RI and jt k Now suppose that we have a solution for which vjt < djt and vk = dk while jt =k. Then, since there exists at least one supply variable that is fractiona l, we can increase or decrease vjt by a small amount. If the cost of the solution is nonincreasing when vjt is decreased we may do so until no supply variables are fractional (at which point the RI becomes one of the second type) or until it is decreased to 0. However, in that case the cost of the solution will also be nonincreasing when vk is decreased either until no supply variables are fractional or until it is decreased to zero. Similarly, if the cost of the solution decreases when vjt is increased we may again do so until no supply variables are fractional (at which point the RI becomes one of the second type and the original RI solution cannot be part of an optimal solution) or until it is increased to djt. Thus the desired result follows. Propositions 3.2 and 3.3 can be utilized to determine candidate demand patterns for each RI of a given type. For th e first RI type, recall that all vjt variables must take values of either 0 or djt. Proposition 3.3 implies that at most TJmax + 1 different candidate demand vectors for the RI (including the zero demand vector) need to be considered, based on an ordering of the demands in (any) nonincreasing order of their -values (since in an RI ( t t ') there can be at most T periods, and within a period we have at most Jmax segments). For the second RI type, recall that we can have at most one demand segment in the RI with 0 < vjt < djt, while all (regular and overtime) produc tion levels are either zero or at capacity and no subcontracting is allowed. Since the total amount produced in the RI must equal the total demand satisfied, if we have a candidate cumulative production level for the RI, then by Propositions 3.2 and 3.3, we can directly establish corresponding vjt

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36 values by filling the candidate cumulative production level with demand segments in nonincreasing order of their values. If there are ties in this ordering we break these ties by selecting demand segments in later time pe riods before demand segments in earlier time periods. This will ensure that if a feas ible solution exists with the given candidate cumulative production levels it will be found. The number of candidate demand vectors for this type of RI is thus equal to the number of possible tota l production quantities in the RI. In general, if the number of periods with regular producti on at capacity and no overtime production is m and the number of periods w ith both regular and overtime production at capacity is n, the total production quantity is equal to mK + n(+1) K for a total of O ( T2) possible demand vectors for the RI. If regular production capacity is an integer multiple of the overtime production capacity, i.e., is integer, then all possible cumulative production levels are integer multiples of K and the number of possible demand vectors for the RI is (T ). The optimal RI cost for a given candidate demand vector can be computed using dynamic programming, as we discuss in the next section. The solution with minimum cost among all candidate demand vectors then provides an optimal solution for that RI. 3.1.3 Optimal Cost Calculation for an RI This section discusses how to compute the optimal RI cost for any given RI and a corresponding candidate demand vector. We c onsider the case in which regular capacity is an integer multiple of overtime capacity in Section 3.1.3.1, while the general case is discussed in Section 3.1.3.2. For a given RI, say ( t t '), let a candidate set of demands to be satisfied be given by the quantities in each demand segment to be satisfied: j tv = t , t 1, j = 1, Jt.

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37 Let the corresponding demand vector, as described in Section 3.1.1, be given by 1J j j D v for = t , t 1. Moreover, we denote th e cumulative demand in periods t ,, by s st D t D for = t , t 1. 3.1.3.1 Regular capacity as integer multiple of overtime capacity In this section, we develop a dynamic pr ogramming method used to find an optimal solution for an RI and a candidate demand v ector when regular production capacity is an integer multiple of overtime capacity (It is straightfo rward to show that if the overtime capacity is an integer multiple of the regular time capacity, a slight modification of the methods developed in the remainder of th is chapter will solve the corresponding problems in the same running time, with then denoting the ratio between overtime and regular capacity). We will construct a laye red network, where the layers correspond to production periods, to track the cumulative supply amounts in each period. Nodes in a layer will represent possible cumulative regula r and overtime production plus subcontracting levels up to the tim e point corresponding to the layer. Given an RI and a candidate demand vector, let ,'1/qDttK denote the number of integer multiples of the capacity parameter K that are required to satisfy a total demand of D ( t t ') and let ,'1 f DttqK denote the remainder. If f > 0 then this candidate demand vector corresponds to an RI of the first type and a quantity equal to f must be produced either in a fractional regular time producti on period, a fractional overtime period, or a subcontracting period. Note that the actual amount of production in a fractional regular time production period can equal wK + f for any w = 0, 1, a

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38 fractional overtime period will contain a production amount equal to f and a subcontracted amount can equal sK + f for any s = 0, q. Let (, u, e ) denote a node in the la yered network, where denotes the current time period (or layer), u denotes the cumulative production as a multiple of K up to and including the period, and e = 1 if the fractional amount has been produced already and is 0 otherwise. Arcs between layer and layer +1 will be of the following types ( = t , t): A1 Zero supply arcs: From node (, u, e ) to node ( + 1, u, e ) Arc costs: 0 A2 Fractional regular production arcs: From node (, u, 0) to node ( + 1, u + w 1) for w = 0, such that u + w q Arc costs: c+1( wK + f ) if ( u + w ) K + f D ( t + 1), otherwise A3 Full capacity regular production arcs: From node (, u, e ) to node ( + 1, u + e ) Arc costs: c+1(K ) if ( u + ) K + ef D ( t + 1), otherwise A4 Fractional overtime production arcs: From node (, u, 0) to node ( + 1, u + 1) if u + q Arc costs: c+1(K ) + o+1( f ) if ( u + ) K + f D ( t + 1), otherwise A5 Full capacity overtime production arcs: From node (, u, e ) to node ( + 1, u + + 1, e ) if u + + 1 q Arc costs: c+1((+1) K ) + o+1( K ) if ( u + + 1) K + ef D ( t + 1), otherwise A6 Subcontracting arcs: From node (, u, 0) to node (+ 1, u + s 1) for s =0, q such that u + s q

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39 Arc costs: g+1( sK + f ) if ( u + s ) K + f D ( t + 1), otherwise A7 Demand satisfaction arcs: From node ( t' q, 1) to the sink Arc costs: '1 1 J t jj tjv Note that we allow parallel arcs in the ne twork. For example, arcs of type A6 for s < are equivalent to arcs of type A2 (although they have different co sts). The shortest path from a single source node ( t ,0, 0) to the sink in the resulting graph provides an optimal solution for the RI and corresponding ca ndidate demand vector. Note that if the shortest path has infinite length, the candidate demand vector is infeasible. To determine the complexity of this RI subproblem, note that the number of nodes is in a layer is O (T ) so that the total number of nodes in the network is O (T2). Since each node has outdegree O (T ) the number of arcs in the network is O (2T3). However, note that in cases where f = 0 (that is, in cases where the demand vector corresponds to an RI of the second type), the arcs of types A2, A4, and A6 are not needed. When these arcs are removed, each node has outdegree at most 1 and the number of arcs in the network to O (T2). The optimal solution for such cases is then given by the shortest path from a source node ( t 0, 1) to the sink. 3.1.3.2 Regular capacity as any positive multiple of overtime capacity In this section we allow regular production capacity to be any positive multiple of overtime capacity. Let m denote the number of periods in which regular production is at capacity and overtime production is not, and n denote the number of periods where regular and overtime production are both at capacity. Then, given an RI ( t t '), a corresponding candidate demand vector, and fixed values for m and n the total demand

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40 satisfied can be expressed as ,'1(1) D ttmKnKf where f may correspond to fractional regular time production, fractiona l overtime production, or a subcontracted amount. Let (, m ', n', e ) represent a node in the network, where is the current period (layer), m is the total number of peri ods, up to and including period in which regular production is at capacity and overtime is not, and n' denotes the number of cumulative full capacity overtime periods We set e = 1 if the fractional amount has been produced already, otherwise it is 0. Arcs between layer and layer +1 will be of the following types ( = t , t): B1 Zero supply arcs: From node (, m ', n ', e ) to node ( + 1, m ', n ', e ) Arc costs: 0 B2 Fractional regular pro duction arcs: (only if 0 f < K ) From node (, m ', n ', 0) to node ( + 1, m ', n ', 1) Arc costs: c+1 ( f ) if m 'K + n'K + f D ( t + 1), otherwise B3 Full capacity regular production arcs: From node (, m ', n ', e ) to node ( + 1, m '+1, n ', e ) Arc costs: c +1(K ) if ( m '+1) K + n 'K + ef D ( t + 1), otherwise B4 Fractional overtime pr oduction arcs: (only if K f < (+1) K ) From node (, m ', n 0) to node ( + 1, m ', n ', 1) Arc costs: c +1( f ) + o+1 ( f K ) if m 'K + n'K + f D ( t + 1), otherwise B5 Full capacity overtime production arcs:

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41 From node (, m ', n e ) to node ( + 1, m ', n '+1, e ) Arc costs: c +1((+1) K ) + o+1( K ) if m 'K + ( n'+1) K + ef D ( t + 1), otherwise B6 Subcontracting arcs: From node (, m ', n ', 0) to node (, m ', n ', 1) Arc costs: g+1( f ) if m 'K + ( n'+1) K + f D ( t + 1), otherwise B7 Demand satisfaction arcs: From node ( t ', m n, 1) to the sink Arc costs: '1 1 J t jj tjv The shortest path from source node (t 0, 0, 0) to the sink in the resulting graph provides an optimal solution for the RI with the corresponding candi date demand vector and values for m and n. To determine the complexity of this RI subproblem, note that the number of nodes is in a layer is O ( T2) so that the total number of nodes in the network is O ( T3). Since each node has outdegree at most 1, the number of arcs in the network is O ( T3) as well. Since, in principle, fo r each candidate demand vector there are O ( T2) potential choices for m and n, the total time required to find an optimal solution for the RI and a corresponding candidate demand vector is O ( T5). However, note that if we consider candidate demand vect ors corresponding to an RI of the second type, the values of m and n are uniquely defined so that an optim al solution for the RI and such a corresponding candidate demand vector is O ( T3).

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42 3.1.4 Complexity of Solution Approach We will next use the results of Sections 3.1.2, 3.1.3.1, and 3.1.3.2 to derive the running time of our algorithm for solving the [CPPPPL]. Proposition 3.4. The [CPPPPL] can be solved in polynomial time in Jmax and T Moreover, if is integral the [CPPPPL] can be solved in pseudo-polynomial time that is superior as a function of T Proof. First consider the case where is integral. For candidate demand vectors corresponding to RIs of the firs t type, the optimal cost can be calculated in at most O (2T3) time and the number of such candidate demand levels to be considered is O ( JmaxT ) for a total of O (2JmaxT4) time. For candidate demand vectors corresponding to RIs of the second type, th e optimal cost can be calculated in at most O (T2) time and the number of such candidate demand levels is O (T ) for a total of O (2T3) time. Since there are O ( T2) RIs, it takes O (2JmaxT6) time to calculate all optimal RI costs. For general the optimal cost for candidate demand vectors corresponding to RIs of the first type can be calculated in at most O ( T5) time and the number of such candidate demand levels to be considered is O ( JmaxT ) for a total of O ( JmaxT6) time. For candidate demand vectors corresponding to RIs of the second type, the optimal cost can be calculated in at most O ( T3) time and the number of such candidate demand levels is O ( T2) for a total of O ( T5) time. Since there are O ( T2) RIs, it takes O ( JmaxT8) time to calculate all optimal RI costs. In both cases, the shortest pa th in the resulting acycli c network containing a node for each RI and an arc for each optimal RI solution can be found in O ( T2) time. Note that for the special case of the [CPPPPL] model considered in Geunes et al. (2006), where = 1 and no overtime or subcontracting options are available, the optimal cost of any RI can be determined in O ( T2). This implies a the worst-case complexity for determining an optimal RI cost of O ( JmaxT3), and a corresponding worst-case problem solution complexity of O ( JmaxT5). While Geunes, Merzifonluolu, Romeijn, and Taaffe (2006) showed this improved complexity result over the O ( JmaxT6) algorithm in Geunes et al. (2006) (who considered only fixed-plus-linear production cost s), our analysis in this

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43 chapter shows that this improved complexity result holds for the more general case of concave production cost functions. In this section, we have provided a polynomial-time solution method for a fixed value of the capacity parameter K In the following section, we generalize our results to account for contexts in which the capacity parameter K is a decision variable with an associated capacity cost. 3.2 Capacity Planning We next consider the case where the manu facturer wishes to determine its optimal internal capacity level for the production horizon. Let ( K ) denote a concave cost function of the capacity parameter K which is a decision variab le. If we consider the case of capacity acquisition with an initial capacity of zero, the function ( K ) characterizes the cost to acquire K units of regular cap acity and associated K units of overtime capacity. The problem formulati on for the capacity planning problem, which we call [CPPPPL( K )], is as follows: [CPPPPL( K )] Minimize 11()tTT tttttttt jtjt ttKcxgyozhir 1J jv Subject to: (3.3) (3.9) and K 0. The objective function minimizes the cap acity, regular time production, overtime production, subcontracting, and i nventory holding costs, le ss revenue from satisfied demand, while the constr aint set is the same as that for [CPPPPL], with the addition of the nonnegativity constraint on K Note that the [CPPPPL( K )] model can also be applied to contexts in which the initial capacity is posi tive instead of zero. If we begin with some

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44 initial capacity KI > 0, replace the nonnegativity constraint with K KI, and allow ( K ) to take negative values for K < 0, then using a variable substitution, the resulting model is structurally identical provided that ( K ) remains concave. We interpret negative values of K to imply a negative cost, or reward, for reducing capacity (through, for example, capacity that is sold or associated capacity costs that are avoided). Whether a firm begins with an initial positive ca pacity or is considering capacity acquisition, the [CPPPPL( K )] model can provide value as a benchmarking tool in order to determine the optimal capacity level during a planning horizon. For problem [CPPPPL( K )], we again minimize a concave function over a polyhedron, which implies that an optimal ex treme point solution exists. The following proposition characterizes the struct ure of extreme point solutions. Proposition 3.5. For every extreme point solution for [CPPPPL( K )] with K > 0 there exists an RI (t t' ) in which all internal producti on levels are either at zero, regular-time capacity, or overtime capacity and there is no subcontracting. In addition, there is no fractional demand satisfaction. That is, x {0, K ( + 1) K }, for all = t ,...,t' ; y = 0, for all = t ,...,t' ; vj {0, dj}, for all = t ,...,t' ; j = 1, J. Proof. The total number of variables in the formulation of [CPPPPL( K )] is 5 T + 1 T t t J (note that we need not consider i0 or iT since these variables can be substituted out of the formulation) so in an extreme point solution 5T + 1 T t t J linearly independent in equalities must be binding. There are T binding balance constraints (3.7) and T binding equality constraint s defining total production (3.5) The remaining 3 T + 1 T t t J binding inequalities in an extreme point solution must come from among the remaining constraints. Any feasible solu tion consists of a sequence of RIs. Let ( x r z y i K ) be an extreme-point solution with K > 0 and let R denote the number of RIs associat ed with this solution, where the ith RI is ( ti,ti+1) (where t1=1 and tR+1= T +1).

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45 Since we know that 110iti for i =1,,R these provide an additional R linearly independent bindi ng equalities (where we previously accounted for constraints (3.6) The remaining 13T t tTJR 1 required binding inequalities for the extreme point solution must then come from among the following constraints: rt K i = 1, R t = ti, ti+1 1, (3.10) rt 0, i = 1, R t = ti, ti+1 1, (3.11) zt K i = 1, R t = ti, ti+1 1, (3.12) zt 0, i = 1, R t = ti, ti+1 1, (3.13) yt 0, i = 1, R t = ti, ti+1 1, (3.14) vjt djt, i = 1, R t = ti, ti+1 1, (3.15) vjt 0, i = 1, R t = ti, ti+1 1, j = 1, ..., Jt. (3.16) When K > 0 and for a given i at most ti+1 ti of each of the (sets of) inequalities (3.10) (3.11) (3.12) (3.13) and (3.14) respectively. Finally, at most of the pair of inequalities 11 i it tt tJ (3.15) (3.16) can be binding. This implies that at most of the constraints corresponding to a given RI i may be binding. It is easy to see that, for a give n RI, the structure of an extreme point solution for this RI is of the type as de scribed in the proposition if and only if the maximum number of inequalities is indeed binding for that RI. Now suppose that for the extreme point solution ( x r z y i K ) this maximum is not achieved by any of the RIs. This implies that no more than 1 113i it tt t iiJtt RJTJttT t t R i t tt t iii i 1 1 1 131 31 binding inequalities exist from among inequalities (3.10) (3.16) which contradicts that ( x r z y i K ) is an extreme point solution. Thus we conclude that an extreme point solution satisfying the conditions of the proposition exists. Proposition 3.6. There are O ( JmaxT5) distinct capacity parameter levels among all candidate optimal extreme point solutions for [CPPPPL( K )]. In addition, when the capacity multiplier is integral, the number of di stinct such capacity parameter levels is O ( JmaxT4). Proof. There are RI choices given by all pairs of the form ( t t' ) with t = 1,,T and t' = t +1,,T +1. As a result of the di scussion in Section 3.1.2 and 1 2 T

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46 Proposition 3.3, within each of the 1 2 T RIs there are O ( JmaxT ) candidate demand vectors that we must consider, for a total of = O ( JmaxT3) candidate combinations of RIs and demand vectors of the form given in Proposition 3.5. (By Propositions 3.2 and 3.3, for any given K and therefore at the optimal value of K in [CPPPPL( K )]when ties exist among multiple -values, by breaking ties based on latest time period first, we consider all potentially optimal extreme points, which implies that there are O ( JmaxT ) candidate demand vectors that must be considered, even in the case in which ties exist among -values.) max1 2 T OJT As a result of Proposition 3.5, in an extreme point solution all demand in at least one RI, say ( t t '), is satisfied by an integer number n of full-capacity regular production periods and an integer number m of full capacity overtime production periods. For a given demand vector we thus have (1),'1 nKmKDtt So, given an RI and candidate demand vector, the total number of di fferent levels of K such that (1),'1 nKmKDtt with n and m nonnegative integers is O ( T2). Since each of the O ( JmaxT3) candidate combinations of RIs and demand vectors may imply O ( T2) capacity levels we have O ( JmaxT5) distinct capacity levels. In the special case where is integral, the total produc tion in any RI of the type given in Proposition 3.5 must be an integer multiple of K up to T ( + 1) K We thus have only O (T ) total production quanti ties to consider for a given demand vector. Multiplying this by the number of potent ial RI and demand vector combinations implies O ( JmaxT4) candidate capacity parameter levels. For each candidate capac ity parameter level K we can then solve a [CPPPPL] problem which yields the following corollary. Corollary 3.1. The problem [CPPPPL( K )]can be solved in polynomial time. Proof. The proof follows from Propositions 3.4 and 3.6 by solving the [CPPPPL] problem with piecewise-linear, concave, and nondecreasing revenue curves for each candidate capacity level. When is integral the overall complexity for solving [CPPPPL( K )] is 2310 maxOJT while for general values of the resulting worst-case complexity is 213 maxOJT While the resulting order of the complexi ty result is a high power of the horizon length T in light of the generality and the number of integrated decisions contained in the model, the polynomial solv ability is remarkable.

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47 3.3 Multiple Subcontractors and Subcontractor Capacities Throughout this chapter, we have assume d that there is a single uncapacitated subcontractor available for subcontracted purchases. Although this is not an unreasonable assumption in light of past l iterature (see e.g., Atamtrk and Hochbaum 2001), we may also consider a generalization of our model that allows for one or more, say M potentially non-identical subcontractor s with or without time-invariant subcontracting capacity limits. Letting denote the amount subcontracted to the ith subcontractor in period t and letting i ty ii tt g y denote the associated (concave) s ubcontracting cost function, we can modify our problem formulations by replacing each yt by in the balance constraints 1M i t iy (3.2) replacing the nonnegativity constraints on each yt by nonnegativity constraints on each and replacing the cost term gt( yt) by in the objective function. i ty 1 M ii tt igy 3.3.1 Uncapacitated Subcontractors The Capacitated Production Planning Problem with Pricing and Uncapacitated Subcontractors [CPPP-US] is formulated as follows: [CPPP-US] Minimize: 111tTMT ii tttt tttt jtjt tit 1 J j p xozgyhirv Subject to: tt M i i t ttDiyxi 1 1 t = 1, 2, T (3.3) (3.5) (3.8) t = 1, 2, T i = 1, 2, M 0,,,, tttt t iDzriy

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48 Since this is a minimization problem of a concave function over a polyhedron, the spanning tree properties of optimal solutions still apply here. The following proposition will determine the RI types we will need to consider in this case. Proposition 3.7. In an extreme point solution, any regeneration interval can have at most one period t with 0 < rt < K (fractional regular production), at most one period t with 0 < zt < K (fractional overtime production), at most one period t with at most one > 0, i = 1, 2, M (subcontracting with at most one subcontractor in at most one period) or at most one period with 0 < vjt < djt, but not any of these two together. t iy The proof of Proposition 3.7 follows fro m the spanning tree property of optimal solutions. Because there is at most one s ubcontractor utilized within an RI, we can effectively include M parallel subcontracting arcs (one for each subcontractor) in the layered networks constructed in Section 3.1. 3. The properties of optimal demand vectors are therefore the same as those in th e single uncapacitated subcontractor case. Proposition 3.8. There exists an O ( MJmax2T6) algorithm to solve the [CPPP-US] problem with piecewis e linear revenue curves when regular production capacity is an integer multiple of overtime capacity. When is any positive scalar, there exists an O ( MJmaxT8) algorithm to solve [CPPP-US] problem with piecewise linear revenue curves. Proof. When is integral, the optimal cost can be calculated in at most O ( MT2( + T )) time for candidate demand vectors corre sponding to RIs of the first type and the number of such candidate demand levels to be considered is O ( JmaxT ) for a total of O ( JmaxT3( + T )) time. For candidate demand v ectors corresponding to RIs of the second type, the optimal cost can be calculated in at most O (T2) time and the number of such candidate demand levels is O (T ) for a total of O (2T3) time. Since there are O ( T2) RIs, it takes O (max{ M2JmaxT5, MJmaxT6, 2T5}) O ( M2JmaxT6) time to calculate all optimal RI costs. For general the optimal cost for candidate demand vectors corresponding to RIs of the first type can be calculated in at most O ( MT5) time and the number of such candidate demand levels to be considered is O ( JmaxT ) for a total of O ( MJmaxT6) time. For candidate demand vectors corre sponding to RIs of the second type, the optimal cost can be ca lculated in at most O ( T3) time and the number of such candidate demand levels is O ( T2) for a total of O ( T5) time. Since there are O ( T2) RIs, it takes O ( MJmaxT8) time to calculate all optimal RI costs.

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49 In both cases, the shortest pa th in the resulting acycli c network containing a node for each RI and an arc for each optimal RI solution can be found in O ( T2) time. 3.3.2 Capacitated Subcontractors In this case we take into acc ount the possibility of finite subcontractor capacities in the multiple subcontractor case. We assume each subcontractor can supply a limited amount of product (or service) in a period, a nd that the capacity of each subcontractor is known and does not vary throughout the planning horizon. Subcontractor capacities can easily be included in the model by adding constraints of the form for all i and t where denotes subcontractor i capacity in any time period. i tyC i iC The Capacitated Production Planning Pr oblem with Pricing and Capacitated Subcontractors [CPPP-CS] is formulated as follows [CPPP-CS] Minimize: 111tTMT ii tttt tttt jtjt titpxozgyhirv 1J j Subject to: tt M i i t ttDiyxi 1 1 t = 1, 2, T (3.3)(3.5) (3.8) it iCy t = 1, 2, T i = 1, 2, M 0,,,, tttt t iDzriy t = 1, 2, T i = 1, 2, M The RI cost computations for this case are also based on the associated spanning tree property. The following proposition define s the regeneration interval types for the case of multiple capacitated subcontractors. Proposition 3.9. In an extreme point solution, an y regeneration interval can have at most one period t with 0 < rt < K (fractional regular pr oduction), at most one period t with 0 < zt < K (fractional overtime production), at most one period t with for at most one subcontractor i (and all other su bcontractor production it iCy 0

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50 quantities at 0 or capacity in all period s), or at most one period with 0 < vjt < djt, but not any of these two together. As in Section 3.1.1, Proposition 3.9 let us classify the RIs that may be associated with extreme points into two types: Fractional supply: There is exactly one positive s upply quantity that is not at capacity (either 0 < xt < K or K < xt < (+1) K or ), and all (period, segment) pairs that are used are filled to capacity ( vjt {0, djt}). it iCy 0 Fractional demand: Any positive regular, overtime production, and subcontracting quantity is at full capacity ( xt {0, K (+1) K } and {0, Ci}), and there exists at most one (period, se gment)-pair in which the segment-capacity is used partially (0 < vjt < djt). i ty 3.3.2.1 Determining candidate demand leve ls for a regeneration interval For fractional supply RIs, all vjt variables must take values of either 0 or djt in an RI. Based on our previous discussions about optimal demand vector propert ies, there exist at most O ( JmaxT ) different candidate dema nd vectors for a regeneration interval when all vjt variables must be 0 or djt. The fractional demand RI type allows at most one segment of a revenue curve within an RI with 0 < vjt < djt. In this case, all production and subcontracting levels are either zero or at capacity. As a result, the number of possible different internal production levels (regular plus overtime) is O (T ) when is an integer, and O ( T2) otherwise, while the possible number of levels of the cumulative subcontracted amount to any single subcontractor i is O ( T ). Since there are M unique subcontractors, for each possible (vector) value of internal pr oduction levels, we need to consider O ( T ) possible values of subcontractor production for each of the M subcontractors. Therefore, considering at most O (TM+1) and O ( TM+2) different demand vectors is sufficient for the integer and general case, respectively.

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51 3.3.2.2 Regular capacity as integer multiple of overtime capacity Given an RI and a candidate demand vector, let KCb ttDqM i i i/ 1',1 denote the number of integer multiples of the capacity parameter K that are required to satisfy a total demand of D ( t t ') where bi is the number of periods in which subcontractor i produces at full capacity In addition, let M i i iCbqKttDf11', denote the remainder. If f > 0 then this candidate demand vector corresponds to a fractional supply RI a nd a quantity equal to f must be produced either in a fractional regular time production peri od, a fractional overtime period, or in a fractional subcontracting period. Note that the actual amount of producti on in a fracti onal regular time production period can be wK + f for any w = 0, 1, a fractional overtime period will contain a production amount equal to f and the fractional subcontracted amount for subsontractor i can be sK + f for any s =0,..., TKCi,/ min Let (, u, r, e ) denote a node in the la yered network, where denotes the current time period (or layer), u denotes the cumulative production as a multiple of K up to and including period r denotes an M -vector of the number of full production periods for each subcontractor that have taken place up to and including period t (the ith element of the vector r indicates the cumulative number of fu ll production periods for subcontractor i ), and e = 1 if the fractional amount has been pr oduced already (and is 0 otherwise). Arcs between layer and layer +1 will be of the following types ( = t , t '): C1 Zero production arcs: From node (, u r e ) to node ( + 1, u r ', e )

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52 (where r = r + l and l is an M -vector containing a 1 in the ith position if subcontractor i produces at capacity in period (and is 0 otherwise)). Arc costs: M i ii iCgl1 1 if for i = 0, 1, M if iibr M i i iCr1 + ef D ( t + 1), and otherwise. C2 Fractional regular production arcs: From node (, u r, 0) to node ( + 1, u + w r 1) for w = 0, if u + w q and for i = 0, 1, M iibr Arc costs: c+1( wK + f )+ M i ii iCgl1 1 if ( u + w ) K + M i i iCr1 +f D ( t + 1), and otherwise. C3 Full capacity regular production arcs: From node (, u r,e) to node ( + 1, u + r ,e) if for i = 0, 1, M iibr Arc costs: c+1(K )+ M i ii iCgl1 1 if ( u + ) K + M i i iCr1 + ef D ( t + 1),and otherwise. C4 Fractional overtime production arcs: From node (, u r,0) to node (+1, u + r 1) if u + q and for i = 0, 1, M iibr Arc costs: c+1(K ) + M i ii iCgl1 1 + o+1( f ) if (u + ) K + M i i iCr1 + f D ( t + 1), and otherwise. C5 Full capacity overtime production arcs: From node (, u r,e) to node ( + 1, u + + 1, r e ) if u + + 1 q and for i = 0, 1, M iibr Arc costs: c+1((+1) K ) + M i ii iCgl1 1 + o+1( K ) if ( u + + 1) K + + ef D ( t + 1), and otherwise. M i i iCr1 C6 Fractional subcontracting arcs for subcontractor m : From node (, u r 0) to node (+1, u+w, r ', 1) for w = 0, TKCm,/ min and if u + w q and for i = 0, 1, M mmrr iibr

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53 Arc costs: M i ii iCgl1 1 + fwKgm 1 if ( u + w ) K + M i i iCr1 + f D ( t + 1), and otherwise. C7 Demand satisfaction arcs: From node ( t' q b 1) to the sink, where b is an M -vector containing bi in the ith position. Arc costs: '1 1 J t jj tjv As in the network for the single subcontract or case, we allow parallel arcs in the network. The shortest path from a single source node to the si nk provides an optimal solution for the RI and corresponding candidate demand vector. To determine the complexity of this RI subproblem, note that the number of nodes is in a layer is O (TM+1) so that the total number of nodes in the network is O (TM+2). Since each node has outdegree O ( M 2T ), the number of arcs in the network is O ( M 2TM+3). Considering all possible subcontracting levels (which is O ( TM)), the shortest path for each regeneration interval of this type can be calculated in O ( M 2T2M+3) time for a given demand vector. When the RI is a fractional demand RI, the arcs of types C2, C4, and C6 are not needed. When these arcs are rem oved, each node has outdegree at most M and the number of arcs in the network to O ( MTM+2). The optimal solution for such cases is then given by the shortest pa th from a source node (t 0, z 1) to the sink, where z is a zero vector with size M 3.3.2.3 Regular capacity as any positive multiple of overtime capacity Let m denote the number of periods in wh ich no overtime is used and regular production is at capacity, and let n denote the number of periods where regular and overtime production are at cap acity. If we again let bi denote the number of periods

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54 where subcontractor i produces at capacity, then total amount of demand satisfied in a regeneration interval ( t t ') can be expressed as 1,'(1)M C i iDttmKnKbCf i In this equation, f may correspond to fractional regular time production, fractio nal overtime production, or a fractional subcontracting amount. Let ( t m ', n ', r, e) represent a node in the network, where t is the current period, m is the number of cumulative regu lar full capacity production periods, n denotes the number of cumulative full capacity overtime periods, and r is the vector of the number of cumulative full capacity subcontra cting periods including period t. We set e = 1 if the fractional amount has been produced already, otherwise it is 0. Arcs between layer and layer +1 will be of the following types ( = t , t ): D1 Zero production arcs: From node (, m ', n ', r, e ) to node ( + 1, m ', n ', r ,e) if for i = 0, 1, M iibr (where r = r + l and l is an M -vector containing a 1 in the ith position if subcontractor l produces at capacity in period (and is 0 otherwise)). Arc costs: M i ii iCgl1 1 if M i i iCr1 D ( t + 1), and otherwise. D2 Fractional regular pro duction arcs: (only if 0 f < K ) From node (, m ', n ', r, 0) to node ( + 1, m ', n ', r ', 1) if for i = 0, 1, M iibr Arc costs: c +1( f ) + M i ii iCgl1 1 if m'K + n K + M i i iCr1 + f D ( t + 1), and otherwise. D3 Full capacity regular production arcs:

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55 From node (, m ', n ', r, e ) to node ( + 1, m '+1, n ', r', e ) if for i = 0, 1, M iibr Arc costs: c+1 (K ) + M i ii iCgl1 1 if ( m '+1) K + n K + M i i iCr1 + ef D ( t + 1), and otherwise. D4 Fractional overtime pr oduction arcs: (only if K f < (+1) K ) From node (, m ', n r, 0) to node ( + 1, m ', n ', r ', 1) if for i = 0, 1, M iibr Arc costs: c+1( f ) + o +1( f K ) + M i ii iCgl1 1 if m'K + n K + + f D ( t + 1), and otherwise. M i i iCr1 D5 Full capacity overtime production arcs: From node (, m ', n r, e ) to node ( + 1, m ', n '+1, r ', e ) if for i = 0, 1, M iibr Arc costs: c +1((+1) K ) + o+1( K ) + M i ii iCgl1 1 if m'K + ( n '+1) K + + ef D ( t + 1), and otherwise. M i i iCr1 D6 Fractional subcontracting arcs for subcontracting i : From node (, m ', n ', r, 0) to node ( +1, m ', n ', r ', 1) if for i = 0, 1, M iibr Arc costs: + ) (1fgm M i ii iCgl1 1 if m'K + ( n '+1) K + M i i iCr1 + f D ( t + 1), and otherwise. D7 Demand satisfaction arcs: From node ( t' m n b 1) to the sink, where b is a M -vector containing bi in the ith position. Arc costs: '1 1 tJ jj tjv

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56 The shortest path from the source node ( t 0, 0, z 0) to the sink in the resulting graph provides an optim al solution, where z is a zero vector with size M The number of nodes is in a layer is O ( TM+2) so that the total number of nodes in the network is O ( TM+3). Since each node has outdegree at most O ( M2), the number of arcs in the network is O ( M2TM+3) as well. For each candidate demand vector there are O ( T2) potential choices for m and n and O ( TM) potential choices for the b vector, and the total time required to find an optimal solution for the RI and a corresponding candidate demand vector is O ( M2T2M+5). When we consider candidate demand vectors corresponding to an RI of the second type, the values of m and n and b are uniquely defined. In addition each node has outdegree at most O ( M ), and the number of ar cs in the network is O ( MTM+3). Therefore we can conclude that the cost of each RI of the second type can be computed in O ( MTM+3) for a given demand vector. The optimal solution for such cases is then given by the shortest path from a source node ( t 0, 1) to the sink. 3.3.2.4 Complexity of solution approach Proposition 3.10. There exists an O ( M2Jmax2T2M+6) algorithm to solve the [CPPPCS] problem with piecewise li near revenue curves, when is a positive integer. When is any positive scalar, there exists an O ( M2JmaxT2M+8) algorithm to solve the [CPPP-CS] problem with piec ewise linear revenue curves. Proof. When regular capacity as integer multiple of overtime capacity, for candidate demand vectors corresponding to fractional production RIs, the optimal cost can be calculated in at most O ( M2T2M+3) time and the number of such candidate demand levels to be considered is O( JmaxT ) for a total of O( M2 JmaxT2M+4. For candidate demand vectors corr esponding to RIs of the second type, the optimal cost can be calculated in at most O ( MTM+2) time and the number of such candidate demand levels is O (TM+1) for a total of O( M2T2M+3) time. Since there are O ( T2) RIs, it takes O (max { M2JmaxT2M+6 M2T2M+5}) M2Jmax2T2M+6 time to calculate all optimal RI costs. For fractional production RIs, the optimal cost can be calculated in at most O ( M2T2M+5) time and the number of such candida te demand levels to be considered is O ( JmaxT ) for a total of O ( M2JmaxT2M+6). For the second type, the optimal cost for

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57 an RI can be calculated in at most O ( MTM+3) time and there are O ( TM+2) candidate demand levels for this case. Since there are O ( T2) RIs, it takes O (max{ M2JmaxT2M+8, MT2M+7}) O ( M2JmaxT2M+8) time to calculate all optimal RI costs. 3.3.3 Capacity Planning with Multiple Subcontractors We consider the case where the manufact urer wishes to determine its optimal internal capacity level for the production hori zon when several subcon tractors exist. We can modify the model in Section 3.2 to account for the case of several subcontractors. For both cases of capacitated and uncapacitated subcontractors, we minimize a concave function over a polyhedron, which implies that an optimal extreme point solution exists. We let [CPPP-US(K )] and [CPPP-CS( K )] denote the capacity planning problems for uncapacitated and capacitated subcontrac tor cases respectively. The following proposition characterizes the struct ure of extreme point solutions. Proposition 3.11. For every extreme point solution for [CPPP-US( K )] and [CPPPCS( K )] with K > 0 there exists an RI ( t t' ) in which all intern al production levels are either at zero, regular-time capacity, or overtime capacity. If subcontractors have no capacity limits, all subcontracting levels are at zero (in the uncapacitated case) or each subcontracting level is either at zero or at capacity. In addition, there is no fractional demand satisfaction. That is, x {0, K ( + 1) K }, for all = t ,...,t' ; vj {0, dj}, for all = t ,...,t' ; j = 1, J. Uncapacitated Subcontractors ([CPPP-US( K )]) i y = 0, for all = t ,...,t' and i =1, M ; Capacitated Subcontractors ([CPPP-CS( K )]) i y = {0, Ci}, for all = t ,...,t' and i =1, M ; Proof. The proof follows the discussion for Pr oposition 3.5. At an extreme point solution, all demand in at least one RI, ( t t ') is satisfied by an integer number n of full-capacity regular production periods and an integer number m of full capacity overtime production periods, and, if subcontractors are capacitated, bi full capacity subcontracting periods for each i Therefore, we can conclude that if

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58 subcontractors have no capacity limits, th e size of the set of candidate capacity levels is same as the single subcontrac tor case. Otherwise, for a given demand vector, we have 1,'1(1)M i i iDttmKnKbC So, given an RI and candidate demand vector, the total number of different levels of K such that 1,'1(1)M i i iDttmKnKbC with n m and bi, i =1, M equal to nonnegative integers is O ( TM+2). Since each of the O ( JmaxT3) candidate combinations of RIs and demand vectors may imply O ( TM+2) capacity levels, we have O ( JmaxTM+5) distinct capacity levels. When is integral, we have O ( TM+1) total production and subcontracting quantities to consider for a given demand vector. Consideri ng potential RI and demand vector combinations implies O ( JmaxTM+4) candidate capacity parameter levels. For each candidate capac ity parameter level K we can solve a [CPPP-CS ( K )] problem which yields the following corollary. Corollary 3.2. The problems [CPPP-US ( K )] and [CPPP-CS ( K )] can be solved in polynomial time. Proof. The proof follows from Propositions 3.10 and 3.11 by solving the [CPPPUS] and [CPPP-CS] pr oblems with piecewise-linear, concave, and nondecreasing revenue curves for each candidate capacity level. For the uncapacitated case, when is integral, the overall complexity for solving [CPPP-US( K)] is 2310 maxOMJT while for general values of the resulting worst-case complexity is 213 maxOMJT For the case of capacitated subcontractors, when is integral, the overall complexity for solving [CPPP-CS( K)] is O ( ) and if is not integral the complexity is O ( ) 10332 max 2 MTJM 1332 max 2 MTJM Tables 3.1 and 3.2 summarize the complex ity results for both integer and general values of The results in these tables indicate th at the increase in complexity when considering multiple uncapacitated subcontractors is only a factor of M However, when the subcontractors are capacitated, the comple xity increases by an additional factor of MT2M when capacity is not a decision variable, and by MT3M when capacity is a decision variable (we provide a proof of the NP-Hardness of the multiple capacitated subcontractor case in the Appe ndix A). Our approach theref ore only solves the problem

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59 with multiple capacitated subcontractors in polynomial time when the number of subcontractors is fixed (although this is still a noteworthy result, i.e., when a producer must consider a small number of capaci tated subcontractors with time-invariant capacities, the problem of setting optimal internal capacity and demand levels remains polynomially solvable). In prac tice, we would expect that the number of subcontractors considered by a producer would typically be lim ited to a reasonably small set of qualified firms. Table 3-1. Complexity results with M subcontractors (integral ) Uncapacitated Subcontractors Capacitated Subcontractors Fixed Internal Capacity O ( MJmax2T6) O ( M2Jmax2T2M+6) Capacity as Decision Variable 2310 maxOMJT O ( ) 10332 max 2 MTJM Table 3-2. Complexity results with M subcontractors (general ) Uncapacitated Subcontractors Capacitated Subcontractors Fixed Internal Capacity O ( MJmaxT8) O ( M2JmaxT2M+8) Capacity as Decision Variable 213 maxOMJT O ( ) 1332 max 2 MTJM 3.4 Conclusions Production planning and control de cisions deal with the acqu isition, utilization, and allocation of resources to satisfy customer demands in the most economical way. Typical factors that affect producti on planning include pricing, capacity levels, production and inventory costs, and overtime and subcontrac ting costs. Past research suggests a hierarchy of optimization models that are ap plicable for these distinct decision making categories. In this chapter we developed a class of models focuse d on integrating these decisions and provided effective so lution methods for this class.

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60 In the first model, the general fixed capacity dynamic production planning problem is extended to account for pricing, tem porary capacity expansion (overtime), and subcontracting options. Although subcont racting has been considered by many researchers, integration of these decisions with overtime opportunities has not been fully discussed. From our perspective, overtime and subcontracting should be considered as separate notions when modeling. Although bot h practices help manuf acturer to increase supply capacity, there are important differen ces between overtime and subcontracting. We also included integrated capacity and pr icing decisions in the model because these two decisions are highly in terrelated and str ongly influence production planning. Our models in this chapter determine a producers optimal price, production, inventory, subcontracting, ove rtime, and internal capacity levels, while accounting for production economies of scale and capacity costs through concave cost functions. We use polyhedral properties and dynamic programming techniques to provide polynomial-time solution approaches for obtaining an optimal so lution for this class of problems when the internal capacity level is time-invariant. A lthough the models in this chapter incorporate several operational decisions, they require piecewise linear revenue curves to obtain polynomial-time solution procedures. In Chapte r 4, we introduce more general concave revenue functions for our procurement planni ng models. In additi on, the case where a single price is set for the enti re horizon is also considered.

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CHAPTER 4 CAPACITATED PRODUCTION PLANNING MODELS WITH PRICE SENSITIVE DEMAND AND GENERAL CONCAVE REVENUE FUNCTIONS Introduction Recent research in operations planning recognizes the impacts that demand and revenue planning measures ha ve on operations-related costs. Measures to manage revenues through pricing directly influence ope rations requirements, and the combination of revenues and operations costs often largely determine an operations overall profitability. Determining pricing and dema nd plans without considering their impact on operations costs can therefore lead to plans that drive high revenues, while at the same time sacrificing potential profits due to associ ated operations-related co sts. When a firm wishes to maximize the profit associated w ith the production and delivery of a good with price-sensitive demand (as opposed to, for ex ample, maximizing market share), it is important to consider both the revenue and cost factors that drive profitability. This has led many operations researchers in recent years to consider models that integrate revenue and operations decisions with a goal of profit maximization. In Chapter 3, we considered integrated capacity, demand and production planning models. In this chapter, we also addre ss a class of combined revenue and operations planning problems which have not been previ ously considered in the literature. In particular, we consider a class of discrete -time finite-horizon opera tions planning models with production capacity limits, where demand for a good is price sensitive. As opposed to the problems discussed in Chapter 3, we onl y consider problem contexts in which there 61

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62 are no temporary capacity expansion mechan isms, such as employing overtime hours and subcontracting. Instead, we focus our at tention on handling more general concave revenue functions. We account for economies of scale in production of the good using a fixed plus variable production co st structure, and also consider linear holding costs in the amount of inventory of the good held at the end of a period. Within this class of problems we consider the case in which price may vary dynamically by period, as well as the case in which a firm wishes to charge a single price throughout the entire planning horizon. To the best of our knowledge, this is the first work that combines price-sensitive demands with general concave revenue f unctions, production economies of scale, and production capacity limits in a single integr ated (and tractable) model with a goal of profit maximization. We restrict ourselves to problem contexts in which production capacity limits are time-invariant, i.e., th e planner faces the same production capacity limit in each period of the planning horizon. Th is chapter contributes to the literature by demonstrating that this problem class can be solved in polynomial time for a broad class of general concave revenue functions, for both the dynamic and static pricing cases. (As we will see, in the static pricing case, a lthough the demand function in a given period can take any general concave form, to achieve polynomial solvability we require that a specific relationship holds between revenue func tions in different periods.) The resulting solution approaches expand the set of availabl e tools for determining how to best match production supply capabilitie s with demands. In addition, the structural properties of optimal solutions lead to in sights on how demand management decisions are interrelated with production economics and capacity limits in complex settings involving combined production and demand planning.

