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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 Copyright 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 emails, 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. TABLE OF CONTENTS 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 31 Complexity results with M subcontractors (integral a) ........ ......................59 32 Complexity results with M subcontractors (general a)...................................59 51 Demand time windows for the example problem shown in Figure 5.1 ...............101 52 Demand (d,) and setup cost values (St) for example problem 1............................112 53 P problem 1Iteration 0. ............................................................................. .... .. 113 54 Problem 1Iteration 1 .................. ................................ ..... ................ 114 55 Problem 1Iteration 2. ............................................ .. ...... ................ 114 56 Demand and setup cost values for example problem 2 .............. ... ...............115 57 P problem 2Iteration 0. ............................................................................. .... .. 115 58 Problem 2Iteration 1 .................. ........................ .. .. ..... .. .......... .. 116 59 Order rejection rates under different cost parameter value settings.......................124 61 Problem sizes in the numerical study ............................................ ...............158 62 Computation times for various problem settings ................................................159 63 Performance evaluation table for heuristic algorithm ................. ................161 LIST OF FIGURES Figure pge 31 Piecewise linear concave revenue function........................ ...................30 32 Network representation of the [CPPP L] ................... .................................... 32 41 Candidate subgradient values and corresponding candidate demand values...........69 42 Illustration of Proposition 4.4 ........... ...... ......... .................. 80 43 An arbitrarily selected nperiod regeneration interval. ...........................................89 51 Fixedcharge network flow representation of the [DFFP] problem ..................101 52 Structure of longest path graph. .................................................. .....................105 53 Impact of dimensions of flexibility on profit. ................................................ 123 54 Profit levels as a percentage of the maximum profitability (FLEX(D, T))............125 61 Supply chain network for 5 facilities and 10 downstream demand points ...........128 62 Description of problem parameters on an example distribution network. .............133 63 Comparison of computation times for various problem settings ......................... 160 64 Performance evaluation for heuristic algorithm .....................................................161 65 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 shortterm 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 polynomialtime solution approaches for obtaining optimal solutions for some of the problems that are not NP Hard. When the problem is NPHard, 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 shortterm 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 advantageous. Therefore, in addition to internal capacity management decisions, our models can be used to deal with traditional "makeorbuy" questions. 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 twostage 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 pricedependent 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 (nonidentical) 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 nondecreasing piecewiselinear 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 polynomialtime 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 discretetime, finitehorizon 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 timeinvariant 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 programmingbased 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 demandtiming 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 customerspecified 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 deliverytiming 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 NPHard (Appendix B), which implies that we cannot reasonably seek an efficient (polynomialtime) 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 NPHard. 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 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 twoechelon 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 largescale 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 pricedependent 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 WagnerWhitin (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 timeinvariant 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 continuoustime model with marketspecific 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, SimchiLevi, and Swann (2005) considered a related dynamicpricing 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 timevarying capacities, but do not consider the equalcapacity case. Recently, Deng and Yano (2006) studied an integrated pricing and production planning problem under timevarying capacities, leading to a solution approach with an exponential running time. They also considered the timeinvariant capacity case, and showed that it is polynomially solvable. Our solution methodology in Chapter 4 differs essentially from theirs with an improved running time. In addition, in Chapter 4 we address interesting insights about the 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 timevarying price case, Chapter 4 generalizes the past methods for constantpriced goods to account for timeinvariant capacities. Kunreuther and Schrage (1973) first considered the problem of setting a single price over an entire planning horizon with an uncapacitated lotsizingbased cost structure (with fixedcharge procurement cost structures and linear holding costs), and provided a heuristic solution approach for this problem. Gilbert (1999) provided a polynomialtime 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 timevarying costs can be solved in polynomial time. Our constantpricing 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 constantpriced goods and economies of scale in procurement (note that Gilbert 2000, also considered a periodic multiproduct planning problem with constantpriced 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 timevarying and constantpriced goods cases) can lead to insights for developing solution methods for more general classes of profit maximization problems with general concave revenue functions and fixedcharge 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 lotsizing 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 demandtiming 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 periodspecific 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. Calosso, Cantamessa, Vu, and Villa (2003) modeled a businesstobusiness electronic negotiation process in a maketoorder 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 simulationbased 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 buildup to manage shortages (see van Mieghem 2003). When shortages are allowed, one should account for the impacts of backlogging or lost sales with related demandshortage 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 nonstationary 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 gametheoretic models related to subcontracting. In addition, Bertrand and Sridharan (2001) study heuristic decision rules for subcontracting in a maketoorder 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 timevarying 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 maketostock 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 singleitem production system in a maketoorder 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 capacityrelated 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 models that address critical tradeoffs between capacity/production costs and increased 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 shortterm 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 longterm internal production capacity, as well as the capacity usage strategy when different forms of short term capacity adjustments are available. 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 (timeinvariant) 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 shortterm capacity adjustments through overtime and subcontracting options. These shortterm 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 shortterm flexible capacity sources. As mechanisms for shortterm 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 piecewiseconcave 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, inhouse production capacity (regular plus overtime) may not be sufficient (as in the call center case), and second, inhouse 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 inhouse 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, they can be used to address traditional "makeorbuy" questions. Ultimately, however, 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 pricedependent demands, economies of scale in production costs, and subcontracting and capacityconstrained 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 orderdependent net revenues (after subtracting any variable fulfillment costs). Beyond applications the model might have in shortterm 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 "bestcase" 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 piecewiselinear 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 polynomialtime solutions. Relative to most of the classical planning models, the worstcase 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 largescale 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 pricedependent 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 shortterm 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 regulartime capacity is twice overtime capacity and a= 2. However, in general we will allow a to take any positive value, and therefore regulartime 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 onetoone 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 regulartime 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 piecewiselinear 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 31. 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 = d1,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 regulartime 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 regulartime 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 worstcase complexity of the algorithm provided by Geunes et al. (2006). The [CPPPpL] can be modeled as a concavecost 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 32. 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 segmentcapacity 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 pvalues (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 pvalues. 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= t1, ..., 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 (t1,0, 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 (t1, 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 (= t1, ..., 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, (faK) 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 (t1, 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 pseudopolynomial 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 worstcase complexity for determining an optimal RI cost of O(Jmax73), and a corresponding worstcase 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 fixedpluslinear 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 polynomialtime 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, regulartime 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 extremepoint 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,...,R1 these provide an additional R1 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 Kand therefore at the optimal value of K in [CPPPpL(K)]when ties exist among multiple pvalues, 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 pvalues.) 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 fullcapacity 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 piecewiselinear, 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 worstcase 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 nonidentical subcontractors with or without timeinvariant 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 [CPPPUS] is formulated as follows: [CPPPUS] 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 [CPPPUS] 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 [CPPPUS] 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 [CPPPCS] is formulated as follows [CPPPCS] 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 segmentcapacity 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, ..., a1, 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 Mvector 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= tl, ..., 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 Mvector 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
