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PAGE 1 OPTIMIZA TION MODELS F OR SOUR CING DECISIONS IN SUPPL Y CHAIN MANA GEMENT By WEI HUANG A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2004 PAGE 2 Cop yrigh t 2004 b y W ei Huang PAGE 3 This w ork is dedicated to m y family PAGE 4 A CKNO WLEDGMENTS Man y p eople ha v e help ed me to complete the w ork in this dissertation and I w ould lik e to thank them all. Man y thanks go to m y advisor H. Edwin Romeijn. F or 4 y ears, he sup ervised and guided m y researc h w ork. This dissertation w ould not ha v e b een p ossible without his help. I am in debt for his insigh tful though ts and detailed suggestions. I learned man y things from him, ab out researc h and ab out life. Sp ecial thanks go to Ra vindra K. Ah uja, Joseph Geunes, and Dolores Romero Morales. Their researc h inspired m uc h of the w ork in this dissertation. They oered man y helpful suggestions and m uc h advice ab out m y researc h. With them, I ha v e shared man y w onderful discussions and memories. Finally I w ould lik e to thank the facult y and m y colleagues at the Departmen t of Industrial and Systems Engineering, Univ ersit y of Florida, for their help and supp ort. 4 PAGE 5 T ABLE OF CONTENTS page A CKNO WLEDGMENTS . . . . . . . . . . . . . . 4 LIST OF T ABLES . . . . . . . . . . . . . . . . 8 LIST OF FIGURES . . . . . . . . . . . . . . . . 10 ABSTRA CT . . . . . . . . . . . . . . . . . . 11 CHAPTER1 INTR ODUCTION . . . . . . . . . . . . . . . 13 1.1 Supply Chain Managemen t and Op erations Researc h . . . 13 1.2 Logistic Net w orks and Co ordination . . . . . . . . 14 1.3 Applications and Algorithms . . . . . . . . . . 16 2 CLASS OF ASSIGNMENT PR OBLEMS . . . . . . . . . 20 2.1 F orm ulation . . . . . . . . . . . . . . . 20 2.2 Greedy Heuristic . . . . . . . . . . . . . 22 2.3 V ery LargeScale Neigh b orho o d Searc h (VLSN) . . . . . 23 2.3.1 In tro duction . . . . . . . . . . . . . 23 2.3.2 VLSN Algorithm . . . . . . . . . . . 24 2.3.3 P enalized VLSN . . . . . . . . . . . . 29 2.4 Branc handPrice Algorithm . . . . . . . . . . 30 2.4.1 In tro duction . . . . . . . . . . . . . 30 2.4.2 Reform ulation of the Assignmen t Problem . . . . 30 2.4.3 Column Generation . . . . . . . . . . . 33 2.4.4 Branc hing . . . . . . . . . . . . . 37 2.5 Summary . . . . . . . . . . . . . . . 39 3 MUL TIPERIOD SINGLESOUR CING PR OBLEM (MPSSP) . . . 41 3.1 In tro duction . . . . . . . . . . . . . . . 41 3.2 Problem and F orm ulation . . . . . . . . . . . 44 3.2.1 T raditional F orm ulation . . . . . . . . . . 44 3.2.2 Assignmen t F orm ulation . . . . . . . . . 47 3.3 Subproblems . . . . . . . . . . . . . . . 49 3.3.1 Domain of the Subproblem . . . . . . . . . 49 3.3.2 Acyclic Case . . . . . . . . . . . . . 54 3.3.3 Cyclic Case . . . . . . . . . . . . . 55 5 PAGE 6 3.4 Greedy Heuristic . . . . . . . . . . . . . 58 3.4.1 Outline of the Greedy Heuristic . . . . . . . 58 3.4.2 PseudoCost F unctions . . . . . . . . . . 59 3.4.3 Sto c hastic Mo del for the Problem Data . . . . . 60 3.4.4 F easibilit y Condition . . . . . . . . . . 61 3.4.5 Asymptotic feasibilit y and optimalit y . . . . . . 63 3.5 Computational Results . . . . . . . . . . . . 65 3.5.1 Generation of Problem Instances . . . . . . . 65 3.5.2 Results . . . . . . . . . . . . . . 68 3.6 Summary . . . . . . . . . . . . . . . 73 4 CONTINUOUSTIME SINGLESOUR CING PR OBLEM (CSSP) . . 81 4.1 In tro duction . . . . . . . . . . . . . . . 81 4.2 Con tin uousTime SingleSourcing Problem . . . . . . 82 4.2.1 F orm ulation . . . . . . . . . . . . . 83 4.2.2 Pricing Problem for LP(SP) . . . . . . . . 86 4.2.3 The CSSPU revisited . . . . . . . . . . 90 4.3 Extensions of the CSSPU . . . . . . . . . . . 92 4.3.1 Pro duction Capacit y Constrain ts . . . . . . . 93 4.3.2 The CSSP with Pro duction Capacit y Expansion . . . 94 4.3.3 The CSSP with In v en tory Capacit y . . . . . . 95 4.3.4 The CSSP with In v en tory Capacit y Expansion . . . 96 4.4 Greedy Heuristic . . . . . . . . . . . . . 101 4.5 Computational Results . . . . . . . . . . . . 103 4.6 Summary . . . . . . . . . . . . . . . 109 5 MUL TIPERIOD FLEXIBLE DEMAND ASSIGNMENT PR OBLEM (MPFD A) 111 5.1 In tro duction . . . . . . . . . . . . . . . 111 5.2 The MPFD A with Plate In v en tory . . . . . . . . . 113 5.2.1 F orm ulation . . . . . . . . . . . . . 113 5.2.2 Subproblem and Greedy Heuristic . . . . . . . 115 5.2.3 Knapsac k Problem with Expandable Items . . . . 118 5.3 The MPFD A with Slab In v en tory . . . . . . . . . 126 5.3.1 Basic F orm ulation . . . . . . . . . . . 126 5.3.2 Mo del Expansions . . . . . . . . . . . 129 5.4 The MPFD A with Plate and Slab In v en tory . . . . . . 139 5.4.1 Basic F orm ulation . . . . . . . . . . . 139 5.4.2 Expansion: Recycle and Purc hase in An y P erio d . . . 142 5.5 Computational Results . . . . . . . . . . . . 144 5.5.1 Generation of Problem Instances . . . . . . . 144 5.5.2 Main Results . . . . . . . . . . . . . 145 5.6 Summary . . . . . . . . . . . . . . . 150 6 CONCLUSIONS AND FUTURE RESEAR CH DIRECTIONS . . . 153 6 PAGE 7 REFERENCES . . . . . . . . . . . . . . . . . 155 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 160 7 PAGE 8 LIST OF T ABLES T able page 3{1 Basic case, acyclic, seasonal demands . . . . . . . . . 73 3{2 Basic case, cyclic, seasonal demands . . . . . . . . . 74 3{3 Throughput constrain ts, acyclic, seasonal demands, 0 = 1 : 3 . . . 74 3{4 Throughput constrain ts, cyclic, seasonal demands, 0 = 1 : 3 . . . 74 3{5 In v en tory capacities, acyclic, seasonal demands, 00 = 1 : 1 . . . 75 3{6 In v en tory capacities, acyclic, seasonal demands, 00 = 1 : 5 . . . . 75 3{7 In v en tory capacities, acyclic, seasonal demands, 00 = 2 . . . . 75 3{8 In v en tory capacities, cyclic, seasonal demands, 00 = 1 : 1 . . . . 76 3{9 In v en tory capacities, cyclic, seasonal demands, 00 = 1 : 5 . . . . 76 3{10 In v en tory capacities, cyclic, seasonal demands, 00 = 2 . . . . 76 3{11 P erishabilit y constrain ts, acyclic, seasonal demands, k = 1 . . . 77 3{12 P erishabilit y constrain ts, acyclic, seasonal demands, k = 2 . . . 77 3{13 P erishabilit y constrain ts, cyclic, seasonal demands, k = 1 . . . 77 3{14 P erishabilit y Constrain ts, cyclic, seasonal demands, k = 2 . . . 78 3{15 Basic case, acyclic, general demands . . . . . . . . . 78 3{16 Basic case, cyclic, general demands . . . . . . . . . . 78 3{17 In v en tory capacities, acyclic, general demands, = 1 : 1, 00 = 1 : 5 . . 79 3{18 In v en tory capacities, cyclic, general demands, = 1 : 1, 00 = 1 : 5 . . 79 3{19 Basic case, acyclic, general demands, m = 10 . . . . . . . 79 3{20 Basic case, cyclic, general demands, m = 10 . . . . . . . 80 3{21 In v en tory capacities, acyclic, seasonal demands, m = 10, 00 = 1 : 1 . 80 3{22 In v en tory capacities, cyclic, seasonal demands, m = 10, 00 = 1 : 1 . . 80 4{1 Results for M = 5 facilities . . . . . . . . . . . . 105 8 PAGE 9 4{2 Results for M = 10 facilities . . . . . . . . . . . 106 4{3 Results for n = 100 retailers, with f i 2 [100 ; 500] . . . . . . 107 5{1 Seasonal factor . . . . . . . . . . . . . . . 145 5{2 Results for MPFD AP . . . . . . . . . . . . . 148 5{3 Results for MPFD AS . . . . . . . . . . . . . 149 5{4 Results of GRASP for MPFD AS . . . . . . . . . . 151 9 PAGE 10 LIST OF FIGURES Figure page 2{1 Cyclic exc hange . . . . . . . . . . . . . . . 25 2{2 P ath exc hange . . . . . . . . . . . . . . . 27 3{1 Acyclic case, T = 6 . . . . . . . . . . . . . . 50 3{2 Cyclic case, T = 8 . . . . . . . . . . . . . . 50 4{1 Appro ximation algorithm . . . . . . . . . . . . 100 10 PAGE 11 Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y OPTIMIZA TION MODELS F OR SOUR CING DECISIONS IN SUPPL Y CHAIN MANA GEMENT By W ei Huang August 2004 Chair: H. Edwin Romeijn Ma jor Departmen t: Industrial and Systems Engineering In this dissertation, w e dev elop ed optimization mo dels and algorithms for sourcing problems arising in supply c hain managemen t. W e fo cused on problems in whic h a set of retailer demands needs to b e assigned to either a set of pro duction or storage facilities or a set of pro duction resources in a dynamic en vironmen t. Ho w ev er, w e rst studied a m uc h more general class of assignmen t problems (AP) in whic h w e assume that the cost functions asso ciated with assignmen ts are separable in the agen ts, but otherwise arbitrary W e dev elop ed three solution metho dologies for solving (AP), including a greedy heuristic, a v ery largescale neigh b orho o d searc h impro v emen t algorithm, and a branc handprice algorithm. W e next studied three sp ecic applications of (AP). In the rst application, w e considered the assignmen t of retailers to facilities o v er a discrete time horizon under dynamic demands and with a linear cost structure in the presence of constrain ts suc h as pro duction capacit y throughput capacit y in v en tory capacit y and p erishabilit y In the second application, w e considered a con tin uoustime mo del in whic h the retailers face a constan t demand rate, while eac h facilit y faces xedc harge pro duction costs. W e deriv ed a decreasing net rev en ue prop ert y that PAGE 12 is satised b y the optimal solution to (the linear relaxation of ) an imp ortan t subproblem, called the pricing problem, for a particular class of (AP). This class encompasses man y v arian ts of this application, including cases with pro duction and in v en tory capacities, and cases with capacit y expansion opp ortunities. In the third application w e considered a man ufacturer that has exibilit y in meeting its demands, that is, eac h of its customers will accept pro duct quan tities within a sp ecied range. W e considered three dieren t kinds of strategies, where w e can sto c k ra w materials only but pro duce justintime; sto c k only end pro ducts but acquire ra w materials justintime; or sto c k b oth ra w materials and end pro ducts. The pricing problem for this application is closely related to the socalled knapsac k problem with expandable items, whic h w e studied in detail and for whic h w e dev elop ed new solution approac hes. W e p erformed extensiv e tests for all applications, and concluded that the dev elop ed heuristics usually pro duce v ery high qualit y solutions in limited time. PAGE 13 CHAPTER 1 INTR ODUCTION 1.1 Supply Chain Managemen t and Op erations Researc h The study of supply c hain managemen t (SCM) started in the late 1980s and has gained a gro wing lev el of in terest from b oth companies and researc hers o v er the past 2 decades. There are man y denitions of supply c hain managemen t. Hau Lee, the head of the Stanford Global Supply Chain Managemen t F orum (1999), giv es a simple and straigh tforw ard denition at the forum w ebsite as follo ws: Supply c hain managemen t deals with the managemen t of materials, information and nancial o ws in a net w ork consisting of suppliers, man ufacturers, distributors, and customers. F rom this denition, w e can see that SCM is not only an imp ortan t issue to man ufacturing companies, but is also relev an t to service and nancial rms. W e can ev en sa y that almost ev ery compan y will face SCM problems and the decisions made regarding SCM will impact the op eration of the en tire business. More and more companies ha v e realized the imp ortance of SCM and are