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63 The remainder of this chapter is organized as follows. Section 4.1 deals with the problem when prices may vary dynamically by time period. Section 4.1.1 provides the problem definition and corres ponding model formulation, wh ile Section 4.1.2 derives a series of properties of optimal solutions th at lead to a polynomialtime solution approach under very general conditions on the revenue functions. Section 4.2 deals with the problem when a constant price is required over the time horizon. The constant price problem is defined and a model is formulated in Section 4.2. 1, while in the remainder of Section 4.2 a polynomial-time solution approach is developed. Section 4.3 then summarizes our results for this chapter. 4.1 Model and Solution Approach with Dynamic Prices 4.1.1 Problem Definition and Model Formulation We consider a T period planning model for a single item with production capacity equal to some positive value K in every period. This capacity may correspond to a production capacity limit, a truckload supply capacity limit, or other supplier limit on production quantity in a period, depending on th e context. As in Chapter 3, demand in period t ( t = 1, T ), which we denote by Dt, is a decision variable, and we assume that any value of demand in period t implies a corresponding unit pri ce. The total revenue in period t is determined by a nondecreasing concave function of demand ()tt R D defined for Dt 0, where the corresponding price as a function of demand equals ()/tt t R DD when Dt > 0 and we assume that 0lim()/tttt DRDD Marginal revenue is nonincreasing in demand, or equivalently demand is nonincreasing in price (and the rate of change of this demand is nonincreasing in revenue; thus since t R is concave, demand is nonincreasing in price). We assume that production in period t incurs a nonnegative fixed order cost of

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64 St (if the amount procured is positive) plus an additional nonnegative variable cost of per unit procured. A nonnegative holding cost of ht is incurred for each unit remaining in inventory at the end of period t and we assume that shortages are not permitted. Therefore, if the decision variable xt denotes the amount pr ocured in period t the amount of inventory remaining at the end of period t equals tc 11tt x D and this quantity must be nonnegative for every period t For ease of notation we define ,T tT thh ,, s usscch u t and ,()()tttttT R DRDhD and we will refer to the latter function as the revenue function throughout this chapter. It is easy to see that the functions Rt are concave and nondecreasing. Our goal is to maximize total revenue less production and holding costs, which results in the following Dynamic Concave-Revenue Production Planning [DCRPP] model. [DCRPP] Maximize 1()T tttttTt t R DSycx Subject to: 1tt 1 x D t = 1, T 0 xt Kyt, t = 1, T yt {0, 1} t = 1, T Dt 0, t = 1, T It is straightforward to veri fy that the above formulation is equivalent to a more standard formulation that explicitly uses inventory variables. The objective function maximizes revenue less production and holding co sts. The first constraint set ensures that all demand is met and that inventory is nonnegative, while the second constraint set limits production in any period t to no more than capacity K if the binary ordering

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65 variable yt equals 1, and to zero otherwise. For the case of piecewise-linear, nondecreasing, and concave re venue functions, Geunes, Ro meijn, and Taaffe (2006) showed that this problem can be solved in polynomial time. Here we provide a more general (and quite different ) solution approach that applies to general concave (nondecreasing) revenue functions. Because we allow the revenue functions to be general concave functions, we need to introduce some additional notation before we proceed. For all t = 1, T let ()t D and ()t D denote the left and right derivatives of the function Rt at D so that the set of subgradients of the function Rt at D is given by ()(),()ttt R DD D Technically, for a concave function, these should be called supergradients since they overestimate a concave function. For ease of presenta tion, however, we employ the commonly used term subgradients to imply supergradients when speaking of concave functions. The value ()t D can be interpreted as the marginal rate of increase in revenue as we increase demand at D while ()t D provides the marginal rate of decrease in revenue as we decrease demand at D Note that (0)t since we assume that the function Rt is defined only for D 0, and (0)t We also define the lim iting slope of the revenue function as In general, the set of subgradients is a nonempty interval in while if Rt is differentiable at D it reduces to the singleton ,()lim()0ttt DDh T {} ()t R D Moreover, since Rt is concave, the subgrad ients are nonincreasing in D in the sense that ()()tt D D whenever 0 DD Finally, it will be convenient to define the inverse of Rt as follows: ()0:()tt R rDrRD

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66 It is easy to see that ()t R r is an interval in {} so that we can express it as ()(),()ttt R rr r The set ()t R r can be interpreted as an interval of demands on which the unit revenue is constant and equal to r Put differently, the value can be interpreted as the marginal rate of increase in demand as we increase the unit revenue at r while provides the marginal rate of decrea se in demand as we decrease the unit revenue at r Alternatively, we can interpret ()tr ()tr ()tr and ()tr as the endpoints of an interval of demand values such th at the revenue curve in period t has slope r for all demand values between th ese endpoints (thus if ()()ttr r these correspond to a linear segment with slope r of the revenue curve). Note that (){}tRr for Since Rt is concave, the sets 0tr () ()t R r are nonincreasing in r in the sense that whenever Moreover, if Rt is strictly concave then the set ()()ttr r 0 rr ()t R r is a singleton for In general, however, we assume that the number of values of r for which is Jt J < which means that the revenue function Rt has only a finite number of linear se gments. We denote these values by rtj for j = 1, Jt and t = 1, T and, in addition, denote the or dered sequence of these values by where ()(0)tr t r ()()ttr )()( )1(......M mrrr T t tJM1 Note that the values r(1), r(M) thus provide the slopes of all lin ear segments of the revenue curves in all periods. 4.1.2 Development of Solution Approach for DCRPP Observe that, for any fixed vector of demands ( D1, D2, DT), the resulting problem is an equal capacity lot-sizing problem, which can be solved in polynomial time (see Florian and Klein (1971), a nd van Hoesel and Wagelmans ( 1996)). In particular, an optimal solution to the [DCRPP] problem exists consisting of sequences of capacity

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67 constrained regeneration intervals. As in Chapter 3, we de fine a regeneration interval (RI) as a sequence of periods s, s + 1, u such that Is-1 = Iu-1 = 0 and It > 0 for t = s , u where It is the inventory remain ing a the end of period t and is denoted by ( s, u ) (with u > s ). A capacity constrained RI is an RI such that the production quantity in every period within the RI except at most one is equal to zero or the capacity limit K We will refer to a period in which production is neither at zero nor at capacity as a fractional production period and to a period in which xt = K as a full production period. These properties and definitions will play an impor tant role in the development of solution methods for the [DCRPP]. Our solution strategy is to find an optimal solution, i.e., a maximum profit solution with at most one fractional production period, for each RI (in the remainder, we therefore consider only capac ity constrained RIs). If we can efficiently solve the problem for each RI we can, sim ilarly to Florian and Klein (1971), apply a shortest path approach for solving the overall problem. 4.1.2.1 Properties of optimal RI demand vectors We will first focus on establishing properties of optimal solutions for a given RI and, in particular, on properties of optimal demand sequences for an RI. The following proposition provides a condition th at the subgradients at optimal demand values in an RI must satisfy. Proposition 4.1. Any set of optimal demand values for RI ( s u ) satisfies or, equivalently, 1()u tt tsRD D) ,...,1 ,...,1max()min().tt tt tsu tsuD (4.1) Proof. Consider a feasible set of demand values Dt, t = s, ..., u for RI (s, u ) for which condition (4.1) does not hold. Then there are two periods j and k in the RI with s j k < u for which ()( j jkD kD (and therefore Dk > 0 since (0)k ).

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68 Now suppose that we increase the demand in period j by > 0 and decrease the demand in period k by the same amount. Since we are within an RI we can do this without changing any procurement quantities and, if is chosen small enough, without causing any inventory levels to become equal to zero. The corresponding change in the objective function value equals ()()()(jjjjkkkkRDRDRDRD ) Noting that 0()()()() lim ()()0jjjjkkkk jjkkRDRDRDRD DD yields that the solution can be strictly improved and can therefore not be optimal. Proposition 4.1 indicates that if we have an optimal demand value for one period in an RI then the candidate set of demands th at must be considered for the remaining periods in the RI can be substantially reduced. Corollary 4.1. An optimal solution for the RI (s u ) satisfies ()tt D Rr for t = s, u and for some ,...,1max()t tsur A somewhat stronger result can be obtained if we consider the important special case where all revenue functi ons are strictly concave. Corollary 4.2. When all revenue functi ons are strictly concave, there exists some ,...,1max()t tsur such that an optimal solution for the RI (s, u ) is given by ()() if (0) 0otherwitt t trrr D se. for t = s, u 1. Corollary 4.2 indicates that if we have identical strictly concave revenue curves in every period, then by the definition of Rt( Dt), the optimal value of marginal revenue for periods 1 and 2 within the same RI (assuming w ithout loss of generality that 1 < 2) will differ by the cost to hold a unit of inventory from period 1 to period 2 (which implies the optimal prices will therefore vary within an RI, even if costs are timeinvariant, assuming non-zero holding costs). For periods in different RIs, however, we

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69 cannot draw such conclusions on the relationship between optimal prices. Thus, the structure of optimal pricing d ecisions is intimately related to the structure of the optimal procurement plan. Dj Dk dka dkb Period j Revenue Curve Dj Dk Rj( Dj) Rk( Dk) Rj( Dj) dja Rk( Dk) Period k Revenue Curve Figure 4-1. Candidate subgr adient values and correspond ing candidate demand values The results of Proposition 4.1 and Corollaries 4.1 and 4.2 are illustrated in Figure 4.1: the horizontal lines at 1r and 2r represent candidate subgradient values and the corresponding candidate demand values associ ated with the revenue curves in two periods ( j and k) within an RI. In the figur e, candidate subgradient value 1r implies unique values of demand in periods j and k ( djb and dkb), while candidate subgradient value 2r implies a unique value of demand in period j ( dja) but a range of values in period k ([0, dka]). This motivates the development of an algorithm for finding an optimal solution for each RI by searching among candidate subgradient values. In particular, we will show that we only need to consider a relatively small number of such values. In the next two sections we will study RI solutions with one fractional procurement period and RI solutions with no fractional pr ocurement period, respectively. djb 1r 2r

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70 4.1.2.2 Characterizing optimal demand in RIs containing one fractional procurement period Recall that any RI contains at most one fractional procurement period. This section discusses how to find the best RI soluti on that contains exactly one fractional procurement period by characterizing all candi date subgradient valu es and corresponding candidate sets of demand values for such solutions that satisfy Proposition 4.1. The following proposition shows that we only need to consider a single candidate subgradient value for an RI and correspondi ng candidate demand values if we fix a particular period within that RI to be the fractional procuremen t period. Note that a set of demand vectors for an RI can of course only be a candidate optimal solution if it can be feasibly procured. Therefore, for RI ( s, u ) we only need to consider demand values Dt, t = s, ,u that satisfy the condition: KsDst t1 for ,...,1. su (4.2) In the following we will not explicitly veri fy this condition. However, any set of candidate demand values that violates condition (4.2) can of course be eliminated from further consideration. Proposition 4. 2. If period (with s < u ) is the fractional procurement period in a solution for RI ( s, u), i.e., 0 < x < K then we only need to consider sets of candidate demand values Dt, t = s, ,u 1 that satisfy ,()()ttTtt D cD or, equivalently, ,(ttTDRc ) for t = s, u Proof. Suppose that we have a solution for which this condition does not hold. We consider two cases:

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71 ,()tt T D c for some t = s, u 1. Then, since 0 < x < K we can increase both x and Dt by some small amount > 0. The resulting increase in objective function value is equal to ,()()ttttT R DRDc Since , 0()() lim ()0ttttT ttTRDRDc Dc the given solution can be strictly im proved so that it cannot be optimal. ,()ttT D c for some t = s, u 1. This immediately implies that Dt > 0 so that we can decrease both x and Dt by some small amount > 0. The resulting increase in objective function value is equal to ,()()TttttcRDRD Since , 0()() lim ()0Ttttt TttcRDRD cD the given solution can be strictly im proved so that it cannot be optimal. Proposition 4.2 says that if period is a fractional procurement period in an RI, then c,T must lie in the set of subgradients of th e revenue curve for each period in the RI. When the revenue function is st rictly concave, then we can interpret Proposition 4.2 as requiring that marginal cost equal marginal re venue. The appropriate marginal cost term, however, is determined by the fractional procurement period. Proposition 4.2 also shows that if we know that period is a fractional procurement period then the value c,T allows us to uniquely determine demand values sati sfying the property for periods in the RI unless c,T corresponds to the slope of a linear segment of the re venue curve. We will next show that we may, in the presence of a fractional proc urement period, restrict our attention to only the left endpoi nts of these linear segments. Proposition 4.3. If period s < u is the fractional procurement period in a solution for RI ( s, u), i.e., 0 < x < K then we only need to consider the set of candidate demand values that is given by ,(ttT ) D c t = s , u Proof. Suppose that we have a solution in which period is the fractional procurement period that satisfies the prope rties in Proposition 4.2 and, moreover,

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72 ,(ttT ) D c for some t = s , u Now consider a change of Dt by and a corresponding change in x by without changing the procurement quantities in any other period. If ,(ttT ) D c the corresponding change in objective function value is by definition equal to 0 as long as the solution remains feasible. The result now follows since either (i) the solution with ,(ttT ) D c is indeed feasible, or (ii) for a smaller value of one of the inventory levels becomes 0 (which means that we obtain a solution with the same objective function value as the current one but with periods t and in different RIs), or (iii) the procurement level in period becomes K (which means that period is no longer a fractional procurement period). Proposition 4.3 says that we need not consider RIs containing both a fractional procurement period and a period whose demand leve l is in the interior of a linear segment of its revenue function. In the special cas e where the revenue curve in a period, say period j in the RI is piecewise linear and c oncave, Proposition 4.3 implies that if a fractional procurement period exists in this RI then without loss of optimality the demand level in period j is at a breakpoint of its piecewise-linear revenue curve. This result is consistent with the resu lts presented in Geunes, Romeijn, and Taaffe (2006). 4.1.2.3 Characterizing optimal demands in RIs containing no fractional procurement period Recall that it is possible for an optimal RI solution to contain no fractional procurement periods, which means that all periods in which procurement occurs are full procurement periods. We will next discuss how to determine candidate RI solutions without a fractional procurement period. Consider such a solution for some RI ( s, u ). Then the total procurement for the RI, and th erefore the total demand satisfied in the RI, must be an integer multiple of the procurement capacity K i.e., it must be equal to fK for some f = 0, 1, u s Given a value of f Proposition 4.1 says that the demands in the RI and the corresponding subgradient value r must satisfy the following set of equations: ()() for ,...,1ttttDrDtsu

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73 1u t tsDfK or, equivalently, by us ing Corollary 4.1: ()() for ,...,1tttrDrtsu 1u t tsDfK (4.3) It is straightforward to show that wh en the optimal RI solution contains no fractional procurement periods, the value of r that satisfies (4.3) must be at least as great as the maximum value of c,T among all periods in the RI such that procurement is at capacity. When strict inequality holds, Equation (4.3) illustrates how capacity serves as a bottleneck for gaining additional profit. For example, given an optimal solution satisfying (4.3) if the revenue curves are strictly c oncave, then the marginal increase in revenue available by increasing capacity exceeds the associ ated marginal supply cost, implying a potential for increased profit (we must, however, account for the associated cost increase as a result of the capacity increase). Given a value of f {1, s u }, a demand vector for an RI satisfying (4.3) provides a candidate for an optimal demand vector for RI ( s, u ). Our approach to finding a solution for all values of f and a given RI is based on th e monotonicity of the intervals ()(),()ttt R rr r Let m1, mM(s,u) denote the index set of the values )(m r (previously defined in Section 4.2.1) corresponding to the periods in RI ( s u ) and let m0 = 0 and mM(s,u)+1 = M +1 with corresponding values (0)()tr and (recall that each value of r(m) corresponds to the slope of a linea r segment of a revenue curve in some period within the RI, and we assume that there are M ( s u ) such values in the RI ( s, (1)(0)M tr

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74 u )). Then the problem decomposes into a sequence of subproblems that alternate between two types. The first type of subproblem considers values )( )(1 j jm m r r r for some j = 0, M ( s, u ), where we know that for such intervals of r we have ()()ttr r for any value of r on the interval, and this is a strictly decreas ing function of r. A candidate set of demands for a given f and j can then be found by using binary search to find a root of the equation 1()u t tsrfC over the interval )( )(1 j jm m r r r which is unique if one exists. Since the functions are decreasing in r on this interval, a root exists if and only if ()tr 1 )( 1 )(1u st m t u st m tj jr fKr The second type of subproblem considers (4.3) for a fixed value )(jm r r for some j = 0, M(s, u) + 1, where we know that )( )(j jm t m trr for at least one period t = s, u 1 (except potentially for j = 0 and j = M(s, u) + 1). Existence of a solution for a given f and j can easily be established by verifying whether 1 )( 1 )( u st m t u st m tj jr fKr Since any set of demands that solves the system has the same objective function value if it is feasible, we will then simply start by initializing for t = s, u. Then, we decrease the demands sequentially in periods t = s, u such that )(jm t trD )( )(j jm t m trr until such a solution is found as follows. Starting with t = s, we check whether )( )(j jm t m trr exceeds the excess demand. If

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75 not, we reduce the demand in period t from )(jm tr to )(jm tr and proceed with period t + 1. If so, we reduce the demand in period t by the excess demand and stop. We can summarize the algorithm for the case of no fractional (NF) procurement periods as follows: Algorithm NF Step 1. Set j = M(s, u) and f = 1. Step 2. If 1 )( 1 )(1u st m t u st m tj jr fKr solve a subproblem of the first type, store the solution, and set f = f+1. If f 0 repeat Step 1, otherwise stop. Step 3. If 1 )( 1 )( u st m t u st m tj jr fKr solve a subproblem of the second type, store the solution, and set f = f+1. If f 0 repeat Step 2, otherwise stop. Step 4. Set j = j. If f us and j 0 go to Step 1, otherwise stop. It is easy to verify that, if the revenue functions are piecewise linear and concave, all candidate solutions can be found by orde ring and considering the slopes of all segments in the RI in decreasing order to fill the capacity for each value of f. This is precisely the approach presented in Geunes, Romeijn, and Taaffe (2006). It is tempting to conclude, based on our an alysis so far, that the optimal demand values (and hence prices) do not depend on the valu es of the fixed order costs. That is, in the previous section, we concluded that th e candidate optimal demand values for an RI depend on the relationship between the variable cost in the fractional procurement period and the revenue functions. Similarly, in th is section we showed that candidate demand values are determined based on the propertie s of the revenue functions and the capacity limits (see Equation (4.3) ). While these factors do determine candidates for optimal demand levels (and prices), obtaining an optim al RI solution require s solving a shortest

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76 path subproblem with arc costs that explicitly account for or der costs. Therefore, while the complexity of the relationship between optimal demand levels and fixed order costs does not permit closed-form expressions that characterize this relationship, it would be incorrect to conclude that the optimal dema nd and price values are independent of these costs. 4.1.2.4 Complexity of overall solution approach This section characterizes the complexity of an overall solution approach for the [DCRPP] problem, based on the results deve loped in the preceding sections. Our approach is based on solving a shortest path problem on a graph that contains T + 1 nodes (1, 2, T + 1), where a directed arc exists co nnecting each node to all higher numbered nodes. The cost of an arc (t, t') equals the optimal RI (t, t') solution value (note that arc cost values are actually net contribution to profit values, and we therefore solve an acyclic longest path problem after determining all optimal RI solutions and label arcs accordingly). Since this acyclic long est path problem can be solved in O(T2) time (Lawler 1976), the bulk of the solution effort lies in determining the optimal arc cost values by determining an optimal RI solution for each of the O(T2) possible RIs. Our approach is to find a collection of potentially optimal demand vectors for each RI using the results of the previous sections. Given each of these candidate demand vectors we can use a dynamic programming approach to dete rmine the best RI solution for that candidate demand vector in O(T2) time (see Florian and Klei n 1971) if one exists. (Note that while this dynamic programming approach will correctly identify whether a candidate demand vector is indeed feasible a slight computational advantage can be obtained by first verifying feasibility in O(T) time using equation (4.2) However, this will not influence the worst-case running time of the overall algorithm.) For a given RI,

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77 we will determine candidate demand vectors given that exactly one fractional procurement period exists in the RI, as well as candidate demand vectors given that no fractional procurement period exists in the RI The profit of the best solution among the corresponding candidate solutions then serves as the arc cost in the longest path problem. The best solution for an RI that contains exactly one fractional procurement period can be found by considering each period in the RI to be the fractional procurement period and finding a corresponding candidate set of demand values using Proposition 4.3, which means that there are O(T) such candidate demand vectors for each RI. Each of these demand vectors can be found by evaluating the function t for each period t in the RI. If such a function evaluation takes O(R) time (for example, if Rt is a piecewise-linear function with O(R) segments), finding a single candidate demand vector corresponding to a given fractional procurement period takes O(RT) time, and finding all T candidate demand vectors therefore takes O(RT2) time. For each of the T candidate demand vectors, we then solve the O(T2) RI subproblem (see Florian and Klein 1971) to determine the optimal procurement plan asso ciated with the candidate demand vector. Finding the best RI solution containing a si ngle fractional procurement period therefore takes O(T3 + RT2) time. We next consider the computational complexity of finding an optimal RI solution when there are no fractional procurement periods in the RI. It is easy to see that the number of candidate demand vectors found by our algorithm is O(T), because we generate at most one demand vector for each value of f. The feasibility test in Steps 1 and 2 of Algorithm NF takes O(T) time and needs to be performed O(JT) times. Finding a solution in Step 1 takes (log)O binary search iterations, where

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78 1,..., 1,...,max(0)min()tTttTt each taking O(RT) time, while finding a solution in Step 2 for a given value of f takes O(T) time. The total time required for Algorithm NF for a given RI is therefore 2( 1 log)OJRT Since there are O(T2) RIs, the [DCRPP] problem can be solved in 45( 1 log)OJRTT in the worst case. Observe that if each periods revenue curve contains only one strictly concave segm ent then Step 2 of Algorithm NF (employed for RI solutions without a fractional procurem ent period) will never be performed and the running time becomes 45(log)ORTT On the other hand, when the revenue curves are all piecewise-lin ear and concave functions then each of the functions t can be evaluated in O(log J) time and Step 1 of Algorithm NF will never be performed. In that case the running time becomes 45OJTT which improves upon the results presented in Geunes, Romeijn, and Taaffe (2005) and Geunes, Merzifonluo lu, Romeijn, and Taaffe (2006). Note also that the in itial sorting of the linear slopes takes log()OJTJT time in preprocessing and should be considered in the overall computational complexity of this problem. 4.1.2.5 Refining the solution approach While the previous sections have provided properties that lead to an effective solution approach, in this section we devel op an additional property that allows us to reduce the time required to determine an optimal procurement plan for an RI for a given sequence of demands. In particular, Propos ition 4.4 next provides a condition under which it is guaranteed that no procurement will take place in a given period. This result can be used to eliminate candidate demand sequences from consideration and speed up

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79 the dynamic programming algorithm to find an RI solution for a given vector of demands by fixing the values of some procurement quantities. Proposition 4.4. Consider demand values Dt, t = s, ..., u, for RI (s, u). Then if ,...,1min()tT suc D we can assume without loss of optimality that xt = 0. Proof. Suppose that we have a solution for RI (s, u) in which ,()tTcD for some period s < u and xt > 0. This implies that D > 0 so that we can decrease both xt and D by where > 0 and sufficiently small, without changing any of the other demands and procurement quantities. The corresponding increase in objective function value is equal to ,()()tTcRDRD Dividing by and taking the limit as goes to zero we obtain that th e rate of increase converges to This means that we can improve the solution and it can therefore not be optimal. Thus, without loss of optimality we have xt = 0. ,()0tTcD This proposition says that, for a given ca ndidate demand vector found using one of the approaches in Sections 4.1.2.2 and 4.1.2.3, we may be able to eliminate certain procurement periods from consideration, which means that a smaller dynamic programming problem needs to be solved to find the profit corre sponding to the demand vector. Moreover, we may even be able to determine in advance that no feasible and potentially optimal solution exists for this demand sequence by verifying a strengthening of feasibility condition (4.2) In particular, for RI (s, u) we only need to consider demand values Dt, t = s, u that satisfy the condition: ,...,1,...,:mintt T j s u j jfor ,...,1.su tsDKtsc D (4.4) We illustrate the usefulness of Proposition 4. 4 in Figure 4.2. In the figure, we graph ()tttt R DRD as a function of Dt for three periods in a potential RI with differentiable revenue curves, and show two consecu tive values of on the vertical axis ( and correspond to consecutive values of when these values are sorted in non-decreasing order). Observe th at in Figure 4.2, for periods j and k, the revenue tTc mTc 1, mTc tTc

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80 functions are strictly concave, while period l contains a linear segment indicated by the flat spot in the graph of ll R D Corollary 4.2 indicates th at any candidate sequence of optimal demands for an RI must correspond to equal values of tt R D across all periods in the RI. Such solutions can be visualiz ed in the figure by drawing a horizontal line across the graphs corresponding to different periods. Consider the dashed horizontal line in the figure where tt R Dr for all periods in the RI and suppose that the implied values of dt shown on the horizontal axes sum to a multiple of K (recall from Proposition 4.2 that an optimal RI solution containing a fractional procurement period will correspond to the case in which the value of r equals some ct,T value; we need only consider values of r falling between ct,T values if a solution to (4.3) exists, i.e., if a feasible set of associated demands add to a multiple of K). Proposition 4.4 now tells us that for any periods in the re generation interval such that ct,T < r, we either procure 0 or K, and that all periods in the regeneration interval such that ct,T > r will be zero procurement periods. 1, mTc mTc Dj Dk Dl r j j R D kk R D ll R D dj dk dl Figure 4-2. Illustration of Proposition 4.4. Finally, we note that, for any candidate de mand vector that is generated using the approach in Section 4.1.2.2, the fractional procurement period is known by construction.

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81 This information can be used to reduce th e size of the dynamic programming formulation that is used to find the corresponding RI solution. 4.2 Model and Solution Approach with a Constant Price 4.2.1 Model Description We next consider the situation in which a management policy or other constraint requires us to set a constant price for the good over the entire pla nning horizon. Suppose that, given a price p, we face a vector of demands D(p) [D1(p), D2(p), DT(p)]. Using the same notation as in Section 4.1.1, we now wish to solve the following Static Concave-Revenue Procurement Planning [SCRPP] problem: [SCRPP] Maximize 1()T ttttT t t p DpSycx Subject to: 11()tt x Dp t = 1, T, 0 xt Kyt, t = 1, T, yt {0, 1} t = 1, T, p 0. We can write this problem more concisel y as an optimization problem in a single decision variable, p, as follows: Maximize 1()()T t t p DpDp Subject to: p 0 where () D p denotes the cost of the optimal procurement plan associated with demand vector D(p). For a given vector D(p), the cost () D p is in fact the optimal

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82 value of the following equal capacity lot si zing problem, which can be determined in O(T3) time (see van Hoesel and Wagelmans 1996): Minimize 1 T tttTt tSycx Subject to: 11()tt x Dp t = 1, T, yt {0, 1} t = 1, T, 0 xt Kyt, t = 1, T. As in Kunreuther and Schrage (1973), Gilbert (1999), and van den Heuvel and Wagelmans (2006), we assume in the remainder of this section that the demand in period t is given by the function Dt(p) = t + td(p), where t, t 0 and d(p) is a nonincreasing and left-continuous function of p. The function d(p) is period independent and is called the demand effect (see Kunreuther and Schrage 1973 and van den Heuvel and Wagelmans 2006). Note that the optimal lot-sizing cost at a given price p then only depends on the vector of demands D(p) through the scalar d(p). With a slight abuse of notation we will denote the corresponding cost by ()dp and view the function as a function of a scalar variable. Letting 1 T t t and 1 T t t the [SCRPP] becomes Maximize ()() p pdpdp Subject to: p 0. However, it will be more convenient to fo rmulate the [SCRPP] as a function of the demand effect d rather than of the price p. The generalized i nverse of the function d expresses the price as a function of demand: ()sup:() p dpdp d This then finally yields the formulation of the [SCRPP] that we will use:

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83 [SCRPP] Maximize ()()() p ddpdd Subject to: d 0. It will sometimes be convenient to write the demand in period t as a function of the demand effect d as follows: Dt(d) = t + td. Van den Heuvel and Wagelmans (2006) show that, if is a piecewise-linear function (as is the case in the absence of cap acities), a problem of the form [SCRPP] can be solved by solving a number of (i) lot-sizing problems (each of which determines the value of (d) for a given demand effect d) and (ii) problems of the form [SCRPP] with replaced by a linear function ( each of which determines the optimal price for a given procurement plan); the number of such problems that must be solved is of the order of the number of breakpoints of Then, following Kunreuther and Schrage (1973) by assuming that problems (ii) can be solved efficiently, van den Heuvel and Wagelmans (2006) derive a polynomial time algorithm fo r the uncapacitated [SCRPP] by showing that the number of breakpoints of is O(T 2). In this section, we will develop our so lution procedure for the capacitated [SCRPP] by proceeding as follows. We first show in Section 4.2.2, that the function is piecewise linear and concave w ithin each interval of a contiguous sequence of demand intervals. In Section 4.2.3 we then show that the number of such intervals is polynomial in the number of time periods T. Finally, in Section 4.2.4, we show that the number of breakpoints of is polynomial in the number of time periods T for each of these intervals. We then combine these results in Section 4.2.5 to obtain an effective algorithm for solving the capacitated [SCRPP].

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84 4.2.2 Linearity of Cost in Demand Effect It is immediate that, as in Section 4.1, an optimal procurement plan solution to the capacitated [SCRPP] exists that consists of a sequence of RIs with an associated specific plan (SP) for each. In the remainder, we will de note an RI with an associated SP (RI-SP) by (s, u, v, F), where (s, u) (with 1 s < u T) denotes the RI, v denotes the fractional (i.e., unconstrained) procurement period s v < u, and F {s, u}\{v} denotes the set of periods in which procurement is equal to the capacity K (so that |F| is the number of full procurement periods within the RI-SP) ; procurement in the remaining periods is equal to zero: 10if if ifu t stFv x DdFKtv Kt F (4.5) Clearly, for a given demand effect d, an RI-SP is only valid if the cumulative quantity procured exceeds the cumulative demand up to each period in the RI and procurement in the fractional procurement period in each RI is in [0, K ]. A sequence of consecutive RI-SPs yields a candidate (i.e., potentially optimal) procurement plan. In this section, we will demonstrate that the cost associated with a given procurement plan (consisting of a sequence of RI-SPs) is linear in the demand effect d on the interval where the sequence of RI-SPs is valid. Using equation (4.5) we can write the total cost associated with the RI-SP as 1 ,, u ttTvvTt tF tsScKScDdFK 1 ,, u ttTvTvvTt tF tsSccKScDd 1 ,, u ttTvTvvTt tF tsSccKSc td

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85 11 ,, , uu ttTvTvvTtvTt tF ts tsSccKScc d which shows that the cost within the RI-SP is linear in d where the RI-SP is valid. This immediately implies that the cost of a procurement plan consisting of a sequence RI-SP1, RI-SPn of RI-SPs is linear in d as well where all RI-SPs in the plan are valid. We next derive the set of values of d for which a given RI-SP ( s u, v F ) is valid. For convenience, let Fs,t F {s , t } denote the set of full pr ocurement periods up to and including period t in the RI. Firstly, we require that the cumulative procurement exceeds cumulative demand for each period in the RI. Distinguishing between periods before the fractional procurement period and later ones, we see that we must have t st sFKDd for ,...,1tsv and 1 ut st ssFKDdFKDd for ,...,1tvu or, equivalently, t st s t sFK d for ,...,1tsv and 1 1 1 1\u st t u tFFK d for ,...,1.tvu Furthermore, the quantity procured in th e fractional procurement period should be feasible, i.e., 10u sDdFKK or, 1 1 111u u t t s s uu tt ssFK FK d Combining these, we obtain that RI-SP ( s u, v F ) is valid if where (,,,)(,,,)LUdsuvFddsuvF

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86 1 1 1 1 11 ,...,1 11\ ,,,maxmax ,0u u st t t L t t uu tvu tt ttFFK FK dsuvF (4.6) and 1 1 ,...,11 ,,,minmin ,t u st U s s tu tsv ssFK FK dsuvF (4.7) Clearly, if no demand effect exists for which RI-SP ( s u, v F ) is valid. Also note that in the absence of capacities we may restrict ourselves to v = s and F = so that (,,,)(,,,)LUdsuvFdsuvF (,,,)(,,,)0LLdsuvFdsus and (,,,)(,,,)UUdsuFdsus i.e., all (relevant) RI-SPs are valid for all d 0. Now consider a complete procurement plan P consisting of a sequence of n consecutive RI-SPs ( sj, sj+1, vj, Fj), with starting periods s1 = 1, s2, sn, and with sn+1 = T + 1, so that the jth RI starts at period sj and ends at period sj+1 1. The cost of this procurement plan is linear in d as long as all RI-SPs ar e valid, i.e., as long as (4.8) 1 1,..., 1,...,max(,,,)min(,,,)LL UL Pj j j jPj j jn jnddssvFdddssv 1 j jFU P If no price exists for which the procurement plan is valid. L Pdd 4.2.3 Characterizing the Structure of the Optimal Cost Function To explore the overall st ructure of the function observe that its value at d is equal to the minimum of the costs of a ll procurement plans that are valid at d. In the absence of procurement capacities, all procur ement plans with fractional procurement in the first period and no full capacity procuremen t periods are valid and, moreover, we may without loss of optimality restrict ourselves to these procuremen t plans. By the results of Section 4.2.2 it then immediat ely follows that the function is the lower envelope of a

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87 set of linear functions of d and is therefore a piecewiselinear and concave function of d. However, in the presence of a finite procurement capacity K in each period, the function instead is the lower envelope of a set of functions that is linear in d on some interval and infinite elsewhere (namely where the co rresponding procurement plan is not valid). This means that the function is still piecewise linear but not necessarily everywhere concave. Instead, it is piecewise linear and concave on each of a sequence of consecutive intervals for d, and within each of these intervals we only change the structure of the optimal procurement plan (as d changes) because it is economically attractive. The endpoints of these intervals correspond to n ecessary changes in the structure of the procurement plan not because it is economically attractive, but rather because we have reached a point where we cannot maintain feas ibility using the current procurement plan if d is increased. In other words, procurem ent in the fractional procurement period has reached K and any further increase in d requires a change in the structure of the procurement plan. We will next characteri ze the endpoints of the intervals of demand effect values d on which is concave. From Equation (4.8) it immediately follows that the candidate endpoints are given by the values and for all RI-SPs ( s u, v F ). Examining equations (,,,)LdsuvF (,,,)UdsuvF (4.6) and (4.7) we may then observe that the di stinct values of these endpoints are given by 1 1 j jt m t j jtmK d for 1,...,1;1,...,;1,..., mttTt T This means that there are O(T3) unique values of m td to consider. By construction, on any interval for d that is between two consecutive values m td the optimal cost function

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88 is piecewise linear and concave. So sorting the values m td in nondecreasing order yields a contiguous sequence of O(T3) intervals covering all d 0 on which is piecewise linear and concave. 4.2.4 Number of Breakpoints of We next study the number of breakpoints of the function To this end, define a procurement subplan for a set of t consecutive periods to consist of a set of consecutive RI-SPs for these periods, and let P(t) be the total number of valid procurement subplans for a set of t consecutive periods. A proc urement (sub)plan of length T then serves as a candidate solution for th e entire problem and P(T) is the total number of procurement plans or candidate solutions that we need to consider. Limiting ourselves to an interval for d on which is piecewise linear and concave, the number of breakpoints of the function is no more than P(T) 1 (because each of the P(T) candidate solutions is a linear function of the demand effect d on some interval). Now consider a particular RI of n periods in length with m periods preceding this RI and T n m periods following this RI, and let PmnT = P(m) P(T m n) denote the number of procurement subpl ans for all periods except the n-period RI (see Figure 4.3). Observe that for any given RI-SP for the n-period RI, there are PmnT total procurement plans (or candidate solutions). A total of n2n-1 RI-SPs exist for the n-period RI; thus there are n2n-1PmnT total procurement plans given the selected n-period RI. This provides an a priori upper bound of n2n-1PmnT 1 breakpoints associated with the given nperiod RI, which is (n2n-1 1)PmnT greater than the PmnT 1 breakpoints that may be associated with different subplans in the first m and last T n m periods. We next

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89 show, however, that instead of contributing an additional (n2n-1 1)PmnT breakpoints to the total, the selected n-period RI only contributes n 1 additional breakpoints. m periods n periods T n m periods T periods Figure 4-3. An arbitrarily selected n-period regeneration interval. Recall that the cost of a given RI-SP (s, u, v, F) as a function of d is equal to: 11 ,, , uu ttTvTvvTtvTt tF ts tsSccKSccd m n if the RI-SP is valid on the current interval for d and infinity otherwise. Therefore, the cost function associated with two RI-SPs (s, u, v, F) and (s, u, v, F ) for a given RI whose fractional procurement period is the sa me have the same slope but (possibly) different intercepts. Thus, such a set of cost functions cannot produce a breakpoint and we only need to consider at most a single li ne for each choice of fractional procurement period in a given RI (where th ere is no line if there is no valid RI-SP for the choice of fractional procurement period). The number of breakpoints associated with a particular n-period RI is therefore at most nPmnT 1. Among procurement plans containing a particular n-period RI, we next consider pairs of procurement plans of the following form: Plan 1 Plan 2mT nf f The -signs imply that both plans are identical for these periods, i.e., we have one choice of the PmnT potential subplans for periods 1, m and T m n + 1, T, and the plans only differ with resp ect to the choice of fractiona l procurement period for the nperiod RI. The -signs correspond to the setup plan in the non-fractiona l periods of the

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90 n-period RI for Plans 1 and 2. The cost functi on for this pair of plans now differs in slope and intercept by the same amount for any choice from among the PmnT potential subplans for the remaining periods, which means that we can apply the result of Lemma 2 in van den Heuvel and Wagelmans ( 2006) to conclude that PmnT 1 of these plans cannot contribute a breakpoint to the lo wer envelope. Given that Pl an 1 contains its fractional procurement period in th e first period of the n-period RI, we can consider n 1 different positions for the fractional period in Plan 2, which means that we lose a total of (n 1) (PmnT 1) breakpoints. (Note that we cannot c onsider all possible pairs of choices for the fractional period in the two plans since that will lead to double-counti ng. That is, when we consider f in the first position of the n-period RI of Plan 1 and in the second position of Plan 2, we effectively eliminate PmnT 1 of these plans from being able to contribute to the lower envelope, so we cannot reconsider the intersection of these eliminated plans with other plans.) The maximum number of breakpoints due to procurement plans containing the n-period RI is therefore equal to 1(1)11mnT mnT mnTnPnPPn 1 The n-period RI therefore contributes at most an additional n 1 breakpoints over those due to the procurement subpl ans associated with the first m and last T m n periods. Since the n-period RI was chosen arbitraril y, we can conclude that any n-period RI contributes at most n 1 additional breakpoints to th e total created by the remaining subplans in the problem. Since there are T n + 1 choices for an n-period RI, the total number of breakpoints of the function (on any interval for which is piecewise linear and concave) is no more than 1 3 1 6 1111 1(1)(TT nnnTnnTnTTTOT )

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91 4.2.5 Solution Approach for SCRPP Van den Heuvel and Wagelmans (2006) de velop a polynomial time algorithm for the uncapacitated case that we can now genera lize in order to solve the [SCRPP] problem in polynomial time. Note that, given any demand effect value d, we can determine a corresponding optimal procurement plan P in polynomial time. Let this procurement plan P consist of a sequence of n consecutive RI-SPs (sj, sj+1, fj, Fj), with starting periods s1 = 1, s2, sn, and with sn+1 = T + 1. Then, for that procurement plan, we can determine an optimal demand effect by solving the corresponding problem [P] Maximize ()()PP p ddpdAB d Subject to: LU P Pddd Where 1 11 ,, 1 1 1.j jj j j j j j jns P ttTvTvvTt jtF ts ns Pv Tt jt sAS c c K S c Bc As in Kunreuther and Schrage (1973) and van den Heuvel and Wagelmans (2006), we assume that the above problem is tractable. For example, if d ( p ) = p or, equivalently, p ( d ) = d [P] becomes the maximization of a univariate concave quadratic function over an interval which can be done an alytically and theref ore simply takes the time required to compute the problem parameters, i.e., in O ( T ) time. We may also reformulate [P] as a problem in the price rath er than the demand variable which yields a univariate convex optimization problem in the common case where both demand and revenue are expressed as a function of price, i.e., where the functions d ( p ) and p ( +

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92 d ( p )), are concave. Alternat ively, if we approximate p ( d ) or d ( p ) as a nonincreasing piecewise-linear function, we obtain a convex program on each of the intervals defining the piecewise-linear function (regardless of the functional form of the function p ( d ) or d ( p ) we are approximating). We will next outline our solution procedure for the capacitated [SCRPP], which draws on the exact algorithm developed by van den Heuvel and Wagelmans (2006) for the uncapacitated [SCRPP] (which in turn employs a heuristic approach proposed by Kunreuther and Schrage 1973, as its starting point). Howeve r, for convenience we will describe the solution proce dure using the demand effect d as a decision variable rather than the price p Kunreuther and Schrage (1973) essent ially showed that if we have an interval of demand effects on which ( d ) is piecewise linear and concave, we can apply the following approach to obtain good uppe r and lower bounds on the optimal demand effect. Suppose we have a valid lower and upper bound on the optimal demand effect, say d and d with the additional property that ( d ) is piecewise linear and concave for ddd Then we find the optimal procurem ent plan for each of these values by solving, in our case, a lot-sizing problem with equal capacities. For each of these procurement plans, we can then, in turn, fi nd an optimal demand effect by solving [P]. We continue to iterate in this manner, updating d and d as the algorithm progresses, until no further improvements are made. Kunreu ther and Schrage (1973) showed that, at any point in this algorithm, the values d and d remain valid lower and upper bounds on the optimal demand effect. Van den Heuvel and Wagelmans (2006) then used a method due to Eisner and Severance (1976) to, in cas es where the heuristic algorithm terminates with dd find a demand effect (with d ddd ) with the property that the optimal

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93 procurement plan given this demand effect is different from the optimal procurement plans at the current lower and upper bounds. Th is new demand effect value can then be used to restart the algorithm by Kunreuther a nd Schrage (1973). Repeating this approach a number of times that is bounded by the number of linear segments of the function ( d ) leads to the optimal valu e for the demand effect. Now recall that, in Section 4.2.3, we s howed that for our problem the function ( d ) is piecewise linear and concave on each of a contiguous sequence of O ( T3) intervals for d (as determined by consecutive m td values). We can therefore apply the above algorithm on each of these intervals, yielding O ( T3) candidate demand effects. In Section 4.2.4 we showed that, in each interval on which is piecewise linear and concave in d its number of breakpoints is O ( T3). We can therefore conclude that we need to solve O ( T3) equalcapacity lot-sizing problems for O ( T3) intervals, each of which can be solved in O ( T3) time. Our overall solution approach for the capacitated problem therefore runs in O ( T9) time. 4.3 Conclusion In this chapter, we developed efficient solution algorithms for discrete-time, finitehorizon procurement planning problems with eco nomies of scale in procurement, pricesensitive demand, and time-invariant procuremen t capacities. The models in this chapter consider general concave revenue functions in each time period, and seek to maximize total revenue less procurement and inventory holding costs. We consider the case in which prices may vary dynamically, as well the case in which a consta nt price is required during the planning horizon. Under mild conditions on the revenue function properties, we provide polynomial-time solution methods for this problem class. The structural

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94 properties of optimal solutions also provide useful managerial insights regarding optimal demand management strategies in complex planning situations. This study fills a gap in the literature and generalizes several recent works on integr ated pricing and production planning. In the next chapte r we recognize that in addition to pricing, firms may also have flexibility in timing of delivery of cust omer demands. Therefore, Chapter 4 focuses on demand management problems where deliver y timing flexibility is also present.