putting forw ard eorts to impro v e the p erformance of their supply c hains. With the globalization of economics, the b oundary b et w een nations are b ecoming less visible in the view of man y in ternational companies, whic h usually op erate a supply c hain at a global lev el. Since suppliers, man ufacturers, distributors, and customers are usually lo cated in dieren t nations in a global supply c hain, the companies can tak e adv an tage of the dierences in c haracteristics of v arious coun tries when designing their pro duction and sourcing strategies. F or example, W almart plans to imp ort $15 billion in go o ds from China in 2004 b ecause dev eloping countries suc h as China usually ha v e signican tly smaller lab or and material costs than 13 PAGE 14 14 coun tries lik e the U.S.A. Another similar example is the trend of outsourcing call cen ters and soft w are dev elopmen t from the U.S.A. to India. This trend of globalization forces companies to (re)design their logistic net w orks and (re)consider their sourcing strategies. An example can b e found in the pap er on supply c hain design at the Digital Equipmen t Corp oration (see Arn tzen et al. [ 10 ]). T o design an ecien t supply c hain, w e need to consider problems at the strategic, tactical, and op erational lev els. F or example, w e need to c ho ose suppliers, decide on the lo cation of pro duction, and c ho ose the shipmen t mo dalit y from the suppliers and to the customers. W e also need to consider the pro duction, storage, and transp ortation requiremen ts for dieren t pro ducts. In general, these problems are large scale problems that are complex in nature. It is almost imp ossible to solv e these problems using only past exp erience and in tuitiv e rulesofth um b. Op erations Researc h has long pro v ed to pro vide a p o w erful to ol for supp orting realw orld decision making in man y areas. Th us, the study of SCM has naturally utilized Op erations Researc h tec hniques from the rst da y By using mathematical mo dels and scien tic approac hes, w e can usually impro v e the p erformance of the supply c hain b y impro ving b oth costs and the lev el of customer satisfaction. With the help of Op erations Researc h tec hniques and the adv ance of computer tec hnologies during the past y ears, soft w are and consultan t rms in area of supply c hain managemen t are commonplace no w. 1.2 Logistic Net w orks and Co ordination In this dissertation w e fo cused our study on ecien t material o w, that is, w e studied the logistic or distribution net w ork in whic h the material o ws tak e place. The ph ysical no des of this net w ork usually consist of suppliers, pro duction facilities, w arehouses, retailers, and end customers. The links among these ph ysical no des are the routes through whic h the material o ws pass, suc h as high w a ys or railroads. In addition, activities can tak e place in eac h of the no des, usually PAGE 15 15 requiring a set of decisions to b e made. F or example, in pro duction facilities w e need to pro vide a pro duction planning metho d that guides the pro duction pro cess. In a w arehouse, w e need to manage the in v en tories using a certain in v en tory con trol metho d. The problems that w e need to deal with in a logistic net w ork can b e divided in to strategic, tactical, and op erational lev el problems. A t the strategic lev el, w e, for example, face the problem of lo cating facilities. A t the tactical lev el, w e ma y consider transp ortation mo dalit y c hoice and sourcing decisions. Finally at the op erational lev el t w o classic mo dels for pro duction and in v en tory con trol are Economic Lot Sizing (ELS) mo del in discrete time and Economic Order Quan tit y (EOQ) mo del in con tin uous time. By solving these problems, w e can exp ect that a b etter p erforming logistics net w ork will b e obtained. Ho w ev er, a basic principle in Op erations Researc h is that the individual optimization of subproblems will usually yield solutions that are sub optimal with resp ect to the en tire system. This tells us that ev en if w e solv e the optimization problems at eac h lev el men tioned ab o v e, in general the p erformance of the logistic net w ork th us obtained will not b e the b est p ossible. Hence, an imp ortan t topic in SCM is that of supply c hain co ordination. The reason b ehind the imp ortance of supply c hain co ordination is that the decisions made at the strategic and tactical lev els can inuence the options a v ailable at the op erational lev el (and vice v ersa). F or example, the lo cation of facilities will certainly limit the t yp es of shipping metho ds that can b e considered, whic h is usually a consideration at the tactical lev el. On the other hand, the desire to con trol transp ortation costs that are faced at the op erational lev el, has an impact on the c hoice of the lo cation of facilities. Th us, decisions ha ving impact on eac h other m ust b e co ordinated in the supply c hain to ensure maximal eectiv eness and eciency Suc h co ordination will exist among not only dieren t lev els of the planning pro cess, but also among dieren t stages of the planning pro cess, suc h PAGE 16 16 as pro duction, storage, and distribution. The mo dels that w e studied in this dissertation will in tegrate sev eral t yp es of co ordinated decisions in the supply c hain in to accoun t. 1.3 Applications and Algorithms The goal of this dissertation is to obtain insigh t in to sev eral applications in supply c hain managemen t and logistic net w ork design and dev elop ecien t algorithms for solving them. F or these problems, w e considered not only the strategic and tactical decisions of selection of facilities and the assignmen t of retailers to facilities, but also the op erational lev el decisions on pro duction and in v en tory planning. Th us, with these mo dels w e aimed at impro ving the p erformance of a logistics net w ork using co ordination. W e can form ulate all applications considered in this dissertation as sp ecial cases of a general class of assignmen t problems. In order to obtain generally applicable results and obtain a b etter understanding of the applications studied in this dissertation, w e started the dissertation with the study of this class of assignmen t problems. In this general class of assignmen t problems, a set of tasks needs to b e assigned to a set of agen ts. As is often the case in practice, w e assume that the costs asso ciated with suc h assignmen ts is separable in the agen ts, that is, the cost of a giv en agen t only dep ends on the tasks assigned to it, but not on the w a y the remaining tasks are assigned to the remaining agen ts. Suc h cost often requires the solution to a socalled subproblem. In addition, w e assume that eac h task can b e assigned to a unique agen t only whic h is often referred to as singlesour cing Suc h a strategy is widely used in practice since it generally reduces crossco ordination planning complexities in facilitates impro v ed ordering, shipping, and receiving co ordination b et w een eac h retailer and its source facilit y W e analyzed the structure of this class of assignmen t problems and prop osed three solution metho dologies for solving it, including a greedy heuristic, a v ery largescale PAGE 17 17 neigh b orho o d (VLSN) searc h algorithm and a branc handprice algorithm. All three algorithms w ere then sp ecialized and w ork ed out in detail for eac h application studied. By rigorously studying the subproblems (needed to ev aluate the costs of a giv en solution) and the pricing problem (whic h is a part of the branc handprice metho d), w e can dev elop ecien t implemen tations of these algorithms. W e can also adapt these algorithms to solv e man y other problems that can b e form ulated as a sp ecial case of the class of assignmen t problems. The logistics net w ork setting in the rst t w o applications is v ery similar. In b oth cases, w e considered a logistics net w ork with a set of facilities and a set of retailers. There is a single t yp e of pro duct and the order of eac h retailer can only b e fullled b y exactly one facilit y The main dierence b et w een the t w o applications is that in the rst application, the MultiP erio d SingleSourcing Problem (MPSSP), w e considered a discrete time horizon and assume linear pro duction and holding costs, while in the second application, the Con tin uousTime SingleSourcing Problem (CSSP), the problem will b e set in con tin uoustime horizon and w e consider xed setup costs in pro duction. Usually w e considered sev eral v arian ts of eac h application in the presence of v arious t yp es of constrain ts. In the absence of capacities, the MPSSP can b e view ed as a co op erativ e m ultisupplier and m ultiretailer v ersion of the classical ELS mo del. Similarly the uncapacitated CSSP can b e view ed as a co op erativ e m ultisupplier and m ultiretailer v ersion of the classical EOQ mo del. In the con text of the MPSSP w e considered t w o t yp e of mo dels: the acyclic mo del for short term pro duction planning and the cyclic mo del for long term pro duction planning. W e incorp orated sev eral t yp es of constrain ts to the mo del including pro duction capacit y constrain ts, throughput capacit y constrain ts, in v en tory capacit y constrain ts, and p erishabilit y constrain ts. W e studied the domain of the subproblem, and used this to prop ose a random data generation PAGE 18 18 mo del under whic h a greedy heuristic is pro v ably asymptotically feasible and optimal. W e furthermore dev elop ed ecien t p olynomialtime algorithms for solving the subproblems in b oth mo dels and th us an ecien t implemen tation of our greedy heuristic and VLSN searc h algorithm. F or the CSSP w e rst studied a sp ecial sub class of assignmen t problems, whic h in turn generalizes all v arian ts of CSSP that w e consider. In general, the CSSP is a largescale nonlinear in teger programming problem. Ho w ev er, w e sho w ed that for this class of problems the optimal solution to (the relaxation of ) the pricing problem satises a socalled decreasing net rev en ue prop ert y (DNRS). Using this prop ert y w e w ere able to solv e the pricing problem or its relaxation in p olynomial time in some cases. Ev en in cases where no p olynomialtime algorithm exists for the pricing problem, the DNRS prop ert y ma y lead to an ecien t heuristic approac h. In particular, w e dev elop ed suc h a heuristic for the CSSP with in v en tory capacit y expansion opp ortunities and pro v ed that it is asymptotically optimal for solving the relaxed pricing problem. The DNRS prop ert y also motiv ates the dev elopmen t of a greedy heuristic for the CSSP itself, as w ell as algorithms for solving the pricing problem in all v arian ts of CSSP studied. These v arian ts include a basic uncapacitated CSSP mo del, as w ell as mo dels accoun ting for pro duction or in v en tory capacities and mo dels with pro duction or in v en tory capacit y expansion opp ortunities. The third application, the m ultip erio d exible demand assignmen t (MPFD A) problem, w as inspired b y a problem faced b y a man ufacturer in the steel industry W e considered a man ufacturer that has exibilit y in meeting its demands, that is, eac h of its customers