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CHAPTER 5 UNCAPACITATED PRODUCTION PLAN NING MODELS WITH DEMAND FULFILLMENT FLEXIBILITY Introduction A basic assumption of traditional producti on planning models is that a firm must satisfy all of the predetermined demand in each prescribed time period. In many practical settings, however, a firm may explicitly make certain demand selection and timing decisions that can exclude some subset of demands. That is, satisfying all potential market or customer requests may not be economi cally attractive to a producer due to high fulfillment costs relative to the associated revenue. As discussed in Chapters 3 and 4 a producer can control its demand characteristics through pricing. In other settings, a supplier may not have a great deal of pri ce setting flexibility, but may wish to be selective in its choice of markets (or custom ers) and the timing of demand fulfillment, because different markets (or customers) and different timing imply different fulfillment costs. In Chapter 3, given a set of potential customer demands for a product over a finite horizon, we developed models that determ ine optimal demand selection, capacity and production planning decisions. In these mode ls, however, customer demands are timeinflexible, in the sense that that the manuf acturer either fulfills the order in some predefined time period, or completely rejects the order. In a number of practical contexts, customers may allow a grace peri od (also called a demand time window) during which a particular demand or order can be satisfied. Lee, etinkaya, and Wagelmans 95

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96 (2001) modeled and solved general lot-si zing problems with demand time windows. Their model still requires, however, that each demand is ultimately satisfied during its predetermined time window. That is, they considered demand-timing flexibility without the benefits of order (demand) selection a nd rejection decisions. The approach we present in this chapter integrates demand se lection and a more general version of demand time windows for lot-sizing problems, providi ng two dimensions of demand planning flexibility. Charnsirisakskul, Griffin, and Keskinocak (2004) consider a similar model that focuses on the economic benefits of lead time flexibility and order se lection decisions in production planning with finite production ca pacities. Their model assumes that each order has a preferred due date and a latest acceptable due da te, after which the customer will not accept delivery. A tardiness penalty is incurred if an order is completed after the preferred period. Our model, on the other hand, allows a market (or customer) to provide any period-specific per unit revenue values for delivery within the acceptable range of delivery dates. This allows customers to specify any subset of acceptable delivery periods, and this subset need not consist of c onsecutive periods, as is the case with past models that consider demand time windows. In addition, Charnsiris akskul et al. (2004) do not provide any tailored solution procedures for exploiting the special structure of the model; rather, they primarily studied the mode l parameter settings under which lead time flexibility is most beneficial, and relied on the CPLEX solver for model solution. We provide a dynamic programming algorith m that provides an optimal solution for the uncapacitated version of the probl em under certain production and holding cost assumptions that we discuss in Section 5.2. Moreover, the heuristic solution approach we

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97 propose works quickly on large instances of the general problem, although the complexity of the problem requires an exponent ial increase in solution time using branchand-bound. In our model, in contrast to prior literatu re, the revenue associat ed with a particular demand may take arbitrary valu es (including negative infini ty to reflect cases where delivery is infeasible) as a resu lt of early or late shipment relative to a preferred delivery period (alternatively, the price difference may reflect a loss of goodwill penalty to the supplier for early/late delivery). In addition, our model allows the s upplier to produce in advance of a preferred delivery period, a nd to choose between early delivery to the customer (with implied reduced revenue) a nd holding the inventory for later delivery (with the associated internal holding costs). We therefore seek to maximize profit as a result of demand acceptance and timing d ecisions, as well as production planning decisions. As with prior work on relate d lot-sizing problems (e.g., Wagelmans, van Hoesel, and Kolen (1992)), the problem we c onsider can be cast as an equivalent flexible uncapacitated facility location pr oblem, where setup periods act as possible facilities and demands in each period act as customers. We use the term flexible because the model we study is different from the traditional plant location models in that we do not require satisfying all demands. Facility location models have been well studied by many researchers, and various effi cient algorithms have been developed for the uncapacitated facility location prob lem (UFLP) (e.g., Er lenkotter (1978)). When the requirement to satisfy all de mand is removed, the resulting profitmaximizing problem is quite different from th e classical uncapacitated facility location model. To our knowledge, no past literatur e has considered this UFLP with demand

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98 satisfaction flexibility. Since the model we co nsider is a generaliza tion of the classical UFLP, we may draw on solution approaches previously developed for the UFLP. The approximation algorithm presented by Shmoys, Tardos and Aardal (1997) provides close to optimal solutions for the UFLP. However, their algorithm depends largely on specialized structural and symmetric cost a ssumptions (that is, in the facility location context supply and demand fac ilities are collocated, and the transportation cost from a location a to location b is the same as that from b to a; such symmetry does not exist in our analogous production planni ng model when cast as a fac ility location problem). Erlenkotter (1978) developed a dual-base d algorithm (DUALOC) to solve the uncapacitated plant location problem. This algorithm has proved to work quite well except for certain special (and somewhat pathol ogical) cases. In this chapter we present a similar dual-based algorithm that also solves some of the problematic instances encountered by Erlenkotters algorithm. Moreover, we have also developed a polynomial-time algorithm for problem instances that obey certain specialized cost structures likely to be found in production planning contexts. The remainder of this chapter is organized as follows. Section 5.1 presents the basic problem definition and formulation. In Section 5.2, we discuss the special cases in which the problem can be solved in polynomial time using a dynamic programming approach. In Section 5.3, we present the dua l-based heuristic for the case of general cost and revenue coefficients. Section 5.4 discusses a set of comp utational tests that validate the effectiveness of our proposed solution ap proach and illustrate the value of the different dimensions of demand-timing and sel ection flexibility. Conclusions for this chapter are summarized in Section 5.5.

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99 5.1 Problem Definition and Model Formulation We next describe and formulate the uncapacitated requirements planning model with demand fulfillment flex ibility. We consider a T -period planning horizon and a set of candidate demands for a single good pr oduced by a supplier. Producing the item in period requires incurring a fixed setup cost and a variable production cost while holding the item in inventory at the supplier at th e end of period t costs per unit. Candidate demand represents a request for units. A candidate demand may correspond to either an individual customer demand or an aggregate market demand, for example. Delivery of candidate demand j in full provides net revenue (in excess of variable delivery costs) of in period (note that if the customer associated with candidate demand j will not accept delivery in period t, then we can set ). We also define the following decision variables. J t tS tc Jj jd jtR t jtR Fraction of demand satisfied in period .jtzj t Number of units produced in period t x t 1, if there is a production setup in period 0, otherwisett y Inventory remaining at the end of period tit We assume without loss of generality that for each candidate demand source j, a customer-specified delivery time window exists during which the customer will accept delivery, beginning in period t1(j) and ending in period t2(j), (equivalently, Rjt = for t < t1(j) and for t > t2(j)). Note that this characterization does not preclude Rjt values of for periods between t1(j) and t2(j) (inclusive), nor doe s it preclude having t1(j) = 1 and t2(j) = T for all demands j. This characterization will be useful, however, when

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100 discussing the special class of problems in Section 5.2 that contain true demand time windows (such as those defined by Charnsirisak skul et al. 2004, a nd Lee et al. 2001). Note that although our model allows partial demand satisfaction (since the zjt variables are continuous), as we will later see, an optimal solution always exists in which all demands are either satisfied in full or are completely rejected. The producer wishes to maximize net prof it during the planning horizon, defined as the total net revenue from order acceptance and delivery-timing decisions, less the total setup, variable production, and holding costs. We formulate the demand fulfillment flexibility problem [DFFP] as follows. [DFFP] Maximize 11 TT S j tjt tttttt jJt t R zSyhic xti Subject to: 1 ttjjt jJixdz t = 1, T, (5.1) 11 T t jtz j J, (5.2) xt Dyt, t = 1, T, (5.3) xt 0, t = 1, T, yt {0, 1}, t = 1, T, 0 zjt 1, j J, t = 1, T. The objective function maximizes revenue less production and holding costs. Constraint set (5.1) represents inventory balance constr aints. As in previous chapters, this balance constraint takes demand as a de cision variable. The second constraint set ensures that we assign produc tion for each selected demand j to some production period t. In constraint set (5.3) the parameter D represents a large number (note that we can set

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101 Jj jdD and effectively retain our uncapac itated production assumption). This constraint set therefore fo rces production in period t to zero if no setup is performed in a period, and allows effectively unc onstrained production otherwise. We can equivalently represent the [DFFP] as a fixed-charge ne twork flow problem using the example shown in Figure 5.1. Th e figure shows a four-period problem with eight potential demands, with the values of t1(j) and t2(j) provided in Table 5.1 below (this example assumes true demand time windows, i.e., positive Rjt values for t1(j) t t2(j) for each j and that Rjt = for t < t1(j) and for t > t2(j)). Table 5-1. Demand time windows for the example problem s hown in Figure 5.1 Demand, j ( t1( j ), t2( j )) Demand, j ( t1( j ), t2( j )) 1 (1, 3) 5 (2, 3) 2 (1, 2) 6 (3, 4) 3 (1, 2) 7 (3, 4) 4 (1, 3) 8 (2, 4) Demand nodes, dj Inventory holding arcs, flow cost =S th Production Source node, supply = Jj jd Production flow arcs, flow cost = St + ctxt if xt > 0, and 0 otherwise. Dummy Source node, supply = Jj jd Dummy Demand node, demand = Jj jd Revenue arcs, revenue = Rjtzjt Period supply nodes Figure 5-1. Fixed-charge network flow representation of the [DFFP] problem. Given any choice of setup peri ods, the result is a simple network flow problem, and therefore an optimal exists that forms a sp anning tree solution (see Ahuja, Magnanti, and Orlin 1993). A spanning tree solution implies th at for an extreme point solution; if we

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102 draw the subnetwork with only those arcs containing positive flows (i.e., the basic variables), no cycles will exist in the resul ting subnetwork. As in Chapter 3, we develop properties specific to the DFFP as a re sult of this spanning tree property: Property 5.1. An optimal solution exists for [DFFP] such that for each candidate demand i, we either satisfy all dj units or we do not satisfy any fraction of demand j. Property 5.2. An optimal solution exists fo r [DFFP] such that for each demand j, if we satisfy any demand, we deliver the entire demand only in one period, i.e., we do not split delivery between periods. Property 5.3 An optimal solution exists for [D FFP] that satisfies the well-known Zero-Inventory Production property, i.e., an optimal solution exists such that we never hold inventory at the end of period t 1 and produce in period t for all t = 2, T. Property 5.1 implies that under this uncapacitated model, a producer needs to only consider solutions in which all orders are simply accepted or rejected, and no order is partially satisfied. Property 5.2 states that we will not split delive ries to a customer among different periods, and Property 5.3 is a well-known result that applies to a number of generalizations of the basic Wagner-Whitin (1958) model. Properties 5.2 and 5.3 imply that an optimal solution exists such that every zjt variable will equal either 0 or 1, and we can replace the la st constraint set 0 zjt 1 in the [DFFP] with zjt {0, 1} without loss of generali ty. These properties therefore a llow a practicing manager to focus on only a subset of all possible decisions for the model without loss of optimality. Despite these properties, when the revenue parameters Rjt can take any arbitrary values, the uncapacitated version is NP-Hard (for a pr oof of this result, please see Appendix B). Under certain additional mild assumptions on costs and revenues, we can, however, solve the problem in polynomial time, as we next discuss.

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103 5.2 Polynomial Time Solution under Special Cost/Revenue Structures We next consider certain practical re gularity assumptions on revenues and costs that will allow us to solve the uncapacitated version of the problem efficiently. We first assume that the revenue for a candidate demand equals the demand quantity multiplied by some base per unit revenue, i.e., revenue equals rjdj, where rj is the per-unit revenue parameter. In addition to specifying a time window for delivery, each customer also specifies a preferred delivery period tp(j) within the window That is, if t1(j) and t2(j) define the earliest and latest periods of the demand time window for demand j, some preferred delivery period tp(j) exists such that t1(j) tp(j) t2(j). Whereas in the general model of the previous section the revenue parameters Rjt corresponding to job j could take any unrelated values in successive pe riods, we now assume that the parameter Rjt is comprised of the per-unit revenue multiplied by demand, rjdj, which is independent of the period of delivery, minus early (late) delivery costs, that depend on how early (late) the demand is delivered relative to its preferre d demand period. This assumption is not unreasonable in a variety of contexts where later than preferred deliveries are penalized and earlier than preferred de liveries result in some additional holding cost. Let R jth R jtb denote the early (late) delivery cost per unit for demand j delivered in periods t1(j), t1(j) + 1, tp(j) 1 (tp(j) + 1, t2(j)). We make the following assumptions regarding production costs and these early and late delivery costs: i. ct ct+1, for all t = 1, T, ii. for all j J, and t < tp(j), R jt R jthh1 iii. ct + bjt ct-1 + bj,t-1, for all j J, and t > tp(j). Observe that assumption (i) also implies that ct + ct+1, i.e., the variable cost of producing a unit a period early is at least as high as waiting a period. Assumptions (i) th

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104 (iii) are often referred to as implying that no speculative motives exist for holding inventory or backlogging (Lee, Cetinkaya, and Wagelmans 2001), i.e., there is no reason to produce earlier than necessa ry due to the anticipation of production cost increases (or to produce later than necessary to take adva ntage of lower penalty costs later). These assumptions not only simplify the problem, but also correspond to what we might expect to see in practical contexts. That is, the net revenue to the producer for demand j (in excess of variable production, and early/late delivery cost s) is increasing in time for t tp(j) and is decreasing thereafte r. Under assumptions (i) (iii) we have the following property: Property 5.4. Given a sequence of setup periods, an optimal solution exists for [DFFP] where if demand j is satisfied, it is satisfied using production in either the most recent setup prior to (and including) period tp(j) or the first setup following period tp(j). We now consider how to extend the Wagner-Whitin (1958) dynamic programming solution for the economic lot-sizing problem for application to the uncapacitated case of the [DFFP] under cost assumptions (i) (iii). The Wagner-Whitin method is based on finding the shortest path on a graph containing T + 1 nodes. Properties 5.1 5.4 imply that we can solve this special case of the problem using a similar approach on a graph containing a similar structure, but with T + 2 nodes, as shown in Figure 5.2. In this shortest path graph, nodes corre spond to setup periods, and a pa th exists that corresponds to every feasible combination of setups. Th erefore, if costs are properly applied to the arcs, finding the shortest pa th in the graph corresponds to finding the optimal production plan. The arc length calculation for the shortest path approach is different from that for the standard Wagner-Whitin (1958) approach, as we next discuss. Since we maximize

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105 net profit, we interpret arc lengths in terms of net contribution to profit, and so we seek the longest path in the graph. The longest path graph contains an arc for each possible sequence of two consecutive se tups. That is, the arc (t, t') implies that consecutive setups occur in periods t and t' (arcs of the form (0, t') imply that the first setup occurs in period t', while the arc (0, T + 1) is a zero profit arc, implyi ng that we do no setups at all and thus reject all demands). We compute the length of each arc (t, t') (with t' > t) in the graph as the maximum contribu tion to profit (before subtra cting setup costs) possible from demands with preferred periods t, t' 1 by using either the setup in period t or t', minus the setup cost in period t (no demands exist with preferred period 0, and we use the convention that the producti on setup cost in period zero is zero and the variable production cost of producing in period zero is infinite). c (3,5) c (3,4) 1 2 3 4 5 c (1,5) c (1,4) c (1,3) c (1,2) c (2,3) c (4,5) c (2,5) c (2,4) 0 c (0,5) c (0,4) c (0,3) c (0,2) c (0,1) Figure 5-2. Structure of longest path graph. We next consider how to compute the maximum possible contribution to profit from demands with preferred periods t, t' 1, assuming the only available setups occur in periods t and t'. Suppose we incur the setup in period t and its corresponding cost St, and that no other setup can occur until period t' (arcs of the form (0, t') have no associated setup cost, i.e., S0 = 0). To offset this setup cost we will satisfy candidate demand j with preferred period t if and only if rj ct, i.e., if the unit revenue from

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106 demand j is at least as great as the unit production cost in period t. Similarly, we will satisfy demand j with preferred period where t < < t', if the following holds: 1 '' 1maxmin;, 0R jt kjtjtjt krchhrcb (5.4) The left-hand side of (5.4) provides the maximum contribution to profit we can obtain by satisfying demand j with preferred period where t < < t', using either the setup in period t or t'. The two arguments within the maximum in (5.4) correspond to producing items to satisfy demand j in periods t and t', respectively. The inner minimum indicates whether the supplier should hold the items in invent ory and deliver in period tp(j) or deliver demand j early. We let Jt(t) denote the set of de mands with preferred period t, and with rj ct. Similarly, let Jt,t'() denote the set of de mands with preferred period (where t t' 1) such that condition (5.4) holds. We therefore apply the following net profit value to arc (t, t'): ,'1 '' 1,' maxmin;, 0t ttjtj jJt R jt kjtjtjt jJ kcttrcd rchhrcb (5.5) Note that all arcs leaving a periods node contain the setup cost associated with that period. Thus, we ensure that we incur all se tup costs associated with a given path. Since a path exists for all possible sequences of setups, and the sum of arc profits on the path equal the maximum total profit from the a ssociated sequence of setups, finding the longest path in the network provides an optimal solution under assumptions (i) (iii). After finding the longest path in the graph we can determine which demands to satisfy using the appropriate setups by ch ecking the elements of the sets Jt(t) and Jt,t'() for all arcs contained in the longest path.

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107 For each (t, t') pair, we search among all candidate demands with t tp(j) t' 1 to determine their attractiveness, and this preprocessing takes O(|J|T2) time, where |J| denotes the cardinality of the set J, or the total number of potential demands. Solving the acyclic longest path problem takes no more than O(T2) time, and so the worst-case complexity of this algorithm is O(|J|T2). 5.3 Dual Based Heuristic Algorithm fo r General Revenue Parameter Values As discussed earlier, the general uncapacita ted case of the problem, in which the revenue parameter values (Rjts) can take any arbitrary values, is NP-Hard. This section presents a dual-ascent-based heuristic for so lving this general version of the problem presented in Section 1, which uses a strong mixed integer programming reformulation of the problem. This reformulation uses the following property of optimal solutions for the uncapacitated version of the problem. Property 5.5. An optimal solution exists for [DFFP] in which the production that satisfies any candidate demand is not split between production periods. Property 5.5 indicates that a supplier w ill not split production for a single order across different production periods, and follo ws from the spanning tree property of optimal solutions discussed in Section 5.2. Given Property 5.5, we can reformulate the [DFFP] in a manner similar to the classical faci lity location problem formulation. In this reformulation, production periods correspond to facilities and demands correspond to customers. We first deco mpose the production variables xt into demand-specific production variables xjt, where xjt denotes the amount of production in period t that satisfies candidate demand j. We also let jt denote the unit variable profit gained by satisfying demand j using production in period t, and define Jt as the set of all demands that can feasibly be produced in period t. The unit profit corresponding to production in

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108 period t for order j equals unit revenue minus variable production and holding/backordering cost, i.e., 121 (),....,() 1max j tt jtjjk krh tc We can then reformulate the [DFFP] as follows. Maximize 11TT j tjttt tjJt x Sy Subject to: j J, (5.6) j jt jtt jtdx )( )(2 1 t = 1, T, j Jt, (5.7) tjjtydx 0jtx 1.0 ty t = 1, T, j Jt. Using our variable definitions along with Property 5.5 it is straightforward to verify that the above formulation for the [DFFP] is equivalent to the one provided in Section 5.2. Note that we can conclude that the revenue parameter jt is nonnegative without loss of generality because we can delete any pairs (j, t) such that jt < 0 without loss of optimality, since an optimal solutio n exists where all such production xjt variables are zero. The above formulation is similar to th e classical UFLP formulation, except that we do not require satisfying all demand. We can therefore draw on approaches successfully applied to the facility location problem in the past. In particular, we develop a dual-based solution method using an approach similar to the one Erlenkotter (1978) developed for the UFLP. Relaxing the binary restrictions on the setup variables, and defining dual variables vj (for j J) and wjt (for t = 1, T, j Jt) corresponding to Constraints (5.6) and (5.7) the dual of the linear programming rela xation of our problem is the following: Minimize Jj jjvd Subject to: jtjjtvw t = 1, T, j Jt, (5.8)

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109 t = 1, T, (5.9) t Jj jtjSwd , j J, t = 1, T. 0jtw 0jv Observe that at optimality we have jjt jtv w ,0 max and we can restate this dual problem more compactly as: Minimize Jj j Subject to: max0, j tj jJP tS t = 1, T, (5.10) 0j j J. where we have made the substitutions j = djvj and Pjt = djjt. To facilitate the description of our solu tion algorithm for this dual problem, we define jjt tjP a ,0 max and we can restate constraint set (5.10) as 0 t Jj tjSa To further simplify the notation used in describing our algorithm, we define Nt as the lefthand side of this restated constraint, i.e., t Jj tj tSaN We can then state that any solution such that Nt 0 for all t and j 0 for all j is dual feasible. The dual objective minimizes the sum of the j variables, and these variables must be nonnegative. Therefore, we know that if the solution with j = 0 for all j J is feasible, it must be optimal. We apply a he uristic approach for solving this dual that begins with a (likely inf easible) solution with all j = 0, and attempts to obtain feasibility by increasing the j variables as little as possible. By duality theory, any feasible dual solution provides an upper bound on the optimal solution values of the [DFFP]. We later use the resulting dual solution to generate a heuristic solution for the [DFFP] using complementary slackness conditions. Observe that since the j variables are all weighted

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110 equally in the objective function, we would like to increase the j values in a way that attempts to restore the maximum amount of feasibility at each step, i.e., for each unit increase in the j variables, which motivates our solu tion procedure. We next describe our dual solution procedure, which is follo wed by numerical examples that help to illustrate our methodology. Dual Solution Procedure Step 0. Set each j to 0. Step 1. Construct a T |J| table with an entry for each demand/period combination, where each row corresponds to a constraint and each column to a candidate demand. Let atj denote the entry in cell (t, j), where atj = max{0, Pjt j}, and define TV as the set of violated constraints, i.e., those such that Nt > 0. Step 2. For each candidate demand j, let kj equal the number of positive atj values in column j of the table that are also in rows corresponding to elements of the set TV (if atj is positive and t TV, then increasing j will reduce the degree of infeasibility). Let kmax = maxi=1,,m{kj}. Note that if we increase j by one unit, then for any positive atj such that tTV, the degree of constraint violation will be reduced by one unit. Step 3. Define K J as the set of candidate demands j such that kj = kmax and let amin = min{atj: jK, tTV, and atj > 0}. Increasing the j variables for those jK will resolve the greatest number of infeas ibilities for a one unit increase in j. We simultaneously increase j values for all jK either until we restore feasibility to at least one constraint or until the set K of candidate demands (i.e., candidate j variables for increase) loses at least one element. Step 5. For each violated constraint tTV, let tk be the number of positive entries (atjs) corresponding to demands in the set K, and note that if we increase j by one for all jK, we will reduce the amount of infeasibility in constraint t by the amount tk (assuming atj 1 for all j K in row t). Calculate the ratio Nt/tk for each t TV, which is the average increase for each j (such that jK) required to make constraint t feasible. Define the minimum ratio among these values as min/Vk tTt M RN t We now increase the j values for jK by min{MR, amin}. After increasing these j values, either the set K is reduced by at least one element (when the minimum equals amin), or one of the violated constraints becomes satisfied (when the minimum equals MR).

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111 Step 6. For each j > 0, we check whether at least one satisfied constraint in which the j variable appears is tight. If not (and all of the satisfied constraints in which j appears are loose), we then decrease the value of j by the maximum amount possible while maintaining feasibility for the currently satisfied constraints. If there is no infeasible constraint, STOP. Otherwise, return to Step 1. Observe that the algorithm does not increase the variables in any predetermined index order; rather we increase them obeying a rule that depends on the number of infeasibilities we can resolve per unit increa se, which is a different approach from that used by Erlenkotter (1978). Candidate colu mns are chosen according to the number of positive occurrences in the table, and, sin ce we choose the columns with the maximum number of entries, we seek to decrease th e infeasibility by the greatest amount at each step. An important issue in this algorithm is determining how much to increase a variable value at each step. In the set K of candidate demands for increasing the corresponding j values, consider a column with an entry equal to amin. Increasing the value of the corresponding j for that column up to amin improves feasibility by an amount of kj. However, after the corresponding j variable hits amin, each additional unit increase in j would improve feasibility by less then kj (because in this case, the atj = max{0, Pjt j} value will remain zero for rows in which atj = amin), which is why we increase j by no more than amin. Although the j values can be increased up to amin, we only continue to increase the js up to amin if all of the infeasible constraints re main infeasible during the increase. Within the set of candidate demands K, increasing each column by one unit would improve feasibility by the same amount for each j variable. However, while increasing the js, if some constraint becomes feasible (and the corresponding kj values therefore

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112 change, which implies the set K itself changes), an additional increase in this same set of variables may not continue to provide the be st improvement (note that a constraint may become feasible before any j value hits amin). We therefore stop the increase in the corresponding j variables if at least one additional constraint becomes tight. This stopping point is determined by the minimum ratio in Step 5. In addition, as a result of the above discussion, while increasing the j variables, we may hit the value amin prior to achieving feasibility of a constraint. Th erefore, the best local improvement is accomplished by increasing j values for the candidate set us ing an increase equal to the minimum between MR and amin. We next illustrate our dual ascent algorithm through two numerical examples. Numerical Example 1 We use an example problem for illustrative purposes containing 4 candidate demands and 6 production periods. Table 5.2 gives the Pjt values set up costs for the example problem. Table 5-2. Demand (dj) and setup cost values (St) for example problem 1. Candidate demand, j Time period, t 1 2 3 4 Setup cost, St 1 30 10 0 0 10 2 30 0 10 0 10 3 30 0 0 10 10 4 0 40 0 0 20 5 0 0 40 0 20 6 0 0 0 40 20 In the first step (Step 0; see Table 5.3) we set all j values to 0 and construct the table described in Step 1 (Table 5.3), where table entries are denoted by atj; recall that atj = max{0, Pjt j}. Table 5.3 also includes the Nt values (the amount of infeasibility for

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113 each constraint t). Table 5.3 (labeled Iteration 0) shows the initial table with the atj values in the shaded portion of the table, and the j values in the bottom row of the table. The first row below the atj values provides the value of kj (the number of positive elements in the corresponding column comput ed in Step 2 of the algorithm) for each candidate demand j. The column labeled Nt indicates the degree of infeasibility for each constraint, while tk provides the number of positive entries corresponding to columns (demands) in the set K, which contains all demands with the maximum number of positive entries per column (K contains only demand 1 in this case, with three positive atj entries in its column). The final column of the table provides the average increase for each j variable with j K such that the corresponding constraint becomes tight, while the minimum element in this column provides the minimum ratio, which is then compared to amin (see Step 3 of the algorithm). In this case, both amin and MR equal 30, and we increase j for j K by 30, and move to iteration 1 (see Step 5 of the algorithm). Table 5-3. Problem 1-Iteration 0. Candidate Demand, j Time Period, t 1 2 3 4 St Nt tk Nt/tk 1 30 10 0 0 10 30 1 30 2 30 0 10 0 10 30 1 30 3 30 0 0 10 10 30 1 30 4 0 40 0 0 20 20 0 5 0 0 40 0 20 20 0 6 0 0 0 40 20 20 0 ki 3 2 2 2 MR = 30 j 0 0 0 0 kmax = maxj=1,,4{kj} = 3 K = {1}; TV = {1, 2, 3}. amin = min{atj: j K and t TV, atj > 0} = 30 min{amin, MR} = 30. Observe that after iteration 0, 1 = 30 is the only positive variable, and it appears in the first three constraints, all of which are now tight. Therefore, we cannot decrease 1

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114 without removing feasibility from at least on e constraint (see Step 5 of the algorithm). During iteration 1 shown in Table 5.4, we find that the set K now contains j = 2, 3, 4, which implies we will increase 2, 3, and 4 by the same amount in this iteration. The amount of increase in these vari ables equals the minimum between amin and MR, which in this case equals 20, which leads to the following Iteration 2 (Table 5.5). Table 5-4. Problem 1-Iteration 1. Candidate Demand, j Time Period, t 1 2 3 4 St Nt tk Nt/tk 1 0 10 0 0 10 0 2 0 0 10 0 10 0 3 0 0 0 10 10 0 4 0 40 0 0 20 20 1 20 5 0 0 40 0 20 20 1 20 6 0 0 0 40 20 20 1 20 ki 2 2 2 MR = 20 j 30 0 0 0 kmax = maxj=1,,4{kj} = 2 K = {2, 3, 4}; TV = {4, 5, 6}. amin = min{atj: j K and t TV, ati > 0} = 40 min{amin, MR} = 20. Table 5-5. Problem 1-Iteration 2. Candidate Demand, j Time Period, t 1 2 3 4 St Nt tk Nt/tk 1 0 0 0 0 10 -10 2 0 0 0 0 10 -10 3 0 0 0 0 10 -10 4 0 20 0 0 20 0 5 0 0 20 0 20 0 6 0 0 0 20 20 0 ki j 30 20 20 20 In iteration 2, we no longer have any viol ated constraints. We then check to determine whether any of the j variables can be decreased w ithout destroying feasibility. Variables 2, 3, and 4 all have positive values in at least one tight constraint, so we do

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115 not decrease these. The variable 1, however, can be decreased by 10 units without removing feasibility from any of the co nstraints and we, therefore, decrease 1 by 10, which leads to a final solution of 1 = 2 = 3 = 4 = 20 and an objective function of 80. Numerical Example 2 We next present a numerical example for which, if the dual ascent approach of Erlenkotter (1978) is followed, the resulting du al solution is not optimal. Our algorithm does, however, produce an optimal dual solutio n. Table5.6 provides the revenue values and setup costs for this example problem, which contains 3 candidate demands and 3 periods. Tables 5.7 and 5.8 show the only tw o iterations required for solving the problem instance. Table 5-6. Demand and setup cost values for example problem 2. Candidate demand, j Time Period, t 1 2 3 Setup cost, St 1 2 0 2 2 2 2 2 0 2 3 0 2 2 2 Table 5-7. Problem 2-Iteration 0. Candidate Demand, j Time Period, t 1 2 3 St Nt tk Nt/tk 1 2 0 2 2 2 2 1 2 2 2 0 2 2 2 1 3 0 2 2 2 2 2 1 ki 3 2 2 MR = 1 j 0 0 0 kmax = maxj=1,,4{kj} = 2 K = {1, 2, 3}; TV = {1, 2, 3}. amin = min{atj: j K and t TV, atj > 0} = 2 min{amin, MR} = 1. The algorithm terminates after iteration 1, since all constraints are tight and every variable appears in at least one tight constr aint. The resulting dual objective equals 3, while the solution using Erlenkotters dual descent procedure is 1 = 0, 2 = 3 = 2, with

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116 an objective function of 4. The optimal so lution in this case is 3 (see Cornujols, Nemhauser and Wolsey 1990). Table 5-8. Problem 2-Iteration 1. Candidate Demand, j Time Period, t 1 2 3 St Nt tk Nt/tk 1 1 0 1 2 0 2 1 1 0 2 0 3 0 1 1 2 0 ki 2 2 2 j 1 1 1 kmax = maxj=1,,4{kj} = 2 K = {1, 2, 3}; TV = {1, 2, 3}. amin = min{atj: j K and t TV, atj > 0} = 2 min{amin, MR} = 1. We note that our dual ascent procedure doe s not guarantee solving the dual of the relaxed problem optimally, nor do we claim th at it is universally better than Erlenkotters (although we should keep in mind that it was developed for a more general location problem class than Erlenkotter 1978 consid ered). On the other hand, as the computational tests that we present later indi cate, it works very well on average, and in particular, on pathological instances id entified previously in the literature. When our algorithm concludes, we have a dual feasible solution, which provides an upper bound on the maximum profit for the [DFFP] and its linear relaxation. Based on this dual solution, we wish to find a corresponding solution for the primal problem using complementary slackness conditions. If that complementary solution is feasible for the relaxation of the [DFFP], then we know this so lution is optimal to the relaxation. If, in addition, this solution satisfies the binary rest rictions for the [DFFP], then it is optimal for the [DFFP]. Otherwise, we can apply a he uristic adjustment procedure solution or branch-and-bound in order to determine a good feasible solution (or an optimal solution

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117 in the case of branch-and-bound). We next consider the complementary slackness conditions required to determine a complementary solution to our dual solution. 5.3.1 Economic Interpretation of the Dual and Complementary Slackness Relationships Having a dual solution in hand, we may utilize the complementary slackness relationships to create a corresponding primal solution. In doing so, we also consider the economic interpretation of the dual multiplier s. The complementary slackness conditions are provided below (recall our previous substitutions j = djvj and Pjt = jtdj). 1. 0* )( 1 *2 j jt t jt jvxd 2. 0 ,0max* tJj jjt jttv dSy 3. 0 ,0max* ** jjt jttjv xyd Considering complementary slackness (CS) condition 1, we can interpret vj, the dual multiplier of the demand constraint (5.6) as the incremental profit gained from satisfying an additional unit of demand j. We can think of this as the true value of satisfying a unit of demand for the firm. Therefore, this value may be also be useful in pricing decisions, i.e., the firm may use this information to ne gotiate the price required to make the demand attractive to the firm. If vj = 0, then an additional unit of demand j has no value to the firm. If vj > 0, then we satisfy all dj units of demand, and an additional unit of demand j can increase the firms profit. In order to ge nerate this additional demand, the firm might consider offering a price disc ount (by discounting up to vj less than the current price) for additional units of demand (e.g., a quantity discount). Note that this complementary slackness condition does not appear in the st andard facility location problem, since all demand must be satisfied in that problem. Observing CS conditions 2 and 3, we see that a setup variable can only be positive if the corresponding dual constraint is tight. We can interpret the quantity

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118 *max0, j tjv as a per unit allocation of the setup cost for demand source t to candidate demand j. If j v > 0, then by CS condition 1 it is optimal to satisfy demand j. If j v < jt and yt > 0, then by CS condition 3, we must have * j tj t x dy and demand source t is used to satisfy demand j tdy units of demand j (and the amount jt j v is in effect allocated to each unit of demand j satisfied using demand source t production). Note that as intuition suggests, it is suboptimal to value a unit of demand j in excess of its maximum per unit revenue, i.e., maxt = 1, T{jt}, which implies 0 vj maxt = 1, T{jt}. In addition, when j v jt for some demand source t, this implies that none of the setup cost for demand source t is allocated to demand j. This also implies that a higher value supply source exists for demand j and demand j will absorb some of the setup cost of this higher value supply source. 5.3.2 Creating a Feasible Primal Solution In this section we consider how to create a feasible solution to the DFFP, beginning with our dual solution and using the comple mentary slackness relationships. If we can find a complementary solution that is feasible for the DFFP, then our algorithm gives an optimal solution. Our heuristic strategy works as follows. Given the dual solution, if tJj tjjtSP,0 max then by complementary slackness, we set yt to 0. Now denote T= as the set of all periods such that tJj tjjtSP,0 max i.e., such that the corresponding dual constraint is tight. For periods in the set T=, by the complementary slackness conditions, we can set yt to any value between zero and one. Let J+ denote the

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119 set of demand such that j > 0 (recall that implies *0jv j > 0). Since these demands have positive value, we sh ould try to satisfy them using some production period t. For each j J+, let tmax(j) denote the element of T= such that the corresponding Pjt is maximum, i.e., tmax(j) = :max{ } j jt tTP P Let Tmax T= denote the set of all tmax(j) values. Under our heuristic approach, the set Tmax represents actual production periods where setups occur. We then set yt = 1 for t Tmax, and set yt = 0 for all remaining periods in the set T=. For j J+, we set max j jtj x d and all other xjt equal to 0. This heuristic approach generates a feasible [D FFP] solution and, for this solution, the corresponding objective value of the [DFFP] instance is equal to max maxt jtj jJ tTPS This heuristic procedure does not guarantee an optimal solution to the [DFFP] problem. However, for certain special cases, optimality is achieved assuming the dual heuristic finds the optimal dual solution also. The following proposition provides a sufficient condition for claiming optimality as a result of a particular characterization of the dual solution. Proposition 5.1. Let the sets Tmax and J+ be defined as above. In the set J+, for each demand j, let nj be the number of periods where the profit Pjt is greater than the dual variable j. If nj is at most one for each demand in set J+, then Tmax is an optimal set of production periods for the [DFFP]. Proof: Consider the demand point j J+. If nj = 0 for all j J+, then for each j J+ we have maxmax{} j t tTP = j and max(jtj tTP ) = 0, which implies j = j + max(jtj tTP ) The heuristic solution value is equal to max maxmax{} j tt tT jJ tTPS = max()tjj t j jJtTjJP = j jJ which equals our dual solution value. Since the heuristic solution is feasible for the [DFFP] and its objective function value equals the dual solution value, we conclude that the heuristic solution is optimal. If nj =1

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120 for each j J+, then maxmax{} j t tTP = j + max(jtj tTP ) then the remainder of the proof is the same as the case where ni = 0. Proposition 5.1 provides a condition that, when satisfied, allows us to verify the optimality of our heuristic solution in a straightforward manner. 5.4 Computational Testing and Results Our model considers supplier flexibility in demand selection and delivery timing in association with production planning decisions. The goa ls of the numerical study we present in this section are twofold. First, we aim to benchmark the performance of our dual-based heuristic solution approach for the [DFFP]. Second, we also wish to compare the relative economic benefits of demand selection and timing flexibility under various settings. That is, we would like to identify characteristics of situations in which demand selection and delivery timing flexibility play an important role, i.e., when it is favorable to reject an order or to offer a discounted price in conjunction with an adjusted lead time. These results provide valuable information to operations managers on the potential value of demand selection and delivery timing flexibility. To simulate various parameters found in manufacturing settings, we utilized two different levels (low and high) for each important cost parameter in our numerical experiments. Each test instance involved 50 production periods and 50 orders, which we refer to generically as demands. To determine the order quantity for each demand, we used a uniform distribution on [10, 100] (we henceforth let U[a, b] denote a uniform distribution on the interval [a, b]). To test the impact of setup costs, we generated data from two distributions. The first represented a low setup cost setting (U [1000, 1500]) and the second represented a high setting (U[3000, 3500]). The two levels of the per unit holding cost at the supplier were U [0, 0.5] (low) and U [0.5, 1] (high). The low and high

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121 levels of production cost we used were U [20, 24] and U[24, 28], respectively. We applied unit revenues for all demands from the same distribution, U[28, 33]. To test the benefits of demand selection, we chose unit revenue values that pr ovide a relatively high contribution to profit under our low production cost setting, and a small contribution to profit under our high production co st setting. For these parameter settings, if we consider only the variable production costs relative to the unit revenue for on-time delivery, the supplier has no incentive to reject any de mand. The optimal selection of demands, however, must consider the impacts of setu p costs and inventory holding costs on total profit, which requires using the [DFFP] mode l. For our test instances, customers pay a lower price if they do not get a requested sh ipment on the preferre d due date. Although our model and solution approach can handle ar bitrary changes in revenue by period, for our test instances, the revenue for an order deliv ered in the associated preferred period is the highest. If the order is delivered before the preferred date, the customer incurs extra holding costs which are reflected back to th e supplier through reduced revenue; similarly, if the order is received afte r the due date, a backorder co st would be incurred. We therefore have that the customer is willing to pay the preferred period revenue less any associated holding/backorder costs, which de pend on the delivery date. For example, if an order requests a preferred due date of period t with associated unit revenue r, and the supplier delivers the order in period t 2, the customer will pay the supplier 2 R trh for each unit. In our numerical study we consid ered two levels for th ese retailer backorder and inventory holding costs (low perperiod holding and backorder costs of U[0, 0.6] and U[0, 0.8], respectively, and high perperiod holding and backorder costs of U[0.6, 1] and U[0.8, 1.4], respectively).

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122 To test the relative benefits of different dimensions of supplier flexibility, we considered four different scenarios: no demand selection or delivery time flexibility (INF), only demand selection flexibility (FLEX(D)), only delivery time flexibility (FLEX(T)), and both demand selection and deliv ery time flexibility (FLEX(D, T)). For each of our 16 combinations of cost leve l settings, we generated 20 random problem instances. For each instance, all four scenar ios were solved to optimality using CPLEX 8.0. We applied our dual-based heuristic to the FLEX(D, T) and FLEX(D) scenarios, for a total of 640 problem instances solved using the he uristic method. The average ratio of the heuristic solution value to the optimal so lution value across all instances tested was 0.9937, while the lowest ratio (worst indivi dual case) was 0.7548, indicating that the heuristic solution method can quickly provide close to optimal solu tions. It took 0.78 seconds on average to find solutions using the dual-based heuristic method, while solution via CPLEX took an average of 2 s econds. Solution via CPLEX is therefore a viable alternative for small to medium size problems due to the relatively tight bounds obtained by the linear programming rela xation of our facility-location based reformulation (we found that in a large per centage of our test problems no gap existed between the LP and the integer optimal solution values; note however, that as problem sizes become very large, CPLEX solution time would be expected to increase exponentially). In addition to the random instances, we also tested our approach using the wellknown facility location instances from the OR library (Beasley 1990). Since the optimal solution seeks the maximum possible profit, and there is no demand satisfaction requirement, we thus need to augment the facility location data to make the problem

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123 instances consistent with our model. Therefore, given an instance from the OR library, we assigned a large amount of revenue to each customer in order to ensure that all customers would be selected. We can then compare the profit (after adding a large revenue to each demand point to the optimal cost) of the optimal solution to the profit incurred in our heuristic solution. Our heur istic algorithm provided an optimal solution in 3 out of the 11 test instances; however, the average ratio of optimal solution to heuristic was 0.9989 (cost values were co mpared, not including the constant revenue part). The results of these tests indicate that the heuristic solution approach we presented provides an effective method for solvin g the uncapacitated requirements planning problem with flexibility in demand. 5.4.1 Analysis of results This section presents analysis of the nume rical results obtained by CPLEX, in order to examine the relative benefits of flexibility at optimality. We considered all four different flexibility scenarios and their impact on potential profit. 0 0.20.40.60.81 INF FLEX(D) FLEX(T) Average Ratio of Profit to Complete Flexibilit y ( FLEX ( D T ) ) Case Figure 5-3. Impact of dimens ions of flexibility on profit. Figure 5.3 shows the average across all pr oblem instances of the ratio of the profit for the associated scenario to the profit of the FLEX(D, T), or complete flexibility case

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124 (the most profitable case).The figure indi cates that on average (across all problem instances tested), delivery timing flexibility is quite beneficial for the supplier. The average increase in profit when allowing demand selection flexibility, however, is quite small, reflecting the fact that the vast majority of orders were attractive to the supplier based on our random problem settings. In order to gain better insight regarding situations in which demand selection flexibility is most beneficial, we next investigate the impacts of cost parameter level settings on both demand selectivity (the percentage of demands rejected) and profitability. Table 5-9. Order rejection rates under different cost parameter value settings. Production Cost Level Setup Cost Level Inventory Holding Cost Level Early/Late Delivery Cost Level Rejection rate High Low High Low High Low High Low FLEX(D, T) 0.077 0.063 0.070 0.021 0.063 0.029 0.058 0.033 FLEX(D) 0.233 0.200 0.214 0.057 0.200 0.072 0.140 0.132 Table 5.9 indicates the average demand rejection percentage (the % of all demands rejected) for both the FLEX (D, T) and FLEX(D ) cases at the various cost level settings. The table indicates that when any of the cost factors increases, the rejection rate increases as well, as we would expect. For each of the cost parameters, the rejection rate rises sharply when we disallow delivery timing flex ibility. That is, when timing flexibility is available the increased number of delivery timing options allows us to find a beneficial delivery period for nearly all demands. When this timing flexibility is not allowed, we are more likely to reject demands that co uld have otherwise c ontributed to profit. Figure 5.4 shows the average profit as a percentage of the maximum profit under the complete flexibility case (F LEX(D, T)), broken down by co st setting level and degree of flexibility. Note that we do not includ e the FLEX(D) case when considering early/late

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125 delivery costs, since these costs are not rele vant when no delivery time flexibility exists; therefore, for this case, the Inflexible Case and the FLEX(D) case are the same. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Flex(T) Flex(D) Inflexible CaseProfit Percentage High Production Cost Low Production Cost 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Flex(T) Flex(D) Inflexible CaseProfit Percentage High Setup Cost Low Setup Cost 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Flex(T) Flex(D) Inflexible CaseProfit Percentage High Supplier Holding Cost Low Supplier Holding Cost 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Flex(T) Inflexible CaseProfit Percentage High Early/Late Delivery Cost Low Early/Late Delivery Cost Figure 5-4. Profit levels as a percentage of the maximum prof itability (FLEX(D, T)) Observe that the shorter the bar in the figur e, the greater the potential benefits of additional flexibility in increasing profits. The results for each category are similar to those shown in Figure 5.3 for the entire set of random test problems (with the exception of early/late delivery cost case, which we disc uss below). Again, the benefits of timing flexibility far outweigh those associated w ith demand selection flexibility, although in some cases, demand flexibility can provide up to a 4% profit increase when compared with the inflexible case (in the high setup cost case, for example). When we consider the different levels of early/late delivery costs, we see that in the inflexible case, the profit percentage behavior is the oppo site of what occurs in the other cases. The reason for this is that high early/late delivery costs effec tively reduce the suppliers flexibility, and

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126 therefore less difference exists between the completely inflexible case and the case of complete flexibility. Note also that the benefits of demand selection flexibility in all of the cases are a function of our choice of rand om parameter generation settings. That is, we can ensure arbitrarily good performance in terms of the benefits of demand selection flexibility by inserting an arbitrarily una ttractive customer/market order. Our goal, however, was to illustrate a more realistic set of scenarios when generating cost and revenue data. 5.5 Conclusions In nearly all of the requirements planning models considered to date, a consistent assumption is that of fixed exogenous de mand which a supplier must meet. In many practical settings, however, suppliers have some dimensions of flexibility that allow them to affect the total demand th ey face. Therefore, we cons ider such planning problems from a different perspective, assuming cer tain demand characteristics are decision variables. In Chapters 2 and 3, we focuse d on the firms flexibility in pricing and demands selection decisions when production capacities present. In many such settings the net profit of an order also depends on the time at which the order is satisfied, and this chapter recognizes that suppliers often have flexibility to adjust order shipment times in addition to demand selection decisions. In Chapter 5, we, therefore, examined optimal levels of demand, production, and inventory for every planning period when flex ibility exists in selecting demands and their delivery timing. We provided a polynomial-time solution method for the model in presence of certain regularity conditions on cost parameters. However, these cost assumptions (sometimes called non-speculativ e motives) are not unr ealistic. Moreover, we developed a heuristic method based on th e dual of the linear programming relaxation

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127 of our model. Computational tests performed on a set of randomly created problem instances showed that the solution approach we developed is quite e ffective and produces close to optimal solutions. Throughout this thesis, we have consid ered various production planning problems involving varying degrees of flexibilities in demand and capacity planning. One underlying assumption for all of these models is deterministic problem settings. Although deterministic models can effectively be used to produce solution methods and beneficial managerial insights, involving un certainty would improve the applicability of the models, while leading to additional complexi ty in solution methods. Therefore, in the next chapter we solely study demand assign ment models with stochastic demand. We consider a two-stage problem w ith a set of supplier facilities and customer demand stages (e.g., retailers).

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CHAPTER 6 DEMAND ASSIGNMENT MODELS UNDER UNCERTAINTY Introduction The previous chapters studied production pl anning models with various degrees of demand and capacity adjustment flexibility. Although customer demand uncertainty is one of the challenges that remain to be tackled in such problems, we first chose to study models involving several integrated pr oduction planning and demand management decisions in the absence of uncertainty. In this chapter, however, we focus on demand management in stochastic demand environments. The models introduced in this chapter involve key decisions in a supply-demand en vironment, including the assignment of customer demands to supplier facilities, and the resulting subproblems may arise in the selection of customer demands when a single facility exists. K1K2K3K4K5 88(,) 11(,) 22(,) 99(,) 1010(,) 55(,) 33(,) 44(,) 77(,) 66(,) K1K2K3K4K5 88(,) 11(,) 22(,) 99(,) 1010(,) 55(,) 33(,) 44(,) 77(,) 66(,) Figure 6-1. Supply chain network for 5 facilities and 10 downstream demand points. 128

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129 In this chapter, we consider a logistics distribution network consisting of supply facilities and retailers. Many firms have mu ltiple supply or source facilities that are located across a geographical region. In addition, the demand of an individual customer or market is often assigned to a single source facility of a supplier. The aim of this chapter is to determine an optimal assignment of customers to sour ce facilities. Each facilitys supply capacity is assumed to be known and each demand point implies a known random and stationary demand distribu tion. Figure 6.1 provides an illustrative example of the networks on which we focus, and involves five facilities and ten demand points. The dashed arcs denote the possibili ty of assignment of a demand point to a source facility, while the red arcs imply the fi nal assignment decision. We assume that the cost of assigning a demand to a facility is facility dependent. We also assume that facilities within the network cannot share capacity, due to either economic or geographical constraints. The decision making procedure works as follows: The demands are assigned to facilities first, based on expected assign ment costs and availa ble capacities of the facilities. Because actual demands are unknow n at this point, after demand realization, the assigned demands may exceed the available ca pacity of the facility In such cases an additional penalty cost is incurred for the capacity overflow. We allow the overflow cost to vary among different facilities. This co st might correspond to a loss of goodwill or the cost of purchasing additional outside resour ces, such as utilizing third-party logistics providers or quick response manufacturers. We wish to minimize the total expected cost, which includes the expected cost of satisfy ing demands by available facilities and the penalty costs for exceeding the capacity of each supply source.

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130 The focus of this study then is to assign a set of retailers to a set of source facilities with minimum expected overfl ow (shortage) and demand assignment costs. From this point of view, our problem is a stochastic version of the Generalized Assignment Problem (GAP) and is NP-Hard as well (for th e proof of this result please see Appendix C). The Generalized Assignment Problem (GAP ) focuses on assigning a set of tasks to a set of resources with minimum total assignmen t cost. Each resource has a capacity and each task must be assigned to one and only one resource, requiring a certain amount of supply resource consumption. A survey of exact and heuristic algorithms for the GAP can be found in Cattrysse and Van Wassenh ove (1992). More recently, Savelsbergh (1997) also proposed a Br anch and Price algorithm. The stochastic version of the GAP has also attracted the attention of researchers in recent years. Uncertainty in the GAP may ar ise in various problem parameters, such as the amount or resource consumption demanded by items, the assignment costs, the presence or absence of individual items or resources, and the resource capacities. Uncertainty is often handled by plugging in the expected values of the stochastic parameters, and solving the resulting deterministic formulation of the problem. Alternatively, one can implement stochastic -programming based approaches (see Birge and Louveaux 1997), which fully integrate th e relative performan ce of any solution under all possible realizations of uncertain para meters. Albareda-Sambo la, van der Vlerk and Areizaga (2002) consider the GAP where only a random subset of the given set of items is required to actually be pro cessed. They assume that the assignment of each item to a resource is decided a priori, and once the ac tual set of items is known, reassignment is possible. Spoerl and Wood (2003) also consider a stochastic GAP with uncertainty in the

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131 amount of resource used by the item-resource assignments. They focus on normally distributed resource usage parameters, and st udy a stochastic programming formulation in which capacity overflow is penalized. Toktas, Yen, and Zabinsky (2005) study capacity uncertainty and treat the resource capacities as random variables. Ahmed and Garcia (2003) and Taaffe, Geunes and Romeijn (2006A) have also examined variants of stochastic GAPs with dynamic demand assignment and capacity decisions. In the literature, a Branch and Price sche me (e.g., Savelsbergh 1997, Freling et al. 2003) is widely used when solving assignme nt problems. We will also employ a Branch and Price method in this chap ter although the pricing subpr oblems we face are quite different from those faced in the standard GAP. When solving the deterministic GAP, Knapsack Problems result as pricing subpr oblems. Similarly, a particular type of Stochastic Knapsack Problem is encountere d in the Branch and Price procedure as a pricing subproblem for the stochastic versio n of the GAP that we study. Stochastic Knapsack Problems are very interesting on thei r own since they deal with uncertainty in parameters of Knapsack Problems. In prior literature, various studies have fo cused on different variants of stochastic knapsack problems. With a goal of maximizi ng the expected value of selected items, Brian, Dean, and Goemans (2004) seek a solu tion for sequentially inserting items until the capacity is eventually exceeded. Our mo dels do not allow such a sequential or online process. Two recent papers by Kleinbe rg, and Rabani, and Ta rdos (1997) and Goel and Indyk (1999) also consider Stochastic Knapsack problems with items that have deterministic values and random sizes. Given a specified overflow probability p, they aim to maximize the total expected value by in serting a set of items whose probability of

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132 violating the knapsacks capacity is at most p. Kleinberg et al. (1997) consider only the case where item sizes have a Bernoulli-type distribution (with only two possible sizes for each item). Stochastic Knapsack problems with deterministic sizes and random values have also been studied by several other res earchers (Carraway, Schmidt, and Weatherford 1993, Henig 1990, Sniedovich 1980 Steinberg and Parks 1979), all of whom focus on fitting a fixed set of items in the knapsack with a maximum probability of achieving some target value. Several heuristics have b een suggested for this case. Another type of Stochastic Knapsack problem is known as the Stochastic and Dynamic Knapsack problem (Kleywegt and Papastavrou 2001, Papastavrou, Rajagopalan and Kleywegt 1996). In such settings, items arrive dynamically according to some stochastic process and the exact characteristics of an item are not known until it arrives, at which point in time the decision maker needs to decide to either accept or reject the item. Stochastic Knapsack problems correspond to so-called Selective Newsvendor Problems when the capacity of the knapsack is also a decision variable. Taaffe, Geunes and Romeijn (2006B) studied the Selective Newsvendor Problem, where they jointly determine the stock level and the customer de mands to select, in order to maximize a suppliers profit. Carr and Lovejoy (2000) examine an inverse newsvendor problem, which optimally chooses a demand distribution based on a fixed capacity. Based on a set of ranked demand portfolios, they determine the amount of demand to satisfy within each portfolio while not exceeding the predefined capacity. Barnhart and Cohn (2000) consider a very similar Stochastic Knapsack problem to the one considered in this chapter. They solve these Stochastic Knapsack Problems heuristically for the case in which demand sizes are normally distributed. In this chapter,

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133 we develop a new exact Branch and Bound algor ithm to solve this class of problems. We also provide a new solution method that solv es a linear relaxation of this Stochastic Knapsack Problem optimally. The remainder of this chapter is organized as follows. In Section 6.1, we describe and formulate our logistics network design problem with demand uncertainty. Section 6.2 summarizes an exact Branch and Price me thod for solving the design problem based on a Column Generation approach, while Section 6.3 focuses on the Stochastic Knapsack Problems, which are pricing subproblems in th e Branch and Price scheme. We introduce a heuristic solution method for this problem in Section 6.4. Section 6.5 provides a numerical study with key results and manageri al insights. Finally, we provide concluding remarks in Section 6.6. 6.1 Problem Definition and Model Formulation We consider a set of demand points J, and a set of facilities I. We let dj denote a random variable for the value of demand j and let Ki denote the capacity of facility i. Supply Sources (Set I ) Demands (Set J ) d 1 ( 1 1 ) d2 ( 2, 2) d3 ( 3, 3) d 4 ( 4 4 ) d5 ( 5, 5) d 6 ( 6 6 ) d 7 ( 7 7 ) d 8 ( 8 8 ) K1 K 2 K3 K4 xij Figure 6-2. Description of problem parameters on an example distribution network.

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134 We then let xij be a binary variable equal to 1 if demand j is assigned to facility i, and 0 otherwise. We define j = E[dj] and 2 j = Var[dj], and let Di denote the random variable for capacity requ irements at facility i, where i jJDd j i jx .We assume statistical independence of customer demands, and note that the expected value of the assigned demand for facility i is i jJ j i j x with a standard deviation equal to 2i jJ j i j x (note that because each xij is binary, we have 2ijij x x ). We assume that the cost to assign a unit of demand j to facility i is equal to Assuming a linear unit overflow cost of Si for facility i, the overflow cost term in the objective function can be denoted by where ijc ()iiiSK .i is the expected overflow at facility i (also known as the loss function), i.e., iiiiii KKDKfDd iDz From this point forward, we assume normality of the customer demand distributions. The normal distribution is not an uncommon distribution for applications where customer demand is continuous and in high volumes. Normality will also be helpful when characterizing the total demand assigned to a facility since total assigned demand to a facility is also a normal random variable. The loss function under normality at facility i can be expressed as In this formula, is the standard loss function, i.e., where ()iiiiKL ()iLz ()iii zLzuzudu 2ij jJ i jij jJKx z i j x and u is the standard normal probability density function.

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135 A feasible solution to our problem is a partition of the set of demands ( j J ) into I facilities. For each facility i, the expected cost for the assignment vector xi can be defined as: 2 20, () ,otherwiseij ij i j jJ ii ijjiji jij jJ jJ jij jJ 0 x jJ Kx gx cxSL x x Using this functional notation the Static Stochastic Assignment Problem [SSAP] can be formulated as: [SSAP] Minimize .()ii iIgx Subject to: 1 ij iIx for all j I, for all i I, j J. 1,0ijx Set Partitioning Formulation [SSAP] can be formulated as a set part itioning problem in a similar way as was done for the Generalized Assignment Problem by Cattryse et al. (1994) and Savelsbergh (1997). We let Li denote the number of subsets of customers that can feasibly be assigned to facility i and let ia denote the thsuch subset (for facility i), i.e., if demand j is an element of subset for facility i and otherwise. We will then call 1 ija 0 ija ia the th column for facility i. Then Static Stochastic Set Partitioning [SSSP] problem can be formulated as: [SSSP]

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136 Minimize 1()iL iii iIgay Subject to: 11iL iji iIay for all j J, (6.1) 11iL iy for all i I, (6.2) 0,1iy for all Li, i I. In this formulation, iy is equal to 1 if column is chosen for facility i, and 0 otherwise. The first set of constraints (6.1) enforces that each demand is assigned to precisely one facility and the second set of constraints (6.2) enforces that at most one feasible assignment is se lected for each facility. 6.2 Branch and Price Scheme The Static Stochastic Assignment Probl em is a nonlinear Integer Programming Problem which can be solved by enumeration, e.g., a Branch and Bound algorithm. On the other hand, a standard Branch and Bound method would require that all columns are available, while the number of columns associated with each facility will be tremendously large (2| J |), which makes such a Branch and Bound algorithm prohibitive. On the other hand, one can solve the problem in principle by using only the subset of columns which are relevant to the optimization of [SSSP]. The Column Generation scheme has been extensively used for this purpose. When the Column Generation and Branch and Bound methods are integrated, the resulting procedure is referred to as Branch and Price (Barnhart et al. 1998). Bran ch and Price is a genera lization of the linear programming (LP) based Branch and Bound scheme specifically designed to handle integer programming (IP) formulations that cont ain a huge number of variables. We next

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137 describe the Branch and Price scheme for our problem, starting with the Column Generation approach. 6.2.1 Column Generation Algorithm Gilmore and Gomory (1961) first introduced the column generation approach in the context of cutting stock problems. We use Column Generation as a pricing scheme to solve the set partitioning problem [SSSP]. Since the number of columns associated with each facility is extremely large, solving the linear relaxation of [SSSP], denoted as [SSSPL], would also be prohibitive. In the application of Column Generation, the problem is solved only for a subset of columns. This solution is then improved (when possible) by intr oducing additional columns to the problem. Instead of pricing out all of the columns associated with nonbasic variables, the column with the most negative reduced cost is determined by solving an optimization problem. When all the nonbasic column s have non-negative reduced cost, we can conclude that the solution provided by the present columns is optimal for the linear programming relaxation [SSSPL]. 6.2.1.1 Column generation for [SSSPL] Step 1. Construct set of columns, L0 such that [SSSPL(L0)] has a feasible solution. Set L = L0. Step 2. Solve [SSSPL(L)] yielding y*(L) Step 3. Find a column (or a set of columns) so th at the new objective value is as least as good as the objective value of y*(L) and add this column (or set of columns) to L. Go to Step 2. Such columns can be determined by solving the pricing problem, which will be explained in Section 6.2.2. If no such column exists, the current solution (y*(L)) is optimal to [SSSPL].

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138 6.2.1.2 Initial columns The column generation scheme starts with a set of columns that ensure a feasible solution. We choose to star t with 3 columns for each facil ity, one column consisting of all ones and one column consisting of all zer os. The heuristic solution method introduced later in Chapter 6.4 provides the third initial column for each facility. This third column also ensures feasibility. 6.2.2 Pricing Problem In the column generation approach, columns are left out of the linear relaxation because there are too many co lumns to handle efficiently, and most of them will have their associated variable equal to zero in an optimal solution anyway. In order to check the optimality of a solution for a given set of columns L, the following optimization problem (pricing problem) fo r each facility is solved. Minimize *()iiqSSKPL iI where is the optimal dual price from the solution to the problem consisting of the set of columns L associated with the constraint set *()iL (6.2) for facility i, and is the value of the optimal solution to the following problem. iqSSKP [SSKP ( i )] Minimize .,iigxu Subject to: for all j J. 1,0ijx where

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139 *2 20, (,) ,otherwiseij ij i j jJ ii ijjjiji jij jJ jJ jij jJ 0 x jJ Kx gxu cuLxSL x x and is the optimal dual price associated with the partitioning constraint juL (6.1) for demand j. If all of the subproblems yield a non-nega tive objective function value, then the current solution is optimal for the relaxed pr oblem. Otherwise, columns corresponding to feasible solutions with negative objective function values would be added to current set of columns. Although the mathematical formul ation for this pricing subproblem is given here, the solution method is explained in detail in Section 6.3. 6.2.3 Branching Scheme The column generation process terminates when no profitable columns are found, and the resulting solution is optimal for [SSSPL]. On the other hand, this solution might not satisfy the integrality requirements, which implies that such a solution is not optimal for [SSSP]. When the [SSSPL] solution that is obtained by column generation does not have a binary optimal solution, applying a standard Br anch and Bound procedure to this problem over the existing columns will not necessarily find an optimal solution to the original problem. Standard branching on the iy variables creates problems along with the Branch and Bound tree (Savelsbergh 1997). The Branch and Price method is used to overcome such difficulties.

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140 Instead of branching on the i y variables, Branch and Price uses a branching rule that corresponds to branching on the original variables xij. Given an optimal solution for [SSSPL], the original assignment variable xij can be calculated as ij iji L x ay When branching, we select the xij variable which is closest to 0.5 (breaking ties arbitrarily). Assume that at the end of the column generation process, the column set L gives the optimal solution to the linear relaxation and xij is selected for bran ching. Let node (0) denote this initial LP relaxa tion solution. We then create two child nodes, node (1) with xij = 1 and node (2) with xij = 0. For child node (1), we then need to delete all existing columns of node (0) that do not assign demand j to facility i and we also assign demand j permanently to facility i. Similarly, for child node (2), all existing columns in the node (0) that assign demand j to facility i are deleted and demand j is removed from the ith pricing problem. The child nodes also inher it the restrictions of assignment variables from their parent node. We apply the same column generation method for child nodes starting with a set of the parent node columns which satisfy these restrictions. So far we have determined how to gene rate child nodes emanating from a parent node. However we have not specified which node will be selected next for branching. For this purpose we have applied depth-first search. Depth-first search is known to get feasible solutions fast since experience with similar problems shows that the feasible solutions are more likely to be found deep in the tree than at nodes near the root. In addition, at each node of the tree we invoke a greedy heuristic to find a feasible solution which will provide an upper bound for the optimal cost. We next describe the details of this greedy heuristic.

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141 6.2.4 Rounding Heuristic Similar to the Branch and Bound pro cess, when good lower and upper bounds exist, the fathoming process will get faster, th us improving the solution efficiency. At any node of the branching tree, the solution reached at the end of the column generation process provides a lower bound for the optimal solution (if it exists) to be found in the tree emanating from that node. Moreover the objective func tion of any solution implying a binary assignment of demands to facilities provides an upper bound. The greedy heuristic explained in this section is developed to obtain such a solution. At the end of the column generation pr ocess at a node, some of the original assignment variables, xij, might be fractional, while each demand should be assigned to exactly one facility in a feasible solution. In order to generate a binary solution, demands partially assigned to different faci lities are reconsidered. We let set f J denote the set of demands which are partially assigned to a facility. For each demand j f J let the set f j F denote the set of facilities with partial assignments, i.e., with some For each j 1 0ijx f J we select a facility k f jF with the minimum assignment cost, i.e., minf jkj ij iFcc f jkF and set 1kjx and for all other facilities we set 0,ij x ik (note that in this stochastic demand setting, there is no notion of an assignment of demands to a facility that violates the f acility capacity a priori; thus we can assign a demand to any facility without considering the violation of facility capacity). 6.3 Static Stochastic Knapsack Problem In the column generation pro cess, in order to find a column with a negative reduced cost, the following pricin g problem needs to be solved for each facility.

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142 [SSKP ( i )] Minimize .,iigxu Subject to: for all j J. 1,0ijx where *2 20, (,) ,otherwiseij ij i j jJ ii ijjjiji jij jJ jJ jij jJ 0 xj J Kx gxu cuLxSL x x In this formulation, the unit assignm ent cost term can be restated as *j ijij juL cc Since we are dealing with a single facility problem, the index i can also be suppressed and the resulting optimi zation problem becomes the following Static Stochastic Knapsack Problem (SSKP) with random weights. [SSKP] Minimize xg Subject to: 0,1jx for all j J where 2 20, () ,otherwisej jj jJ jjj jj jJ jJ jj jJ 0 x jJ Kx gx cxSL x x This problem can also be viewed as a resource allocation problem where the planner needs to choose among possible demands, and each demand requires some quantity of a resource such as space in a warehouse, truck capacity, or machine time in a

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143 manufacturing environment. In the deterministic knapsack problem, there is a collection of objects, each with a given weight and va lue. The objective is to choose the set of objects with maximum value within the capacity limits. When the weights of the objects are allowed to be random, this problem is know n in the literature as a Static Stochastic Knapsack Problem. We next discuss this generic Static Stochastic Knapsack Problem. We denote the capacity of the facility by K and let xj be the binary decision variable for selecting demand j with random weight dj. The parameter j c denotes the unit cost for satisfying demand j Demands are assumed to be normally distributed with mean j and variance 2 j and we let D denote the random variable for the total selected demand (i.e., j j jJDd x ). Excluding the solution where all decision variables equal zero, the following restricted problem [SSKP(R)] must be solved. If the optimal objective value of this problem is less than zero, then the associated solution is optimal to the original problem. Otherwise, the solution where all the decision variables are equal to zero (with zero objective f unction value) is optimal. [SSKP(R)] Minimize 2 j jj jj jJ jJcxSLzx Subject to: 2 jj jj jJ jJ x zx K (6.3) 1j jJx 0,1jx for all j J, (6.4) z, free. (6.5)

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144 The objective here is to minimize the exp ected cost, which includes the expected cost of selected objects and the penalty fo r exceeding capacity. Since we exclude the solution where all decision variables equal zero, the standard normal variable z can be used in the analysis in place of 2 j j jJ j j jJKx x Constraint (6.3) ensures that this identity holds (although we can easily substitute this cons traint out of the problem, we will find it more convenient to work with the formulation shown above). Since the second term in the objectiv e function is non-separable in the xj values, we have not found it possible to develop a pseudo-polynomial dynamic programming algorithm similar to that commonly used for the standard Knapsack Problem. Romeijn, Geunes, and Taaffe (2006) developed a solution method for a class of nonlinear nonseparable continuous knapsack problems. The Static Stochastic Knapsack problem has a similar structure, with the exception of the added complex ity introduced by the nonlinearity of the capacity constraint. The solution method in that study is primarily based on a characterization of the necessary Karush-Kuhn-Tucker (KKT) optimality conditions. We can follow a similar appr oach and provide an efficient solution mechanism for the continuous version of this problem, which will then serve as a starting point for a Branch and Bound algorithm. We next describe our solution approach for a linear relaxation of the [SSKP(R)]. 6.3.1 Linear Relaxation of Restricted St atic Stochastic Knapsack Problem In this section we focus on solving the linear relaxation of the restricted Stochastic Knapsack Problem, [SSKP(R)L]. The following is the mathematical model for this case: [SSKP(R)L]

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145 Minimize 2 j jj jj jJ jJcxSLzx Subject to: 2 jj jj jJ jJ x zx K (6.6) for all j J, 0jx 1 z, free The KKT conditions are necessary but not su fficient for this problem (sufficiency does not apply here because the objective function is not convex; see Bazaara, Sherali and Shetty 1993; necessity of the KKT conditions for this problem follows by solving (6.6) for z, substituting the result in the objective func tion, and the fact that an interior point x with 0 < x < 1 exists; note that using this approach provides the same KKT conditions we derive in the following sect ion). We next state the KKT optimality conditions for [SSKP(R)L] 6.3.1.1 KKT conditions The KKT optimality conditions for the problem class [SSKP(R)L] can be written as: (1) 22 22() 22jj j jj jj jj jJ jJcSLz zv xx j jw for all j J (2) 2() 0jj jJLz xS z (3) vj( xj 1) = 0, for all j J (4) wjxj = 0, for all j J (5) vj, wj 0, for all j J (6) free,

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146 (7) for all j J 1jx (8) for all j J 0jx (9) z free. 6.3.1.2 Analysis of KKT conditions In this section, we develop candidate solutions for optimality based on the KKT conditions. Recalling the definition of the standard normal loss function, where zzzzLc) ( 1cz z the first KKT condition can be restated as: 2 22j c j j jj jj jJcz S z S z z x jv w for all j J The second KKT condition enforces either 2 j j jJ x or z zL S )( to equal zero at optimality. Assuming nonzero variance values, 20jj jJx is only possible if all demands are rejected, and as we previ ously noted, we consider that case separately as a candidate solution with zero objectiv e function value. For all other cases 0 )( z zL S must hold, implying () 1cLz SSzS z z Assuming a nonnegative unit overflow cost, this forces > 0 in an optimal solution. Note that = 0 implies = 0, which only holds if z = and this only makes sense if expected demand is (which is not possible since we require nonnegative expected demand). Substituting in place of the first KKT condition now becomes: cz zSc 2 22j c j j jj jJSz cSz vw x j j for all j J.

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147 At optimality, KKT condition (1) must hold for any demand j In this equation, values of j v and j w are directly influenced by the j x values. Each j x takes a value of 0, 1, or a fractional value (between 0 and 1) in any solution. We can therefore re-examine the first KKT condition for a given j for these three cases. i. xj= 0: KKT conditions (3), (4), and (5) imply that = 0 and KKT condition (1) then becomes iv 0 iw 2 22j c j jj jj jJSz cSz w x implying: 2 22c jj j j j jJcSz Sz x ii. xj= 1: KKT conditions (3), (4), and (5) imply that and 0jv j w = 0. KKT condition (1) then becomes 2 22j c j jj jj jJSz cSz v x implying: 2 22c jj j j j jJcSz Sz x iii. 0 < xj < 1: KKT conditions (3), (4), and (5) imply that and 0jv j w = 0. KKT condition (1) then becomes 2 22j c jj j j jJSz cSz x implying: 2 22c jj j j j jJcSz Sz x As a result of (i) (iii), any given value of z implies an ordering of demands based on the ratio 2 c j j j jcSz rt We index these ratios in nondecreasing order, i.e., 123... J rtrtrtrt The ratio rtj provides a measure of th e attractiveness of each demand, and we will refer this ratio as the desirability ratio The negative of the assignment cost, cj, can be viewed as unit revenue for demand j In such situations,

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148 c jcSz can be seen an adjusted term for expected marginal profit where can be viewed as expected marginal overflow cost, since S is the unit overflow cost, while can be viewed as probability of ove rflow. Therefore, given a value of z, a high mean to variance ratio improves the attractiveness of a demand. This ratio may provide useful managerial insights on the cla ss of static stochastic knapsack problems. For instance, consider a setting where a decision maker needs to select among outstanding demands and must achieve a given service level (w hich implies an associated z value). She can then utilize the ratio zSc cz 2 c j j j jcSz rt as a heuristic selection measure for ranking candidate customer demands. The KKT-based solution approach is primarily based on the ranking among demands for given z values. In the solution algorithm, we consider all possible z values in two complementary sets. The first set consists of z values for which there may exist ties among the desirability ratios of demands. Ties among rt values of items i and j only exist at the values of zij such that 2 cc ii jiji j ijcSzcSz 2 j This also implies that 2 2 2 2 jiij ijjjii ij cSS cc z and such a zij value can only be valid if 1 02 2 2 2 jiij ijjjiiSS cc Assume that there are M such valid zij values, and denote this set by Z We let denote the kth ordered value in the set Z i.e., In addition let the set kz ][ ]2[]1[...Mzzz c Z denote the z values outside of this list; note that for z values in

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149 the set c Z there must be a strict ordering among demands according to the rtj ratio values. We next develop a solution algorithm to investigate all possible solutions that satisfy the KKT conditions. 6.3.1.3 KKT based algorithm In this algorithm, we consider all poss ible values that the decision variable z can take at optimality. Based on these values, a ll possible candidate solutions satisfying the KKT conditions are determined. At the end of the algorithm, the candidate solution with the minimum objective function value is chosen. If this solution is nonnegative or there is no candidate solution satisfying the KKT c onditions, then the solution that rejects all demands is optimal to [SSKPL]. Otherwise, the candidate solution chosen at the end of this algorithm is optimal for [SSKPL]. We consider two types of z values, the set of Z z (where ties exist among rtj values) and the set of c Z z (where a strict ordering exists among rtj values). 1. z values in the set Z A z value in the set Z implies an ordering among rtj values such that there exist ties among at least two rtj values. Given a value of ][ kzz for k = 1, M, we can order the demands based on their rtj values as follows: Such an ordering suggests that demands m m +1, n -1, n have the same desirability ratio. We let denote this ratio such that and let 12 11 1... ,... ...mmnnnrtrtrtrtrtrtrtrt J n krt 11,...,k mmnrtrtrtrtrt ][ k R denote the

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150 set of demands [ m ], [ n ]. Similarly, we let denote the set of demands [1], [ m 1] and let the set denote the set of demands [ n +1], [ J]. ][ 0 kR ][ 1 kR Any possible solution with ][ kzz will either include at least one fractional value for variables in the set ][ k R or not. We next explore these two cases. a. There is at least one fractional demand in the set []k R If any demand in the set ][ k R is fractional, we have that [] 22k k j j jJSz rt x and so [] [1][2] [1] [1] [] 2... ... 2k mn jj jJSz rtrtrt rtrt x I Based on the KKT conditions, this further lead s to the conclusion that 0jx for 0 k j R and for We can also determine the values of 1jx 1kjR j x for demands in the set by solving an optimization problem. Noting that the condition kR 1222kkk jj j k jR jRSz x rt must hold, the objective function can then be written as 1() () 2kkk k jjj jj k jR jRSz cxcSLz rt and the capacity constraint as 1() 2kkk k jj j k jR jRSz xKz rt The optimization problem when z = zk now becomes Minimize 1() ,( 2kkk kk jj j j j j k jR jRSz xjRcxcSLz rt ) Subject to: 1() 2kkk k jj j k jR jRSz xKz rt

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151 12 22() 2kkk j jj k jR jRSz x rt 0 xj 1, j kR This is a simple linear program, and will result in at most two fractional variables at optimality. The solution of this optimization problem (if feasible), together with for 0jx 0 k j R and for 1jx 1 k j R creates a candidate solution which satisfies the KKT conditions. b. There is no fractional demand in the set kR When the decision variables associat ed with the demands in the set ][ k R take binary values in the optimality, there will be at most one fractional demand which is either in the set 0 k R or in the set 1 k R in the optimal solution. We de note the fractional demands index in the ordered list by [ g], and this demand will be either in the set or in Since demand [ g] is fractional and not in the set ] kR0 kR1 [ k R KKT condition (1) le ts us state that [] [] 22k g j j jJSz rt x and [] [1][2] [1] [1] [] 2... ... 2k g gJ jj jJSz rtrtrt rtrt x In this case, we let the set denote the demands [1], [ g-1] and let the set denote the demands [ g+1], [ J ]. For these sets, we set k gR0 k gR1 0jx 0 k g j R and 1jx 1 k g j R In order to fully characterize a candidate solution, we also need the value of the fractional demand The value of the fractional demand can be determined by solving the following equations. ][ gx 1[][] []() 2k gk k g g jR gSz xKz rt j (6.7)

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152 12 2 [][] []() 2k gk 2 g g jR gSz x rt j (6.8) 0 x[g] 1 (6.9) If 1 12 2 [] [] 2 [] []() () 2 2 01k k g gk k k j j jR jR g g ggSz Sz Kz rt rt then there exists a unique solution for this problem, i.e., 1[] [] []() 2k gk k j jR g g gSz Kz rt x Otherwise, no such solution with x[g] fractional serves as a candida te for an optimal solution. 2. z values in the set Zc We next consider values of z such that ][ ]1[ k kzzz for k = 2, M We will now investigate candidate KKT solutions where z is in any one of these intervals. The ranking among demands is the same for all possible z values within the interval Therefore, when [1][](,kkzz ) ][ ]1[ k kzzz a strict preference ordering among demands can be obtained for any arbitrary z value in this set, and we let denote this arbitrarily selected value. For we let ),1( kkz ),1( kkz (1,) 2 ckk j j j jcSz rt denote the desirability ratio for demand j. When at most one of the xj values can be fractional in the optimal solution. Assume the gth demand in the list is the fractional demand with the ratio ][ ]1[ k kzzz (1,) [] 22kk g j j jJSz rt x The following ordering among desirability ratios of demands

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153 must hold for this case: (1,) [1][2] [1] [1] [] 2... ... 2kk g gJ jj jJSz rtrtrt rtrt x In this case, we use the notation to denote the set of demands [1], [g-1] and to denote the set of demands [g+1], [J]. Based on KKT condition (1), we can also deduce that for and ),1( 0 kk gR ),1( 1 kk gR 0jx (1,) 0 kk gjR 1jx for In order to completely define a candidate solution, we also n eed to determine the exact values of z and x[g]. The following optimization problem is designed for this purpose. (1,) 1 kk gjR Minimize (1,) (1,) 1122 [][][] [][]()kk kk g gg jj gg j jR jRcxcSLzx Subject to: (1,) (1,) 112 [][] [][]kk kkgg jgg j jR jR 2 x zx K (6.10) 0 x[g] 1, zk-1 z zk. The constraint (6.10) re, the problem above is actually a one dimensional optimization problem. Appendix D shows that the objective function of this problem is either concave or convex in x[g] on the entire interval defined by zk-1 z zk. Therefore, considering only the stationary points of the cost function is sufficient to obtain a candidate solution (if one exists on the interval ). We also need to check two critical values of z, where the one-dimensional function in x[g] changes from convex to concave (or concave to convex). These critical points and points where the derivative of the cost function is zero provide candidate solutions for this case. The formulas for such critical points are given in Appendix D. We can determine the value of x[g] where the derivative of the objective function is zero on an interval (if one ex ists) by applying bisection

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154 search. We also require and []01gx 1 (1,) 1[][] [1] [] 22 [][]k kkgg j jR kk gg j jRKx zz x for feasibility, and therefore for this solution to be a candidate for an optimal solution. For the cases where and [1]zz [] M zz we can apply the same approach as long as we can provide lower and upper bounds on z values without loss of optimality. Complexity of the KKT Based Algorithm In the algorithm, we first assume that z takes values from the set Z. We already assumed that there are M such valid z values, where M is O (|J|2). In this set, we let kz denote the kth ordered value, i.e., Any possible solution with will either include at least one frac tional value for variables in the set ][ ]2[]1[...Mzzz ][ kzz ][ k R (defined in the previous secti on), or not. For the first case we solve an LP which consists of at least two decision variables. We let the time required to solve this LP be denoted as O( ). For the second case, for each possi ble fractional demand (of which there are O(|J|)) we only need check whether the following inequality holds. 1 12 2 [] [] 2 [] []() () 2 2 01k k g gk k k j j jR jR g g ggSz Sz Kz rt rt When z is not in the set Z, we consider O(|J|2) intervals for z. For each interval, we consider each demand as a candidate for taking a fractional value in the optimal solution. For each demand, we solve a one dimensiona l optimization problem on the interval [0, 1]. To solve these problems we determine critical points (as explained in Appendix D). We then apply a bisection search over the inte rval [0, 1] to determine the point where the derivative of the objective function is zero. If we want the final interval of uncertainty in

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155 this search to be of length then the complexity of the bisection search becomes log(1/ ). The overall complexity for this problem thus becomes O(|J|2(( +|J|log(1/ ))). 6.3.2 Branch and Bound Scheme The algorithm above provides us the optimal solution for the linear relaxation of the Static Stochastic Knapsack problem. On the other hand, in order to find valid columns for our problem, the original pric ing problem must be solved optimally. Therefore, we apply a Branch and Bound pr ocedure to obtain a binary solution. Given a fractional solution, we need to select a demand to branch on. As opposed to common intuition which branches on fractional items, we select the item by examining the desirability rankings. We utilize the ratio 2 c j j j jcSz rt as a desirability measure for each demand, where z* is the optimal z value given by the KKT-based algorithm for solving the relaxation. The dema nd with the highest desirability ratio is selected for branching. Although this item ge nerally does not take a fractional value in the linear relaxation solution, it is likely to equal one in an optimal solution, and resolving the values of such variables early in the branch and bound process can speed up the branch and bound process. This branching technique has also been used for regular knapsack problems where the item with highe st desirability ra tio (value/resource requirement) is first se lected for branching. Depth first search is employed when determining the node sequence in branching. In addition, to speed up the algorithm we em ploy the following greedy heuristic to obtain feasible solutions to provid e upper bounds for the optimal objective function value. Because the solution of the relaxed problem i nvolves at most two fractional variables, we

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156 can round up and down the fractional variables to obtain solutions for the binary problem. This approach provides us at l east two and at most four diffe rent solutions with close-tooptimal objective function values. In addition, it should also be noted here that feasibility is not an issue when rounding, since we a llow capacity overflow at a penalty cost. 6.4 Preference Order Greedy Heuristic In this section, we present a heuristic method which will provide a benchmark for our Branch and Price algorithm. This me thod first assigns demands to facilities according to a minimum cost criteria. Such a solution is then improved by applying local search in an iterative manner. We next explain the heuristic in detail. Step 1. For each demand j, determine the facility m with minimum assignment cost, i.e., and set minmj ij iIc c 1mjx Step 2. Let Ji denote the set of demands assigned to facility i. Let 2i iij jJ i j jJB z for each facility i with at least one demand assigned. Step 3. For each demand j, calculate the desirability ratio: 2 c ijiij j jcSz rt where zi is the z value associated with the facility i to which demand j is assigned. Sort demands in increasing order of rtj and let LIST denote this list. Step 4. Determine the demand q that has the minimum desirability ratio, i.e., minqj jJrtrt Step 5. Calculate the cost decrease for each facility when demand q is added to their supply list, i.e., If there is no cost decrease for any facility, remove demand q from the LIST and go back to Step 3. iJ q Step 6. Determine the facility f with maximum cost decrease and assign demand q to facility f. Update each Ji list. Go back to Step 2.