sp ecies a range of demands, and will accept pro duct quan tities within that range. W e considered three dieren t kinds of pro duction and in v en tory holding strategies, where w e can sto c k ra w materials only but pro duce justintime, sto c k only end pro ducts but acquire ra w materials justintime, or PAGE 19 19 sto c k b oth. In the MPFD A the pricing problem can b e solv ed b y solving a (set of ) knapsac k problem(s) with expandable items (KPEI), whic h w as previously studied b y Balakrishnan and Geunes [ 11 ]. W e further studied the KPEI problem and sho w ed that the optimal solution to the linear relaxation of KPEI satises some in teresting prop erties. W e then used these prop erties to dev elop a p olynomialtime algorithm for the relaxed KPEI, and an ecien t heuristic approac h for KPEI. W e further sho w ed that this heuristic is asymptotically optimal when the problem data are randomly generated with nite exp ectation. F or all three applications, w e p erformed extensiv e tests on sets of randomly generated problem instances. These tests indicate that the branc handprice algorithm can b e eectiv ely used for smaller problem instances and/or to obtain a go o d b ound on the qualit y of a heuristic solution for in termediate size problems. F or larger problem instances, our heuristic algorithms can usually obtain v ery high qualit y results in limited time. The outline of this dissertation is as follo ws. In Chapter 2 w e studied a general class of assignmen t problems and prop osed three solution metho dologies for solving it, including a greedy heuristic approac h, a VLSN searc h algorithm, and the branc handprice metho d. The next three c hapters, Chapters 3 5 dealt with the three supply c hain optimization applications describ ed ab o v e: the MPSSP the CSSP and the MPFD A, resp ectiv ely Finally w e concluded this dissertation in Chapter 6 with some concluding remarks and directions for future researc h. PAGE 20 CHAPTER 2 CLASS OF ASSIGNMENT PR OBLEMS All applications that w e studied in this dissertation are sp ecial cases of a class of assignmen t problems. In this c hapter, w e giv e the form ulation of this class of assignmen t problems, generalizing the class of con v ex capacitated assignmen t problems (CCAP) that w as studied b y Romero Morales [ 44 ]. W e then discuss three t yp es of algorithms for solving suc h problems, including a greedy heuristic approac h, a v ery largescale neigh b orho o d searc h algorithm, and a branc handprice metho dology In the next c hapters, w e then sp ecialize eac h of these approac hes to particular assignmen t problems encoun tered in supply c hain applications. 2.1 F orm ulation Consider an assignmen t problem in whic h w e need to partition a set of tasks f 1 ; : : : ; N g in to M agen ts at minim um cost. In man y applications, the costs are separable in the agen ts, that is, the cost of an agen t can b e determined when the set of tasks assigned to that agen t is kno wn, and th us is indep enden t of the w a y in witc h the remaining tasks are assigned to the remaining agen ts. W e also assume that there is a singlesourcing constrain t, that is, eac h task can only b e assigned to exactly one agen t. W e therefore consider the follo wing class of assignmen t problems: minimize M X i =1 H i ( x i ¢ ) sub ject to (AP) M X i =1 x ij = 1 j = 1 ; : : : ; N x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N 20 PAGE 21 21 x i ¢ 2 X i i = 1 ; : : : ; M : In this form ulation, x ij denotes a binary decision v ariable that tak es the v alue 1 if task j is assigned to agen t i and 0 otherwise, and x i ¢ = ( x i 1 ; : : : ; x iN ). F urthermore, the function H i : X i R is the cost of agen t i as a function of the tasks assigned to agen t i Finally sets X i represen t the feasibilit y of subsets of tasks for agen t i and are assumed to b e con v ex. As sho wn in later c hapters, ev aluating a function H i often in v olv es solving an optimization subproblem. If functions H i are linear and sets X i are eac h formed b y a single knapsac k constrain t, this sp ecial case of (AP) b ecomes the w ellkno wn generalized assignmen t problem (GAP). It th us follo ws that (AP) is a NPHard problem in general, since the GAP is kno wn to b e NPHard. W e next iden tify a sp ecial case of (AP) in whic h w e can relax the binary constrain ts to nonnegativit y constrain ts: Lemma 1 When the functions H i ar e c onc ave and X i = R N for i = 1 ; : : : ; M (AP) is e quivalent to the fol lowing pr oblem: minimize M X i =1 H i ( x i ¢ ) subje ct to M X i =1 x ij = 1 j = 1 ; : : : ; N x ij 0 i = 1 ; : : : ; M ; j = 1 ; : : : ; N : Pro of: The transp ose of the matrix of co ecien ts of the rst constrain t set of (AP) forms what Nemhauser and W olsey [ 38 ] (Section I I I.1) refer to as an interval matrix that is, a (0 ; 1) matrix in whic h all ones o ccur consecutiv ely within eac h a column. Nemhauser and W olsey [ 38 ] also sho w that suc h a matrix (along with its transp ose) is totally unimo dular (TU). It com bined with the fact that if a PAGE 22 22 matrix A is TU, then the p olyhedron x 2 R n+ : Ax = 1 (if nonempt y) only has in tegral extreme p oin ts (Nemhauser and W olsey [ 38 ]). This implies that R(AP) con tains all in tegral extreme p oin ts. Because a conca v e minimization problem on a closed con v ex set has an optimal extreme p oin t solution, w e kno w that the relaxed problem has a binary optimal solution, whic h is th us also an optimal solution to (AP). In the remainder of this c hapter w e will describ e dieren t solution approac hes to solving (AP). 2.2 Greedy Heuristic Since in general (AP) is NPHard, one approac h that w e pursued is a heuristic approac h to solv e problems from this class. Martello and T oth [ 34 ] prop osed a greedy heuristic for the GAP whic h is a sp ecial case of (AP). In the Bin P ac king Problem, whic h is also a sp ecial case of (AP), there is a widely used heuristic approac h, namely the b est t heuristic, whic h is also a t yp e of greedy heuristic. Similar to these approac hes w e will consider a greedy heuristic for (AP). In the greedy heuristic w e will assign tasks to agen ts one b y one. In eac h step of the algorithm, w e mak e an assignmen t according to a certain rule. This rule should b e dened in suc h a w a y that it exploits prop erties of functions H i Because functions H i are not sp ecied in (AP), w e will only giv e a outline of the greedy heuristic here and dev elop more detailed algorithms for eac h application considered in later c hapters. W e dene a function G ( I ; J ) that represen ts a rule for c ho osing a pair consisting of an agen t from set I and a task from set J This assignmen t is then made, and the corresp onding task is eliminated from further consideration. More formally the outline of the greedy heuristic is: PAGE 23 23 Greedy heuristic Step 0. Set I = f 1 ; : : : ; M g J = f 1 ; : : : ; N g and x G = 0. Step 1. Let ( i; j ) = G ( I ; J ), set x Gij = 1, and set J = nf j g Step 2. If J = ; stop and w e ha v e a solution x G If I = ; stop and w e ha v e a partial solution x G Otherwise return to step 1. 2.3 V ery LargeScale Neigh b orho o d Searc h (VLSN) 2.3.1 In tro duction Using the greedy heuristic w e can, in man y cases, nd a reasonable solution of (AP) v ery rapidly But there ma y also b e cases in whic h the greedy heuristic can only nd a lo w qualit y feasible solution, or cannot ev en nd a feasible solution at all. So it is v ery natural to pursue an impro v emen t heuristic approac h to impro v e the solution of (AP) obtained b y using the greedy heuristic, and also to obtain a feasible solution of (AP) when the greedy heuristic fails to nd a feasible solution. W e will pursue a widely used impro v emen t approac h, namely lo cal searc h, to impro v e the initial (full or partial) solution obtained b y the greedy heuristic. In the discussion b elo w, w e will rst fo cus, for ease of exp osition, on the case where w e ha v e a feasible solution to the (AP) a v ailable. Generally a lo cal searc h algorithm starts with an initial solution and rep eatedly replaces it b y an impro v ed solution within a socalled neigh b orho o d of the curren t solution un til some termination criterion is satised. T o start a neigh b orho o d searc h algorithm w e need to dene a neigh b orho o d structure of the solution space. One of the most widely used neigh b orho o ds is the 2exc hange neigh b orho o d, in whic h the neigh b orho o d of a giv en solution con tains all solutions that can b e obtained b y in terc hanging the assignmen t of exactly t w o tasks curren tly assigned to t w o distinct agen ts while retaining feasibilit y In particular, if task j 1 is assigned to agen t i 1 and task j 2 is assigned to agen t i 2 w e ma y c hec k if assigning task j 2 to agen t i 1 and task j 1 to agen t i 2 impro v es the qualit y of the solution and main tains PAGE 24 24 the feasibilit y of the solution. If no impro v emen t is found for an y of the p ossible exc hanges of this t yp e, the curren t solution is lo cally optimal. The 2exc hange neigh b orho o d is simple and ecien t in man y cases. Ho w ev er, to obtain ev en b etter solutions w e will pursue a generalization of the 2exc hange neigh b orho o d searc h, called v ery largescale neigh b orho o d (VLSN) searc h, instead. The c hoice of neigh b orho o d structure determines the qualit y of the nal solution obtained. In general, disregarding computational eort, the larger the neigh b orho o ds, the b etter the nal solution is exp ected to b e. Ho w ev er, this will generally come at the exp ense of an increase in computation time. Th us, the use of larger neigh b orho o ds can not guaran tee a more eectiv e algorithm unless the neigh b orho o d can b e searc hed in a v ery ecien t manner. Thompson [ 48 ], Thompson and Orlin [ 49 ], Thompson and Psaraftis [ 50 ] rst prop osed the idea of VLSN searc h b y implicitly (rather than explicitly) en umerating and ev aluating all the neigh b ors in a v ery large neigh b orho o d. Ah uja et al. [ 1 5 ] ha v e rened this algorithm b y dev eloping fast net w orkbased algorithm that signican tly sp eed up the neigh b orho o d searc h. They ha v e applied this algorithm successfully to v arious problems, suc h as the capacitated minim um spanning tree problem [ 6 ], the quadratic assignmen t problem [ 3 ], the com bined througheet assignmen t problem [ 2 ]. The same tec hnique has b een used to solv e set partitioning problems suc h as v ehicle routing problems [ 50 27 22 ], minim um mak espan mac hine sc heduling [ 24 ] and other sc heduling problems [ 50 ]. Next w e in tro duce the basic idea of VLSN algorithm for the set partitioning problem based on (AP). 2.3.2 VLSN Algorithm In its original form, VLSN considers a v ery largescale neigh b orho o d in whic h the neigh b ors of a solution are all solutions that can b e reac hed through a socalled cyclic exc hange. Let x b e a feasible solution of (AP). Supp ose that w e ha v e 2 R M tasks j 1 ,. . j R eac h of whic h is assigned to a distinct agen t, that PAGE 25 25 is, task j r is assigned to agen t i r or x i r j r = 1 for r = 1 ; : : : ; R A cyclic exc hange is a m ultitask exc hange that reassigns task j r to agen t i r +1 for r = 1 ; : : : ; R ¡ 1 and task j R to agen t i 1 (Figure 2{1 ) and lea v es the assignmen t of all other tasks unc hanged. W e sa y that agen ts i 1 . i R b elong to this cyclic exc hange. Let x 0 b e the solution after the cyclic exc hange. Then w e ha v e 1 8 6 2 7 1 7 1 1 1 2 1 0 1 4 1 6 1 3 3 4 1 5 5 9 Figure 2{1: Cyclic exc hange x 0i r ¢ = 8><>: x i r ¢ ¡ e j r + e j R for r = 1 x i r ¢ ¡ e j r + e j r ¡ 1 for r = 2 ; : : : ; R where e j 2 R N is the j th unit v ector. If x 0 is feasible for (AP) w e call this cyclic exc hange feasible. The dierence in total cost b et w een x and x 0 is equal to: R X r =1 [ H i r ( x 0i r ¢ ) ¡ H i r ( x i r ¢ )] : Since up to M tasks can b elong to a cyclic exc hange, the neigh b orho o d of an y feasible solution will usually b e v ery large and w e can not explicitly en umerate all neigh b ors. Instead w e use the concept of impro v emen t graph [ 49 ] to implicitly searc h the neigh b orho o d. The impro v emen t graph is dened with resp ect to a feasible solution x and denoted b y G ( x ). The graph G ( x ) has N no des, eac h corresp onding to a task. PAGE 26 26 Supp ose task j 1 is assigned to agen t i 1 and task j 2 is assigned to agen t i 2 A directed arc ( j 1 ; j 2 ) represen ts that task j 1 is assigned to i 2 the agen t whic h curren tly p erforms the task j 2 while sim ultaneously the task j 2 is deassigned from i 2 W e dene the cost of arc ( j 1 ; j 2 ) as the c hange in costs of agen t i 2 due to these c hanges, that is, a ( j 1 ; j 2 ) = H i 2 ( x i 2 ¢ ¡ e j 2 + e j 1 ) ¡ H i 2 ( x i 2 ¢ ) : W e include arc ( j 1 ; j 2 ) in the impro v emen t graph only when tasks j 1 and j 2 b elong to t w o distinct agen ts and the assignmen t to agen t i 2 after the c hange is feasible. W e call a directed cycle in the impro v emen t graph G ( x ) subsetdisjoint if the tasks corresp onding to the no des in the directed cycle b elong to dieren t agen ts. It has b een sho wn that (i) there is a onetoone corresp ondence b et w een cyclic exc hange with resp ect to x and subsetdisjoin t directed cycles in the impro v emen t graph G ( x ), and (ii) the cost of suc h a cycle is equal to the cost dierence b et w een the t w o corresp onding solutions [ 49 ]. Th us if w e can nd a negativ ecost subsetdisjoin t cycle in G ( x ), w e nd an impro v ed solution in the neigh b orho o d of x Since the problem of nding a negativ ecost subsetdisjoin t cycle is an NPcomplete problem [ 49 ], w e will use an ecien t heuristic to solv e it appro ximately One v ery ecien t heuristic metho d has b een dev elop ed b y Ah uja et al. [ 6 ]. This heuristic is a mo dication of the w ellkno wn lab elcorrection algorithm for the shortest path problem, where eac h directed path main tained b y the algorithm is a subsetdisjoin t path. Once a negativ ecost subsetdisjoin t cycle is found, w e will p erform the cyclic exc hange and replace the original solution x b y the mo died solution x 0 Then w e need to up date the impro v emen t graph so that it corresp onds to the neigh b orho o d of x 0 Note that arcs from or to all no des corresp onding to items assigned to those agen ts that b elong to the cyclic exc hange need to b e up dated. Based on feasibilit y c hec ks and arc cost computations with resp ect to the new solution x 0 suc h up dates PAGE 27 27 ma y c hange arc costs, add arcs, and delete arcs. Then, based on this up dated impro v emen t graph, w e can further impro v e our solution un til some stopping criterion is satised. By adding additional no des and arcs to the impro v emen t graph, the neigh b orho o d can b e expanded ev en further b y also including socalled path exc hanges. A path exc hange is similar to a cyclic exc hange except that after task j R is deassigned from agen t i R it is assigned to an agen t other than i 1 (Figure 2{2 ). The impro v e1 8 6 2 7 1 7 1 1 1 2 1 0 1 4 1 6 1 3 3 4 1 5 5 9 Figure 2{2: P ath exc hange men t graph is expanded b y adding a pseudo no de p as w ell as an agen t no de, sa y [ i ], for eac h agen t. Then, arcs of the form ( p; j ) are added; suc h an arc corresp onds to deassigning task j from the agen t to whic h it is curren tly assigned, sa y i and is only included in the impro v emen t graph if this deassignmen t is feasible. The costs of this arc are a ( p; j ) = H i ( x i ¢ ¡ e j ) ¡ H i ( x i ¢ ) : In addition, arcs of the form ([ i ] ; p ) are added with arc cost 0. Finally arcs of the form ( j ; [ i ]) are added b et w een a task no de j and agen t no de i ; suc h an arc corresp onds to assigning task j to agen t i and is only included in the impro v emen t graph if task j is not curren tly assigned to agen t i and this assignmen t is feasible. PAGE 28 28 The costs of this arc are a ( j ; [ i ]) = H i ( x i ¢ + e j ) ¡ H i ( x i ¢ ) : It has b een sho wn that in suc h an impro v emen t graph whenev er w e nd a negativ ecost subsetdisjoin t cycle con taining pseudo no de p it means that w e nd a negativ e cost path exc hange (see Ah uja et al. [ 5 ]). Then w e can apply suc h a path exc hange to impro v e our curren t solution and decrease the total cost. Ab o v e, w e discussed ho w to up date the impro v emen t graph when there are only cyclic exc hanges. When b oth cyclic and path exc hanges are considered, w e can again sp eed up the up dating pro cess b y only up dating some of the arcs and arc costs of the impro v emen t graph instead of constructing the impro v emen t graph from scratc h. Ho w ev er, the up dating pro cess when b oth t yp es of exc hanges are considered is more complex. Note that the arcs b et w een those tasks assigned to agen ts whic h are not aected b y the p erformed exc hange remain unc hanged. All agen ts to whic h tasks that o ccur in the negativ ecost subsetdisjoin t cycle are assigned are aected, and all tasks assigned to these aected subsets are aected as w ell. It then follo ws that only the costs of arcs in the follo wing three categories need to b e c hanged: Arcs from the pseudo no de p to the aected task no des. Arcs from all task no des to the aected agen t no des. Arcs b et w een all task no des and the aected task no des. No w w e can summarize the ab o v e discussion and giv e the outline of the VLSN algorithm:VLSN Step 0. Initialize the impro v emen t graph G ( x ) based on an initial feasible solution x PAGE 29 29 Step 1. If a negativ ecost subsetdisjoin t cycle can not b e found in the impro v emen t graph G ( x ), stop. Otherwise go to next step. Step 2. Obtain a b etter solution x 0 b y p erforming the exc hange. Up date the impro v emen t graph to G ( x 0 ) and set x = x 0 Return to step 1. Eac h time when some arc costs need to b e computed in the VLSN algorithm, either when w e construct the initial impro v emen t graph or during the subsequen t up dates, w e will need to rep eatedly ev aluate the functions H i Although this seems a relativ ely easy task since the assignmen t v ectors are giv en, this ev aluation often in v olv es the solution of an optimization subproblem. It is therefore critical to b e able to solv e these subproblems ecien tly W e will dev elop ecien t algorithms for solving these subproblems for eac h of the applications studied in this dissertation. 2.3.3 P enalized VLSN In the rst step of the VLSN algorithm an initial feasible solution is needed to start the algorithm. Suc h a solution can, for example, b e pro vided b y a heuristic suc h as the one describ ed in Section 2.2 But in some cases a heuristic only pro vides a partial assignmen t, that is, a solution in whic h some tasks are not assigned to agen ts due to the constrain ts. W e can construct an infeasible solution from this partial solution b y assigning those unassigned tasks to arbitrary agen ts without considering the constrain ts, or in suc h a w a y that the constrain t violations are minimized. Then w e can use a rst phase VLSN algorithm called the p enalized VLSN algorithm to con v ert this infeasible assignmen t to a feasible solution b y making all assignmen ts feasible but p enalizing the infeasibilit y of the assignmen ts to agen ts. In particular, supp ose that the assignmen t x i ¢ to agen t i is infeasible and the violation of the constrain ts can b e measured b y ¢ i ( x i ¢ ). F or example, when a knapsac k constrain t is violated, ¢ i ( x i ¢ ) will b e the amoun t of the assigned v olume o v er the knapsac k capacit y Then w e p enalize the cost of subset i b y PAGE 30 30 adding L ¢ i ( x i ¢ ) to H i where L is a v ery large n um b er. If no neigh b or of the curren t solution is feasible, an iteration of the p enalized VLSN algorithm should nd a solution with reduced infeasibilit y If, ho w ev er, one of the neigh b ors of the curren t solution is feasible, it is v ery lik ely that the cycle in the impro v emen t graph corresp onding to this neigh b or will b e detected since this cycle has a negativ e cost of v ery large magnitude. Th us w e ma y exp ect to eliminate the infeasibilit y using the p enalized VLSN algorithm. Then, the resulting feasible solution ma y b e used as a starting p oin t for the regular VLSN algorithm. 2.4 Branc handPrice Algorithm 2.4.1 In tro duction W e ha v e prop osed heuristics including a greedy heuristic and a VLSN algorithm for (AP). F or man y cases of the applications studied in this dissertation these heuristics, esp ecially the VLSN algorithm, will giv e us a high qualit y solution. But still there are situations, for example, when the qualit y of the solutions obtained b y using these heuristics are not satisfactory or when a precise gap need to b e obtained, w e w an t to solv e (AP) to optimalit y W e kno w that the branc handb ound metho d can solv e in teger problems including (AP) to optimalit y Ho w ev er, this requires an ecien t algorithm to solv e the relaxed problem obtained b y relaxing the in tegralit y constrain ts to optimalit y F or (AP), this relaxation is still v ery hard to solv e since in general the ob jectiv e function of this problem is not linear. Th us the traditional branc handb ound metho d can not solv e (AP) ecien tly In the next subsection, w e therefore reform ulate (AP) as a set partitioning problem. In the remainder of this section, w e then discuss a branc handprice algorithm for (AP) based on this reform ulation. 