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157 6.5 Numerical Study We have conducted a numerical study to de monstrate the efficiency of the Branch and Price algorithm and the Preference Order Gr eedy Heuristic. All tests were performed on a PC with an Intel Pentium M 1.86 GHz processor with 1GB RAM. All Linear Programming problems were solv ed using Dash Xpress solver. Our computational tests draw on problem data based on the GAP problem instances generated by Cattryse, Salomon and Va n Wassenhove (1991). In our [SSAP] computational tests, we consider negative assignment costs (to reflect revenues) and follow a profit maximization approach. In our numerical study, each demand has an associated base revenue value drawn from DU (5, 20) and a facility specific revenue adjustment value drawn from DU (0, 5) (We let DU(x, y) denote the discrete uniform distribution with lower bound x and upper bound y). In the GAP instances, capacity requirements for demand-resource pairs are ta ken from discrete uniform distributions, DU (5, 25). GAP data involves facility-specific capacity requirements for each demand, while [SSAP] requires mean and standard de viation information for each demand. The expected value for demand j (independent from facility information) is determined using 12*(0,1)*I ij i jUm I where mij denotes the GAP data for the capacity requirements of demand j from resource i, and mij is drawn from DU (5, 25). We used the capacity data from the GAP instances where the capacity for resource i is computed as 10.8J ij j im K I In order to examine the relative performa nce of the Branch and Price algorithm and the Preference Order Greedy Heuristic under various problem settings, we created

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158 problem instances with low and high levels of demand variation and overflow cost. For low levels of standard deviations, we set *0.1jj and for high levels we set *0.5jj In addition, overflow costs, whic h do not exist in GAP instances, are randomly generated using a discrete uniform di stribution, with DU (20, 25) for low levels and DU (30, 35) for high levels. For our experiments we used a variety of problem sizes. The num ber of facilities |I| equals 5, 8, and 10, while the ratio J n I is set to 3, 4, 5 and 6, which determines the number of demands |J|. The following table illustrates th e different problem sizes used in our numerical study. For each de mand/facility combination, 5 base problem instances are generated involving capacity, mean, and revenue information. For each base instance, 4 problem instances are generated addressing low and high levels of standard deviation values and overflow costs. Therefore, 240 instances are ge nerated in total. Table 6-1. Problem sizes in the numerical study Number of Facilities (| I |) Number of Demands (| J |) n = 3 n = 4 n = 5 n = 6 5 15 30 40 40 8 20 24 48 50 10 25 32 30 60 Our computational study indicated that the run time required for the heuristic method is negligible compared to the Bran ch and Price run times, since the longest heuristic run time faced is 0.062 seconds. Th erefore, if the heuristic method provides competitive results with the Branch and Pr ice algorithm, it is more desirable to implement, especially for large problem instances.

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159 We have tried different pricing approaches to speed up the algorithm. In the first case, the pricing problems are solved optima lly for each facility at every iteration of the column generation process. On the other hand, the Branch and Bound method to solve the pricing problems often requires a great d eal of time after finding a close-to-optimal feasible solution. In the second case, therefore, we did not solve the pricing problems optimally at each iteration of the column ge neration procedure. Instead, for all the facilities we employed a heuristic method, which provides results in shorter running times. In this heuristic method, during Branch and Bound, we prune nodes when the difference between best binary solution and the relaxed solution is within 10% of the relaxed solution. At the end of this pr ocess, if there are no outstanding candidate columns to add to the column list, we solve the pricing problems optimally until we get the first candidate column. Table 6-2. Computation times for various problem settings Problem Size Computation Time in Seconds | I | |J | Solve pricing problems optimally at every step Solve pricing problems heuristically at the beginning 5 15 30.9 9.1 5 20 38.01 10.5 5 25 361.7 78.7 5 30 350.2 225.9 8 24 842.4 112.2 8 32 1297.8 173.8 8 40 1581.8 223.1 8 48 1693.2 339.8 10 30 1768.8 398.3 10 40 1943.1 659.7 Table 6.2 provides a comparison of aver age computation times of each problem size when these two approaches are applied. When the problem size exceeds 10x50, the required computation time gets very long (m ore than 2 hours for an instance) for both pricing options. Therefore, we did not involve these cases in our numerical analysis.

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160 Figure 6.3 summarizes the average run times for various problem sizes and different pricing mechanisms. Examining Fi gure 6.3, one can de duce that the problem size has a huge influence on the running time of the Branch and Price algorithm. For instance, when the number of facilities is 5, the problem can be solved in around 9 seconds; on the other hand, when the number of facilities increases to 8, the running time increases to almost 2 minutes. We also not e that in our problem generation procedure, when the number of facilities increases by F, the number of demands also increases by nF. We have observed that the running time of the algorithm becomes undesirably long beyond the 10 facility and 50 demand case. 0 100 200 300 400 500 600 700 5 9.08125 10.50795 78.6639 225.8672 8 112.19225173.82585 223.1461 339.77265 10 398.2601 659.6718 n =3 n =4 n =5 n =6 Figure 6-3 Comparison of computation times for various problem settings We next investigate the relative performa nce of the heuristic algorithm compared to the Branch and Price method. We also ex amine the effects of demand variation and overflow cost on the running time of the heuristic method. Figures 6.4 and 6.5 summarize the results presented in Table 6.3. The results presented in Table 6.3 denote the ratio Heuri stic objective / Branch and Price objective

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161 for each problem size. The results show that the heuristic method provides solutions with objective values equal to 84% of the optimal objective function value on average. Figure 6.4 shows that the heuristic algorithms perf ormance improves when the problem size is very large. Since the Branch and Price al gorithm does not produce fast results for very large instances, the heuristic method can effectively be used for such cases. Table 6-3. Performance evaluati on table for heuristic algorithm Standard Deviation Overflow Cost Number of Facilities Number of Demands Low High Low High Average 5 15 0.72 0.79 0.87 0.64 0.76 5 20 0.89 0.91 0.94 0.86 0.90 5 25 0.84 0.86 0.90 0.80 0.85 5 30 0.86 0.86 0.93 0.79 0.86 8 24 0.80 0.80 0.87 0.73 0.80 8 32 0.80 0.76 0.85 0.71 0.78 8 40 0.92 0.92 0.96 0.88 0.92 8 48 0.97 0.99 1.00 0.96 0.98 10 30 0.72 0.75 0.82 0.64 0.73 10 40 0.79 0.77 0.88 0.69 0.78 Average 0.83 0.84 0.90 0.77 0.84 0.00 0.20 0.40 0.60 0.80 1.00 1.20 5 0.76 0.90 0.85 0.86 8 0.80 0.78 0.92 0.98 10 0.73 0.78 n =3 n =4 n =5 n =6 Figure 6-4. Performance eval uation for heuristic algorithm The results given in Table 6.3 allow us to conclude that the va riation in standard deviation values does not change the effectiveness of the heur istic solution. On the other

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162 hand, Table 6.3 and Figure 6.5 indicate that when the overflow cost is higher, the heuristic solution method results in relatively undesirable solu tions. For instance, for the low levels of overflow cost, the heuristic method is competitive with the Branch and Price method, resulting in objective functi on values which are 90% of the optimal objectives on average. On the other hand, wh en overflow costs are hi gh, this percentage decreases to 77%. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 5 0.87 0.64 0.940.860.900.800.93 0.79 8 0.87 0.73 0.850.710.960.881.00 0.96 10 0.82 0.64 0.880.69 OC-low OC-high OC-low OC-high OC-low OC-high OC-low OC-high n =3 n =4 n =5 n =6 Figure 6-5. Performance comparis on for high and low overflow costs 6.6 Conclusion In this chapter we considered a two-st age logistics network design problem. In contrast to the rest of this thesis, this chapter assumes uncertain customer demand, although we sacrifice the more complex operatio nal cost structures considered in the prior chapters. Given a set of supply sour ces (e.g. warehouses or production facilities) and a set of uncertain demands (e.g customer s or markets), we provide models to assign demands to available resources in the best pos sible way. The problems presented in this study have broad application since efficiency of supply chains is highly dependent on network design. Models that account for uncertainty improve the quality of solution

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163 when compared to deterministic models. Th e objective is to minimize expected cost, which includes the expected cost of satisfying demands by available resources and penalty costs for exceeding the capacity of each supply source. We employ a Branch and Price approach, which is commonly used to solve large scale assignment problems. The associated pricing problems can be viewed as Stochastic Knapsack problems, which are both practically and theoretically inte resting in their own right. Our solution methodology for the class of Stochastic Knaps ack problems is also novel and can be used in many other contexts. We also discuss an alternative heuristic solution method for our network design problem and a num erical study that demonstrat es the performance of the Branch and Price algorithm and the suggested heuristic method.

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CHAPTER 7 CONCLUSION Production planning and control de cisions deal with the acqu isition, utilization, and allocation of resources to satisfy customer demands in the most economical way. Typical factors that affect producti on planning include pricing, capacity levels, production and inventory costs, and overtime and subcontrac ting costs. Past research suggests a hierarchy of optimization models that are ap plicable for these distinct decision making categories. In nearly all of the models considered to date, a consistent assumption employed is that of fixed exogenous demand, which a supplier mu st meet, along with pre-specified capacity limits. In certain contexts, however, suppliers have some dimensions of flexibility that allow them to affect the total de mand they face and to determine the supply capacity. This thesis provides set of models that d eal with a suppliers capability to manage its demand, its internal supply level, its inventory, produc tion and subcontracting levels, and its demand-supply network design. We fo cused primarily on single-stage problems, which generalize classical requirements plan ning models by addre ssing various sources of demand and capacity flexibility a firm might have. In some of the models, pricing serves as an implicit mechanism for effec tively selecting demand levels. We also discussed a different interpretation of the mode ls where we considered the problem in the context of demand selection and timing decisions In the last chapter, on the other hand, we dealt with a two-stage supply network desi gn problem. In contrast to previous 164

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165 studies, our models incorporat ed uncertainty in demand in the allocation of demands to facilities. Chapter 3 provides profit-maximizing produc tion planning models for determining optimal demand and internal production capacity levels under price-sensitive deterministic demands with subcontracting a nd overtime options. The models determine a producers optimal price, production, inventor y, subcontracting, overtime, and internal capacity levels, while accounting for producti on economies of scale and capacity costs through concave cost functions. Although subc ontracting has been considered by many researchers, integration of these decisions with overtime opportunitie s has not been fully discussed. Although both practices help the manufacturer to increase supply capacity, there are important differences between over time and subcontracting. We also include capacity acquisition decisions in our models. We use pol yhedral properties and dynamic programming techniques to provide polynom ial-time solution approaches for obtaining an optimal solution for this class of problem s when the internal capacity level is timeinvariant. Future research may also incorpor ate lead time flexibility into these models by allowing the producer to choose the period in which it satisfies each demand, which would provide extra flexib ility on the demand management side. In addition, subcontractor selection may provide an in teresting avenue for further research. Coordinating decisions with a subcontractor may also contribute to increased revenues and profitability for both the subcontractor and manufacturer. Therefore, supply chain coordination through agreements with subcont ractors provides an additional promising research direction to pursue.

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166 Chapter 4 provided an integrated model th at simultaneously considers setting prices and procurement planning decisions under cap acity limits and production economies of scale. In this chapter we provided effective solution methods for discrete-time, finitehorizon production planning probl ems with economies of scale in procurement, pricesensitive demand, and time-invariant produc tion capacities. We considered general concave revenue functions in each time peri od, and sought to maximize total revenue less procurement and inventory hol ding costs. For both the dynamically varying price case and the constant price case, we have shown that these problems can be solved in polynomial time for a very general class of concave revenue functions. This work fills a gap in the literature and generalizes seve ral recent works on in tegrated pricing and procurement planning. In additi on, the structural properties of optimal solutions that lead to efficient solution methods also serve to sharpen intuition regarding optimal demand management strategies in complex planning si tuations. Natural directions for future research include heuristic approaches fo r solving these problems under time-varying capacities, as well as generalizations to handle multiple products that share procurement capacities. In Chapter 5, we have integrated a se t of demand planning de cisions in production planning models to take advantage of dema nd timing flexibility. In addition to demand selection decisions, our work r ecognizes that suppliers often have flexibility to adjust order shipment times. In many such setti ngs, the net profit of an order depends on the time at which the order is satisfied. Th e models in Chapter 5 accounted for these demand-timing decisions as well as the de mand selection decisions. We provided a polynomial-time solution method for the model in presence of certain regularity

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167 conditions on cost parameters. However, th ese cost assumptions (sometimes called nonspeculative motives) are not unrealistic. Mo reover, we developed a heuristic method based on the dual of the linear programming relaxation of our model. Computational tests performed on a set of randomly created problem instances showed that the solution approach we developed is quite effective and produces close to optimal solutions. Although we discussed flexibility in demand ma nagement in this chapter, we did not account for finite production capacity limitati ons. Demand management can play an even more significant role when a manufact urer has limited production capacity. Hence, future research may consider capacity restrictions in models with both demand timing and demand selection flexibility. In Chapter 6 we consider demand pla nning under uncertain environments. We study two-echelon problems containing an upstream supply echelon and a downstream demand echelon. The upstream echelon might correspond to production or warehouse facilities, while the downstream echelon might be retail sites. Our models in Chapter 6 aim to determine assignments of customer demands to existing f acilities, accounting for costs associated with meeting demands and overflow costs associated with loss of goodwill or subcontracting. We assume the s upplier has several facilities, each of which can produce or distribute a given product. The supplier faces demands from a number of customer demands for the product and wishes to determine which faci lity will serve each demand. Our approach assumes that the s upplier uses a single-sourcing strategy where each demand is assigned to a single facility. We developed an efficient heuristic algorithm and a Branch and Price algorith m for this supply chain network design problem. The subproblems that arose in the Br anch and Price scheme can be viewed as

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168 Stochastic Knapsack problems, which also have various applications in supply chains. The Stochastic Knapsack Problems that we consider are similar to regular Knapsack Problems, except that the item sizes are random variables in our case. We also provided a tailored exact solution method for this class of problems. Future research may consider involving mo re complex operational cost structures in these distribution network design problems, such as production costs or inventory holding costs. In addition, allowing capacity sharing flexibility among different facilities and allowing multi-sourcing for customer de mands would improve the applicability of our models. Furthermore, this thesis ha s only considered the case where customer demands are normally distributed. Developing solution methods for other demand distributions would provide more general re sults and potential a dditional managerial insights. Although this thesis provided solu tion methods to determine the best possible customer demand allocation to available re sources of a firm, capacity adjustment flexibility is not considered in the decisi on making process. A more general study may also consider integrated demand and capaci ty management, when customer demand is stochastic. Integrated demand and capacity management is a worthwhile research direction to pursue in a variety of application settings. In this thesis, we have primarily focused on integrated demand and capacity planning for production planning settings. On the other hand, there exist a wide ra nge of applications where efficient demand and capacity management is a must, e.g., healthcare system s, airline operations, etc. In healthcare operations, capacity and demand matching play a vital role in order to provide the best service to patients at minimum cost. In su ch cases, one needs to analyze and understand

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169 the capacity and demand issues to spot the oc currences of idle capacity or waiting lists. Demand and capacity management in airline operations has also a huge influence on an airline firms profits. Considering the current competitive environment among airline companies, providing new strategies for the best match between airline capacities and customer demands is a valuable research direction.

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APPENDIX A NP HARD PROOF FOR THE CAPA CITATED PRODUCTION PLANNING PROBLEM WITH PRICING AND CA PACITATED SUBCONTRACTORS We want to show that for a fixed T, the problem is NP-Hard for an arbitrary number of capacitate d subcontractors, M. Let gmt(ymt) denote the production cost function for the subcontractor m in period t. We consider cases in which M T, and where the production cost functions are such that the in ternal production cost function is strictly dominated by all subcontractor production cost functions in every period, i.e., pt(x) > Maxm=1,,M{gmt(x)} for all x 0 and for all t = 1, T. We consider special cases such that the revenue function in every period cons ists of two linear segments, where the slope of the first segment equals rt for x dt and the slope of the second segment equals zero for x dt, where dt is some integer demand parameter for t = 1, T. We for these special cases that assume rtdt > pt(dt) so that an optimal solution satisfies dt units of demand in period t. Suppose that the constant capacity level for every subcontractor is different, i.e., Ci Cm for all pairs of subcontractors i and m. Now consider the special case in which there are exactly T subcontractors, and suppose that gtt(y) < gmt(y) for all 0 < y Ct, t = 1, T, and 1 m M and m t, (A1) i.e., the subcontractor with index t in period t has an associated cost function that dominates all other subcontractor cost functions in that peri od for any production quantity less than or equal to the capacity of the tth subcontractor. Assume further that holding costs are zero, for = 1, T, and 1 1 t t t tdC 170

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171 T j jjj T j j mzg zg1 1 for = 1, T, and for any = 1, T, m and any non-zero vector (z+1, z+2, zT) such that 0 zj Cj for < j T. (A2) Assumptions (A1) and (A2) ensure that an optimal solution exists such that any production in any period strictly uses the tth subcontractor. Since none of the subcontracting capacities are equal, we have the equivalent of a capacitated lot sizing problem with different capacities, which is NP-Hard (please see Florian et al. 1980).

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APPENDIX B NP HARD PROOF FOR THE UNCAPA CITATED PRODUCTI ON AND LOCATION PLANNING MODEL WITH DEMA ND FULLFILLMENT FLEXIBILITY Here we show that the [DFFP] is NP-Hard when the rjt parameters can be arbitrarily ordered in successive periods for any order j. We begin by considering the following generic uncapacitated facility location problem with flexible demands (UFLF), where we have a set I of facilities available to serve the demands of a set J of customers. Si denotes a fixed facility cost for facility i I, cij denotes the variable net revenue for units produced at facility i I and distributed to customer j J, and customer j requests dj units. The decision variable xij denotes the amount pr oduced at facility i and distributed to customer j, while the binary variable yi equals 1 if facility i is open, and equals zero otherwise. The goal is to maximize profit through the choice of which customers will be accepted (by satisfying any portion of their demands) and which facilities will serve the accepted customers. Maximize ijij ii iIjJ iIcxSy Subject to: ij jJ x Miyi, i I, ij iI x dj, j J, xij 0, i I, j J, yi {0, 1}, i I. 172

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173 For every j let kj = maxiI{cij} and let sij = kj cij, i.e., cij = kj sij (note that sij 0). We can write the above objective as j ijij ii iIjJ iIksxSy = j ij ijij ii jJiIiIjJ iIkxsxS y Consider a special case of this pr oblem such that all values of kj are sufficiently large enough to ensure that ij iI x = dj in any optimal solution. The resulting problem can be equivalently written as Minimize ii ijij jj iI iIjJ jJSy sxkd Subject to: ij jJ x Miyi, i I, ij iI x = dj, j J, xij 0, i I, j J, yi {0, 1}, i I. The above is a standard uncapacitated facili ty location problem, which is a special case of the facility location problem with flexible demands, implying that the latter is NPHard. We next consider the special case of [DFFP] in which t1(j) = 1 and t2(j) = T for all j = 1, J. Let xtj equal the production in period t used to satisfy demand j in period ; then 11 MT t m t m x x Given this definiti on we then have that 1 T jjt jtdzx Also, note that It, the inventory at the end of period t, is equal to all produc tion in periods 1, t that is used for demands in periods t + 1, T, i.e., 111JtT tj jk tix k We can rewrite [DFFP] equivalently as:

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174 Maximize 111 11111 TJT JTJtT j tj ttttt jt tj j jktrxSycxhx j k Subject to: 11 MT tm m x Mtyt, t = 1, T, 11 TT j t t x dj, j J, xtj 0, t, = 1, T, j J, yt {0, 1} t = 1, ..., T. Let for t denote the net revenue associated with satisfying demand j in period using production in period t. We can then write the above objective as: 1 tjjt k ktcrch Maximize 11 1 TJT T tjtj tt tjttcxSy Considering the variables xtj, observe that assuming production for demand j occurs in period t, we can determine the unique period in which demand satisfaction will occur by computing 1 ,..., ,...,max maxtj jt k j kt kt kt tT tTcrchrh 1c which is the net revenue that result s if we assign demand j to production in period t. Thus, we need only define variables xtj since given a demand-productionperiod pair, we can quickly determine the appropriate value of We therefore have the equivalent formulation of [DFFP]: Maximize 111 TJT tjtj tt tjtcxSy 1 J tj j x Mtyt, t = 1, T, 1 T tj t x dj, j = 1, J, xtj 0, t = 1, T, j = 1, J,

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175 yt {0, 1}, t = 1, T. Given any instance of UFLF (which we showed to be NP-Hard), we can create an equivalent instance of [DFFP] as follows: For each facility i I we create an equivalent period t, setting the setup cost St in the period equal to the fixed facility cost Si. Each customer corresponds to a single demand with dj in the above model equal to dj in the UFLF. Set ct = 0 for t = 1, T, and set ht to a very large number for t = 1, T. Given the set of cij values from the UFLF problem, let t(i) denote the period corresponding to facility i and let m(j) denote the demand corresponding to customer j; as a result of the choice of ct and ht values, we will set rm(j)t(i) = cij. We will then have ij timjc c and the resulting solution to the above problem is equivalent to therefore solves the UFLF. This implies that since the UFLF is NP-Hard, the [DFFP] is also NP-Hard.

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APPENDIX C NP HARD PROOF FOR THE STATIC STOCHASTIC ASSIGNMENT PROBLEM For this problem, a set of demand points J, and a set of facilities I are considered. Let dj denote a random variable for the value of demand j and let Ki denote the capacity of facility i. In addition, we define xij as a binary variable equal to 1 if demand j is assigned to facility i, and 0 otherwise. We let j = E[dj] and 2 j = Var[dj], and let Di denote the random variable for the ca pacity requirements at facility i, where Due to the statistical independen ce of customer demands, the expected value of the assigned demand for facility i is i jJDd j i jxj i j i jJ x with a standard deviation equal to 2i jJ j i j x (since each xij is binary, 2 ijij x x ). The cost to assign a unit of demand j to facility i is equal to The parameter Si denotes a linear unit overflow cost for facility i, and the overflow cost term in the objective function can be denoted by where is the expected overflow at facility i (also known as the loss function), i.e., We formulate the static stochastic assignment problem as follows. ijc ()iiiSK .i iiiiiii KKDKfDd D1 Minimize .()ii iIgx Subject to: ij iIx for all j I, for all i I, j J, 1,0ijx 176

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177 where .0,0 () ,otherwiseij ii ijjijiii jJ x jJ gx cxSK Consider the special case of this problem where each dj has a discrete degenerate probability distribution where Pr()1jjda in other words, for given xij values, Also assume that all aj and Ki values are integer for this special case. We can then rewrite the loss function as Pr( )1ij i j jJDax 0iiK if j iji jJaxK and if ii jij jJKax i K j iji jJaxK If 0iiK for all facilities, we will refer solution as a capacity feasible solu tion, and all other so lutions as capacity infeasible solutions. We assume that there exists at least one capacity feasible solution for this special case. Assume in this special case that all Si values take very large values, which precludes a capacity infeasible solution at optimality. To achieve this, let and ijijjcca max{} j iIijc c In this case, j jJc denotes an upper bound on th e objective function of any capacity feasible so lution. We then set 1, ij jJSci I Therefore, the objective function of any capacity feasible solution will be less than the objective function of any capacity infeasible solution. Since there exists at least one capacity feasible solution, we need only consider solutions that satisfy the inequality j iji jJaxK at each facility, and can thus expl icitly impose this set of constraints without loss of optimality.

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178 The cost function (gi) then becomes () ii ijjij jJgxcx when these constraints are enforced. The new formulation for this instance is as following: Minimize ijjij iIjJcx Subject to: 1 ij iIx for all j I, j iji jJaxK for all i I, for all i I, j J. 1,0ijx The special case of this problem formulation when all Ki are equal is equivalent to the batch loading problem, which is NP-Hard (see Dobson and Nambimadom 2001), which implies that our SSAP is also NP-Hard.

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APPENDIX D CHARACTERISTICS OF THE OBJECTIVE FUNCTION OF THE DEMAND ASSIGNMENT PROBLEM WITH A SINGLE DECISION VARIABLE In Section 6.3.1.3 we need to so lve the following optimization problem: Minimize (1,) (1,) 1122 [][][] [][]()kk kk g gg jj gg j jR jRcxcSLzx Subject to: (1,) (1,) 1122 [][] [][]kk kkgg jgg j jR jR x zx K 0 x[g] 1, zk-1 z zk. We next prove that the objectiv e function for this problem, i.e., (1,) (1,) 112 [][][] [][]()kk kk 2 g gg jj gg j jR jRcxcSLzx is either concave or convex in x[g] on the entire interval defined by zk-1 z zk where (1,) 1[][]kk g gj jRKx z and (1,) 122 [][]kk g gj jRx The following equations will be useful in our further analysis. ()1Lzzzz 1dLz z dz () dz z dz 2 dz zz dz 179

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180 2 1/2 [] 1/2 []2g gd dx 1/22 [][] []2 2gg gz dz dx We next derive the first and second deri vatives of the objec tive function with respect to x[g]. 2 [] [] 1/2 [][] 1/2 [] []() () 2g g gg ggdx dLz dz cS L dz dx dx z 1/22 2 [] [][] [] 1/2 [][] 1/2 []2 11 22g ggg gg gdx z cSz zzz dx 1/22 [] [][] [][] 1/2 []2 2c g gg gg gdx zz cS dx 2 [] [] [][] [] 1/2 [] 2g g c gg g gdx z cSz dx While the second derivative can be stated as: 1/2 [] [] 1/2 2 [] [] [] [] [] [] 2 [] []2gg g ggg g g g gdzxdx dz z dz dx dx dzx dx Sz dx dx 1/22 2 [][] [] 1/2 1/2 1/22 2 [][][] []2 22 2 22ggg ggg gz zz z z Sz

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181 22 2 [][][][] 2 1/221/21/2 []1 4gggg gSzz z Letting A = (1,) 12kk j jR and (1,) 1kk j jRB this can be written: 2 22 2 [] [][] [] 1/2 2 [] 3/2 2 2 [] []() 1 4g gg g g g gdx Sz zz dx 22 22 [][][] [] [] 2 [][][] 3/2 2 []() 1 4ggg gg ggg gAx Sz Bxz 22 2 [][] [] 2 [] 3/22 []() 1 4gg g g gSzA Bz This term is negative if and only if 2 [] 2 [] 22 [][]4 1g g ggA zB So the function is concave in x[g] for any z such that (1,) 1 (1,) 12 [] 2 2 [] 22 [] []4 1kk kkgj jR gj jR ggK z Otherwise, the function is convex. This implies that given an x[g], we can compute the critical value zc( g)= (1,) 1 (1,) 12 [] 2 22 [] []4 1kk kkj jR g j jR ggK If this critical value is negative, then convexity holds in x[g] for any z If this value is positive, then the profit function is concave in x[g] for all cz(g)z(g)z c and is convex otherwise. If the interval [ zk-1, zk.] does not contain cz(g) or cz(g) then the problem is either concave or convex on the entire interval. If it contains only cz(g) then the function transitions from convex to concave on this interval. If it contains cz(g) then the function transitions from concave to convex concave on this interval.

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188 Taaffe, K., Geunes, J., and Romeijn, H. E., 2006B. Integrated marketing and operational decisions: The selective newsvendor probl em. Working paper, Industrial and Systems Engineering Department, University of Florida, Gainesville, FL. Ten Kate, H. A., 1994. Towards a better understanding of order acceptance. International Journal of Production Economics 37(1), 139-152. Thomas, J., 1970. Price-production deci sions with deterministic demand. Management Science 16 (11), 747-750. Toktas B, Yen, J. W., and Zabinsky, Z. B., 2006. Addressing cap acity uncertainty in resource-constrained assignment problems. Computers and Operations Research 33-3, 724 -745. Van den Heuvel, W., and Wagelmans, A. P. M., 2006. A polynomial time algorithm for a deterministic joint pricing and inventory model. European Journal of Operational Research 170 (2), 463-480. Van Hoesel, C. P. M., and Wagelmans, A. P. M., 1996. An O ( T3) algorithm for the economic lot-sizing problem w ith constant capacities. Management Science 42 (1), 142-150. Van Mieghem, J. A., 1999. Coordinating i nvestment, production, and subcontracting. Management Science 45, 954-971. Van Mieghem, J. A., 2003. Capacity mana gement, investment and hedging: review and recent developments. Manufacturing & Service Operations Management 5(4), 269-302. Van Mieghem, J.A., and Rudi, N., 2002. Newsvendor networks: Inventory management and capacity investment with discretionary activities. Manufacturing Service Operations Management 4 (4), 313-335. Wagelmans, A. P. M., van Hoesel, S., and Kolen, A., 1992. Economic lot sizing: An O ( n log n)-algorithm that runs in linear time in theWagner-Whitin case. Operations Research 40, 145-156. Wagner, H., and Whitin, T., 1958. Dynamic version of the economic lot size model. Management Science 5, 89-96. Zangwill, W., 1969. A backlogging model and a multi-echelon model of a dynamic economic lot size production system Management Science 15, 506-527.

PAGE 201

BIOGRAPHICAL SKETCH Yasemin Merzifonluo lu was born on October 1, 1979, in Ankara, Turkey, and has lived there until she came to United States fo r the graduate school. She has obtained her B.S. degree in industrial engineering from Bilk ent University in Ankara, Turkey, in July 2002. In August 2002, she started her Ph.D. st udy in Gainesville, Florida. She got her M.S. degree in industrial and systems engineer ing from the University of Florida in May 2005. Her main research interests lie in supply chain management and operations research. She has worked jointly on her di ssertation with Professor Joseph Geunes and Professor Edwin Romeijn. During her Ph.D., she has served as a teaching assistant for an undergraduate course on quality control and a graduate level course on supply chain management for several semesters. She also ta ught a senior level e ngineering class, Lean Production Systems (EIN4401), as an instructor. At the Universi ty of Florida, in addition to working on theoretical research and teaching, she also served as a consultant for Merck & Co., Inc. 189

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Full Text

OPTIMIZATION MODELS FOR INTEGRATED PRODUCTION, CAPACITY AND
REVENUE MANAGEMENT

By

YASEMIN MERZIFONLUOGLU

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006

by

Yasemin Merzifonluoglu

To my family.

ACKNOWLEDGMENTS

I would like to thank all those people who have helped to make this thesis possible.

Firstly, I would like to express my sincere gratitude to Dr. Joseph Geunes for being

the perfect supervisor for me. He has become more of a mentor and a friend to me than a

professor. He has had always time to discuss my research ideas, to listen my problems,

to answer my e-mails, and to carefully edit my writing. Thanks to his relieving attitude

and his understanding, I have lived through many difficulties in the last four years.

I also consider myself very lucky to work with Dr. Edwin Romeijn. He has an

amazing eye for detail and his interesting research ideas helped to shape this thesis. I

also want to thank Dr. Elif Akcali and Dr. Selcuk Erengui for their helpful comments and

for participating in my dissertation committee.

I would like to express my warmest gratitude to my parents, Nurhan and Nuifer

Merzifonluoglu for being courageous and patient enough when I was making life

changing decisions and for teaching me that I would never be alone regardless of the

place that I live. I am the person that I am today thanks to their endless faith and trust in

me and my abilities. I would like to thank my brother, Abdurrahman Merzifonluoglu, for

his unmatched friendship and guidance in life as a big brother. I could not have imagined

reaching this goal without their unconditional love and support. Lastly, I would like to

express my appreciation to Eray Uzgoren for giving me the extra motivation and support

to complete this thesis.

page

A C K N O W L E D G M E N T S ................................................................................................. iv

L IST O F T A B L E S ......................................................................... ............ ix

LIST OF FIGURES ................................... ...... ... ................. .x

ABSTRACT ........ .............. ............. ...... ...................... xi

CHAPTER

1 IN TR OD U CTION .............................. ...... .. .... .. ......... ................

1.1 Integrated Capacity, Demand and Production Planning Models with
Subcontracting and Overtime Options................................. ..................3
1.2 Capacitated Production Planning Models with Price Sensitive Demand
and General Concave Revenue Functions ................................................5
1.3 Uncapacitated Production Planning Models with Demand Fulfillment
F lex ib ility ............................. ...... ...... ......... ......................... 7
1.4 Demand Assignment Models under Uncertainity............... ..................8
1.5 Research Scope and Thesis Outline .............................. .................9

2 LITERATURE REVIEW ......................................................... .............. 11

2.1 R equirem ents Planning ......................................................... .................. 11
2.2 Pricing/Demand Management with Production Planning..........................12
2.3 Capacity Planning ......................................... .... ........ .. ........ .... 16
2.4 Subcontracting .................................... ................... ......... 17
2.5 O vertim e Planning ........................................................ .............. 18

3 INTEGRATED CAPACITY, DEMAND AND PRODUCTION PLANNING
MODELS WITH SUBCONTRACTING AND OVERTIME OPTIONS .............20

Introduction ......................................... ... ......... ......... .................. 20
3.1 Model and Solution Approach with Fixed Procurement Capacity ............25
3.1.1 Problem Definition and Model Formulation for Single
Uncapacitated Subcontractor ......................................................25
3.1.2 Determining Candidate Demand Vectors for an RI.....................33
3.1.3 Optimal Cost Calculation for an RI............................................ 36

3.1.3.1 Regular capacity as integer multiple of overtime
capacity ........................ ........ ..... ...... .... ............... 37
3.1.3.2 Regular capacity as any positive multiple of overtime
capacity .................................... ....... ........... 39
3.1.4 Complexity of Solution Approach.............................................. 42
3.2 Capacity Planning ....................... ....... .. ........ .................. .. 43
3.3 Multiple Subcontractors and Subcontractor Capacities...........................47
3.3.1 Uncapacitated Subcontractors.................................. ...................47
3.3.2 Capacitated Subcontractors.................. ... .............49
3.3.2.1 Determining candidate demand levels for a regeneration
interval ......................................... ... .... ............... 50
3.3.2.2 Regular capacity as integer multiple of overtime
capacity ........................ ........ ..... ...... .... ............... 51
3.3.2.3 Regular capacity as any positive multiple of overtime
capacity ........................................ .... . ......... 53
3.3.2.4 Complexity of solution approach.................. ............56
3.3.3 Capacity Planning with Multiple Subcontractors ........................57
3.4 C conclusions .............................................................................. 59

4 CAPACITATED PRODUCTION PLANNING MODELS WITH PRICE
SENSITIVE DEMAND AND GENERAL CONCAVE REVENUE
F U N C T IO N S ..................................................... ................ 6 1

Introduction ............................ ............................. ...................... 61
4.1 Model and Solution Approach with Dynamic Prices .............................63
4.1.1 Problem Definition and Model Formulation...............................63
4.1.2 Development of Solution Approach for DCRPP........................66
4.1.2.1 Properties of optimal RI demand vectors.........................67
4.1.2.2 Characterizing optimal demand in RIs containing one
fractional procurem ent period........................................70
4.1.2.3 Characterizing optimal demands in RIs containing no
fractional procurement period..................... ........... 72
4.1.2.4 Complexity of overall solution approach.........................76
4.1.2.5 Refining the solution approach............ ................78
4.2 Model and Solution Approach with a Constant Price ............................81
4.2.1 M odel Description .................................................. 81
4.2.2 Linearity of Cost in Demand Effect..................................84
4.2.3 Characterizing the Structure of the Optimal Cost Function...........86
4.2.4 Number of Breakpoints of ........................... ...............88
4.2.5 Solution Approach for SCRPP.................... .............. ............... 91
4 .3 C o n clu sio n ................................................. ................ 9 3

5 UNCAPACITATED PRODUCTION PLANNING MODELS WITH
DEMAND FULFILLMENT FLEXIBILITY .................................................95

Introduction .................................... .............. .................. ..... .... 95
5.1 Problem Definition and M odel Formulation................... ....................99

5.2 Polynomial Time Solution under Special Cost/Revenue Structures........103
5.3 Dual Based Heuristic Algorithm for General Revenue Parameter
V values ................... ..................... ......... ..................................... 107
5.3.1 Economic Interpretation of the Dual and Complementary
Slackness R relationships ............. .......................... .................117
5.3.2 Creating a Feasible Primal Solution ...........................................118
5.4 Com putational Testing and Results ............................... ................120
5.4.1 A analysis of results............................................... .................. 123
5.5 Conclusions ................................... .................................. 126

6 DEMAND ASSIGNMENT MODELS UNDER UNCERTAINTY.................. 128

Introduction ............... ........ ........... .............................................. 128
6.1 Problem Definition and Model Formulation.............................133
6.2 Branch and Price Schem e ............................................. ............... 136
6.2.1 Column Generation Algorithm .......................................... 137
6.2.1.1 Column generation for [SSSPL].....................................137
6.2.1.2 Initial colum ns ...................................... ............... 138
6.2.2 Pricing Problem ........................................ ........ ............... 138
6.2.3 B ranching Schem e ............................................ ............... 139
6.2.4 Rounding Heuristic .......... ................................. .............. 141
6.3 Static Stochastic Knapsack Problem .................................... ............... 141
6.3.1 Linear Relaxation of Restricted Static Stochastic Knapsack
P ro b lem ................................................................................. 14 4
6.3.1.1 K K T conditions..................................... ............... 145
6.3.1.2 Analysis of KK T conditions ...........................................146
6.3.1.3 KKT based algorithm ............................... ...... ...... 149
6.3.2 Branch and Bound Schem e ............. ................. .................... 155
6.4 Preference Order Greedy Heuristic ............. ................. ....................156
6.5 Numerical Study ................................. ... ...................... 157
6 .6 C on clu sion .......................................................................... 162

7 C O N CLU SIO N .................................................................... 164

APPENDIX

A NP HARD PROOF FOR THE CAPACITATED PRODUCTION PLANNING
PROBLEM WITH PRICING AND CAPACITATED SUBCONTRACTORS.. 170

B NP HARD PROOF FOR THE UNCAPACITATED PRODUCTION AND
LOCATION PLANNING MODEL WITH DEMAND FULFILLMENT
F L E X IB IL IT Y .......................................................................... ..................... 172

C NP HARD PROOF FOR THE STATIC STOCHASTIC ASSIGNMENT
P R O B L E M ......................................................... 176

D CHARACTERISTICS OF THE OBJECTIVE FUNCTION OF THE
DEMAND ASSIGNMENT PROBLEM WITH A SINGLE DECISION
V A R IA B L E ....................................................................... 179

LIST OF REFEREN CES .................................................................. ............... 182

B IO G R A PH ICA L SK ETCH ............ .................................................... .....................189

LIST OF TABLES

Table page

3-1 Complexity results with M subcontractors (integral a) ........ ......................59

3-2 Complexity results with M subcontractors (general a)...................................59

5-1 Demand time windows for the example problem shown in Figure 5.1 ...............101

5-2 Demand (d,) and setup cost values (St) for example problem 1............................112

5-3 P problem 1-Iteration 0. ............................................................................. .... .. 113

5-4 Problem 1-Iteration 1 .................. ................................ ..... ................ 114

5-5 Problem 1-Iteration 2. ............................................ .. ...... ................ 114

5-6 Demand and setup cost values for example problem 2 .............. ... ...............115

5-7 P problem 2-Iteration 0. ............................................................................. .... .. 115

5-8 Problem 2-Iteration 1 .................. ........................ .. .. ..... .. .......... .. 116

5-9 Order rejection rates under different cost parameter value settings.......................124

6-1 Problem sizes in the numerical study ............................................ ...............158

6-2 Computation times for various problem settings ................................................159

6-3 Performance evaluation table for heuristic algorithm ................. ................161

LIST OF FIGURES

Figure pge

3-1 Piecewise linear concave revenue function........................ ...................30

3-2 Network representation of the [CPPP L] ................... .................................... 32

4-1 Candidate subgradient values and corresponding candidate demand values...........69

4-2 Illustration of Proposition 4.4 ........... ...... ......... .................. 80

4-3 An arbitrarily selected n-period regeneration interval. ...........................................89

5-1 Fixed-charge network flow representation of the [DFFP] problem ..................101

5-2 Structure of longest path graph. .................................................. .....................105

5-3 Impact of dimensions of flexibility on profit. ................................................ 123

5-4 Profit levels as a percentage of the maximum profitability (FLEX(D, T))............125

6-1 Supply chain network for 5 facilities and 10 downstream demand points ...........128

6-2 Description of problem parameters on an example distribution network. .............133

6-3 Comparison of computation times for various problem settings ......................... 160

6-4 Performance evaluation for heuristic algorithm .....................................................161

6-5 Performance comparison for high and low overflow costs...............................162

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

OPTIMIZATION MODELS FOR INTEGRATED PRODUCTION, CAPACITY AND
REVENUE MANAGEMENT

By

Yasemin Merzifonluoglu

August 2006

Chair: Joseph Geunes
Major Department: Industrial and Systems Engineering

This thesis provides new planning models for making synchronized decisions on

capacity, demand management and production/inventory planning in supply chains.

These models focus on the tradeoffs between capacity costs, production costs, costs for

assigning customer demands to different supply resources and revenues associated with

satisfying customer demands. Within this class of models, we study various degrees of

flexibility on the part of a supplier of goods, including flexibility in demand and capacity

management. We consider integrated production, capacity, and pricing planning

problems, where a good's price may change throughout a planning horizon, as well as

contexts in which a constant price is required for the entire horizon. We also consider

production planning models in which a supplier may not have a great deal of price setting

flexibility, but may wish to be selective in its choice of markets (or customers) and the

timing of demand fulfillment, as a result of the unique fulfillment costs associated with

different markets (or customers). We also investigate the role of capacity planning in

these contexts, including capacity acquisition problems that require setting the supplier's

best capacity level for an entire planning horizon. We examine subcontracting and

overtime as mechanisms for short-term capacity flexibility. We also consider logistics

supply network design problems that determine the best allocation of downstream

demands to upstream facilities in uncertain demand environments. We used polyhedral

properties and dynamic programming techniques to provide polynomial-time solution

approaches for obtaining optimal solutions for some of the problems that are not NP-

Hard. When the problem is NP-Hard, we proposed very efficient heuristic solution

approaches which are developed considering particular features of the problems. We also

employed a Branch and Price method for the large scale nonlinear assignment problems.

CHAPTER 1
INTRODUCTION

Our research provides a new set of planning models for simultaneously making

decisions on capacity, demand management (pricing, demand selection, demand

assignment) and production/inventory planning. Although these decisions are extremely

interrelated in practice, coordination among them has not been fully addressed in the

literature. Therefore, the models we present generally concentrate on the critical

tradeoffs between capacity costs, production costs, costs associated with assigning

customer demands to different supply facilities (e.g., transportation cost) and the

revenues associated with satisfying customer demands.