2.4.2 Reform ulation of the Assignmen t Problem Similar to the GAP [ 14 ] [ 47 ], the CCAP [ 44 ], and the MPSSP [ 25 ], w e can form ulate the (AP) as a set partitioning problem. Let K i b e the n um b er of subsets PAGE 31 31 of tasks that can b e feasibly assigned to agen t i for i = 1 ; : : : ; M Let v ector y k i ¢ = ( y k i 1 ; : : : ; y k iN ) denote the k th feasible subset of tasks for agen t i that is, y k ij is 1 if task j is an elemen t of the k th feasible subset for agen t i and 0 otherwise. W e call y k i ¢ the k th column for agen t i Then w e can reform ulate the (AP) to a set partitioning problem as follo ws: minimize M X i =1 K i X k =1 H i ( y k i ¢ ) ki sub ject to (SP) M X i =1 K i X k =1 y k ij ki = 1 j = 1 ; : : : ; N K i X k =1 ki = 1 i = 1 ; : : : ; M ki 2 f 0 ; 1 g i = 1 ; : : : ; M ; k = 1 ; : : : ; K i : V ariable ki is equal to 1 if column k is c hosen for agen t i and 0 otherwise. The adv an tage of this form ulation is that it is an in teger linear programming problem, while in general the original form ulation is an in teger nonlinear programming problem, whic h will b e hard to solv e ev en after relaxing the in tegralit y constrain ts unless the functions H i are con v ex. It is clear that there is a onetoone corresp ondence b et w een feasible solutions to (AP) and (SP). W e denote the problem obtained b y relaxing the in tegralit y constrain ts in (AP) b y R(AP), and denote the linear programming relaxation of (SP) b y LP(SP). Eac h feasible solution to LP(SP) can b e transformed to a feasible solution of R(AP) through the relationship: x ij = K i X k =1 y k ij ki : Ho w ev er, the rev erse is generally not true. That is, not ev ery feasible solution to R(AP) corresp onds to a feasible solution to LP(SP). PAGE 32 32 Let V R ( AP ) and V LP ( S P ) b e the optimal ob jectiv e v alues of R(AP) and LP(SP) resp ectiv ely Romero Morales [ 44 ] sho ws that if the functions H i are all con v ex the optimal solution v alue of the LP relaxation of (SP) is nev er w orse than the optimal solution v alue of the con tin uous relaxation of (AP), that is, V R ( AP ) V LP ( S P ) : The follo wing prop osition sho ws that the opp osite relationship b et w een these v alues holds in case all functions H i are conca v e: Prop osition 2 If the functions H i ar e al l c onc ave we have V R ( AP ) V LP ( S P ) : Pro of: W e ha v e seen that eac h feasible solution of LP(SP) can b e transformed to a feasible solution to R(AP). If the functions H i are conca v e, w e th us ha v e that V R ( AP ) = M X i =1 H i ( x i ¢ ) = M X i =1 H i K i X k =1 y k i ¢ ki M X i =1 K i X k =1 H i ¡ y k i ¢ ¢ ki = V LP ( S P ) : These results imply that, when the functions H i are all con v ex, a tigh ter lo w er b ound to (AP) can b e obtained from the optimal solution of LP(SP) than directly from the optimal solution of R(AP). In con trast, when the functions H i are all conca v e the lo w er b ound obtained from LP(SP) will nev er b e b etter than that of R(AP). Despite this seemingly negativ e result, it ma y still b e v ery attractiv e to use the set partitioning form ulation (SP) rather than the original form ulation (AP) due to the ab o v emen tioned dicult y in solving ev en the relaxation R(AP) to optimalit y By using the reform ulation w e transfer the nonlinear problem to a linear problem, whic h ma y b e m uc h easier to solv e as long as w e can ecien tly ev aluate the functions H i for an y giv en assignmen t. PAGE 33 33 Based on the reform ulation (SP) and b y com bining the branc handb ound metho d with a column generation approac h to solving LP(SP) w e can dev elop a socalled branc handprice algorithm that can solv e the (AP) ecien tly for man y cases of the applications studied in this dissertation. This approac h has b een review ed b y Barnhart et al. [ 12 ]. The branc handprice metho d solv es the relaxed problem using the column generation metho d. In this metho d, sets of columns are omitted from the relaxed problem if there are to o man y v ariables to handle explicitly and ecien tly and if it can b e exp ected that most of them will ha v e v alue equal to zero in an optimal solution. After solving suc h a reduced relaxed problem, called the master problem, w e need to c hec k the optimalit y of the solution with resp ect to the full problem. A subproblem, called the pricing problem, is then solv ed to try to iden tify one or more columns that could en ter the basis if added to the master problem. If suc h columns are found, the expanded master problem is reoptimized, and this pro cedure is then rep eated un til no more columns can b e found. Branc hing o ccurs as usual when no columns can b e priced out to en ter the basis and the corresp onding solution to the master problem do es not satisfy the in tegralit y conditions. 2.4.3 Column Generation In order to apply the standard branc handb ound metho d to (SP) all columns need to b e presen t. Ho w ev er, in general the total n um b er of columns will increase exp onen tially in the size of the problem. This mak es the standard branc handb ound metho d quite unattractiv e from a computational p oin t of view. In con trast, emplo ying the column generation metho d to solv e the relaxed problem at a no de of the branc handb ound tree eliminates this dra wbac k. In particular, the column generation metho d usually only considers a v ery small subset of columns and iterativ ely adds protable columns un til the optimal solution of the relaxed PAGE 34 34 problem is ac hiev ed. The resulting algorithm is called the branc handprice algorithm. The column generation metho d considers the problem (SP) with only a limited set of columns, sa y K T o start the metho d, w e require an initial set of columns that con tains a feasible solution to LP(SP). W e ma y for example, use a heuristic suc h as a greedy heuristic with or without applying the VLSN metho d (see Section 2.2 ) to obtain a feasible solution to (SP). This solution then yields a set of initial columns for (SP), sa y K 0 (Belo w w e will discuss ho w to pro ceed if no heuristic solution to (SP) is a v ailable.) The idea b ehind column generation is then to solv e LP(SP( K 0 )) to optimalit y and emplo y the dual optimal solution to v erify whether the optimal solution to LP(SP( K 0 )) is in fact an optimal solution to the full problem LP(SP). If this is not the case, w e generate additional columns that can impro v e this solution, add these to the set K 0 and rep eat the pro cedure un til the optimal solution to LP(SP) is obtained. The usual approac h to nd the new column is to consider the dual problem D(SP) to LP(SP): maximize N X j =1 u j ¡ M X i =1 v i sub ject to: N X j =1 y k ij u j ¡ v i H i ( y k i ¢ ) k = 1 ; : : : ; K i ; i = 1 ; : : : ; M (2.1) u i free j = 1 ; : : : ; N v i free i = 1 ; : : : ; M : The dual problem of LP(SP( K )), sa y D(SP( K )), is similar to D(SP) except that with resp ect to constrain t set ( 2.1 ) it only includes the constrain ts corresp onding to columns in the subset K W e denote the optimal solution to problem D(SP( K )) b y f u ¤ ( K ) ; v ¤ ( K ) g and the optimal solution to LP(SP( K )) b y ¤ ( K ). W e sa y ¤ is an PAGE 35 35 extension of ¤ ( K ) if ¤ is obtained from ¤ ( K ) b y setting the v alue of all column v ariables not in the set K to zero. Then if f u ¤ ( K ) ; v ¤ ( K ) g satises all constrain ts in D(SP), the extension of ¤ ( K ) will b e an optimal solution to LP(SP). Otherwise, an y violated constrain t from constrain t set ( 2.1 ) in D(SP) will yield a column that is lik ely to impro v e the curren t solution to LP(SP) if added to K The remaining question is ho w to ecien tly c hec k the feasibilit y of the dual solution f u ¤ ( K ) ; v ¤ ( K ) g for problem D(SP). This can b e done implicitly b y nding the most violated constrain ts from ( 2.1 ) for eac h agen t i b y solving the follo wing socalled pricing problems: minimize H i ( z ) ¡ N X j =1 u ¤j ( K ) z j + v ¤ i ( K ) sub ject to: z j 2 f 0 ; 1 g j = 1 ; : : : ; N z 2 X i : W e denote the pricing problem for agen t i and column set K b y PP i ( K ). Since the structure of the pricing problem for dieren t agen ts is iden tical, w e will for clarit y of notation often ignore the index i in the pricing problem throughout this dissertation. If the optimal solutions to all pricing problems ha v e a nonnegativ e v alue, then all dual constrain ts in D(SP) are satised and w e ha v e obtained an optimal solution to LP(SP). Otherwise, an y feasible solution to a pricing problem with negativ e ob jectiv e function v alue corresp onds to a column that could en ter the basis if added to K (when using the simplex metho d starting from the solution ¤ ( K )). Put dieren tly if w e add this column to LP(SP( K )) the reduced cost of the corresp onding v ariable is negativ e at ¤ ( K )). There are sev eral strategies for adding new columns. W e could only add the solution among all optimal solutions to the M pricing problems whose ob jectiv e function v alue is most negativ e. PAGE 36 36 Alternativ ely w e could add the optimal solutions to all pricing problems whose v alue is negativ e, or w e ma y ev en add all or some of the solutions with negativ e ob jectiv e function v alue found during the solution of the pricing problems. There is no theoretical result concerning the p erformance of dieren t strategies for selecting the column(s) to add, and w e will decide exp erimen tally whic h strategy to follo w. With new columns b eing generated in eac h iteration, the size of LP(SP( K )) ma y b ecome v ery large. W e ma y therefore also include metho ds to delete columns from K that do no longer seem promising. One w a y is to set a threshold on the reduced cost. W e ma y then, in eac h iteration, eliminate columns whose reduced cost is larger than the preset threshold. Alternativ ely w e migh t only use that strategy for eliminating columns whenev er the total n um b er of columns exceeds some preset limit. The column generation metho d for solving LP(SP) can b e summarized as follo ws: Column Generation Step 0. Set K = K 0 Step 1. Solv e LP(SP( K )), and denote the optimal primal solution b y ¤ ( K ) and the corresp onding optimal dual solution b y f u ¤ ( K ) ; v ¤ ( K ) g Step 2. Solv e PP i ( K ) for eac h agen t i If all these pricing problems ha v e a nonnegativ e optimal ob jectiv e function v alue, stop and extend ¤ ( K ) to obtain an optimal solution to LP(SP). Otherwise, generate a new column (or set of columns) and add it to subset K Eliminate some columns if desired. Return to step 1. F rom the ab o v e discussion, w e kno w that in the column generation metho d w e need to v ery frequen tly solv e a pricing problem: M pricing problems need to b e solv ed in eac h iteration, and it ma y tak e man y iterations for the metho d to obtain the optimal solution to LP(SP). Th us, the eectiv eness of the column generation PAGE 37 37 metho d relies hea vily on our abilit y to solv e the pricing problem ecien tly W e will dev elop ecien t solution metho ds for solving the pricing problems asso ciated with sev eral applications of (AP) in the remainder of this dissertation. As w e ha v e stated b efore, the initial set of columns K 0 can b e obtained from solution of the greedy heuristic or VLSN metho d. In case the greedy heuristic only nds a partial solution, the p enalized VLSN ma y help us to nd a feasible solution. Ho w ev er, there will still exist cases in whic h ev en the p enalized VLSN cannot nd a feasible solution. In suc h cases, w e ma y further apply the idea of p enalization to the column generation metho d, yielding a p enalized column generation metho d. In particular, w e will create an initial set of columns K 0 from an infeasible solution in the same w a y as a starting solution for the p enalized VLSN w as obtained. W e ma y then let the function v alue H i ( y k i ¢ ) = ¡1 for ev ery infeasible column y k i ¢ 2 K 0 Since the column generation metho d is an exact metho d and it will reac h an optimal solution if one exists, it is certain that after some iterations all v ariables corresp onding to the infeasible columns will b e 0 in the optimal solution for the master problem, that is, a feasible solution for LP(SP) is obtained. 