Within this class of models, we consider various degrees of flexibility in demand

management. In most production and inventory planning models in the operations

literature, demand is exogenously determined (possibly characterized by some probability

distribution). On the other hand, some past studies exist that assume that demand is

completely endogenous. In such models, a manufacturing department might produce

according to a predefined production schedule, and marketing and sales departments then

attempt to realize sales according to this output plan. The practical reality, however, is

typically neither of these two extremes, because demand is neither absolutely exogenous

nor endogenous. To more closely reflect the complexities in operations practices, our

models consider demand and capacity decisions together, which results in optimal

demand levels that are influenced by capacity and production costs, and vice versa.

We examine demand management decisions through pricing, demand selection and

demand assignment models. We consider dynamic pricing problems where an item's

price may change throughout a production horizon, as well as contexts in which a

constant price is required for the entire horizon. The pricing models we provide permit

us to establish the best demand schedule based on the supplier's available resources and

cost structure in conjunction with customer responses to prices. Some of our pricing

models may also be interpreted as demand selection problems, where a supplier must

decide whether or not a particular customer order is economically attractive enough to

accept and produce. In these demand selection problems, we typically assume that

customers offer a price and provide a required shipping date for their orders. In practice,

however, customers may also be flexible in terms of both shipping date and price, and in

these cases, an item's price may be a function of the actual delivery time. We therefore

provide solution methods for cases in which the price associated with a customer's order

may depend on the delivery time. In addition to pricing and demand selection models,

the decision maker often faces the problem of assigning potential customer markets to

available resources in the most profitable way. Although assignment problems have been

studied extensively in the operations literature, demand uncertainty has not been fully

addressed in such settings. We therefore provide contributions in the area of assignment

problems under demand uncertainty.

In addition to flexibility on the demand side, we also focus on capacity adjustments

as a mechanism for matching supply with demand, including capacity acquisition

problems that determine the best production capacity for an entire planning horizon. As

mechanisms for short-term capacity flexibility, we also examine subcontracting and

overtime options as two distinct choices for managing capacity. With the recent increase

in contract manufacturing, production planning models that include subcontracting and

outsourcing decisions have received a significant amount of attention in the literature.

On the other hand, overtime decisions in conjunction with capacity and demand planning

models have not been completely examined in the operations literature. Our models also

provide an option for a producer's output to be completely subcontracted instead of using

internal production, in cases where the subcontracting option is more economically

In the broadest sense, our research primarily considers production and inventory

planning contexts, where various pricing and capacity issues also play important roles in

maximizing a firm's profit. In addition to single stage models, our work also considers

two-stage supply chain problems, where demand assignment under uncertainty is the

main focus. This introductory chapter provides an overview of the thesis.

1.1 Integrated Capacity, Demand and Production Planning Models with
Subcontracting and Overtime Options

In this study, we outline two basic models for capacity, demand, and production

planning; the first case assumes a fixed capacity level for the manufacturer, and assumes

that this capacity is exogenously predetermined. The resulting model and solution

approach lay the groundwork for the case in which capacity is a decision variable.

We consider a manufacturer producing a good to satisfy price-dependent demand

over a finite number of time periods. The objective is to determine the production

(regular and overtime) schedule, inventory quantities, subcontracting, and demand levels

in order to maximize net profit.

In this problem context, we allow internal production to consist of regular and

overtime production. Since, in practice, the available "regular" production hours are

often a bottleneck, a manufacturer may benefit from extra working hours, or by adding

temporary workers. The total production output in such cases therefore consists of

regular production plus any overtime and the associated cost functions are often concave

in total output, reflecting economies of scale in production. We therefore assume that the

cost of regular production is a nondecreasing concave function of total output from

regular time production. Overtime cost is an incremental cost (over regular time

production cost) and is also a concave and nondecreasing function of the overtime

production level in a period.

A producer either supplies demand using internal production (regular or overtime),

or purchases finished products from a subcontractor (or simultaneously utilizes both

internal and external resources). We first assume a single subcontractor without capacity

limits, where cost function is concave for each subcontractor. We have also considered

multiple (non-identical) subcontractors, both with and without subcontracting capacity

limits in order to generalize our approach.

The revenue functions associated with satisfying demands may vary among

periods, and is characterized by a non-decreasing piecewise-linear function of the

demand satisfied in a period. We also allow linear inventory holding cost for the

inventory remaining at the end of each period.

This problem minimizes a concave cost function over a set of network flow

constraints, and therefore an optimal extreme point solution exists. In any extreme point

solution, the basic variables create a spanning tree in the network. Our suggested

dynamic programming solution primarily relies on the spanning tree structure of extreme

point solutions in this network. We use dynamic programming methods to provide

polynomial-time solution algorithms for obtaining an optimal solution for this class of

problems.

Chapter 3 also focuses on capacity acquisition decisions. We consider the case

where a manufacturer attempts to determine its optimal internal capacity level for the

planning horizon. We consider a concave capacity cost function for internal production

capacity, which is a decision variable. We again have the minimization of a concave

function over a polyhedron, which implies that an optimal extreme point solution exists.

We characterize extreme point properties of the associated polyhedron, which permit

considering a polynomial number of distinct capacity levels in order to determine an

optimal capacity level. For each of the candidate capacity levels associated with extreme

point solutions, we can then solve a fixed capacity problem. We also consider the

capacity acquisition problem under various assumptions regarding the number and

capacity levels of subcontractors.

1.2 Capacitated Production Planning Models with Price Sensitive Demand and
General Concave Revenue Functions

Chapter 4 continues to consider discrete-time, finite-horizon operations planning

models with capacity limits, where demand for a good is price sensitive. In contrast to

the problem discussed in Chapter 3, we do not consider the availability temporary

capacity expansion mechanisms, such as employing overtime hours and subcontracting

(we continue to assume time-invariant production capacities, however). Instead, we

focus our attention on handling more general concave revenue functions.

We account for economies of scale in production of a good using a fixed plus

variable production cost structure. We also consider linear holding costs in the amount of

inventory held in a period. Within this class of problems we consider both the case in

which price may vary dynamically by period, as well as the case in which a single price is

chosen for the entire planning horizon.

In the dynamic pricing problem class, we consider a finite horizon planning model

for a single item with production capacity equal to some positive value in every period.

Demand in a period is a decision variable, and we assume that any value of demand in a

period implies a corresponding unit price. The total revenue in a period is determined by

a nondecreasing concave function of demand, where the corresponding price as a

function of demand equals total revenue divided by demand, i.e., "revenue/demand." We

assume that production in any period requires incurring an order cost plus an additional

variable cost. A holding cost is incurred for each unit remaining in inventory at the end

of period. The amount of inventory remaining at the end of a period must be

nonnegative. Our goal is to maximize total revenue less production and holding costs. In

Chapter 4, we show that we can solve this model in polynomial time using a dynamic-

programming-based approach.

We also consider the problem setting where a manufacturer requires setting a single

price for a good over the entire planning horizon. For this case, we assume that the

demand in a period is given by a nonincreasing function of price. We define total

revenue in a period as the product of price and total demand. The suggested algorithm

runs in polynomial time under some mild assumptions on the revenue functions. The

solution method along with the complexity results is explained in Chapter 4.

1.3 Uncapacitated Production Planning Models with Demand Fulfillment
Flexibility

In addition to pricing and demand selection decisions, Chapter 5 recognizes the

flexibility manufacturers often have to adjust order shipment times for a given set of

production orders. In many such settings the net profit of an order depends on the time at

which the order is satisfied. Our model accounts for these demand-timing decisions as

well as order acceptance decisions, along with their production and inventory planning

implications. For this class of problems, we do not account for production capacity

limits.

We consider a discrete planning horizon and a set of candidate demands (or orders)

for a single good produced by a supplier. We consider a fixed plus linear production cost

structure and linear holding costs. As in the previous demand selection type models, each

candidate demand represents a request for a fixed amount. Net revenue for satisfying a

candidate demand depends on the delivery period.

We assume without loss of generality that for each candidate demand source, a

customer-specified delivery time window exists during which the customer will accept

delivery. The producer wishes to maximize net profit during the planning horizon,

defined as the total net revenue from order acceptance and delivery-timing decisions, less

the total setup, variable production, and holding costs. Interestingly, this problem may

also be considered as a facility location problem, where the supplier wishes to maximize

his profit with full flexibility to choose customer demands.

This problem is proved to be NP-Hard (Appendix B), which implies that we cannot

reasonably seek an efficient (polynomial-time) solution method for this problem in its

most general form. Under certain mild assumptions on costs and revenues, we can,

however, solve the problem in polynomial time. In particular, under specialized cost

assumptions that imply that no speculative motives exist for holding inventory or

backlogging (i.e., there is no reason to produce earlier than necessary due to the

anticipation of production cost increases, or to produce later than necessary to take

advantage of lower penalty costs later), our problem can be solved in polynomial time by

employing a dynamic programming methods.

As mentioned earlier, the general uncapacitated case of the problem, in which the

revenue parameter values can take any arbitrary values, is NP-Hard. However, the dual

of the LP relaxation of the problem leads to a very efficient heuristic procedure.

Although the procedure does not guarantee an optimal solution to every instance of the

problem, for certain special cases, optimality is achieved. In addition, this dual approach

provides a range of managerial insights about the problem setting. We also tested the

relative benefits of different dimensions of supplier flexibility, considering possible

combinations of demand selection and timing flexibility.

1.4 Demand Assignment Models under Uncertainity

In Chapter 6 we introduce demand uncertainty to our planning models. Although

and solution methods. Chapter 6, therefore, solely considers the problem settings where

the planner determines the best allocation of customer demands to available resources.

We consider a two-echelon model with an upstream supply echelon and a downstream

demand echelon. The upstream echelon might be manufacturing or warehouse facilities,

while the downstream echelon might be retail sites. Each resource's supply capacity is

assumed to be known and each retailer site implies a known probability distribution of

demand. For such a network, our models deal with the best assignment of demands to

available resources. In addition, we assume that the cost of assigning a demand to a

resource is resource specific.

Customer demands are assigned to supply facilities based on their expected

assignment costs and the available capacities of the resources. After demand realization,

the assigned demands may exceed the available capacity of one or more resources. In

such cases an additional penalty cost is incurred, which is associated with the capacity

shortage. We wish to minimize the total expected cost, which includes the expected cost

of satisfying demands using available resources and the penalty costs for exceeding the

capacity of each supply source.

The assignment problem introduced in Chapter 6 is a nonlinear Integer

Programming Problem which can be solved by enumeration, e.g., a Branch and Bound

algorithm. Branch and Price is commonly used when solving such large-scale

assignment problems. Branch and Price is a generalization of the linear programming

(LP) based Branch and Bound scheme, specifically designed to handle integer

programming (IP) formulations that contain a huge number of variables. In Chapter 6,

we described the Branch and Price scheme for our problem starting with a Column

Generation scheme. The pricing problems encountered in the Branch and Price scheme

are Static Stochastic Knapsack problems which are interesting theoretically and

practically on their own. We also present an efficient and novel solution method for the

linear relaxation of the associated Static Stochastic Knapsack problem.

1.5 Research Scope and Thesis Outline

In general, we focus on developing efficient solution methods for operations

management problems involving production/inventory planning, as well as demand and

capacity management decisions. We consider dynamic pricing and capacity adjustment

problems under various subcontractor assumptions, production and inventory planning

problems with general concave revenue functions, uncapacitated production planning

problems with flexibility in demand timing and two stage demand assignment models

under uncertainty. A wealth of past research on production planning models seeks to

meet prescribed demands at minimum cost. In many contexts, demand comes from

independent sources, not all of which are necessarily profitable. We consider such

planning problems from a different perspective, assuming certain demand characteristics

are decision variables. This thesis, therefore, primarily aims to fill gaps in literature by

providing optimization models involving integrated production, capacity and demand

management in the supply chains. Chapter 2 provides an introductory literature review

for Chapters 3, 4, and 5. In Chapter 3, we discuss the production planning problem

where demand is price dependent and various types of capacity adjustments are available,

including capacity acquisition, overtime, and subcontracting. In Chapter 4, we present a

similar pricing and capacitated production planning model which fills a gap in the

literature by considering price-dependent demand with general concave revenue

functions and fixed production capacities. In Chapter 5, we introduce demand timing

flexibility into our production and demand management models, where an item's price

may also change according to the actual delivery date. Chapter 6 recognizes the

importance of demand uncertainty and supply chain network design by providing a

planning model that assigns customer demands to available resources in the most

profitable way. Finally, Chapter 7 concludes with a summary and results.

CHAPTER 2
LITERATURE REVIEW

The focus in Chapters 3, 4, and 5 is production and inventory planning with

integrated demand and capacity decisions. In this chapter, we provide an introductory

literature review for these integrated decisions. Since the focus of Chapter 6 differs

slightly from these chapters, the related literature will be discussed within that chapter.

We classify the relevant literature for Chapter 3, 4, and 5 into five categories:

requirements planning, capacity planning, subcontracting, overtime and pricing/demand

management. These are the primary elements of the models we will present, and for the

most part represent distinct research streams.

2.1 Requirements Planning

Wagner and Whitin (1958) first modeled the classical uncapacitated economic lot-

sizing problem (ELSP), which addresses the tradeoff between setup and holding costs

under dynamic, deterministic demand. Since their original work appeared, many

generalizations of the basic problem have been studied (e.g., Zangwill 1969, Love 1972,

Thomas 1970, Afentakis and Gavish 1986). The capacitated version of the dynamic

requirements planning problem has also been well researched (see Florian and Klein

1971, Baker, Dixon, Magazine and Silver 1978). This past research on dynamic

requirements planning problems assumes demands and capacities are predetermined. In

these models, demands must be filled as they occur, or in models allowing backlogging,

demands can be met later periods. In either case, these models assume that all demand

must be met during the time horizon, using the (fixed) capacity levels available to the

firm.

2.2 Pricing/Demand Management with Production Planning

The first integrated dynamic lot sizing and pricing analysis was provided by

Thomas (1970). He generalized the Wagner-Whitin (1958) model by characterizing

demand in each of a finite number of time periods as function of price, treating each

period's price as a decision variable. Geunes, Romeijn, and Taaffe (2006) considered a

more general form of this model with time-invariant production capacities and piecewise-

linear, nondecreasing, and concave revenue functions for each period. Our dynamic-

pricing models in Chapter 3 generalize this work to the case where capacity decisions are

taken into consideration. Chapter 4 also generalizes this work by considering more

general concave revenue functions. To address contexts with demand selection and

production economies of scale, Geunes, Shen, and Romeijn (2004) also considered

integrated production planning and market selection decisions in a continuous-time

model with market-specific constant and deterministic demand rates. Loparic, Pochet

and Wolsey (2001) also considered a related problem where the manufacturer maximizes

profit, and does not require satisfying all demand, but sets lower bounds on inventory to

account for safety stock requirements. Their model, however, assumed that only one

demand source exists in every period and revenue gained from this demand is

proportional to the satisfied demand. Biller, Chan, Simchi-Levi, and Swann (2005)

considered a related dynamic-pricing problem in which revenue is concave and

nondecreasing in the demand satisfied, procurement capacity limits vary with time, and

procurement costs are linear in the procurement volume. They note that the addition of

setup costs to the model would result in a dynamic programming approach with solution

time that grows exponentially in the size of the problem under time-varying capacities,

but do not consider the equal-capacity case. Recently, Deng and Yano (2006) studied an

integrated pricing and production planning problem under time-varying capacities,

leading to a solution approach with an exponential running time. They also considered

the time-invariant capacity case, and showed that it is polynomially solvable. Our

solution methodology in Chapter 4 differs essentially from theirs with an improved

relationships between optimal prices in different periods, and the relationships between

optimal price vectors and production plans.

In addition to providing solution methods for the time-varying price case, Chapter 4

generalizes the past methods for constant-priced goods to account for time-invariant

capacities. Kunreuther and Schrage (1973) first considered the problem of setting a

single price over an entire planning horizon with an uncapacitated lot-sizing-based cost

structure (with fixed-charge procurement cost structures and linear holding costs), and

provided a heuristic solution approach for this problem. Gilbert (1999) provided a

polynomial-time solution method for this problem under the assumption of stationary

costs. Van den Heuvel and Wagelmans (2006) subsequently showed that the more

general version of the problem with time-varying costs can be solved in polynomial time.

Our constant-pricing model in Chapter 4 generalizes this work to account for time-

invariant procurement capacities. Our work requires a much more general

characterization of the properties of optimal solutions, and provides the first solution

method for combined pricing and capacitated procurement planning with constant-priced

goods and economies of scale in procurement (note that Gilbert 2000, also considered a

periodic multi-product planning problem with constant-priced goods that share

procurement capacity, although procurement costs were linear in the amount of an item

procured). Moreover, the structural properties of optimal solutions that we provide (both

in the time-varying and constant-priced goods cases) can lead to insights for developing

solution methods for more general classes of profit maximization problems with general

concave revenue functions and fixed-charge cost structures. Bhattacharjee and Ramesh

(2000) also considered the pricing problem for perishable goods, assuming demand can

be characterized as a function of price. They studied structural properties of the optimal

profit function, and provided heuristic methods to solve the problem.

In addition to demand selection and pricing flexibility, firms may have flexibility

in delivery timing of the selected demands. In a number of practical contexts, customers

may allow a grace period (also called a demand time window) during which a particular

demand or order can be satisfied. Lee, Cetinkaya, and Wagelmans (2001) modeled and

solved general lot-sizing problems with demand time windows. Their model still

requires, however, that each demand is ultimately satisfied during its predetermined time

window. That is, they considered demand-timing flexibility without the benefits of order

(demand) selection and rejection decisions. The approach we present in Chapter 5

integrates demand selection and a more general version of demand time windows for lot-

sizing problems, providing two dimensions of demand planning flexibility.

Chamsirisakskul, Griffin, and Keskinocak (2004) considered a similar model that focuses

on the economic benefits of lead time flexibility and order selection decisions in

production planning with finite production capacities. Their model assumes that each

order has a preferred due date and a latest acceptable due date, after which the customer

will not accept delivery. A tardiness penalty is incurred if an order is completed after the

preferred period. Our models in Chapter 5, on the other hand, allow a market (or

customer) to provide any period-specific per unit revenue values for delivery within the

acceptable range of delivery dates. This allows customers to specify any subset of

acceptable delivery periods, and this subset needs not consist of consecutive periods, as

in the case with past models that consider demand time windows. In addition,

Chamsirisakskul et al. (2004) did not provide any tailored solution procedures for

exploiting the special structure of the model; rather, they primarily studied the model

parameter settings under which lead time flexibility is most beneficial, and relied on the

CPLEX solver for model solution. Moodie (1999) also considered pricing and lead time

negotiation strategies as a mechanism for influencing demand with time- and price-

sensitive customers under fixed capacity using a simulation model.

Recent operations management literature also discusses additional mechanisms

for affecting demand in order to increase net profit after subtracting operations costs.

Crandall and Markland (1996) classified several demand management approaches for

service industries, including capacity management and general demand influencing

strategies. Iyer et al. (2003) use postponement (with an associated customer

reimbursement) as a mechanism for managing demand surges under limited capacity.

negotiation process in a make-to-order environment where the firm determines the jobs it

will bid on (or accept) using a goal programming approach. Recent literature on

available to promise (ATP) functions (e.g., Pibernik 2005, Chen, Zhao, and Ball 2002)

considers order acceptance in a rolling fashion, based on a production or supply chain

system's constraints and resource availability levels. Order acceptance decisions have

also been addressed in scheduling contexts, where the machine utilization level and

expected lead times (based on the set of previously accepted jobs) drive acceptance

decisions (e.g., see Ten Kate 1994, for a simulation-based approach for this scheduling

problem class).

2.3 Capacity Planning

Production capacities specify the abilities and limitations of a firm in producing

outputs. The capacity expansion literature is concerned with determining the size,

timing, and location of additional capacity installations. Luss (1982), Love (1973), and

Li and Tirupati (1994) provide examples of work on dynamic capacity expansion and

reduction. These studies do not, however, consider more detailed dynamic production

decisions; moreover, demand values are predefined (possibly according to some time-

based function) and are not price dependent.

When demand is uncertain and capacity is expensive, capacity may be insufficient

to meet demand. In these situations, firms employ strategies such as pricing, backlogging

or advance inventory build-up to manage shortages (see van Mieghem 2003). When

shortages are allowed, one should account for the impacts of backlogging or lost sales

with related demand-shortage penalties in the model. Manne (1961) considered settings

with backlogging where only capacity expansion is allowed. Van Mieghem and Rudi

(2002) also considered capacity additions using a growing stochastic demand model with

backlogging.

In most operations management models, demand is primarily treated as exogenous

and may contain some associated uncertainty. Another extreme might be the case in

which demand is completely treated as endogenous. For example, manufacturing

produces according to a plan based on its production capabilities and marketing and sales

do what is necessary to realize sales according to that output plan. The practical reality is

typically somewhere between these two extreme cases. An example of work that

considers such effects is Cachon and Lariviere (1999), who consider situations in which

demand is influenced by a scarcity of capacity. Kouvelis and Milner (2002) also present

a two stage model that addresses the effects of demand and supply uncertainties on

capacity expansion decisions. In our models, we also consider demand and capacity

decisions together. Because of this, optimal demand levels are influenced by capacity

costs (and vice versa) and are neither completely endogenous nor exogenous.

2.4 Subcontracting

Subcontracting and outsourcing have been the subjects of a number of recent

studies due to their increased use in practice. Gaimon (1994) presents a model that

investigates subcontracting as an alternative to capacity expansion. She also examines

the effects of using subcontracting on pricing services. Lee et al. (1997), Logendran and

Puvanunt (1997), and Logendran and Ramakrishna (1997) considered subcontracting

models for cellular manufacturing and flexible manufacturing systems. Atamturk and

Hochbaum (2001) consider the tradeoffs between capacity acquisition, subcontracting,

and production and inventory decisions (with production economies of scale) to satisfy

non-stationary deterministic demand over a finite horizon. Our work in Chapter 3

generalizes their results to account for capacitated overtime availability and price-

dependent demand. Our model also permits characterizing the impacts of multiple,

capacitated subcontractors, whereas Atamtirk and Hochbaum (2001) focused on a single

uncapacitated subcontractor. Coordination issues related to subcontracting, capacity and

investment decisions are discussed by van Mieghem (1999), Kamien and Li (1990), and

Cachon and Harker (2002), who also present game-theoretic models related to

subcontracting. In addition, Bertrand and Sridharan (2001) study heuristic decision rules

for subcontracting in a make-to-order manufacturing system in an effort to maximize

utilization while minimizing tardy deliveries.

2.5 Overtime Planning

Models for overtime planning have been addressed in a number of contexts.

Kunreuther and Morton (1974) developed a production planning model that considers

overtime, lost sales, simple subcontracting, undertime and backlogging, when production

costs are linear in volume. Dixon et al. (1983) considered another model that deals with

the size and timing of replenishments for an item with time-varying demand. In their

model, regular time and overtime production options (the latter at a cost) are available

and production capacities can also vary with time, but are not decision variables. They

provided a heuristic approach for minimizing cost. Adshead and Price (1989) described

simulation experiments of an actual make-to-stock shop to examine the impact of

changes in the decision rules used to control overtime on cost performance. Ozdamar

and Birbil (1998) considered capacitated lot sizing and facility loading with overtime

decisions and setup times, minimizing total tardiness on unrelated parallel processors.

They developed hybrid heuristics involving search techniques such as simulated

annealing, tabu search, and genetic algorithms. Dellaert and Melo (1998) addressed a

stochastic single-item production system in a make-to-order environment to determine

the optimal size of a production lot and minimize the sum of setup costs, holding costs

for orders that are finished before their promised delivery dates, penalty costs for orders

that are not satisfied on time (and are therefore backordered), and overtime costs. Pinker

19

and Larson (2003) developed a model for flexible workforce management in

environments with uncertainty in the demand for labor.

CHAPTER 3
INTEGRATED CAPACITY, DEMAND AND PRODUCTION PLANNING MODELS
WITH SUBCONTRACTING AND OVERTIME OPTIONS

Introduction

The primary function of a production planning system is to determine how to best

meet demand utilizing a firm's production capacity. Production planning systems found

in practice typically take a set of prescribed demands and predefined capacity levels as

input, and determine how to meet the prescribed demands at a minimum cost without

violating capacity limits. These prescribed demand levels often result from a market's

response to the price of a good (when the producer has a degree of monopoly power),

while the predefined capacity levels are a consequence of a producer's capacity

investments. Pricing and capacity investment decisions therefore impose a set of

constraints within which the production planning system must work. As a result, overall

production system performance is affected not only by the production planning decisions

themselves, but also by demand- and capacity-related decisions that typically precede the

production planning process. Achieving maximum performance from a given production

system thus requires an ability to determine the best match between supply capacity and

demand, based on the system's operating and capacity costs and the market response to

price. To address this problem, this thesis provides modeling and solution techniques for

integrated capacity, demand, and production planning decisions.

Current practice typically addresses capacity, production/inventory, and pricing

decisions separately as part of a hierarchy of decisions. Capacity decisions are often

made by strategic operations managers, while sales and pricing decisions are made by

marketing and sales departments, and production/inventory decisions are made by

production planners. Traditional approaches in the operations literature reflect this

practice by sequentially considering these decisions according to their relative

importance, or based on the length of the associated planning horizon for each decision

type. Using such an approach, each successive optimization model imposes constraints

on the model at the next level in the hierarchy (Graves 2002). Generally, the last link in

this decision process is production/inventory planning, and for these decision problems,

demand and capacity are taken as fixed parameters that are exogenous to the model.

Recent literature has begun to recognize the importance of considering demand and

capacity level decisions in production planning. For example, Bradley and Arntzen

(1999) recently discussed the benefits of simultaneous consideration of capacity and

production decisions. Similarly, Geunes, Romeijn and Taaffe (2006) provided analytical

models that consider capacitated production and pricing decisions together in order to

maximize a producer's profit. This chapter takes a further step by providing planning

revenues by simultaneously considering capacity, pricing (demand management), and

production/inventory planning decisions under economies of scale in production costs

over a finite planning horizon. The pricing component of the models allows a supplier to

selectively determine the demand levels it will satisfy. On the capacity side, in addition

to setting an internal base capacity level (which is time invariant), our models allow two

types of short-term capacity adjustments, through subcontracting and capacity-

constrained overtime. When a firm can influence its demand levels through pricing

strategies, these decisions impact the optimal amount of medium- to long-term internal

production capacity, as well as the capacity usage strategy when different forms of short-

A great number of traditional production planning models take demand as

predetermined and exogenous, whereas we consider pricing decisions that determine

demand levels. Similarly, traditional models typically take some initial starting capacity

as given, and consider capacity adjustments that have an associated variable cost, such as

a hiring or layoff cost per worker. For example, classical aggregate planning models use

linear programming techniques to make long term aggregate production, inventory, and

personnel planning decisions (e.g. Manne 1961, Holt, Modigliani, Muth, and Simon

1960). In contrast, our models determine an optimal level of fixed (time-invariant)

internal capacity for a horizon before the start of that horizon, where a capacity cost

function exists that is concave in the amount of capacity acquired. The producer may

then draw on flexible short-term capacity adjustments through overtime and

subcontracting options. These short-term capacity adjustments allow a firm to use a

"chase strategy" for meeting demand fluctuations; alternatively the producer may use

inventory as a mechanism for applying a "level" strategy that does not employ short-term

flexible capacity sources.

As mechanisms for short-term capacity flexibility, we focus on subcontracting and

overtime as two distinct choices. Although both of these alternatives can be used to

provide extra capacity in the short run, in practice these are two completely distinct

options and should therefore be modeled distinctly. In particular, the cost and capacity

structures typically dictate that overtime is only used after regular internal capacity has

been exhausted, while subcontracting can also be used instead of internal production.

When output requirements cannot be met during regular working hours, employees may

be scheduled to work overtime hours. In these cases, the production cost is structured in

such a way that, although economies of scale in production will apply to both regular and

overtime production, regular internal production is always utilized before using overtime

options. Moreover, the amount of available overtime capacity is typically proportional to

the amount of regular internal capacity. For instance, in telephone call centers, the

amount of overtime is often limited to 50% of the total amount of regular production time

(Gans and Zhou 2002). We provide a scheme for modeling overtime cost that leads to

total internal production cost that is piecewise-concave in the production level in a

period. This scheme permits modeling a very general class of total production cost

functions while retaining analytical tractability.

As an alternative to overtime, excess demand may be subcontracted.

Subcontracting is not an uncommon practice in a variety of contexts and may be utilized

for two reasons (Day 1956). First, in-house production capacity (regular plus overtime)

may not be sufficient (as in the call center case), and second, in-house production (in

particular overtime production) may actually be more expensive than subcontracting.

While we may expect that typically the marginal cost of subcontracting is higher than

(regular) marginal in-house production cost (since otherwise one would subcontract all

demand, resulting in complete outsourcing), the models we consider in this chapter do

permit complete internal production (make) or complete outsourcing (buy). Therefore,

our models are most appropriate for combinations of make/buy decisions.

The primary contribution of this chapter is the integration of various important

planning decisions in single, tractable model under a fairly general set of assumptions on

cost structures and production dynamics. To our knowledge, no model currently exists in

the literature that simultaneously determines optimal capacity and demand levels with

dynamic price-dependent demands, economies of scale in production costs, and

subcontracting and capacity-constrained overtime options. Our approach for modeling

demand permits application of the model more broadly to contexts in which pricing does

not apply, but a supplier can accept or reject production orders based on order-dependent

net revenues (after subtracting any variable fulfillment costs).

Beyond applications the model might have in short-term operations planning, it can

also provide substantial value through its ability to determine an optimal fit between

demand levels and supply capacities. That is, the model can provide a benchmark for the

ideal fit between demand and capacity levels in a production system, given the

production system's operations cost structures, capacity options, and constraints.

Therefore, even in contexts that do not lend themselves to dynamic pricing or order

acceptance/rejection decisions, strategic decision makers can use the model to gain a

better understanding of how current capacity and demand conditions deviate from a

"best-case" scenario. The extent of such deviations can then be analyzed to develop new

strategies for demand and capacity management.

As we later show, solving our model requires minimizing a concave function over a

polyhedron (as discussed in Section 3.1, this concavity is ensured by employing

piecewise-linear and concave revenue curves within each planning period). We

characterize important extreme point properties for this polyhedron that permit using

dynamic programming techniques to provide polynomial-time solutions. Relative to

most of the classical planning models, the worst-case performance is typically a relatively

high order polynomial function of the planning horizon length. Given the number of

simultaneous decisions addressed by the model, however, the resulting polynomial

solvability is a reasonably powerful result that permits solution of large-scale problems in

reasonably fast computing time.

The organization of this chapter is as follows. Section 3.1 describes the modeling

and solution approach used for the capacitated production planning and pricing problem

when the capacity level is exogenously determined. In Section 3.2, the capacity planning

problem is addressed, extending the models of Section 3.1 to the case in which capacity

is a decision variable. Section 3.3 summarizes our results for various assumptions on

subcontractor parameters, such as an uncapacitated versus capacitated subcontractor and

single versus multiple subcontractors. Finally Section 3.4 concludes with a summary.

3.1 Model and Solution Approach with Fixed Procurement Capacity

3.1.1 Problem Definition and Model Formulation for Single Uncapacitated
Subcontractor

In this section we consider our basic model with a fixed capacity level that is

exogenously predetermined. The resulting model and solution approach lay the

groundwork for the case in which capacity is a decision variable, which we consider in

Section 3.2. We consider a manufacturer producing a good to satisfy price-dependent

demand over a finite number of time periods, T. The manufacturer can affect demand

through pricing (implying some degree of relative monopoly for the good) and can draw

on overtime and subcontracting as mechanisms for short-term capacity flexibility. The

objective is to determine the production (regular and overtime) schedule, inventory

quantities, subcontracting, and demand levels (through pricing), in order to maximize net

profit.

In practice, the available regular production hours a producer has are often a

capacity bottleneck. A manufacturer may often benefit from employing extra working

hours, or by adding temporary workers. We therefore distinguish between regular and

overtime production, the latter of which can be viewed as a temporary production

capacity increase in a period. We assume that the amount of available overtime capacity

is often proportional to the amount of regular internal capacity. We will denote the ratio

between regular and overtime production capacity by a positive constant a. We expect

that, in practice, a will often be integral and small, say between 1 and 3. For example, if a

firm regularly utilizes two shifts per day, but has the option of adding a third, then

regular-time capacity is twice overtime capacity and a= 2. However, in general we will

allow a to take any positive value, and therefore regular-time capacity can be any

positive multiple of overtime capacity (thus there is no loss of generality here in defining

this ratio a between regular and overtime capacity). We next define aK and K as the

total amount of available regular and overtime capacity, respectively, in any time period,

where K is a positive constant. The total regular and overtime capacity in a period

therefore equals (a+ 1)K.

Let xt denote the total internal production quantity, consisting of regular production

rt plus any overtime production zt. Letpt(xt) denote a corresponding base internal

production cost function in period t. In addition to this base cost, output produced during

overtime in period t incurs an incremental overtime cost given by the function ot(zt). That

is, when no overtime is used in period t then zt = 0 and xt rt and the production cost

equals pt(rt); if overtime is used in period t, then zt > 0 and xt rt + zt and the production

cost equalspt(rt + zt) + ot(zt). The producer may also purchase finished products from a

subcontractor. Letyt denote this subcontracted quantity in period t and let gt(yt) be

subcontracting cost function. We assume the subcontractor's capacity is unlimited,

although we discuss the implications of limited subcontractor capacity in Section 3.3.

We assume that the cost functions pt, ot, and gt are all nonnegative and nondecreasing. In

addition, we also assume that they are concave, representing the presence of economies

of scale. Note, however, that due to the presence of production capacities and overtime

costs, the total internal production cost function depends on the amount of regular

capacity and is not necessarily concave in the total quantity produced. Finally, for

convenience and without loss of generality, we assume that pt(0) = o(O) = gt(O) = 0 for all

t.

We assume that there is a one-to-one correspondence between price and demand in

any period (except possibly when the price equals zero), where demand is a downward-

sloping function of price (see Gilbert (1999) and Geunes et al. (2006)). That is, a

quantity of demand satisfied in period t, say Dt, implies a unique value of the price in

period t. We therefore work directly with demand values as decision variables rather

than prices. The revenue in period t is assumed to be a nondecreasing and concave

function of the demand satisfied in period t, and is denoted by R,(Dt). The concavity of

R,(Dt) is consistent with standard economics models that assume decreasing marginal

revenue in output (see Gilbert 1999). For convenience and without loss of generality, we

assume that Rt(0) = 0 for all t. Let it be the decision variable denoting the inventory level

at the end of period t and let h be the corresponding nonnegative per unit inventory

holding cost.

Our Capacitated Production Planning Problem with Pricing [CPPP] can now be

formulated as follows:

[CPPP]

Minimize T p(xt) + o,(z) +g (yt)+hit)- C Rt(Dt) (3.1)

Subject to: it + X + y, -i, = D,, t = 1, 2,... ,T, (3.2)

r, < aK, t = 1, 2, ... T, (3.3)

zt K t = 1, 2, ... T, (3.4)

x, = r, +z,, t = 2, ... T, (3.5)

yt,it,r,,z,,D, >_0, t= 1, 2, ... T,

i, = i, = 0. (3.6)

The objective function (3.1) minimizes production, overtime, subcontracting and

holding costs less revenue from satisfied demand (thus the negative of the objective

function value provides the net profit). Constraint set (3.2) represents inventory balance

constraints. Note that this balance constraint takes demand as a decision variable.

Constraint sets (3.3) and (3.4) ensure that regular production is limited by capacity level

aK and overtime production is limited by capacity level K. Constraint set (3.5) implies

that internal production consists of both regular and overtime production.

Note that the [CPPP] model does not explicitly require regular capacity to be

exhausted before overtime capacity. Suppose, however, that we have a feasible solution

to the [CPPP] for which, in some period t, some overtime is utilized and regular capacity

is not exhausted, i.e., z, > 0 and rt < cK Then we can define an alternative solution with

z~ = (r, + z, aK) and r,'= max {aK, r, + z, (which is also feasible, because

x' = r,'+ z' = r, + z, = x,). This new solution is at least as profitable as the first since

x' = x, and z <_ z,, and the incremental overtime cost function ot is nonnegative and

nondecreasing. Thus we can assume without loss of optimality that the facility employs

overtime production only if regular-time capacity is exhausted.

The [CPPP] minimizes the difference between concave functions. This problem is

a difficult global optimization problem in general (Horst and Tuy 1990, Geunes et al.

2006). However, if the revenue functions in each period are piecewise-linear we can

exploit the concavity of the remainder of the objective function. We will discuss a setting

in which such a revenue function arises naturally. However, in general this choice of

revenue function form may serve as a close approximation to the actual revenue function.

In the remainder of this chapter, we therefore assume that the revenue function in

period t consists of Jr consecutive linear segments with widths dt and positive slopes rt

(where, for convenience, we define Jmax = max- 1, {J,}). The concavity of the revenue

function dictates that the slopes are decreasing; moreover, we add a final segment having

slope zero, beginning at some upper bound on the total possible demand level (see Figure

3.1). This upper bound may occur, for example, at the demand value that results when

the price is set to the variable cost (while further reducing the price below such a value

may increase total revenue, such solutions will never be profitable, and need not be

considered; similarly, since costs increase for demands satisfied beyond this upper bound

on demand in our model, such solutions will never be optimal for our model).

R,(D,)

-- -------

~D ,
d,

Figure 3-1. Piecewise linear concave revenue function.

We modify our model for such revenue function structures by letting v,, denote the

decision variable for the amount of demand satisfied within the/ h segment of the revenue

curve in period t (note that we only need such variables for the first Jt such segments,

since the (Jt+l)st segment implies zero additional revenue at additional cost). We thus

reformulate the problem with piecewise linear revenue functions, which we refer to as

[CPPPpL], as follows:

[CPPPpL]
SMinimizeT J(t rv
MinimizSubjetto: (p(x,) +y,-i,= t=)+ ,2,...,T, (3.7)

Subject to: ,,+x, +y i, = t 1, 2, T, (3.7)

0 < vt < djt, t = 1, 2, ..., Tj = 1, ..., Jt, (3.8)

(3.3)-(3.6), and

yt, it, rt, zt > 0, t = 1, 2, ..., T. (3.9)

The objective function of [CPPPPL] is concave in the decision variables. Observe

that for demand segments within a period, the model will naturally select those segments

with higher slopes first. We therefore need not explicitly impose constraints specifying

that a segment's v,, variable can only be positive if v,-1,t = d-1,t. Note that this problem can

also be interpreted as an equivalent order selection problem, where a manufacturer

produces goods to satisfy demands corresponding to different orders over a finite number

of time periods. In this context we allow offering different prices for different orders,

where each order provides (possibly) a unique price. In this interpretation, each (/, t)

segment corresponds to a customer order, where dt is the quantity associated with order

in period t, and r, is the unit revenue. In addition, vjt is the decision variable for the

amount delivered corresponding to order in period t.

Observe that the special case in which a = 1 and ot(zt) = oc for zt > 0 and for all t

represents contexts in which no overtime options exist and regular-time capacity equals

K. If, in addition gt(yt) = co foryt > 0 and for all t, then no subcontracting will be utilized,

and the resulting model generalizes the problem studied by Geunes, et al. (2006) to

account for general concave production costs (they considered only fixed plus linear

production costs). Thus the [CPPPpL] generalizes the capacitated production planning

and pricing problem considered by Geunes, et al. (2006) in two important ways. First,

we allow the additional options of subcontracting and capacitated overtime, whereas their

model considered only a regular-time capacity limit. Second, the production cost

function takes a general concave form (as do the overtime and subcontracting cost

functions). Moreover, the solution procedure we provide in this chapter improves upon

the worst-case complexity of the algorithm provided by Geunes et al. (2006).

The [CPPPpL] can be modeled as a concave-cost network flow problem as depicted

in Figure 3.2. The flow on the arcs from periods to demand segments (or customer

orders) represents demand satisfaction, and each has capacity dt. The flow on arc (r, t)

represents regular production, (o, t) overtime production, (s, t) subcontracting, and the

flow on arc (t, t+1) denotes the inventory carried from period t to period t+ 1. There is no

capacity for subcontracting while regular production arcs have capacity aK and overtime

arcs have capacity K.

D

Production r_ : o) Production (s) Subcontractor

xt .... Periods

Orders

Dummy
Source

Figure 3-2. Network representation of the [CPPPpL].

Since the [CPPPpL] minimizes a concave cost function over a set of network flow

constraints, an optimal extreme point solution exists. In any extreme point solution, the

basic variables create a spanning tree in the network (Ahuja, Magnanti, and Orlin 1993).

Before proceeding, we provide an important definition of the concept of a Regeneration

Interval (RI) as provided by Florian and Klein (1971).

Regeneration Interval: Given a feasible production plan, a Regeneration Interval
(RI) (t, t) is a sequence of consecutive periods t, t + 1, ..., t'- 1 with i = i,, = 0
and i, >0 for r= t, t+ 1, ..., t'- 2 (where 1 t < t'<.T+1).

The following proposition states important characteristics of those RIs that can be

contained in an extreme point solution.

Proposition 3.1. In an extreme point solution for [CPPPPL], any RI can have at
most one period t with 0 < rt < aK (fractional regular production), at most one
period t with 0 < zt < K (fractional overtime production), at most one period with yt

> 0 (subcontracting), or at most one period with 0 < jt < djt, but not any of these
together.

Proof. Proof of this result follows from the spanning tree property by showing that
any solution that violates the conditions of the proposition results in an undirected
cycle in the network.

Any solution to the [CPPPpL] can be decomposed into a sequence of RIs. More

importantly, any extreme point solution to the [CPPPpL] can be decomposed into a

sequence of RIs that satisfy the characterization in Proposition 3.1. Therefore, if we can

find the minimum cost associated with each possible RI while observing the structure of

Proposition 3.1, then the overall problem can be solved using an acyclic shortest path

graph containing a path corresponding to every sequence of possible RIs.

Using Proposition 3.1 we can classify the RIs that may be associated with extreme

points into two types:

Fractional supply: There is exactly one positive supply quantity that is not at
capacity (either 0 < xt < aK, or aK < x < (a+l)K, oryt > 0), and all (period,
segment) pairs that are used are filled to capacity (vr e {0,4dt}).

Fractional demand: Any positive regular and overtime production is at full
capacity and no subcontracting takes place (xt e {0, aK, (a-+l)K} andyt= 0), and
there exists at most one (period, segment)-pair in which the segment-capacity is
used partially (0 < vjt < dj).

In order to efficiently determine the minimal cost of these types of RIs, we will

next discuss how to identify candidate demand levels for a given RI in polynomial time.

3.1.2 Determining Candidate Demand Vectors for an RI

In the previous section we have seen that we need to consider at most two types of

RIs associated with extreme point solutions. For each RI type, we will characterize a

candidate set of demand levels such that at least one of these demand levels provides an

optimal solution for the given RI type.

We first reformulate our problem in a more compact form, which allows us to

characterize a key property of optimal demand levels for any given RI. We can define

the inventory level in period t as the cumulative production amount less satisfied demand,

i.e., i, = x, +Z yV ., v,, which allows us to reformulate the problem as

follows:

Minimize T I(c (xt) + st (yt)+o, (z,))- ZT: P1jt

Subject to: x' + ,- I J > t=,..,T,

(3.3)-(3.6), (3.8), and

y,r,,z, >0, t= 1, ..., T,j= 1, ..., Jr,

where we have used the following set of redefined cost functions and revenue

coefficients:

c, (x,) =p, (x,) + x,1h,

S,(yt) =gt(yt) + y, ,Th1

P, = r _t + h .

Note that the functions ct and st inherit the properties of p and gt, i.e., they are

nonnegative, nondecreasing, and concave, and c,(O) = st(O) = 0 for all t. The pt values

will play an important part in our solution approach, as the following propositions

illustrate.

Proposition 3.2. For each RI, we only need to consider solutions for which vkr > 0
(where period ris in the RI) implies that vt = dt for all (j, t) such that pjt>Pkr and
period t is in the RI.

Proof. Consider an RI containing periods t and rand suppose that we have a
corresponding solution for which vt < djt and Vkr> 0 while pjt>Pkr. Then a small but
positive amount of demand can be shifted from period s kth segment to period t's
jth segment without changing any supply levels. The change in the objective

function per unit of shifted demand is equal to pk pt < 0, which implies that the
original solution cannot be optimal for the RI. This implies the desired result.

Proposition 3.3. For each RI of the first type [fractional supply], we only need to
consider solutions for which vk, > 0 (where period ris in the RI) implies that v, =
dt for all (/, t) such that pjt >pkr and period t is in the RI.

Proof. The desired result follows immediately from Proposition 3.2 for all (/, t)
such that period t is in the RI and pjt pkr. Now suppose that we have a solution for
which vjt < dt and vkr = dk, while pjt =pk. Then, since there exists at least one
supply variable that is fractional, we can increase or decrease vt by a small amount.
If the cost of the solution is nonincreasing when vt is decreased we may do so until
no supply variables are fractional (at which point the RI becomes one of the second
type) or until it is decreased to 0. However, in that case the cost of the solution will
also be nonincreasing when vk, is decreased either until no supply variables are
fractional or until it is decreased to zero. Similarly, if the cost of the solution
decreases when vt is increased we may again do so until no supply variables are
fractional (at which point the RI becomes one of the second type and the original
RI solution cannot be part of an optimal solution) or until it is increased to dt. Thus
the desired result follows. 0

Propositions 3.2 and 3.3 can be utilized to determine candidate demand patterns for

each RI of a given type. For the first RI type, recall that all vj variables must take values

of either 0 or dt. Proposition 3.3 implies that at most TJmax + 1 different candidate

demand vectors for the RI (including the zero demand vector) need to be considered,

based on an ordering of the demands in (any) nonincreasing order of their p-values (since

in an RI (t, t') there can be at most T periods, and within a period we have at most Jmax

segments).

For the second RI type, recall that we can have at most one demand segment in the

RI with 0 < vt< djt, while all (regular and overtime) production levels are either zero or at

capacity and no subcontracting is allowed. Since the total amount produced in the RI

must equal the total demand satisfied, if we have a candidate cumulative production level

for the RI, then by Propositions 3.2 and 3.3, we can directly establish corresponding v,,

values by filling the candidate cumulative production level with demand segments in

nonincreasing order of their p-values. If there are ties in this ordering we break these ties

by selecting demand segments in later time periods before demand segments in earlier

time periods. This will ensure that if a feasible solution exists with the given candidate

cumulative production levels it will be found. The number of candidate demand vectors

for this type of RI is thus equal to the number of possible total production quantities in

the RI. In general, if the number of periods with regular production at capacity and no

overtime production is m and the number of periods with both regular and overtime

production at capacity is n, the total production quantity is equal to maK + n(a+1)K for a

total of O(72) possible demand vectors for the RI. If regular production capacity is an

integer multiple of the overtime production capacity, i.e., a is integer, then all possible

cumulative production levels are integer multiples of K and the number of possible

demand vectors for the RI is O(aT).

The optimal RI cost for a given candidate demand vector can be computed using

dynamic programming, as we discuss in the next section. The solution with minimum

cost among all candidate demand vectors then provides an optimal solution for that RI.

3.1.3 Optimal Cost Calculation for an RI

This section discusses how to compute the optimal RI cost for any given RI and a

corresponding candidate demand vector. We consider the case in which regular capacity

is an integer multiple of overtime capacity in Section 3.1.3.1, while the general case is

discussed in Section 3.1.3.2.

For a given RI, say (t, t'), let a candidate set of demands to be satisfied be given by

the quantities in each demand segment to be satisfied: vJt r= t, ..., t' 1,j = 1, ..., J.

Let the corresponding demand vector, as described in Section 3.1.1, be given by

D, = ', vl, for r= t, ..., t'- 1. Moreover, we denote the cumulative demand in periods

t,...,rby D(t,r)= D, for = t, ..., t'- 1.

3.1.3.1 Regular capacity as integer multiple of overtime capacity

In this section, we develop a dynamic programming method used to find an optimal

solution for an RI and a candidate demand vector when regular production capacity is an

integer multiple a of overtime capacity (It is straightforward to show that if the overtime

capacity is an integer multiple of the regular time capacity, a slight modification of the

methods developed in the remainder of this chapter will solve the corresponding

problems in the same running time, with a then denoting the ratio between overtime and

regular capacity). We will construct a layered network, where the layers correspond to

production periods, to track the cumulative supply amounts in each period. Nodes in a

layer will represent possible cumulative regular and overtime production plus

subcontracting levels up to the time point corresponding to the layer.

Given an RI and a candidate demand vector, let q = LD(t,t'- 1)/ K denote the

number of integer multiples of the capacity parameter K that are required to satisfy a total

demand ofD(t, t'-l) and let f = D(t,t'- 1) qK denote the remainder. Iff> 0 then this

candidate demand vector corresponds to an RI of the first type and a quantity equal tof

must be produced either in a fractional regular time production period, a fractional

overtime period, or a subcontracting period. Note that the actual amount of production in

a fractional regular time production period can equal wK +f for any w = 0, ..., a- a

fractional overtime period will contain a production amount equal tof and a

subcontracted amount can equal sK +f for any s = 0, ..., q.

Let (z, u, e) denote a node in the layered network, where r denotes the current time

period (or layer), u denotes the cumulative production as a multiple of K up to and

including the period, and e = 1 if the fractional amount has been produced already and is

0 otherwise. Arcs between layer rand layer r+1 will be of the following types (r= t-1,

..., t' 2):

Al Zero supply arcs:

From node (r, u, e) to node (r+ 1, u, e)

Arc costs: 0

A2 Fractional regular production arcs:

From node (r, u, 0) to node (r+ 1, u + w, 1) for w = 0, ..., a- such that u + w < q

Arc costs: c,+i(wK +J) if (u + w)K +f > D(t, r+ 1), oc otherwise

A3 Full capacity regular production arcs:

From node (r, u, e) to node (r+ 1, u + a, e)

Arc costs: c,+(aK) if(u + a)K+ ef > D(t, r+ 1), co otherwise

A4 Fractional overtime production arcs:

From node (r, u, 0) to node (r+ 1, u + a, 1) ifu + a < q

Arc costs: c,~+(aK) + r+i(f) if (u + a)K +f > D(t, r+ 1), oc otherwise

A5 Full capacity overtime production arcs:

From node (r, u, e) to node (r+ 1, u + a+ 1, e) if u + a+ 1 < q

Arc costs: c,+i((a+l)K) + o+(K) if (u + a+ 1)K+ ef > D(t, r+ 1), oo otherwise

A6 Subcontracting arcs:

From node (r, u, 0) to node (r+ 1, u + s, 1) for s=0, ..., q such that u + s < q

Arc costs: g+, (sK+f) if (u + s)K +f > D(t, r+ 1), oc otherwise

A7 Demand satisfaction arcs:

From node (t'-l, q, 1) to the sink

t'-1 J,
Arc costs: -1 pjVj
z=t j=1

Note that we allow parallel arcs in the network. For example, arcs of type A6 for s

< a are equivalent to arcs of type A2 (although they have different costs). The shortest

path from a single source node (t-1,0, 0) to the sink in the resulting graph provides an

optimal solution for the RI and corresponding candidate demand vector. Note that if the

shortest path has infinite length, the candidate demand vector is infeasible. To determine

the complexity of this RI subproblem, note that the number of nodes is in a layer is

O(aT) so that the total number of nodes in the network is O(caf). Since each node has

outdegree O(aT) the number of arcs in the network is O(a273). However, note that in

cases where= 0 (that is, in cases where the demand vector corresponds to an RI of the

second type), the arcs of types A2, A4, and A6 are not needed. When these arcs are

removed, each node has outdegree at most 1 and the number of arcs in the network to

O(aT2). The optimal solution for such cases is then given by the shortest path from a

source node (t-1, 0, 1) to the sink.

3.1.3.2 Regular capacity as any positive multiple of overtime capacity

In this section we allow regular production capacity to be any positive multiple of

overtime capacity. Let m denote the number of periods in which regular production is at

capacity and overtime production is not, and n denote the number of periods where

regular and overtime production are both at capacity. Then, given an RI (t, t'), a

corresponding candidate demand vector, and fixed values for m and n the total demand

satisfied can be expressed as D(t,t'-1)= maK + n(a + 1)K + f wherefmay correspond to

fractional regular time production, fractional overtime production, or a subcontracted

amount.

Let (r, m', n', e) represent a node in the network, where ris the current period

(layer), m' is the total number of periods, up to and including period r, in which regular

production is at capacity and overtime is not, and n' denotes the number of cumulative

full capacity overtime periods. We set e = 1 if the fractional amount has been produced

already, otherwise it is 0. Arcs between layer rand layer r+1 will be of the following

types (= t-1, ..., t'-2):

B1 Zero supply arcs:

From node (r, m', n', e) to node (r+ 1, m', n', e)

Arc costs: 0

B2 Fractional regular production arcs: (only if 0 f< aK)

From node (r, m', n', 0) to node (r+ 1, m', n', 1)

Arc costs: c, (/) if m'aK + n'K+f >D(t, z+ 1), oc otherwise

B3 Full capacity regular production arcs:

From node (r, m', n', e) to node (r+ 1, m'+l, n', e)

Arc costs: c,~+(aK) if (m'+l) aK + n'K + ef > D(t, z+ 1), oc otherwise

B4 Fractional overtime production arcs: (only if acK _f< (a+1)K)

From node (r, m', n', 0) to node (r+ 1, m', n', 1)

Arc costs: c,+(f) + o, (f-aK) ifnm'aK + n'K+f >D(t, z+ 1), co otherwise

B5 Full capacity overtime production arcs:

From node (r, m', n', e) to node (r+ 1, m', n'+l, e)

Arc costs: c,+l((a+l)K) + o,+(K) if m'aK+ (n'+l)K + ef > D(t, r+ 1), 0o

otherwise

B6 Subcontracting arcs:

From node (r, m', n', 0) to node (r, m', n', 1)

Arc costs: gl (f) if m'aK+ (n'+l)K +f > D(t, r+ 1), oc otherwise

B7 Demand satisfaction arcs:

From node (t'-1, m, n, 1) to the sink

t'- J,
Arc costs: pJ,,
z=t J=1

The shortest path from source node (t-1, 0, 0, 0) to the sink in the resulting graph

provides an optimal solution for the RI with the corresponding candidate demand vector

and values for m and n. To determine the complexity of this RI subproblem, note that the

number of nodes is in a layer is O(7T) so that the total number of nodes in the network is

O(7f). Since each node has outdegree at most 1, the number of arcs in the network is

0(7f) as well. Since, in principle, for each candidate demand vector there are O(T1)

potential choices for m and n, the total time required to find an optimal solution for the RI

and a corresponding candidate demand vector is 0(T5). However, note that if we

consider candidate demand vectors corresponding to an RI of the second type, the values

ofm and n are uniquely defined so that an optimal solution for the RI and such a

corresponding candidate demand vector is 0(7).

3.1.4 Complexity of Solution Approach

We will next use the results of Sections 3.1.2, 3.1.3.1, and 3.1.3.2 to derive the

running time of our algorithm for solving the [CPPPpL].

Proposition 3.4. The [CPPPpL] can be solved in polynomial time in Jmax and T
Moreover, if a is integral the [CPPPpL] can be solved in pseudo-polynomial time
that is superior as a function of T.

Proof. First consider the case where a is integral. For candidate demand vectors
corresponding to RIs of the first type, the optimal cost can be calculated in at most
O(a273) time and the number of such candidate demand levels to be considered is
O(JmaxT) for a total of O(2Jm.axT4) time. For candidate demand vectors
corresponding to RIs of the second type, the optimal cost can be calculated in at
most O(af) time and the number of such candidate demand levels is O(aT) for a
total of O(acfZ) time. Since there are O(T2) RIs, it takes O(2Jmaxt6) time to
calculate all optimal RI costs.

For general a, the optimal cost for candidate demand vectors corresponding to RIs
of the first type can be calculated in at most 0(75) time and the number of such
candidate demand levels to be considered is O(JmaxT) for a total of O(JmaxT6) time.
For candidate demand vectors corresponding to RIs of the second type, the optimal
cost can be calculated in at most 0(73) time and the number of such candidate
demand levels is O(7T) for a total of 0(7T) time. Since there are O(7T) RIs, it takes
O(Jmax78) time to calculate all optimal RI costs.

In both cases, the shortest path in the resulting acyclic network containing a node
for each RI and an arc for each optimal RI solution can be found in O(7T) time.

Note that for the special case of the [CPPPpL] model considered in Geunes et al.

(2006), where a = 1 and no overtime or subcontracting options are available, the optimal

cost of any RI can be determined in 0(71). This implies a the worst-case complexity for

determining an optimal RI cost of O(Jmax73), and a corresponding worst-case problem

solution complexity of O(JmaxT). While Geunes, Merzifonluoglu, Romeijn, and Taaffe

(2006) showed this improved complexity result over the O(JmaxT6) algorithm in Geunes et

al. (2006) (who considered only fixed-plus-linear production costs), our analysis in this

chapter shows that this improved complexity result holds for the more general case of

concave production cost functions.

In this section, we have provided a polynomial-time solution method for a fixed

value of the capacity parameter K. In the following section, we generalize our results to

account for contexts in which the capacity parameter K is a decision variable with an

associated capacity cost.

3.2 Capacity Planning

We next consider the case where the manufacturer wishes to determine its optimal

internal capacity level for the production horizon. Let (K) denote a concave cost

function of the capacity parameter K, which is a decision variable. If we consider the

case of capacity acquisition with an initial capacity of zero, the function (K)

characterizes the cost to acquire aK units of regular capacity and associated K units of

overtime capacity. The problem formulation for the capacity planning problem, which

we call [CPPPpL(K)], is as follows:

[CPPPpL(K)I

Minimize O(K) + T (c, (x,)+ g, (y,)+o,) ,)+h i,)- _IT, 1 ,rv

Subject to: (3.3)-(3.9), and

K> 0.

The objective function minimizes the capacity, regular time production, overtime

production, subcontracting, and inventory holding costs, less revenue from satisfied

demand, while the constraint set is the same as that for [CPPPpL], with the addition of the

nonnegativity constraint on K. Note that the [CPPPpL(K)] model can also be applied to

contexts in which the initial capacity is positive instead of zero. If we begin with some

initial capacity KI > 0, replace the nonnegativity constraint with K > -KI, and allow (K)

to take negative values for K < 0, then using a variable substitution, the resulting model is

structurally identical provided that AK) remains concave. We interpret negative values

of K to imply a negative cost, or reward, for reducing capacity (through, for example,

capacity that is sold or associated capacity costs that are avoided). Whether a firm begins

with an initial positive capacity or is considering capacity acquisition, the [CPPPpL(K)]

model can provide value as a benchmarking tool in order to determine the optimal

capacity level during a planning horizon.

For problem [CPPPpL(K)], we again minimize a concave function over a

polyhedron, which implies that an optimal extreme point solution exists. The following

proposition characterizes the structure of extreme point solutions.

Proposition 3.5. For every extreme point solution for [CPPPPL(K)] with K > 0
there exists an RI (t, t) in which all internal production levels are either at zero,
regular-time capacity, or overtime capacity and there is no subcontracting. In
addition, there is no fractional demand satisfaction. That is,

x, e {0, aK, (a+ 1)K}, for all = t,...,t'-1;

y,= 0, for all r= t,...,t'l;

v,, {0, dj}, for all r= t,...,t'-l;j = 1, ..., J.

Proof. The total number of variables in the formulation of [CPPPpL(K)] is 5 T+
= J, (note that we need not consider io or iT since these variables can be
substituted out of the formulation), so in an extreme point solution 5T+ T,=
linearly independent inequalities must be binding. There are Tbinding balance
constraints (3.7) and Tbinding equality constraints defining total production (3.5).
The remaining 3T+ =1J, binding inequalities in an extreme point solution must
come from among the remaining constraints. Any feasible solution consists of a
sequence of RIs. Let (x, r, z, y, i, K) be an extreme-point solution with K > 0 and
let R denote the number of RIs associated with this solution, where the ith RI is
(t,,t,1+) (where tl=l and tR+=1T+I).

Since we know that it = 0 for i=1,...,R-1 these provide an additional R-1
linearly independent binding qualities (where we previously accounted for
constraints (3.6). The remaining 3T + J R +1 required binding inequalities
for the extreme point solution must then come from among the following
constraints:

rt: aK, = 1, R, t= t,, ..., to+- 1, (3.10)

rt> 0, i= 1, ...,R, Rt= t,, ., t+- 1, (3.11)

zt < K, i = 1, ...,R, t = t,, ..., t,+- 1, (3.12)

zt > 0, i = 1, ..., R, t = t,, ..., t,+- 1, (3.13)

Yt > O, i = 1, ...,R, t= t,, ..., t,+- 1, (3.14)

vjt < djt, i= 1, ..., R, t = t,, ..., t 1- 1, (3.15)

vt> = l, ...,R,t=t,, ..., t 1,j= ...,Jt. (3.16)

When K > 0 and for a given i, at most t,+1 t, of each of the (sets of) inequalities
t'+l J of
(3.10)-(3.11), (3.12)-(3.13), and(3.14), respectively. Finally, at most ,'J, of
the pair of inequalities (3.15)-(3.16) can be binding. This implies that at most
3(t, + 1t, J, of the constraints corresponding to a given RI i may be
binding. It is easy to see that, for a given RI, the structure of an extreme point
solution for this RI is of the type as described in the proposition if and only if the
maximum number of inequalities is indeed binding for that RI. Now suppose that
for the extreme point solution (x, r, z, y, i, K) this maximum is not achieved by any
of the RIs. This implies that no more than
S(3(t t+ 1 J, 1)= 3T + T J, R binding inequalities exist from
among inequalities (3.10)-(3.16), which contradicts that (x, r, z, y, i, K) is an
extreme point solution. Thus we conclude that an extreme point solution satisfying
the conditions of the proposition exists. 0

Proposition 3.6. There are O(Jmaxt) distinct capacity parameter levels among all
candidate optimal extreme point solutions for [CPPPpL(K)]. In addition, when the
capacity multiplier a is integral, the number of distinct such capacity parameter
levels is O(JmaxaT4).

Proof. There are T RI choices given by all pairs of the form (t, t) with t =

1,...,Tand t'= t+l,...,T+1. As a result of the discussion in Section 3.1.2 and

Proposition 3.3, within each of the T RIs there are O(JmaxT) candidate

demand vectors that we must consider, for a total of 0 Jm1xT = O(Jmax)
2
candidate combinations of RIs and demand vectors of the form given in Proposition
3.5. (By Propositions 3.2 and 3.3, for any given K-and therefore at the optimal
value of K in [CPPPpL(K)]-when ties exist among multiple p-values, by breaking
ties based on latest time period first, we consider all potentially optimal extreme
points, which implies that there are O(JmaxT) candidate demand vectors that must
be considered, even in the case in which ties exist among p-values.)

As a result of Proposition 3.5, in an extreme point solution all demand in at least
one RI, say (t, t'), is satisfied by an integer number n of full-capacity regular
production periods and an integer number m of full capacity overtime production
periods. For a given demand vector we thus have naK + m(a + 1)K = D(t,t'- 1). So,
given an RI and candidate demand vector, the total number of different levels of K
such that naK + m(a + 1)K = D(t,t'-1) with n and m nonnegative integers is O(T2).
Since each of the O(Jmax3) candidate combinations of RIs and demand vectors
may imply O(T2) capacity levels we have O(JmaxT5) distinct capacity levels.

In the special case where a is integral, the total production in any RI of the type
given in Proposition 3.5 must be an integer multiple of Kup to T(a+ 1)K. We thus
have only O(aT) total production quantities to consider for a given demand vector.
Multiplying this by the number of potential RI and demand vector combinations
implies O(JmaxaT4) candidate capacity parameter levels. 0

For each candidate capacity parameter level K we can then solve a [CPPPpL]

problem which yields the following corollary.

Corollary 3.1. The problem [CPPPpL(K)]can be solved in polynomial time.

Proof. The proof follows from Propositions 3.4 and 3.6 by solving the [CPPPpL]
problem with piecewise-linear, concave, and nondecreasing revenue curves for
each candidate capacity level. When a is integral the overall complexity for
solving [CPPPpL(K)] is O(J; a3T10) while for general values of a the resulting
worst-case complexity is O(JxjT13").

While the resulting order of the complexity result is a high power of the horizon

length T, in light of the generality and the number of integrated decisions contained in the

model, the polynomial solvability is remarkable.

3.3 Multiple Subcontractors and Subcontractor Capacities

Throughout this chapter, we have assumed that there is a single uncapacitated

subcontractor available for subcontracted purchases. Although this is not an

unreasonable assumption in light of past literature (see e.g., Atamtiurk and Hochbaum

2001), we may also consider a generalization of our model that allows for one or more,

say M, potentially non-identical subcontractors with or without time-invariant

subcontracting capacity limits.

Letting y; denote the amount subcontracted to the ith subcontractor in period t and

letting g; (y;) denote the associated (concave) subcontracting cost function, we can

modify our problem formulations by replacing each yt by m y; in the balance

constraints (3.2), replacing the nonnegativity constraints on each yt by nonnegativity

constraints on each yt, and replacing the cost term gt(yt) by I g; (Y;) in the objective

function.

3.3.1 Uncapacitated Subcontractors

The Capacitated Production Planning Problem with Pricing and Uncapacitated

Subcontractors [CPPP-US] is formulated as follows:

[CPPP-US]

Minimize: T (p (x,) + o, (z,) + g (y:) +hi,) -+ 1 rTv,

Subject to: i, + x, + y -i = D, t= 1, 2, ...T,

(3.3)-(3.5), (3.8)

yY i,r,z, Dt > 0, t = 1, 2, ..., T, i= 1, 2, ..., M

Since this is a minimization problem of a concave function over a polyhedron, the

spanning tree properties of optimal solutions still apply here. The following proposition

will determine the RI types we will need to consider in this case.

Proposition 3.7. In an extreme point solution, any regeneration interval can
have at most one period t with 0 < rt< aK (fractional regular production), at most
one period t with 0 < zt< K (fractional overtime production), at most one period t
with at most one y, > 0, i = 1, 2, ..., M (subcontracting with at most one
subcontractor in at most one period) or at most one period with 0 < vt< dct, but not
any of these two together.

The proof of Proposition 3.7 follows from the spanning tree property of optimal

solutions. Because there is at most one subcontractor utilized within an RI, we can

effectively include M parallel subcontracting arcs (one for each subcontractor) in the

layered networks constructed in Section 3.1.3. The properties of optimal demand vectors

are therefore the same as those in the single uncapacitated subcontractor case.

Proposition 3.8. There exists an O(MJmaax26) algorithm to solve the [CPPP-US]
problem with piecewise linear revenue curves when regular production capacity is
an integer multiple of overtime capacity. When a is any positive scalar, there
exists an 0 (MJmaxJ8) algorithm to solve [CPPP-US] problem with piecewise linear
revenue curves.

Proof. When a is integral, the optimal cost can be calculated in at most O(MaI2(a
+ T)) time for candidate demand vectors corresponding to RIs of the first type and
the number of such candidate demand levels to be considered is O(JmaxT) for a total
of O(Jmaxa73(a + T)) time. For candidate demand vectors corresponding to RIs of
the second type, the optimal cost can be calculated in at most O(afI) time and the
number of such candidate demand levels is O(aT) for a total of O(af7i) time.
Since there are O(f2) RIs, it takes O(max{MaZJmaxf MaoZaxt6, a2T6}) <
O(Ma2Jmaxf6) time to calculate all optimal RI costs.

For general a, the optimal cost for candidate demand vectors corresponding to RIs
of the first type can be calculated in at most 0(Mf6) time and the number of such
candidate demand levels to be considered is O(JmaxT) for a total of O(MJmax,6)
time. For candidate demand vectors corresponding to RIs of the second type, the
optimal cost can be calculated in at most 0(7f) time and the number of such
candidate demand levels is 0(7T) for a total of 0(76) time. Since there are 0(T1)
RIs, it takes O(MJmax,8) time to calculate all optimal RI costs.

In both cases, the shortest path in the resulting acyclic network containing a node
for each RI and an arc for each optimal RI solution can be found in 0(T2) time.

3.3.2 Capacitated Subcontractors

In this case we take into account the possibility of finite subcontractor capacities in

the multiple subcontractor case. We assume each subcontractor can supply a limited

amount of product (or service) in a period, and that the capacity of each subcontractor is

known and does not vary throughout the planning horizon. Subcontractor capacities can

easily be included in the model by adding constraints of the form y\ < C' for all i and t,

where C' denotes subcontractor i capacity in any time period.

The Capacitated Production Planning Problem with Pricing and Capacitated

Subcontractors [CPPP-CS] is formulated as follows

[CPPP-CS]

Minimize: T ((x,) +o, (z,) + ;g (y ) +hti,) -'1 T r

Subject to: i +x + y i = D,, t= 1, 2, ..., T,

(3.3)-(3.5), (3.8)

y < C t = 1, 2,... ,T, i = 1, 2, ...,M

y,, irt z,, D > 0, t = 1, 2, T, i = 1, 2, ..., M

The RI cost computations for this case are also based on the associated spanning

tree property. The following proposition defines the regeneration interval types for the

case of multiple capacitated subcontractors.

Proposition 3.9. In an extreme point solution, any regeneration interval can have
at most one period t with 0 < rt < aK (fractional regular production), at most one
period t with 0 < z < K (fractional overtime production), at most one period t with
0 < y\ < C' for at most one subcontractor i (and all other subcontractor production

quantities at 0 or capacity in all periods), or at most one period with 0 < vt < dt, but
not any of these two together.

As in Section 3.1.1, Proposition 3.9 let us classify the RIs that may be associated

with extreme points into two types:

Fractional supply: There is exactly one positive supply quantity that is not at
capacity (either 0 < xt< aK, or aK < x< (a+l)K, or 0 < y1 < C'), and all (period,
segment) pairs that are used are filled to capacity (vjt e {0,4})).

Fractional demand: Any positive regular, overtime production, and
subcontracting quantity is at full capacity (xt e {0, aK, (ca+-)K} and y, e {0, C')),
and there exists at most one (period, segment)-pair in which the segment-capacity
is used partially (0 < vjt < dj).

3.3.2.1 Determining candidate demand levels for a regeneration interval

For fractional supply RIs, all vj variables must take values of either 0 or dat in an RI.

Based on our previous discussions about optimal demand vector properties, there exist at

most O(JmaxT) different candidate demand vectors for a regeneration interval when all vjt

variables must be 0 or dt.

The fractional demand RI type allows at most one segment of a revenue curve

within an RI with 0 < vjt< dt. In this case, all production and subcontracting levels are

either zero or at capacity. As a result, the number of possible different internal

production levels (regular plus overtime) is O(aT) when a is an integer, and O(f2)

otherwise, while the possible number of levels of the cumulative subcontracted amount to

any single subcontractor i is O(T). Since there are Munique subcontractors, for each

possible (vector) value of internal production levels, we need to consider O(T) possible

values of subcontractor production for each of the M subcontractors. Therefore,

considering at most O(ac7M1) and O(7A+2) different demand vectors is sufficient for the

integer a and general case, respectively.

3.3.2.2 Regular capacity as integer multiple of overtime capacity

Given an RI and a candidate demand vector, let q= (D(t, t'-l) -1 bC')/ K

denote the number of integer multiples of the capacity parameter K that are required to

satisfy a total demand ofD(t, t'-l) where b, is the number of periods in which

subcontractor i produces at full capacity. In addition, let f = D(t,t'- )- qK bC'

denote the remainder. Iff> 0 then this candidate demand vector corresponds to a

fractional supply RI and a quantity equal tofmust be produced either in a fractional

regular time production period, a fractional overtime period, or in a fractional

subcontracting period. Note that the actual amount of production in a fractional regular

time production period can be wK +ffor any w = 0, ..., a-1, a fractional overtime period

will contain a production amount equal tof and the fractional subcontracted amount for

subsontractor i can be sK +f for any s=0,..., minjC' / K T}

Let (r, u, r, e) denote a node in the layered network, where r denotes the current

time period (or layer), u denotes the cumulative production as a multiple of K up to and

including period r, r denotes an M-vector of the number of full production periods for

each subcontractor that have taken place up to and including period t (the ith element of

the vector r indicates the cumulative number of full production periods for subcontractor

i), and e = 1 if the fractional amount has been produced already (and is 0 otherwise).

Arcs between layer rand layer r+1 will be of the following types (r= t-l, ..., t'-2):

C1 Zero production arcs:

From node (r, u, r, e) to node (r+ 1, u, r', e)

(where r' = r + 1, and I is an M-vector containing a 1 in the ith position if

subcontractor i produces at capacity in period r(and is 0 otherwise)).

Arc costs: ,l1, g C') if r,' D(t, r+

1), and oc otherwise.

C2 Fractional regular production arcs:
From node (r, u, r, 0) to node (r+ 1, u + w, r', 1) for w = 0, ..., a- ifu +w < q
and r, < b, for i= 0, 1, ..., M.

Arc costs: c (wK +f)+ lg' 1 (') if(u + w)K + r C' +f >D(t, r+ 1),
and oc otherwise.

C3 Full capacity regular production arcs:

From node (r, u, r,e) to node (r+ 1, u + a, r',e) if r < b, for i = 0, 1, ..., M.

Arc costs: c,(aK)+ l, (C') if (u + a)K+ 'C' + ef >D(t, r+ 1),and
oc otherwise.

C4 Fractional overtime production arcs:

From node (r, u, r,0) to node (r+1, u + a, r', 1) if u + a< q and r < b for i = 0, 1,
..., M

Arc costs: c, 1(aK) + M g (C')+ o,,(f) if(u + a)K+ rM C' +f > D(t, r+
1), and oc otherwise.

C5 Full capacity overtime production arcs:

From node (r, u, r,e) to node (r+ 1, u + a+ 1, r', e) ifu + a+ 1 < q and r, < b,
for i = 0, 1, ...,M.

Arc costs: c, ((a+1)K) +M, lg ,(C)+ o+ (K) if (u + a+ )K+ + 'C' +ef
> D(t, r+ 1), and oc otherwise.

C6 Fractional subcontracting arcs for subcontractor "m":

From node (r, u, r ) to node (r+1, u+w, r', 1) for w = 0, ..., minCCm /K T} and
rm =r ifu + w

Arc costs: l g: (C')+ gm,1 (wK + f) if (u + w)K + m C' +f > D(t, r+
1), and oc otherwise.

C7 Demand satisfaction arcs:

From node (t'-l, q, b, 1) to the sink, where b is an M-vector containing bi in the ith
position.

t'- J,
Arc costs: p,,J I.
1=t J=1

As in the network for the single subcontractor case, we allow parallel arcs in the

network. The shortest path from a single source node to the sink provides an optimal

solution for the RI and corresponding candidate demand vector. To determine the

complexity of this RI subproblem, note that the number of nodes is in a layer is O(a7 1)

so that the total number of nodes in the network is O(a 7+2). Since each node has

outdegree O(M2aT), the number of arcs in the network is O(M2aa +3). Considering all

possible subcontracting levels (which is 0(7M)), the shortest path for each regeneration

interval of this type can be calculated in O(M 2aT2 3) time for a given demand vector.

When the RI is a fractional demand RI, the arcs of types C2, C4, and C6 are not

needed. When these arcs are removed, each node has outdegree at most M and the

number of arcs in the network to O(McaI+2). The optimal solution for such cases is then

given by the shortest path from a source node (t-1, 0, z 1) to the sink, where z is a zero

vector with size M.

3.3.2.3 Regular capacity as any positive multiple of overtime capacity

Let m denote the number of periods in which no overtime is used and regular

production is at capacity, and let n denote the number of periods where regular and

overtime production are at capacity. If we again let b, denote the number of periods

where subcontractor i produces at capacity, then total amount of demand satisfied in a

regeneration interval (t, t') can be expressed as

Dc (t,t') = maK +n(aC +1)K + -1 bC' + f. In this equation,fmay correspond to

fractional regular time production, fractional overtime production, or a fractional

subcontracting amount.

Let (t, m', n', r, e) represent a node in the network, where t is the current period, m'

is the number of cumulative regular full capacity production periods, n' denotes the

number of cumulative full capacity overtime periods, and r is the vector of the number of

cumulative full capacity subcontracting periods including period t. We set e = 1 if the

fractional amount has been produced already, otherwise it is 0. Arcs between layer rand

layer r+1 will be of the following types (r= t-1, ..., t'-2):

D1 Zero production arcs:

From node (r, m', n', r, e) to node (r+ 1, m', n', r',e) if r < b, for i = 0, 1, ..., M

(where r' = r + 1, and I is an M-vector containing a 1 in the ith position if

subcontractor I produces at capacity in period r(and is 0 otherwise)).

Arc costs: _M1/g1,, (C') if 1 rC' > D(t, r+ 1), and o otherwise.

D2 Fractional regular production arcs: (only if 0
From node (r, m', n', r, 0) to node (r+ 1, m', n', r', 1) if r, < b, for i = 0, 1, ..., M.

Arc costs: c,+ (f) + Aj,g+ (C') if m'aK + 'K+ K+ r' C' +f > D(t, r+ 1), and

oc otherwise.

D3 Full capacity regular production arcs:

From node (r, m', n', r, e) to node (r+ 1, m'+l, n', r', e) if r, < b, for i = 0, 1,

...,M

Arc costs: c,~ (K) + gl, (C')if (m'+l) aK + n'K+ -'< C' +ef >D(t, r+

1), and oo otherwise.

D4 Fractional overtime production arcs: (only if aK
From node (r, m', n', r, 0) to node (r+ 1, m', n', r', 1) if r' < b, for i= 0, 1, ..., M.

Arc costs: ci() + o,+(f-aK) + 1,g' (C')if m'aK + n'K + 'C' +f >

D(t, r+ 1), and oc otherwise.

D5 Full capacity overtime production arcs:

From node (r, m', n', r, e) to node (r+ 1, m', n'+l, r', e) if b < b for i= 0, 1,

.

Arc costs: c,+i((a+l)K) + o,i(K) + IMg' 1(C)if m'aK+ (n'+l)K+

ZM C' +ef > D(t, r+ 1), and oc otherwise.

D6 Fractional subcontracting arcs for subcontracting i:

From node (r, m', n', r, 0) to node (r+1, m', n', r', 1) if r, < b, for i= 0, 1, ...,M

Arc costs: g',(f)+ l g'+, (C) ifm'aK + (n'+l)K+ ZI C' +f >D(t, r+

1), and oo otherwise.

D7 Demand satisfaction arcs:

From node (t'-1, m, n, b, 1) to the sink, where b is aM-vector containing b, in the
ith position.
Arc costs: -- Y: IPyrI .

The shortest path from the source node (t-1, 0, 0, z, 0) to the sink in the resulting

graph provides an optimal solution, where z is a zero vector with size M. The number of

nodes is in a layer is 0(7M+2) so that the total number of nodes in the network is O(7'+3).

Since each node has outdegree at most 0(M2), the number of arcs in the network is

O(M2 P+3) as well. For each candidate demand vector there are O(T2) potential choices

for m and n and 0(7P) potential choices for the b vector, and the total time required to

find an optimal solution for the RI and a corresponding candidate demand vector is

O(M2AT2Y5). When we consider candidate demand vectors corresponding to an RI of the

second type, the values ofm and n and b are uniquely defined. In addition each node has

outdegree at most 0(M), and the number of arcs in the network is O(MTM+3). Therefore

we can conclude that the cost of each RI of the second type can be computed in

O(MY7l+3) for a given demand vector. The optimal solution for such cases is then given

by the shortest path from a source node (t-1, 0, 1) to the sink.

3.3.2.4 Complexity of solution approach

Proposition 3.10. There exists an O(M2Jmaxa'T7 6) algorithm to solve the [CPPP-
CS] problem with piecewise linear revenue curves, when ais a positive integer.
When a is any positive scalar, there exists an O(M2JmaxT2~+8) algorithm to solve
the [CPPP-CS] problem with piecewise linear revenue curves.

Proof. When regular capacity as integer multiple of overtime capacity, for
candidate demand vectors corresponding to fractional production RIs, the optimal
cost can be calculated in at most O(M2 aT22+3) time and the number of such
candidate demand levels to be considered is O(JmaxT) for a total of O(M2
JmaxaT" 4. For candidate demand vectors corresponding to RIs of the second type,
the optimal cost can be calculated in at most O(MaI7M+2) time and the number of
such candidate demand levels is O(a7cf+1) for a total of O(Ma2T2+3) time. Since
there are O(T2) RIs, it takes O(max {/2JmaxaT2+6 M2I2M +5}) < AM2Jmaxa T+6
time to calculate all optimal RI costs.

For fractional production RIs, the optimal cost can be calculated in at most
O(M2 T2+5) time and the number of such candidate demand levels to be considered
is O(JmaxT) for a total of O(2A2Jmax 12+6). For the second type, the optimal cost for

an RI can be calculated in at most O(Ml"+3) time and there are ((T1+2) candidate
demand levels for this case. Since there are O(T7) RIs, it takes
O(max{M2JmaxT'+8, MT2+7}) < O(M2JmaxT28) time to calculate all optimal RI
costs.

3.3.3 Capacity Planning with Multiple Subcontractors

We consider the case where the manufacturer wishes to determine its optimal

internal capacity level for the production horizon when several subcontractors exist. We

can modify the model in Section 3.2 to account for the case of several subcontractors. For

both cases of capacitated and uncapacitated subcontractors, we minimize a concave

function over a polyhedron, which implies that an optimal extreme point solution exists.

We let [CPPP-US(K)] and [CPPP-CS(K)] denote the capacity planning problems for

uncapacitated and capacitated subcontractor cases respectively. The following

proposition characterizes the structure of extreme point solutions.

Proposition 3.11. For every extreme point solution for [CPPP-US(K)] and [CPPP-
CS(K)] with K > 0 there exists an RI (t, t) in which all internal production levels
are either at zero, regular-time capacity, or overtime capacity. If subcontractors
have no capacity limits, all subcontracting levels are at zero (in the uncapacitated
case) or each subcontracting level is either at zero or at capacity. In addition, there
is no fractional demand satisfaction. That is,

x,e {0, aK, (a+ 1)K}, for all = t,...,t'-1;

v,, e {0, d}, for all r= t,...,t'-l;j = 1, ..., J,

Uncapacitated Subcontractors ([CPPP-US(K)])

y = 0, for all r= t,...,t'- and i =1, ..., M;

Capacitated Subcontractors ([CPPP-CS(K)])

y = {0, C'}, for all r= t,...,t'- and i =1, ...,M;

Proof. The proof follows the discussion for Proposition 3.5. At an extreme point
solution, all demand in at least one RI, (t, t') is satisfied by an integer number n of
full-capacity regular production periods and an integer number m of full capacity
overtime production periods, and, if subcontractors are capacitated, b, full capacity
subcontracting periods for each i. Therefore, we can conclude that if

subcontractors have no capacity limits, the size of the set of candidate capacity
levels is same as the single subcontractor case. Otherwise, for a given demand
vector, we haveD(t,t'-1)= maK + n(a + )K + 1 bC' So, given an RI and
candidate demand vector, the total number of different levels of K such that
D(t, t'- 1) = maK + n(a + 1)K + =bC' with n, m and b,, i=, ...,M equal to
nonnegative integers is 0(7M+2). Since each of the O(Jmaxf7) candidate
combinations of RIs and demand vectors may imply 0(7M+2) capacity levels, we
have O(Jmax7M5) distinct capacity levels.

When a is integral, we have O(ca 7 1) total production and subcontracting
quantities to consider for a given demand vector. Considering potential RI and
demand vector combinations implies O(Jmaxa7+I4) candidate capacity parameter
levels. m

For each candidate capacity parameter level Kwe can solve a [CPPP-CS (K)]

problem which yields the following corollary.

Corollary 3.2. The problems [CPPP-US (K)] and [CPPP-CS (K)] can be solved in
polynomial time.

Proof. The proof follows from Propositions 3.10 and 3.11 by solving the [CPPP-
US] and [CPPP-CS] problems with piecewise-linear, concave, and nondecreasing
revenue curves for each candidate capacity level. For the uncapacitated case, when
a is integral, the overall complexity for solving [CPPP-US(K)] is O(MJ x a3 10)
while for general values of a the resulting worst-case complexity is O(MJTj13).
For the case of capacitated subcontractors, when a is integral, the overall
complexity for solving [CPPP-CS(K)] is O(M2J2ax3T3M+10) and if a is not
integral the complexity is O(M2JaxT3M'13 )

Tables 3.1 and 3.2 summarize the complexity results for both integer and general

values of a. The results in these tables indicate that the increase in complexity when

considering multiple uncapacitated subcontractors is only a factor ofM. However, when

the subcontractors are capacitated, the complexity increases by an additional factor of

MT12 when capacity is not a decision variable, and by MT3 when capacity is a decision

variable (we provide a proof of the NP-Hardness of the multiple capacitated

subcontractor case in the Appendix A). Our approach therefore only solves the problem

with multiple capacitated subcontractors in polynomial time when the number of

subcontractors is fixed (although this is still a noteworthy result, i.e., when a producer

must consider a small number of capacitated subcontractors with time-invariant

capacities, the problem of setting optimal internal capacity and demand levels remains

polynomially solvable). In practice, we would expect that the number of subcontractors

considered by a producer would typically be limited to a reasonably small set of qualified

firms.