2.4.4 Branc hing The general outline of the branc handprice metho d for (AP) is the same as the w ellkno wn branc handb ound metho d. Ho w ev er, there are t w o ma jor dierences. First, the branc handprice metho d applies to the (SP) where the column generation metho d is used to solv e the relaxation of (SP). Second, when the optimal solution of LP(SP) is fractional and branc hing is therefore needed to obtain an optimal in teger solution, w e branc h according to the original v ariables x ij instead of the column v ariables ki [ 12 ]. In the remainder of this section, w e will discuss the details of the branc hing and b ounding steps of the branc handprice in details. PAGE 38 38 Applying the standard branc handb ound metho d w e w ould set some fractional ki to 0 or 1 to create t w o branc hes of the curren t no de in the branc handb ound tree. Since ki represen ts a particular column, this means that this column is either excluded or xed in the solution. Ho w ev er, in the former case it is quite p ossible that the optimal solution to the i th pricing problem is giv en b y this column. In that case, in order to solv e the relaxation of the subproblem, w e need to nd the second b est solution to this pricing problem. A t a depth of n in the branc handb ound tree w e ma y ev en need to nd the n th b est solution. Although this is no problem in theory ecien t algorithms for nding the n th b est solution to the pricing problem are unlik ely to exist. Therefore, w e instead c ho ose to branc h on fractional v alues of the original v ariables x ij instead of the set partitioning v ariables ki [ 12 ]. After solving the relaxed problem and transforming to x w e can branc h on fractional v ariables x ij F or example, w e could branc h on the v ariable whose v alue is closest to 0.5 or some other suitably c hosen v alue. As there are no theoretical results concerning the c hoice of branc hing strategy w e will decide on the appropriate v alue b y exp erimen tation. If w e branc h b y setting x ij = 1, all existing columns in the master problem that do not assign task j to agen t i are deleted and w e x v ariable z j = 1 in the i th pricing problem. On the other hand, if x ij = 0 all existing columns in the master problem that assign task j to subset i are deleted and w e x v ariable z j = 0 in the i th pricing problem. Note that, in general, the structure of the pricing pricing problem will remain (virtually) the same throughout the branc handb ound tree, p erhaps at the exp ense of a sligh t mo dication of the functions H i It is easy to see that when some z j = 0 the pricing problem will remain the same since w e can simply ignore the v ariable. In the case some z j = 1, ho w ev er, some mo dication of the function H i ma y b e needed, esp ecially when the function H i is nonlinear since otherwise only some constan t terms will b e added to the function. PAGE 39 39 But for the applications discussed in this dissertation these mo dications will not c hange the structure of the pricing problem, ev en in the second case when functions H i are nonlinear since the structure of the pricing problem exists in a broad class of problems. When computational time is limited so that it is undesirable for the branc handprice algorithm to explore the c hild no des of the searc h tree, w e ma y c ho ose to terminate the algorithm after nding the optimal solution to LP(SP) at the ro ot no de. In order to nd an in tegral solution to LP(SP) w e can resolv e the master problem with the in tegralit y constrain ts enforced to obtain a heuristic solution for (SP) and th us (AP). This heuristic algorithm will b e v ery useful esp ecially when the ob jectiv e function of (AP) is nonlinear and b oth the greedy heuristic and VLSN algorithm cannot pro vide a feasible solution. Although this heuristic algorithm do es not guaran tee that a feasible solution is found, it still is a v ery promising approac h in general since if it nds a feasible solution, this solution will b e at least as go o d as the solution obtained b y the greedy or VLSN heuristics (assuming that none of the columns corresp onding to an initial feasible solution w ere deleted b y the column generation metho d). 2.5 Summary In this c hapter, w e ha v e discussed a general class of assignmen t problems (AP) as w ell as a set partitioning reform ulation (SP) of suc h problems. W e ha v e prop osed three approac hes for solving suc h problems, including a greedy heuristic, a VLSN impro v emen t metho d, and a branc handprice algorithm. Ho w ev er, due to the generalit y of the problems discussed in this c hapter these descriptions only pro vide a general framew ork for solving particular instances of (AP). Therefore, in the next c hapters w e will discuss sev eral t yp es of problems from the class (AP), and study the issues related to the structure and prop erties of the functions H i that remain. In particular, w e will construct sp ecic greedy heuristics, algorithms PAGE 40 40 for ecien t ev aluation of the functions H i and solution approac hes for the pricing problems for v arious applications of assignmen t problems o ccurring in supply c hain optimization. PAGE 41 CHAPTER 3 MUL TIPERIOD SINGLESOUR CING PR OBLEM (MPSSP) 3.1 In tro duction This c hapter (as w ell as the next one) explores a class of tactical pro duction and distribution net w ork design problems for impro ving crossfacilit y planning in supply c hains. In this problem class, m ultiple pro duction facilities pro duce a single item, whic h subsequen tly m ust b e transp orted from eac h facilit y to a set of retailers to satisfy customer demands. The goal is allo cate eac h retailer's demand to a serv er facilit y while minimizing systemwide a v erage pro duction, holding, and assignmen t costs p er unit time. T ypical singleitem pro duction planning problems consider only a single pro duction facilit y and a single retailer whose demand needs to b e satised. The t w o most widely studied and applied mo dels in this area are the Economic Lot Sizing (ELS) problem [ 53 ], and the Economic Order Quan tit y (EOQ) mo del [ 30 ]. The former mo dels the pro duction planning and distribution problem in discrete time, and allo ws for a timev arying demand stream at the retailer. The latter is a con tin uoustime mo del that assumes that the retailer faces a constan t demand rate. In b oth mo dels the goal is to nd a pro duction plan that minimizes total pro duction and in v en tory holding costs. In the EOQ mo del, v ariable pro duction costs are assumed to b e constan t o v er time, and can th us b e ignored since they are indep enden t of the pro duction planning decisions. Similarly man y singlefacilit y singleretailer mo dels ignore costs related to shipping items to the retailer, although certain extensions of the basic mo dels that include transp ortation costs ha v e recen tly b een studied [ 15 16 31 33 ]. This c hapter considers a more broadly applicable setting where m ultiple upstream or pro duction facilities can pro duce an item, and collectiv ely need 41 PAGE 42 42 to meet the demands of m ultiple do wnstream facilities or retailers. F or this problem class, w e need to sim ultaneously plan pro duction in eac h of the facilities to minimize total pro duction and in v en tory holding costs, as w ell as an y additional costs (suc h as transp ortation costs) asso ciated with satisfying retailer demand. In particular, w e will study generalizations of the m ultip erio d singlesourcing problem (MPSSP), a mo del that can b e view ed as a generalization of the singleitem, singlefacilit y ELS mo del and w as in tro duced as a to ol for ev aluating logistics distribution net w ork designs with resp ect to costs in a dynamic en vironmen t b y Romeijn and Romero Morales [ 43 42 41 ] and Romero Morales [ 44 ]. Moreo v er, this mo del is a sp ecial case of the general class of assignmen t problems (AP) that w e studied in Chapter 2 (as w ell as of the class of CCAP studied in Romero Morales [ 44 ]. In this problem, w e consider a logistics distribution net w ork consisting of facilities and retailers. Pro duction and storage tak es place at the facilities, and the retailers' demand patterns for a single pro duct are assumed kno wn. This mo del is suitable for tactical use, with a xed starting p erio d and horizon, as w ell as for more strategic purp oses, b y assuming that the planning p erio d is a t ypical future one, and will rep eat itself o v er time. In the latter case, this means that the mo del is cyclic in nature. W e assume that there is no transp ortation b et w een the facilities. In addition, w e do not allo w for in v en tories at the retailers. This situation is t ypical in, for instance, the fo o d and b ev erage industry where the retailers often are sup ermark ets and restauran ts, whic h usually ha v e v ery limited storage capacit y Although in practice a giv en retailer ma y o ccasionally receiv e shipmen ts from m ultiple pro duction facilities, practical distribution systems often emplo y a singlesourcing strategy that is, a strategy in whic h retailers are deliv ered b y a single facilit y The singlesourcing strategy oers sev eral practical adv an tages, including reduced managerial co ordination complexit y and a decreased need for information systems in tegration b et w een source facilities. W e th us consider PAGE 43 43 con texts in whic h this singlesourcing strategy is explicitly imp osed based on managerial and administrativ e requiremen ts, as w ell as service considerations. The decisions that need to b e made are (i) the assignmen t of retailers to facilities, (ii) the timing of pro duction, and (iii) the lo cation and size of in v en tories. As in the basic MPSSP in tro duced b efore, w e assume that eac h facilit y has kno wn, nite, and p ossibly timev arying, pro duction capacit y Ho w ev er, in con trast to earlier mo dels, w e allo w v arious additional t yp es of constrain ts. T o accoun t for the fact that w arehouse capacities are limited, w e include ph ysical in v en tory capacit y constrain ts. Throughput capacit y constrain ts are included to accoun t for situations in whic h op erational constrain ts limit the amoun t of pro ducts that can o w through a facilit y in a particular p erio d. Finally man y go o ds are p erishable, either due to a ph ysically limited lifetime, or due to fashion considerations. T o accoun t for this, w e allo w for a constrain t on the n um b er of p erio ds that a go o d is stored at a facilit y b efore b eing transp orted to the retailer. Since ev en the problem of determining whether there exists a feasible solution to the basic MPSSP with pro duction capacities only is NPcomplete [ 35 45 ], w e fo cus on heuristic approac hes to this problem. That is, w e will dev elop a greedy heuristic and apply the VLSN algorithm. F reling et al. [ 25 ] studied the branc handprice algorithm for MPSSP W e will prop ose a sto c hastic mo del on the problem data, and deriv e explicit asymptotic feasibilit y conditions for the case where the retailer demands exhibit a common seasonalit y pattern. In addition to suggesting an approac h for randomly generating problem instances, w e also presen t a greedy heuristic, and pro v e that it is asymptotically feasible and optimal in a probabilistic sense when the n um b er of retailers gro ws large, under this sto c hastic data mo del. W e next emplo y the V ery LargeScale Neigh b orho o d (VLSN) Searc h heuristic to impro v e the qualit y of the greedy solution for nite problem sizes. W e also presen t ecien t solution metho ds for the subproblems. PAGE 44 44 Related literature to date has fo cused mainly on static mo dels [ 28 13 23 ]. Duran [ 20 ] studies a dynamic mo del for the planning of pro duction, b ottling, and distribution of b eer, but fo cuses on the pro duction pro cess, and Chan, Muriel and Simc hiLevi [ 18 ] study a dynamic, but uncapacitated, distribution problem in an op erational setting. 