Table 3-1. Complexity results with M subcontractors (integral a)

Table 3-2.

Uncapacitated Capacitated
Subcontractors Subcontractors
Fixed Internal O(MJm.6axa') O(M2J axo, T216)
Capacity
Capacity as Decision O(MJ2 a3T1) O(M J a T3M10)
Variable

Complexity results with M subcontractors (general a)
Uncapacitated Capacitated
Subcontractors Subcontractors
Fixed Internal O(MJmaaxf) O(M2Jm ax M+8)
Capacity
Capacity as O(MJmT13) O(MJ2JmaxT3M 13)
Decision Variable

3.4 Conclusions

Production planning and control decisions deal with the acquisition, utilization, and

allocation of resources to satisfy customer demands in the most economical way. Typical

factors that affect production planning include pricing, capacity levels, production and

inventory costs, and overtime and subcontracting costs. Past research suggests a

hierarchy of optimization models that are applicable for these distinct decision making

categories. In this chapter we developed a class of models focused on integrating these

decisions and provided effective solution methods for this class.

In the first model, the general fixed capacity dynamic production planning problem

is extended to account for pricing, temporary capacity expansion (overtime), and

subcontracting options. Although subcontracting has been considered by many

researchers, integration of these decisions with overtime opportunities has not been fully

discussed. From our perspective, overtime and subcontracting should be considered as

separate notions when modeling. Although both practices help manufacturer to increase

supply capacity, there are important differences between overtime and subcontracting.

We also included integrated capacity and pricing decisions in the model because these

two decisions are highly interrelated and strongly influence production planning.

Our models in this chapter determine a producer's optimal price, production,

inventory, subcontracting, overtime, and internal capacity levels, while accounting for

production economies of scale and capacity costs through concave cost functions. We use

polyhedral properties and dynamic programming techniques to provide polynomial-time

solution approaches for obtaining an optimal solution for this class of problems when the

internal capacity level is time-invariant. Although the models in this chapter incorporate

several operational decisions, they require piecewise linear revenue curves to obtain

polynomial-time solution procedures. In Chapter 4, we introduce more general concave

revenue functions for our procurement planning models. In addition, the case where a

single price is set for the entire horizon is also considered.

CHAPTER 4
CAPACITATED PRODUCTION PLANNING MODELS WITH PRICE SENSITIVE
DEMAND AND GENERAL CONCAVE REVENUE FUNCTIONS

Introduction

Recent research in operations planning recognizes the impacts that demand and

revenue planning measures have on operations-related costs. Measures to manage

revenues through pricing directly influence operations requirements, and the combination

of revenues and operations costs often largely determine an operation's overall

profitability. Determining pricing and demand plans without considering their impact on

operations costs can therefore lead to plans that drive high revenues, while at the same

time sacrificing potential profits due to associated operations-related costs. When a firm

wishes to maximize the profit associated with the production and delivery of a good with

price-sensitive demand (as opposed to, for example, maximizing market share), it is

important to consider both the revenue and cost factors that drive profitability. This has

led many operations researchers in recent years to consider models that integrate revenue

and operations decisions with a goal of profit maximization.

In Chapter 3, we considered integrated capacity, demand and production planning

models. In this chapter, we also address a class of combined revenue and operations

planning problems which have not been previously considered in the literature. In

particular, we consider a class of discrete-time finite-horizon operations planning models

with production capacity limits, where demand for a good is price sensitive. As opposed

to the problems discussed in Chapter 3, we only consider problem contexts in which there

are no temporary capacity expansion mechanisms, such as employing overtime hours and

subcontracting. Instead, we focus our attention on handling more general concave

revenue functions. We account for economies of scale in production of the good using a

fixed plus variable production cost structure, and also consider linear holding costs in the

amount of inventory of the good held at the end of a period. Within this class of

problems we consider the case in which price may vary dynamically by period, as well as

the case in which a firm wishes to charge a single price throughout the entire planning

horizon. To the best of our knowledge, this is the first work that combines price-sensitive

demands with general concave revenue functions, production economies of scale, and

production capacity limits in a single integrated (and tractable) model with a goal of

profit maximization. We restrict ourselves to problem contexts in which production

capacity limits are time-invariant, i.e., the planner faces the same production capacity

limit in each period of the planning horizon. This chapter contributes to the literature by

demonstrating that this problem class can be solved in polynomial time for a broad class

of general concave revenue functions, for both the dynamic and static pricing cases. (As

we will see, in the static pricing case, although the demand function in a given period can

take any general concave form, to achieve polynomial solvability we require that a

specific relationship holds between revenue functions in different periods.) The resulting

solution approaches expand the set of available tools for determining how to best match

production supply capabilities with demands. In addition, the structural properties of

optimal solutions lead to insights on how demand management decisions are interrelated

with production economics and capacity limits in complex settings involving combined

production and demand planning.

The remainder of this chapter is organized as follows. Section 4.1 deals with the

problem when prices may vary dynamically by time period. Section 4.1.1 provides the

problem definition and corresponding model formulation, while Section 4.1.2 derives a

series of properties of optimal solutions that lead to a polynomial-time solution approach

under very general conditions on the revenue functions. Section 4.2 deals with the

problem when a constant price is required over the time horizon. The constant price

problem is defined and a model is formulated in Section 4.2.1, while in the remainder of

Section 4.2 a polynomial-time solution approach is developed. Section 4.3 then

summarizes our results for this chapter.

4.1 Model and Solution Approach with Dynamic Prices

4.1.1 Problem Definition and Model Formulation

We consider a T period planning model for a single item with production capacity

equal to some positive value K in every period. This capacity may correspond to a

production capacity limit, a truckload supply capacity limit, or other supplier limit on

production quantity in a period, depending on the context. As in Chapter 3, demand in

period t (t = 1, ..., T), which we denote by Dt, is a decision variable, and we assume that

any value of demand in period t implies a corresponding unit price. The total revenue in

period t is determined by a nondecreasing concave function of demand R,(D,) defined for

Dt > 0, where the corresponding price as a function of demand equals R,(D,)/D, when Dt

> 0 and we assume that lim ,0 RW(D,)/D < o. Marginal revenue is nonincreasing in

demand, or equivalently demand is nonincreasing in price (and the rate of change of this

demand is nonincreasing in revenue; thus since R, is concave, demand is nonincreasing

in price). We assume that production in period t incurs a nonnegative fixed order cost of

St (if the amount procured is positive) plus an additional nonnegative variable cost of c,

per unit procured. A nonnegative holding cost of ht is incurred for each unit remaining in

inventory at the end of period t and we assume that shortages are not permitted.

Therefore, if the decision variable xt denotes the amount procured in period t, the amount

of inventory remaining at the end of period t equals 1x- D, and this quantity

must be nonnegative for every period t. For ease of notation we define h,, = th ,

c,, = c, + h,,,, and R,(D,) = R(D,) + h,,D,, and we will refer to the latter function as the

revenue function throughout this chapter. It is easy to see that the functions Rt are

concave and nondecreasing. Our goal is to maximize total revenue less production and

holding costs, which results in the following Dynamic Concave-Revenue Production

Planning [DCRPP] model.

[DCRPP]

Maximize ZT(R(Dt)-- S cx

Subject to: =l =l > t= 1, = ., T,

0 < xt < Kyt, t = 1, ..., T,

yt E t{, 1} t= 1, ..., T,

Dt > t = 1,..., T.

It is straightforward to verify that the above formulation is equivalent to a more

standard formulation that explicitly uses inventory variables. The objective function

maximizes revenue less production and holding costs. The first constraint set ensures

that all demand is met and that inventory is nonnegative, while the second constraint set

limits production in any period t to no more than capacity K if the binary ordering

variable yt equals 1, and to zero otherwise. For the case of piecewise-linear,

nondecreasing, and concave revenue functions, Geunes, Romeijn, and Taaffe (2006)

showed that this problem can be solved in polynomial time. Here we provide a more

general (and quite different) solution approach that applies to general concave

(nondecreasing) revenue functions.

Because we allow the revenue functions to be general concave functions, we need

to introduce some additional notation before we proceed. For all t = 1, ..., T, let p (D)

and p (D) denote the left and right derivatives of the function Rt at D, so that the set of

subgradients of the function Rt at D is given by R, (D) = [p (D), p (D)]. Technically, for

a concave function, these should be called "supergradients" since they overestimate a

concave function. For ease of presentation, however, we employ the commonly used

value p (D) can be interpreted as the marginal rate of increase in revenue as we increase

demand at D while p (D) provides the marginal rate of decrease in revenue as we

decrease demand at D. Note that p, (0) = oo, since we assume that the function Rt is

defined only for D > 0, and p ,(0) < o. We also define the limiting slope of the revenue

function as p (oo) = lim p (D) > h, > In general, the set of subgradients is a nonempty

interval in 91 u {oc}, while if R, is differentiable at D it reduces to the singleton {R,(D) .

Moreover, since Rt is concave, the subgradients are nonincreasing in D in the sense that

p, (D') < p (D) whenever D' > D > 0. Finally, it will be convenient to define the inverse

of 8R, as follows: 8R,(r) {D > 0:r e 8R,(D)}.

It is easy to see that cR,(r) is an interval in i u {oc} so that we can express it as

8Rt(r) =[-,p(r),p (r) The set &R (r) can be interpreted as an interval of demands on

which the unit revenue is constant and equal to r. Put differently, the value p (r) can be

interpreted as the marginal rate of increase in demand as we increase the unit revenue at r

while p, (r) provides the marginal rate of decrease in demand as we decrease the unit

revenue at r. Alternatively, we can interpret p (r) and P (r) as the endpoints of an

interval of demand values such that the revenue curve in period t has slope r for all

demand values between these endpoints (thus if p (r) # ,b (r) these correspond to a

linear segment with slope r of the revenue curve). Note that tR,(r)= {oc} for

0 < r < p (oo). Since Rt is concave, the sets 8R,(r) are nonincreasing in r in the sense that

(r') < p~ (r) whenever r' > r > 0. Moreover, ifRt is strictly concave then the set 8R,(r)

is a singleton for p (oo) < r < pt (0). In general, however, we assume that the number of

values of r for which p (r) < p3 (r) is Jt < J< oo, which means that the revenue function

Rt has only a finite number of linear segments. We denote these values by rt, forj = 1,

.., Jt and t = 1, ..., T, and, in addition, denote the ordered sequence of these values by

r1) <... < r()... < rM) where ,M = Jt Note that the values r(), .., thus

provide the slopes of all linear segments of the revenue curves in all periods.

4.1.2 Development of Solution Approach for DCRPP

Observe that, for any fixed vector of demands (D1, D2, ..., Dr), the resulting

problem is an equal capacity lot-sizing problem, which can be solved in polynomial time

(see Florian and Klein (1971), and van Hoesel and Wagelmans (1996)). In particular, an

optimal solution to the [DCRPP] problem exists consisting of sequences of capacity

constrained regeneration intervals. As in Chapter 3, we define a regeneration interval

(RI) as a sequence of periods s, s + 1, ..., u-1 such that Is1 = 1,u-1 = 0 and It > 0 for t = s,

..., u-2, where It is the inventory remaining a the end of period t, and is denoted by (s, u)

(with u > s). A capacity constrained RI is an RI such that the production quantity in

every period within the RI except at most one is equal to zero or the capacity limit K. We

will refer to a period in which production is neither at zero nor at capacity as a factional

production period and to a period in which xt = K as afullproduction period. These

properties and definitions will play an important role in the development of solution

methods for the [DCRPP]. Our solution strategy is to find an optimal solution, i.e., a

maximum profit solution with at most one fractional production period, for each RI (in

the remainder, we therefore consider only capacity constrained RIs). If we can efficiently

solve the problem for each RI we can, similarly to Florian and Klein (1971), apply a

shortest path approach for solving the overall problem.

4.1.2.1 Properties of optimal RI demand vectors

We will first focus on establishing properties of optimal solutions for a given RI

and, in particular, on properties of optimal demand sequences for an RI. The following

proposition provides a condition that the subgradients at optimal demand values in an RI

must satisfy.

Proposition 4.1. Any set of optimal demand values for RI (s, u) satisfies

u-l
f aR(D,) 0 or, equivalently,
t s

max p (D,)< min p,(D,). (4.1)
t=s, ,u-\ t=s, ,u-l

Proof. Consider a feasible set of demand values Dr, t = s, ..., u-l, for RI (s, u) for
which condition (4.1) does not hold. Then there are two periods and k in the RI
with s p- (Dk) (and therefore Dk > 0 since Pk (0) = oo).

Now suppose that we increase the demand in period by e> 0 and decrease the
demand in period k by the same amount. Since we are within an RI we can do this
without changing any procurement quantities and, if E is chosen small enough,
without causing any inventory levels to become equal to zero. The corresponding
change in the objective function value equals
(R,(D( + ) R (D))-(R,(D,) R,(D, )). Noting that

m (R (DJ +E) -RJ(D)) (Rk(Dk) Rk(Dk ))
hm = D=p,(D )-p,(D,)>O

yields that the solution can be strictly improved and can therefore not be optimal.

Proposition 4.1 indicates that if we have an optimal demand value for one period in

an RI then the candidate set of demands that must be considered for the remaining

periods in the RI can be substantially reduced.

Corollary 4.1. An optimal solution for the RI (s, u) satisfies D, e OR,(r) for t = s,
...,u-1 and for some ~ > max p (oC).
t=s, ,u-1

A somewhat stronger result can be obtained if we consider the important special

case where all revenue functions are strictly concave.

Corollary 4.2. When all revenue functions are strictly concave, there exists some
r > max p (oo) such that an optimal solution for the RI (s, u) is given by
t=s, ,u1-

D,= 0t (-) = tA (-) if O otherwise.

Corollary 4.2 indicates that if we have identical strictly concave revenue curves in

every period, then by the definition of Rt(Dt), the optimal value of marginal revenue for

periods rl and r2 within the same RI (assuming without loss of generality that Tl < r2)

will differ by the cost to hold a unit of inventory from period T1 to period z2 (which

implies the optimal prices will therefore vary within an RI, even if costs are time-

invariant, assuming non-zero holding costs). For periods in different RIs, however, we

cannot draw such conclusions on the relationship between optimal prices. Thus, the

structure of optimal pricing decisions is intimately related to the structure of the optimal

procurement plan.

Period Period k
R,(D) Revenue Curve Rk(Dk Revenue Curve

D, Dk
aR,(D) 8aRk(Dk)

4 4db Dj dka dkb Dk

Figure 4-1. Candidate subgradient values and corresponding candidate demand values

The results of Proposition 4.1 and Corollaries 4.1 and 4.2 are illustrated in Figure

4.1: the horizontal lines at T1 and r2 represent candidate subgradient values and the

corresponding candidate demand values associated with the revenue curves in two

periods (i and k) within an RI. In the figure, candidate subgradient value f1 implies

unique values of demand in periods and k (djb and dkb), while candidate subgradient

value r2 implies a unique value of demand in period (dja) but a range of values in period

k ([0, dka]). This motivates the development of an algorithm for finding an optimal

solution for each RI by searching among candidate subgradient values. In particular, we

will show that we only need to consider a relatively small number of such values. In the

next two sections we will study RI solutions with one fractional procurement period and

RI solutions with no fractional procurement period, respectively.

4.1.2.2 Characterizing optimal demand in RIs containing one fractional
procurement period

Recall that any RI contains at most one fractional procurement period. This section

discusses how to find the best RI solution that contains exactly one fractional

procurement period by characterizing all candidate subgradient values and corresponding

candidate sets of demand values for such solutions that satisfy Proposition 4.1.

The following proposition shows that we only need to consider a single candidate

subgradient value for an RI and corresponding candidate demand values if we fix a

particular period within that RI to be the fractional procurement period. Note that a set of

demand vectors for an RI can of course only be a candidate optimal solution if it can be

feasibly procured. Therefore, for RI (s, u) we only need to consider demand values Dr, t

= s, ...,u-1 that satisfy the condition:

s D, <(r- s + 1)K for r= s,...,u 1. (4.2)

In the following we will not explicitly verify this condition. However, any set of

candidate demand values that violates condition (4.2) can of course be eliminated from

further consideration.

Proposition 4. 2. If period r(with s < r< u) is the fractional procurement period in
a solution for RI (s, u), i.e., 0 < x,< K, then we only need to consider sets of
candidate demand values Dr, t = s, ...,u 1 that satisfy

p+ (D,) < p, (D,)

or, equivalently, D, e 8R,(cr) for t = s, ..., u-1.

Proof. Suppose that we have a solution for which this condition does not hold.
We consider two cases:

p, (D,) > c,, for some t = s, ..., u 1. Then, since 0 < x, < K, we can increase both
x, and Dt by some small amount E > 0. The resulting increase in objective function
value is equal to R,(D, +s)-R,(D,)- sc,,. Since

SR(D, + s) R,(D,) sc c
lim = P (D) c, > 0

the given solution can be strictly improved so that it cannot be optimal.

p,(D,) 0 so that
we can decrease both x, and Dt by some small amount e> 0. The resulting increase
in objective function value is equal to sc ,, -(R,(D,) R,(D )). Since

E c,, (R,(D,) R,(D, E))
lim = =c,-pt (DC) )>0

the given solution can be strictly improved so that it cannot be optimal. 0

Proposition 4.2 says that if period ris a fractional procurement period in an RI,

then cT must lie in the set of subgradients of the revenue curve for each period in the RI.

When the revenue function is strictly concave, then we can interpret Proposition 4.2 as

requiring that marginal cost equal marginal revenue. The appropriate marginal cost term,

however, is determined by the fractional procurement period. Proposition 4.2 also shows

that if we know that period ris a fractional procurement period then the value c,T allows

us to uniquely determine demand values satisfying the property for periods in the RI

unless C,T corresponds to the slope of a linear segment of the revenue curve. We will

next show that we may, in the presence of a fractional procurement period, restrict our

attention to only the left endpoints of these linear segments.

Proposition 4.3. If period s < z< u is the fractional procurement period in a
solution for RI (s, u), i.e., 0 < x,< K, then we only need to consider the set of
candidate demand values that is given by D, = Ap (cr,), t = s, ..., u-1.

Proof. Suppose that we have a solution in which period ris the fractional
procurement period that satisfies the properties in Proposition 4.2 and, moreover,

D, > (cr) for some t = s, ..., u-1. Now consider a change of Dr by E and a
corresponding change in x,by E without changing the procurement quantities in
any other period. If E < D, (c ,) the corresponding change in objective
function value is by definition equal to 0 as long as the solution remains feasible.
The result now follows since either (i) the solution with e = D, (c ,) is indeed
feasible, or (ii) for a smaller value of E one of the inventory levels becomes 0
(which means that we obtain a solution with the same objective function value as
the current one but with periods t and rin different RIs), or (iii) the procurement
level in period becomes K (which means that period ris no longer a fractional
procurement period). *

Proposition 4.3 says that we need not consider RIs containing both a fractional

procurement period and a period whose demand level is in the interior of a linear segment

of its revenue function. In the special case where the revenue curve in a period, say

period, in the RI is piecewise linear and concave, Proposition 4.3 implies that if a

fractional procurement period exists in this RI then without loss of optimality the demand

level in period is at a breakpoint of its piecewise-linear revenue curve. This result is

consistent with the results presented in Geunes, Romeijn, and Taaffe (2006).

4.1.2.3 Characterizing optimal demands in RIs containing no fractional
procurement period

Recall that it is possible for an optimal RI solution to contain no fractional

procurement periods, which means that all periods in which procurement occurs are full

procurement periods. We will next discuss how to determine candidate RI solutions

without a fractional procurement period. Consider such a solution for some RI (s, u).

Then the total procurement for the RI, and therefore the total demand satisfied in the RI,

must be an integer multiple of the procurement capacity K, i.e., it must be equal tofK, for

some= 0, 1, ..., u -s. Given a value off Proposition 4.1 says that the demands in the

RI and the corresponding subgradient value r must satisfy the following set of equations:

p,(D,)< r< p,(D,) fort =s,...,u -1

Dt=s = fK

or, equivalently, by using Corollary 4.1:

S(r)

ZI D= fK (4.3)

It is straightforward to show that when the optimal RI solution contains no

fractional procurement periods, the value ofr that satisfies (4.3) must be at least as great

as the maximum value of c,r among all periods rin the RI such that procurement is at

capacity. When strict inequality holds, Equation (4.3) illustrates how capacity serves as a

bottleneck for gaining additional profit. For example, given an optimal solution

satisfying (4.3), if the revenue curves are strictly concave, then the marginal increase in

revenue available by increasing capacity exceeds the associated marginal supply cost,

implying a potential for increased profit (we must, however, account for the associated

cost increase as a result of the capacity increase).

Given a value off { 1, ..., s u}, a demand vector for an RI satisfying (4.3)

provides a candidate for an optimal demand vector for RI (s, u). Our approach to finding

a solution for all values off and a given RI is based on the monotonicity of the intervals

cR!(r) [, (r),p, (r)] Let mi, .., mM(s,u) denote the index set of the values r(m)

(previously defined in Section 4.2.1) corresponding to the periods in RI (s, u) and let mo =

0 and m(,,,u)+ = M+1 with corresponding values r') = p, (oo) and r(M) = p (0) (recall

that each value of r(M corresponds to the slope of a linear segment of a revenue curve in

some period within the RI, and we assume that there are M(s, u) such values in the RI (s,

u)). Then the problem decomposes into a sequence of subproblems that alternate

between two types.

The first type of subproblem considers values r('') < r < r(m'') for some = 0, ..

M(s, u), where we know that for such intervals ofr we have p (r)= (r) for any value

ofr on the interval, and this is a strictly decreasing function ofr. A candidate set of

demands for a given andj can then be found by using binary search to find a root of the

equation

Zu>, (r) = fC

over the interval r(mj) < r
functions (r) are decreasing in r on this interval, a root exists if and only if

l t=s t i-y
The second type of subproblem considers (4.3) for a fixed value r = r m) for some

j=0, ..., M(s, u) + 1, where we know that p, (r m))
s, ..., u 1 (except potentially forj = 0 andj = M(s, u) + 1). Existence of a solution for a

givenfandj can easily be established by verifying whether

U p r,)) fK -< s" t Since any set of demands that solves the system has

the same objective function value if it is feasible, we will then simply start by initializing

D, =P 3r(m') for t = s, ..., u-1. Then, we decrease the demands sequentially in periods t

= s, ..., u- such that /;(r(mi))<,jr(mi)) until such a solution is found as follows.

Starting with t = s, we check whether p, r(m)r -(r(m,) exceeds the excess demand. If

not, we reduce the demand in period t from (r(m j) to p t r\mj) and proceed with

period t + 1. If so, we reduce the demand in period t by the excess demand and stop.

We can summarize the algorithm for the case of no fractional (NF) procurement

periods as follows:

Algorithm NF

Step 1. Setj = M(s, u) andf= 1.

Step 2.If ts r(r p tij)) the solution, and setf =f+. Iff> 0 repeat Step 1, otherwise stop.

Step 3.If r,, PK r V P t(r ), solve a subproblem of the second type,
store the solution, and setf =f+l. Iff> 0 repeat Step 2, otherwise stop.

Step 4. Setj =j-1. If f< u-s andj > 0 go to Step 1, otherwise stop.

It is easy to verify that, if the revenue functions are piecewise linear and concave,

all candidate solutions can be found by ordering and considering the slopes of all

segments in the RI in decreasing order to fill the capacity for each value off. This is

precisely the approach presented in Geunes, Romeijn, and Taaffe (2006).

It is tempting to conclude, based on our analysis so far, that the optimal demand

values (and hence prices) do not depend on the values of the fixed order costs. That is, in

the previous section, we concluded that the candidate optimal demand values for an RI

depend on the relationship between the variable cost in the fractional procurement period

and the revenue functions. Similarly, in this section we showed that candidate demand

values are determined based on the properties of the revenue functions and the capacity

limits (see Equation (4.3)). While these factors do determine candidates for optimal

demand levels (and prices), obtaining an optimal RI solution requires solving a shortest

path subproblem with arc costs that explicitly account for order costs. Therefore, while

the complexity of the relationship between optimal demand levels and fixed order costs

does not permit closed-form expressions that characterize this relationship, it would be

incorrect to conclude that the optimal demand and price values are independent of these

costs.

4.1.2.4 Complexity of overall solution approach

This section characterizes the complexity of an overall solution approach for the

[DCRPP] problem, based on the results developed in the preceding sections. Our

approach is based on solving a shortest path problem on a graph that contains T+ 1 nodes

(1, 2, ..., T+ 1), where a directed arc exists connecting each node to all higher numbered

nodes. The cost of an arc (t, t') equals the optimal RI (t, t') solution value (note that arc

"cost" values are actually net contribution to profit values, and we therefore solve an

acyclic longest path problem after determining all optimal RI solutions and label arcs

accordingly). Since this acyclic longest path problem can be solved in 0(72) time

(Lawler 1976), the bulk of the solution effort lies in determining the optimal arc cost

values by determining an optimal RI solution for each of the 0(72) possible RIs.

Our approach is to find a collection of potentially optimal demand vectors for each

RI using the results of the previous sections. Given each of these candidate demand

vectors we can use a dynamic programming approach to determine the best RI solution

for that candidate demand vector in 0(72) time (see Florian and Klein 1971) if one exists.

(Note that while this dynamic programming approach will correctly identify whether a

candidate demand vector is indeed feasible, a slight computational advantage can be

obtained by first verifying feasibility in 0(T) time using equation (4.2). However, this

will not influence the worst-case running time of the overall algorithm.) For a given RI,

we will determine candidate demand vectors given that exactly one fractional

procurement period exists in the RI, as well as candidate demand vectors given that no

fractional procurement period exists in the RI. The profit of the best solution among the

corresponding candidate solutions then serves as the arc cost in the longest path problem.

The best solution for an RI that contains exactly one fractional procurement period

can be found by considering each period in the RI to be the fractional procurement period

and finding a corresponding candidate set of demand values using Proposition 4.3, which

means that there are 0(T) such candidate demand vectors for each RI. Each of these

demand vectors can be found by evaluating the function bA for each period t in the RI. If

such a function evaluation takes O(R) time (for example, ifRt is a piecewise-linear

function with O(R) segments), finding a single candidate demand vector corresponding to

a given fractional procurement period takes O(RT) time, and finding all T candidate

demand vectors therefore takes O(RT7) time. For each of the T candidate demand

vectors, we then solve the O(T2) RI subproblem (see Florian and Klein 1971) to

determine the optimal procurement plan associated with the candidate demand vector.

Finding the best RI solution containing a single fractional procurement period therefore

takes 0(7 + RT2) time.

We next consider the computational complexity of finding an optimal RI solution

when there are no fractional procurement periods in the RI. It is easy to see that the

number of candidate demand vectors found by our algorithm is 0(T), because we

generate at most one demand vector for each value off The feasibility test in Steps 1

and 2 of Algorithm NF takes 0(T) time and needs to be performed O(JT) times. Finding

a solution in Step 1 takes O(log A) binary search iterations, where

A = maxt, ,r /p (0) mint1, ,r Pt (o), each taking O(RT) time, while finding a solution in

Step 2 for a given value off takes 0(T) time. The total time required for Algorithm NF

for a given RI is therefore O((J + 1 + RlogA)T2).

Since there are 0(f2) RIs, the [DCRPP] problem can be solved in

O((J + 1 + RlogA)T4 +T5) in the worst case. Observe that if each period's revenue

curve contains only one strictly concave segment then Step 2 of Algorithm NF (employed

for RI solutions without a fractional procurement period) will never be performed and the

running time becomes O((Rlog A)T4 + T). On the other hand, when the revenue curves

are all piecewise-linear and concave functions then each of the functions p, can be

evaluated in O(log J) time and Step 1 of Algorithm NF will never be performed. In that

case the running time becomes O(JT4 +T ) which improves upon the results presented

in Geunes, Romeijn, and Taaffe (2005) and Geunes, Merzifonluoglu, Romeijn, and

Taaffe (2006). Note also that the initial sorting of the linear slopes takes O(JTlog(JT))

time in preprocessing and should be considered in the overall computational complexity

of this problem.

4.1.2.5 Refining the solution approach

While the previous sections have provided properties that lead to an effective

solution approach, in this section we develop an additional property that allows us to

reduce the time required to determine an optimal procurement plan for an RI for a given

sequence of demands. In particular, Proposition 4.4 next provides a condition under

which it is guaranteed that no procurement will take place in a given period. This result

can be used to eliminate candidate demand sequences from consideration and speed up

the dynamic programming algorithm to find an RI solution for a given vector of demands

by fixing the values of some procurement quantities.

Proposition 4.4. Consider demand values Dt, t = s, ..., u-1, for RI (s, u). Then if
ct, > min ,_. pr (DZ) we can assume without loss of optimality that x, = 0.

Proof. Suppose that we have a solution for RI (s, u) in which c,, > pr(D,) for
some period s < r< u and x, > 0. This implies that D,> 0 so that we can decrease
both xt and Dby E, where E > 0 and sufficiently small, without changing any of the
other demands and procurement quantities. The corresponding increase in objective
function value is equal to c,, (R,(D,) R,(D, E)). Dividing by E and taking the
limit as E goes to zero we obtain that the rate of increase converges to
c,, p (D) > 0. This means that we can improve the solution and it can therefore
not be optimal. Thus, without loss of optimality we have xt = 0. u

This proposition says that, for a given candidate demand vector found using one of

the approaches in Sections 4.1.2.2 and 4.1.2.3, we may be able to eliminate certain

procurement periods from consideration, which means that a smaller dynamic

programming problem needs to be solved to find the profit corresponding to the demand

vector. Moreover, we may even be able to determine in advance that no feasible and

potentially optimal solution exists for this demand sequence by verifying a strengthening

of feasibility condition (4.2). In particular, for RI (s, u) we only need to consider demand

values Dt, t = s, ..., u-1 that satisfy the condition:

_D,

We illustrate the usefulness of Proposition 4.4 in Figure 4.2. In the figure, we

graph cR,(D,) = R,(D,) as a function of Dr for three periods in a potential RI with

differentiable revenue curves, and show two consecutive values of ct,, on the vertical

axis (Cm, and Cm+l., correspond to consecutive values of c,, when these values are sorted

in non-decreasing order). Observe that in Figure 4.2, for periods and k, the revenue

80

functions are strictly concave, while period I contains a linear segment indicated by the

flat spot in the graph of Ri(D,). Corollary 4.2 indicates that any candidate sequence of

optimal demands for an RI must correspond to equal values of R,(D,) across all periods

in the RI. Such solutions can be visualized in the figure by drawing a horizontal line

across the graphs corresponding to different periods. Consider the dashed horizontal line

in the figure where R'(D) = r for all periods in the RI and suppose that the implied

values of d, shown on the horizontal axes sum to a multiple of K (recall from Proposition

4.2 that an optimal RI solution containing a fractional procurement period will

correspond to the case in which the value of r equals some ct, value; we need only

consider values of r falling between c,T values if a solution to (4.3) exists, i.e., if a

feasible set of associated demands add to a multiple of K). Proposition 4.4 now tells us

that for any periods in the regeneration interval such that cT < r, we either procure 0 or

K, and that all periods in the regeneration interval such that c,T > r will be zero

procurement periods.

R(D) R'(Dk ) RI(D, )

II I
c r............ ... ............ t ..... ..... ............ .... .......

I I I I
I I I
III
I I------ I------- I--- ---
dj Dj dk Dk di Di

Figure 4-2. Illustration of Proposition 4.4.

Finally, we note that, for any candidate demand vector that is generated using the

approach in Section 4.1.2.2, the fractional procurement period is known by construction.

This information can be used to reduce the size of the dynamic programming formulation

that is used to find the corresponding RI solution.

4.2 Model and Solution Approach with a Constant Price

4.2.1 Model Description

We next consider the situation in which a management policy or other constraint

requires us to set a constant price for the good over the entire planning horizon. Suppose

that, given a price, we face a vector of demands D(p) [Di(p), D2(p), ..., Dr(p)].

Using the same notation as in Section 4.1.1, we now wish to solve the following Static

Concave-Revenue Procurement Planning [SCRPP] problem:

[SCRPP]

Maximize ZTl(pD,(p) Sy- cTx,)

Subject to: Zt X= > Dt (p), t= 1,..., T,

0 < xt < Kyt, t = 1, ..., T,

S {0, 1} t= 1, ...,T,

p>0.

We can write this problem more concisely as an optimization problem in a single

decision variable, p, as follows:

Maximize pyTID(p)-F(D(p))

Subject to: p > 0

where F(D(p)) denotes the cost of the optimal procurement plan associated with

demand vector D(p). For a given vector D(p), the cost F(D(p)) is in fact the optimal

value of the following equal capacity lot sizing problem, which can be determined in

O(7') time (see van Hoesel and Wagelmans 1996):

Minimize ST(Sly, +c,rxI)

Subject to: =x > D(p), t= 1,..., T,

SE t0, 1 t= 1,..., T,

0 < xt < Kyt, t= 1, ..., T.

As in Kunreuther and Schrage (1973), Gilbert (1999), and van den Heuvel and

Wagelmans (2006), we assume in the remainder of this section that the demand in period

t is given by the function Dr(p) = at + Ptd(p), where at, it > 0 and d(p) is a nonincreasing

and left-continuous function ofp. The function d(p) is period independent and is called

the demand effect (see Kunreuther and Schrage 1973 and van den Heuvel and

Wagelmans 2006). Note that the optimal lot-sizing cost at a given pricep then only

depends on the vector of demands D(p) through the scalar d(p). With a slight abuse of

notation we will denote the corresponding cost by F(d(p)) and view the function F as a

function of a scalar variable. Letting a = Tla, and f = lT,, the [SCRPP] becomes

Maximize ap + ppd(p) F(d(p))

Subject to: p > 0.

However, it will be more convenient to formulate the [SCRPP] as a function of the

demand effect d rather than of the price p. The generalized inverse of the function d

expresses the price as a function of demand:

p(d) = sup{p:d(p) >d} .

This then finally yields the formulation of the [SCRPP] that we will use:

[SCRPP]

Maximize ap(d) + pdp(d) F(d)

Subject to: d> 0.

It will sometimes be convenient to write the demand in period t as a function of the

demand effect d as follows: D,(d) = at + /id.

Van den Heuvel and Wagelmans (2006) show that, if F is a piecewise-linear

function (as is the case in the absence of capacities), a problem of the form [SCRPP] can

be solved by solving a number of (i) lot-sizing problems (each of which determines the

value of F(d) for a given demand effect d) and (ii) problems of the form [SCRPP] with F

replaced by a linear function (each of which determines the optimal price for a given

procurement plan); the number of such problems that must be solved is of the order of the

number of breakpoints ofF. Then, following Kunreuther and Schrage (1973) by

assuming that problems (ii) can be solved efficiently, van den Heuvel and Wagelmans

(2006) derive a polynomial time algorithm for the uncapacitated [SCRPP] by showing

that the number of breakpoints of F is O(T2).

In this section, we will develop our solution procedure for the capacitated [SCRPP]

by proceeding as follows. We first show, in Section 4.2.2, that the function F is

piecewise linear and concave within each interval of a contiguous sequence of demand

intervals. In Section 4.2.3 we then show that the number of such intervals is polynomial

in the number of time periods T. Finally, in Section 4.2.4, we show that the number of

breakpoints of F is polynomial in the number of time periods Tfor each of these

intervals. We then combine these results in Section 4.2.5 to obtain an effective algorithm

for solving the capacitated [SCRPP].

4.2.2 Linearity of Cost in Demand Effect

It is immediate that, as in Section 4.1, an optimal procurement plan solution to the

capacitated [SCRPP] exists that consists of a sequence of RIs with an associated specific

plan (SP) for each. In the remainder, we will denote an RI with an associated SP (RI-SP)

by (s, u, v, F), where (s, u) (with 1 < s < u < T) denotes the RI, v denotes the fractional

(i.e., unconstrained) procurement period s < v < u, and F c- {s, ..., u-1 }\{v} denotes the

set of periods in which procurement is equal to the capacity K (so that IFI is the number

of full procurement periods within the RI-SP); procurement in the remaining periods is

equal to zero:

0 if tvFu{v)
x= D, (d)-FK if t=v (4.5)
K if teF

Clearly, for a given demand effect d, an RI-SP is only valid if the cumulative

quantity procured exceeds the cumulative demand up to each period in the RI and

procurement in the fractional procurement period in each RI is in [0, K]. A sequence of

consecutive RI-SPs yields a candidate (i.e., potentially optimal) procurement plan. In this

section, we will demonstrate that the cost associated with a given procurement plan

(consisting of a sequence of RI-SPs) is linear in the demand effect d on the interval where

the sequence of RI-SPs is valid.

Using equation (4.5) we can write the total cost associated with the RI-SP as

tF(St +c,,K)+S,+cVr (:D t (d)- FK)

=Z ( +F (ct-,, cr)K)+S +cryzY D (d)

= t,, St+(c-,,T- )K)+S +c, YU(a, +7,d)

(= F (S, +(c, c,)K +S,)+c, b'at)+(cv, i,) d

which shows that the cost within the RI-SP is linear in d where the RI-SP is valid. This

immediately implies that the cost of a procurement plan consisting of a sequence RI-SP1,

.., RI-SP, of RI-SPs is linear in d as well where all RI-SPs in the plan are valid.

We next derive the set of values of d for which a given RI-SP (s, u, v, F) is valid.

For convenience, let F't F r{s, ..., t} denote the set of full procurement periods up to

and including period t in the RI. Firstly, we require that the cumulative procurement

exceeds cumulative demand for each period in the RI. Distinguishing between periods

before the fractional procurement period and later ones, we see that we must have

F" K> D,-(d) fort=s,...,v-land

Ft K+(uD, (d)-FK) > D,(d) fort=v,...,u-l

Fst K-I a,
or, equivalently, d < t -" for t =s,...,v -1 and

F\Fst K 1 a
d >-_I fort =v,...,u -1.
r=t+1r'

Furthermore, the quantity procured in the fractional procurement period should be

-i F K- l- a, ( F +l)K-J K- a,
feasible, i.e., O < DD (d)- F K =S' t S'= t

Combining these, we obtain that RI-SP (s, u, v, F) is valid if

dL(s,u,v,F) <:d

S F\F \' K-+l 1 la, F K- ~1Ul a,
dL(s,u,v,F)=max max (4.6)
v, ,u-1 =t+ t =t+18 t

and

m F'' K- ( F +1)K- a,
d(s,u,v,F)=min min t -K (4.7)
t S -1 8

Clearly, if dL(s,u,v,F) > dU(s,u,v,F) no demand effect exists for which RI-SP (s, u,

v, F) is valid. Also note that in the absence of capacities we may restrict ourselves to v =

s and F = 0 so that dL(,u,v,F) = dL(s,u,s,0)= 0 and dU(s,u,r,F)= dU(s,u,s,0)= o, i.e.,

all (relevant) RI-SPs are valid for all d > 0.

Now consider a complete procurement plan P consisting of a sequence of n

consecutive RI-SPs (s,, s,j+, v,, F), with starting periods si = 1, S2, ..., Sn, and with Sn+1 = T

+ 1, so that thejth RI starts at period s, and ends at period s,j+ 1. The cost of this

procurement plan is linear in das long as all RI-SPs are valid, i.e., as long as

d -- maxdL(sjs ,vF) d d< min dL(sj, ,VjF). (4.8)
j=1, ,n j=, n

If dL > dp no price exists for which the procurement plan is valid.

4.2.3 Characterizing the Structure of the Optimal Cost Function

To explore the overall structure of the function F, observe that its value at d is

equal to the minimum of the costs of all procurement plans that are valid at d. In the

absence of procurement capacities, all procurement plans with fractional procurement in

the first period and no full capacity procurement periods are valid and, moreover, we may

without loss of optimality restrict ourselves to these procurement plans. By the results of

Section 4.2.2 it then immediately follows that the function F is the lower envelope of a

set of linear functions of d and is therefore a piecewise-linear and concave function of d.

However, in the presence of a finite procurement capacity K in each period, the function

F instead is the lower envelope of a set of functions that is linear in d on some interval

and infinite elsewhere (namely where the corresponding procurement plan is not valid).

This means that the function F is still piecewise linear but not necessarily everywhere

concave. Instead, it is piecewise linear and concave on each of a sequence of consecutive

intervals for d, and within each of these intervals we only change the structure of the

optimal procurement plan (as d changes) because it is economically attractive. The

endpoints of these intervals correspond to necessary changes in the structure of the

procurement plan not because it is economically attractive, but rather because we have

reached a point where we cannot maintain feasibility using the current procurement plan

if d is increased. In other words, procurement in the fractional procurement period has

reached K and any further increase in d requires a change in the structure of the

procurement plan. We will next characterize the endpoints of the intervals of demand

effect values don which F is concave.

From Equation (4.8) it immediately follows that the candidate endpoints are given

by the values dL(s,u,v,F) and dU(s,u,v,F) for all RI-SPs (s, u, v, F). Examining

equations (4.6) and (4.7) we may then observe that the distinct values of these endpoints

are given by

mK a
d"~= =t i form= ,...,r-t+l;r =t+,...,T;t=,...,T .

This means that there are 0(f8) unique values of d, to consider. By construction,

on any interval for d that is between two consecutive values d, the optimal cost function

F is piecewise linear and concave. So sorting the values d, in nondecreasing order

yields a contiguous sequence of 0(7') intervals covering all d > 0 on which F is

piecewise linear and concave.

4.2.4 Number of Breakpoints of F

We next study the number of breakpoints of the function F. To this end, define a

procurement subplan for a set of t consecutive periods to consist of a set of consecutive

RI-SPs for these periods, and let P(t) be the total number of valid procurement subplans

for a set of t consecutive periods. A procurement (sub)plan of length Tthen serves as a

candidate solution for the entire problem and P(T) is the total number of procurement

plans or candidate solutions that we need to consider. Limiting ourselves to an interval

for don which F is piecewise linear and concave, the number of breakpoints of the

function F is no more than P(T) 1 (because each of the P(T) candidate solutions is a

linear function of the demand effect d on some interval).

Now consider a particular RI ofn periods in length with m periods preceding this

RI and T- n m periods following this RI, and let PmnT = P(m) x P(T m n) denote

the number of procurement subplans for all periods except the n-period RI (see Figure

4.3). Observe that for any given RI-SP for the n-period RI, there are PmnT total

procurement plans (or candidate solutions). A total of n2"1 RI-SPs exist for the n-period

RI; thus there are n2"nlPm,, total procurement plans given the selected n-period RI. This

provides an a priori upper bound of n2" -PmnT 1 breakpoints associated with the given n-

period RI, which is (n2"1 1)Pmn greater than the PmnT 1 breakpoints that may be

associated with different subplans in the first m and last T- n m periods. We next