3.2 Problem and F orm ulation 3.2.1 T raditional F orm ulation Let M denote the n um b er of facilities, N the n um b er of retailers, and T b e the giv en planning horizon. The facilities p erform b oth a storage and a pro duction task. With resp ect to the latter, they face nite pro duction capacities in eac h p erio d, giv en b y b it ( i = 1 ; : : : ; M ; t = 1 ; : : : ; T ). The demand of retailer j in p erio d t for a single pro duct is giv en b y d j t ( j = 1 ; : : : ; N ; t = 1 ; : : : ; T ). The pro duction costs are assumed linear, with the unit pro duction costs at facilit y i in p erio d t denoted b y p it ( i = 1 ; : : : ; M ; t = 1 ; : : : ; T ). The singlesourcing asp ect of the mo del requires that the demand of a retailer is satised b y a single facilit y only The total transp ortation costs for supplying retailer j b y facilit y i throughout the planning horizon is giv en b y c ij P Tt =1 c ij t ( d j t ), where c ij t is an arbitr ary transp ortation cost function ( i = 1 ; : : : ; M ; j = 1 ; : : : ; N ; t = 1 ; : : : ; T ). The unit in v en tory holding costs at facilit y i in p erio d t are giv en b y h it ( i = 1 ; : : : ; M ; t = 1 ; : : : ; T ). All parameters are nonnegativ e b y denition. The decisions to b e made are the assignmen t of retailers to facilities, as w ell as the pro duction quan tities and in v en tory lev els at the facilities. T o this end, let x ij b e equal to 1 if retailer j is assigned to facilit y i and 0 otherwise. In addition, let y it represen t the pro duction quan tit y at facilit y i in p erio d t and I it the quan tit y in storage at facilit y i at the end of p erio d t Hereafter x 2 R M £ N will denote the v ector with comp onen ts x ij and similarly for y ; I 2 R M £ T PAGE 45 45 The basic MPSSP can no w b e form ulated as follo ws: minimize T X t =1 M X i =1 p it y it + M X i =1 N X j =1 c ij x ij + T X t =1 M X i =1 h it I it sub ject to y it b it i = 1 ; : : : ; M ; t = 1 ; : : : ; T (3.1) N X j =1 d j t x ij + I it = y it + I i;t ¡ 1 i = 1 ; : : : ; M ; t = 1 ; : : : ; T (3.2) M X i =1 x ij = 1 j = 1 ; : : : ; N (3.3) I i 0 = 0 i = 1 ; : : : ; M (3.4) x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N (3.5) y it ; I it 0 i = 1 ; : : : ; M ; t = 1 ; : : : ; T : The constrain ts ( 3.1 ) mo del the pro duction capacit y constrain ts, and ( 3.2 ) mo del the in v en tory balance constrain ts. Constrain ts ( 3.3 ) (together with ( 3.5 )) enforce the singlesourcing constrain ts, that is, that eac h retailer is assigned to exactly one facilit y Constrain ts ( 3.4 ) mo del the presence of initial in v en tory whic h is generally assumed to b e zero. A p ositiv e in v en tory can of course easily b e incorp orated. Note that the nonnegativit y of the in v en tory holding costs implies that, without loss of optimalit y the ending in v en tories will b e equal to zero: I iT = 0 for i = 1 ; : : : ; M as is common in standard economic lot sizing problems. Despite the undesirable endofstudy eect, this ma y b e an adequate mo deling of a shortterm, op erational system. Ho w ev er, in longerterm, tactical or strategic, studies ev aluating the p erformance of a logistics net w ork, it ma y b e undesirable to x in adv ance the starting and ending in v en tories. W e therefore consider a v arian t of our mo dels that assumes that the planning horizon of T p erio ds represen ts a t ypical future planning cycle that will rep eat itself. In that case, it is reasonable to imp ose PAGE 46 46 the follo wing constrain ts, replacing ( 3.4 ): I i 0 = I iT i = 1 ; : : : ; M : ( 3.4 0 ) The mo dels using constrain ts ( 3.4 ) will b e called acyclic whereas w e will call the mo dels using constrain t ( 3.4 0 ) cyclic In the remainder of this c hapter, w e will assume that there are no sp eculativ e motiv es in the pro duction and in v en tory costs, that is, p i [ t +1] p it + h it for all i = 1 ; : : : ; M ; t = 1 ; : : : ; T ¡ 1 in the acyclic case, and in addition for t = T in the cyclic case, where [ t ] ( t ¡ 1) mo d T + 1. In particular, this means that, capacities p ermitting, demand should alw a ys b e satised b y pro duction in recen t p erio ds, a v oiding holding in v en tories as m uc h as p ossible. In con trast with earlier mo dels, w e will allo w for v arious additional t yp es of capacit y constrain ts that often pla y a role in practice. (i) Thr oughput c onstr aints Op erational limitations often constrain the quan tit y of go o ds that can b e handled at a particular facilit y during a giv en time p erio d. Assuming that the nite thr oughput c ap acity at facilit y i in p erio d t is giv en b y r it w e can mo del the corresp onding constrain ts as N X j =1 d j t x ij r it i = 1 ; : : : ; M ; t = 1 ; : : : ; T : (3.6) (ii) Physic al inventory c onstr aints A nite storage capacit y of I it at facilit y i in p erio d t can easily b e mo deled b y including the follo wing constrain ts: I it I it i = 1 ; : : : ; M ; t = 1 ; : : : ; T : (3.7) It will b e con v enien t to dene I i 0 = 0 for i = 1 ; : : : ; M (iii) Perishability c onstr aints PAGE 47 47 T o accoun t for the p erishable nature of go o ds, w e ma y constrain the n um b er of p erio ds that a go o d is stored at a facilit y In particular, denoting the maxim um n um b er of p erio ds that a pro duct can b e stored b y k w e obtain the follo wing constrain ts in the acyclic case: I it min( T ;t + k ) X = t +1 N X j =1 d j x ij i = 1 ; : : : ; M ; t = 1 ; : : : ; T : (3.8) In the cyclic case, these constrain ts b ecome: I it t + k X = t +1 N X j =1 d j [ ] x ij i = 1 ; : : : ; M ; t = 1 ; : : : ; T : ( 3.8 0 ) P erishabilit y constrain ts ha v e b een considered in in v en tory con trol, but they can hardly b e found in the literature on in tegrated pro duction and distribution planning. A notable exception is My ers [ 37 ] who prop oses a linear programming mo del to determine the maxim um demand that a compan y dealing with p erishabilit y issues can accommo date. W e will refer to the MPSSP with all capacit y constrain ts added as (P), and denote its LPrelaxation b y (LP). 3.2.2 Assignmen t F orm ulation W e can reform ulate the MPSSP as an assignmen t problem as follo ws [ 43 42 ]: minimize M X i =1 H i ( x i ¢ ) sub ject to (AP 1 ) N X j =1 d j t x ij r it i = 1 ; : : : ; M ; t = 1 ; : : : ; T (3.9) M X i =1 x ij = 1 j = 1 ; : : : ; N x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N PAGE 48 48 where H i ( z ), for z 2 R N+ is the optimal solution v alue of the follo wing subproblem in the acyclic case: minimize N X j =1 c ij z j + T X t =1 p it y t + T X t =1 h it I t sub ject to (H Ai ) y t b it t = 1 ; : : : ; T N X j =1 d j 1 z j + I 1 = y 1 (3.10) N X j =1 d j t z j + I t = y t + I t ¡ 1 t = 2 ; : : : ; T (3.11) I t min 0@ I it ; min( T ;t + k ) X = t +1 n X j =1 d j z j 1A t = 1 ; : : : ; T (3.12) y t ; I t 0 t = 1 ; : : : ; T : In the cyclic case, w e obtain the subproblem (H Ci ) b y replacing ( 3.10 ){( 3.12 ) b y N X j =1 d j t z j + I t = y t + I [ t ¡ 1] t = 1 ; : : : ; T I t min I it ; t + k X = t +1 n X j =1 d j [ ] z j t = 1 ; : : : ; T : Notice that for an y giv en assignmen t z the rst term of the ob jectiv e function is constan t and th us can b e eliminated from the ob jectiv e function. W e will often simply refer to the subproblem (H i ), where the con text dictates whether this denotes (H Ai ) or (H Ci ) (or b oth). Clearly the subproblem (H i ) ma y b e infeasible for some z 2 R N+ whic h w ould yield H i ( z ) = 1 Alternativ ely w e ma y imp ose additional constrain ts on (AP 1 ) to ensure that the corresp onding subproblem (H i ) is feasible for all feasible assignmen ts x As w e will see later, it often turns out to b e computationally more ecien t to deal with the infeasibilit y of (H i ) in the former w a y Ho w ev er, it nev ertheless is of indep enden t in terest to explicitly PAGE 49 49 c haracterize the feasible assignmen t v ectors. These c haracterizations will help us design a probabilistic mo del for the problem parameters for computational testing. In addition, some of the c haracterizations will suggest ecien t solution approac hes for the subproblems. In the remainder of this section, w e will study suc h feasibilit y conditions for b oth the acyclic and the cyclic v arian ts of MPSSP W e will refer to the set of v ectors z 2 R N+ for whic h H i ( z ) < 1 (or, equiv alen tly (H i ) is feasible), as the domain of H i 3.3 Subproblems 3.3.1 Domain of the Subproblem The subproblems (H i ) are actually capacitated minim um cost net w ork o w problems after eliminate the rst constan t term when the assignmen t z is giv en. The corresp onding graphs con tain a single supply no de (denoted b y S ), as w ell as T demand no des (denoted b y 1 ; : : : ; T ). Demand no de t has demand equal to the total demand in p erio d t of all retailers assigned to facilit y i that is, P Nj =1 d j t z j and the supply no de has supply equal to the sum of all demands: P Tt =1 P Nj =1 d j t z j There are (pro duction) arcs of the form (0 ; t ) for all t = 1 ; : : : ; T with costs p it and capacities b it In addition, in the acyclic case there are (inv en tory) arcs of the form ( t; t + 1) for all t = 1 ; : : : ; T ¡ 1, with costs h it and capacities min I it ; P min ( T ;t + k ) = t +1 P Nj =1 d j z j In the cyclic case, there are in v entory arcs of the form ( t; [ t + 1]) for all t = 1 ; : : : ; T with costs h it and capacities min I it ; P t + k = t +1 P Nj =1 d j [ ] z j Figures 3{1 and 3{2 illustrate the graphs corresp onding to b oth v arian ts of the problem, where the arc lab els denote the capacities (using the notation of Theorems 3 and 4 ), and the arc costs are omitted. The follo wing theorems c haracterize the domain of the function H i for b oth the acyclic and cyclic case. The results of these theorems w ere already pro v en b y Romero Morales [ 44 ]; ho w ev er, our alternativ e pro ofs are more concise due to the fact that the net w ork o w structure of the subproblems is emplo y ed. PAGE 50 50 S 1 2 3 4 5 6 1 2 3 4 5 6 b 1 b 2 b 3 b 4 b 5 b 6 1 2 3 4 5 S t t Figure 3{1: Acyclic case, T = 6 1 3 5 7 2 8 4 6 S 1 2 3 4 5 6 7 8 S t t 1 2 3 4 5 6 7 8 b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 Figure 3{2: Cyclic case, T = 8 PAGE 51 51 Theorem 3 In the acyclic c ase, the domain of the function H i c onsists of al l z 2 R n+ satisfying + r X t = N X j =1 d j t z j + r X t = b it + I i; ¡ 1 = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ (3.13) + r X t = + k N X j =1 d j t z j + r X t = b it = 2 ; : : : ; T ¡ k ; r = k ; : : : ; T ¡ : (3.14) Pro of: F or con v enience, w e will dene D t = N X j =1 d j t z j t = 1 ; : : : ; T I t = min 0@ I it ; min ( T ;t + k ) X = t +1 N X j =1 d j z j 1A t = 1 ; : : : ; T : Theorem 6.12 in Ah uja et al. [ 4 ] giv es a general necessary and sucien t condition for feasibilit y of capacitated net w ork o w problems. F or our problem, these conditions reduce to: X t 2 S D t X t 2 S b it + X t : t 62 S ; t +1 2 S I t for all S f 1 ; : : : ; T g (3.15) ¡ X t 2 S D t X t : t 2 S ; t +1 62 S I t for all S f 1 ; : : : ; T g : (3.16) It is clear that ( 3.16 ) is redundan t. With resp ect to ( 3.15 ), note that w e can restrict ourselv es to subsets S con taining consecutiv e in tegers, since all other conditions can b e obtained b y adding the conditions for suc h subsets. This means that (H Ai ) is feasible if and only if + r X t = D t + r X t = b it + I ¡ 1 = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ : Returning to the original notation, this yields the condition + r X t = N X j =1 d j t z j + r X t = b it + min 0@ I i; ¡ 1 ; min ( T ; + k ¡ 1) X t = N X j =1 d j t z j 1A = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ PAGE 52 52 whic h is equiv alen t to the pair of conditions + r X t = N X j =1 d j t z j + r X t = b it + I i; ¡ 1 = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ (3.17) + r X t = N X j =1 d j t z j + r X t = b it + min( T ; + k ¡ 1) X t = N X j =1 d j t z j = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ : (3.18) No w note that conditions ( 3.17 ) are precisely conditions ( 3.13 ). F or = 1 ; : : : ; T ¡ k and r = k ; : : : ; T ¡ condition ( 3.18 ) can b e rewritten as + r X t = + k N X j =1 d j t z j + r X t = b it (3.19) while for > T ¡ k or r < k it can easily b e seen to b e redundan t. Finally for = 1 ( 3.19 ) is implied b y condition ( 3.17 ) for = 1. This completes the pro of. A similar result holds for the domain of the function H i in the cyclic case. Theorem 4 In the cyclic c ase, the domain of the function H i c onsists of al l z 2 R n+ satisfying + r X t = N X j =1 d j [ t ] z j + r X t = b i [ t ] + I i [ ¡ 1] = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ 2 (3.20) T X t =1 N X j =1 d j t z j T X t =1 b it (3.21) + r X t = + k N X j =1 d j [ t ] z j + r X t = b i [ t ] = 1 ; : : : ; T ; r = k ; : : : ; T ¡ 2 : (3.22) Pro of: The pro of is analogous to the pro of of Theorem 3 where w e in terpret the term c onse cutive in a cyclic manner. Similarly to Theorem 3 dene D t = N X j =1 d j t z j t = 1 ; : : : ; T I t = min I it ; t + k X = t +1 N X j =1 d j [ ] z j t = 1 ; : : : ; T : PAGE 53 53 Using again Theorem 6.12 in Ah uja et al. [ 4 ], the necessary and sucien t conditions for feasibilit y for (H Ci ) reduce to: X t 2 S D t X t 2 S b it + X t : t 62 S ; [ t +1] 2 S I t for all S f 1 ; : : : ; T g (3.23) ¡ X t 2 S D t X t : t 2 S ; [ t +1] 62 S I t for all S f 1 ; : : : ; T g : (3.24) It is clear that ( 3.24 ) is redundan t. With resp ect to ( 3.23 ), note that w e can restrict ourselv es to subsets S con taining in tegers that are consecutiv e in a cyclic manner, since all other conditions can b e obtained b y adding the conditions for suc h subsets. Consecutiv e in a cyclic manner means that 1 follo ws T so that, for instance, f T ¡ 1 ; T ; 1 ; 2 ; 3 g is a set of consecutiv e in tegers. This means that (H Ci ) is feasible if and only if T X t =1 D t T X t =1 b it + r X t = D [ t ] + r X t = b i [ t ] + I [ ¡ 1] = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ 2 : Returning to the original notation, the rst condition coincides with ( 3.21 ). The second set of conditions yield + r X t = N X j =1 d j [ t ] z j + r X t = b i [ t ] + min I i [ ¡ 1] ; + k ¡ 1 X t = N X j =1 d j [ t ] z j = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ 2 whic h is equiv alen t to the pair of conditions + r X t = N X j =1 d j [ t ] z j + r X t = b i [ t ] + I i [ ¡ 1] = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ 2 (3.25) + r X t = N X j =1 d j [ t ] z j + r X t = b i [ t ] + + k ¡ 1 X t = N X j =1 d j [ t ] z j = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ 2 : (3.26) PAGE 54 54 No w note that conditions ( 3.25 ) are precisely conditions ( 3.20 ). F or = 1 ; : : : ; T and r = k ; : : : ; T ¡ 2 condition ( 3.26 ) can b e rewritten as + r X t = + k N X j =1 d j [ t ] z j + r X t = b i [ t ] (3.27) while for r < k it can easily b e seen to b e redundan t. This completes the pro of. W e ha v e c haracterized the domain of the subproblems for b oth acyclic and cyclic cases. Next w e will describ e ho w these subproblems can b e solv ed ecien tly since w e need to solv e a large amoun t of the subproblems in VLSN to compute the arc costs in the impro v emen t graph. 3.3.2 Acyclic Case As sho wn in Section 3.2.2 the problem (H Ai ) is actually a capacitated minim um cost net w ork o w problem in a directed acyclic graph. By the assumption that there are no sp eculativ e motiv es in the pro duction and in v en tory costs, the optimal solution will a v oid using the in v en tory arcs as m uc h as p ossible. Therefore, w e will pro duce the demand of eac h demand no de as late as p ossible, thereb y using as little in v en tory as p ossible. The optimal c andidate in v en tories, for no w disregarding the in v en tory capacit y constrain ts, can then b e found recursiv ely as follo ws: I t = 8><>: 0 for t = T max 0 ; P Nj =1 d j ;t +1 z j + I t +1 ¡ b i;t +1 for t = T ¡ 1 ; : : : ; 0 : (3.28) If all in v en tory lev els satisfy the capacit y constrain ts (including I 0 = 0), this solution is feasible and therefore optimal. On the other hand, if one or more in v en tory capacities are violated, the problem do es not ha v e a feasible solution. This follo ws directly from the fact that, b y construction, the solution giv en ab o v e uses as little in v en tory as p ossible. (Note that the o ws on the arcs of the form (0 ; t ) are simply equal to P Nj =1 d j t z j + I t ¡ I t ¡ 1 whic h satisfy the capacit y constrain ts PAGE 55 55 b y construction.) The running time of this pro cedure is O ( T ). Note that, in the case with pro duction capacities only w e ma y c hec k feasibilit y of the assignmen ts x i ¢ for facilit y i in O ( T ) time b y observing that only T of the constrain ts ( 3.13 )( 3.14 ) are relev an t. In the presence of other capacities, c hec king feasibilit y directly is computationally more exp ensiv e than applying the recursion ( 3.28 ) and c hec king for feasibilit y of the candidate solution. In addition to the running time of the pro cedure, w e also need to compute the demands, whic h tak es O ( N T ) time when computed from scratc h to initialize the impro v emen t graph. Ho w ev er, after nding an impro v ed neigh b or b y using VLSN, up dating all demands for facilit y i tak es only O ( T ) time, since at most one retailer lea v es the facilit y and at most one retailer en ters the facilit y 3.3.3 Cyclic Case In the cyclic case, note that without loss of optimalit y w e can assume that min t =1 ;::: ;T I t = 0 : W e prop ose to solv e the problems (H Ci ) b y for eac h t = 1 ; : : : ; T xing I t = 0 and treating p erio d t as the `last' planning p erio d. W e can then use the bac kw ard recursion from Section 3.3.2 to obtain T candidate solutions. If at least one of these is feasible for (H Ci ), the c heap est one among these is the optimal solution. Otherwise, (H Ci ) is infeasible. The complexit y of this pro cedure is O ( T 2 ). In the remainder of this section, w e will sho w that in the absence of ph ysical in v en tory and p erishabilit y constrain ts, w e can actually impro v e this complexit y result, and solv e the problem (H Ci ) in O ( T ) time. F or ease of exp osition, w e will alw a ys consider p erio ds in a cyclic manner, that is, 1 ; : : : ; T ; 1 ; : : : etc. F or example, t = 5 ; : : : ; 2 will mean that t tak es on the v alues 5 ; 6 ; : : : ; T ; 1 ; 2. PAGE 56 56 As in Theorem 4 dene D t = N X j =1 d j t x ij t = 1 ; : : : ; T : Then dene the partial sums of residual capacit y starting at the base p erio d 1, as follo ws: ¢ t = t X =1 b ¡ t X =1 D t = 1 ; : : : ; T : It is clear that the problem (H Ci ) is feasible if and only if ¢ T 0, that is, total supply is no smaller than total demand. The follo wing lemma nds a dieren t base p erio d, with resp ect to whic h all residual capacities are nonnegativ e. Lemma 5 L et s 2 arg min t =1 ;::: ;T ¢ t Dene ¢ 0t = t X = s +1 b ¡ t X = s +1 D t = 1 ; : : : ; T : Then ¢ 0t 0 for al l t = 1 ; : : : ; T Pro of: It is easy to see that ¢ 0t = ¢ t + ¢ T ¡ ¢ s 0 for t = 1 ; : : : ; s ¢ 0t = ¢ t ¡ ¢ s 0 for t = s + 1 ; : : : ; T : The follo wing theorem no w sho ws that, without loss of optimalit y w e can c ho ose p erio d s as dened in Lemma 5 to b e the `last' planning p erio d. Theorem 6 L et s 2 arg min t =1 ;::: ;T ¢ t Then, without loss of optimality, we c an assume that I s = 0 Pro of: W e will sho w the result b y con tradiction. Let ( y ¤ ; I ¤ ) b e an optimal solution to the problem (H Ci ). Without loss of optimalit y w e ma y assume that there exists at least one t suc h that I ¤ t = 0. No w supp ose that I ¤ s = > 0. Let t 1 b e the last p erio d b efor e p erio d s suc h that I ¤ t 1 = 0, and let t 2 b e the rst p erio d after PAGE 57 57 p erio d s suc h that I ¤ t 2 = 0 (it ma y b e that t 1 = t 2 ). Since I ¤ t 1 = 0 and I ¤ s = w e kno w that s X t = t 1 +1 y ¤ t = s X t = t 1 +1 D t + that is, the total pro duction in p erio ds t 1 + 1 ; : : : ; s exceeds the total demand in these p erio ds b y Similarly since I ¤ t 2 = 0, w e kno w that t 2 X t = s +1 y ¤ t = t 2 X t = s +1 D t ¡ that is, the total pro duction in p erio ds s + 1 ; : : : ; t 2 falls short of the total demand in these p erio ds b y No w note that, b y Lemma 5 t X = s +1 b t X = s +1 D for all t = s + 1 ; : : : ; t 2 that is, the total pro duction capacit y in p erio ds s + 1 ; : : : ; t is at least equal to the demand in these p erio ds for all t = s + 1 ; : : : ; t 2 W e can no w conclude that w e can nd another feasible solution b y decreasing the aggregate pro duction in the p erio ds t 1 + 1 ; : : : ; s b y decreasing I ¤ s to zero, and increasing the aggregate pro duction in p erio ds s + 1 ; : : : ; t 2 b y Since the pro duction and in v en tory costs exhibit no sp eculativ e motiv es, the cost of this mo died solution is no w orse than the solution w e started with. Th us, w e can assume without loss of optimalit y that I s = 0. Since the v alue of s in Theorem 6 can b e computed in O ( T ) time, and since the optimal solution to (H Ci ) giv en that I s = 0 can b e computed in O ( T ) time, (H Ci ) can b e solv ed in O ( T ) time. Note that, in the case with pro duction capacities only w e ma y c hec k feasibilit y of the assignmen ts x i ¢ for facilit y i in O (1) time b y observing that only the constrain t ( 3.21 ) is relev an t. In the presence of other PAGE 58 58 capacities, c hec king feasibilit y directly is computationally more exp ensiv e than applying the O ( T ) metho d outlined ab o v e. 3.4 Greedy Heuristic 3.4.1 Outline of the Greedy Heuristic Martello and T oth [ 34 ] prop osed a greedy heuristic for the Generalized Assignmen t Problem. This greedy heuristic w as impro v ed b y Romeijn and Romero Morales [ 40 ], who also sho w ed that their impro v emen t of the heuristic is asymptotically feasible and optimal with probabilit y one under a v ery general sto c hastic mo del for the problem instances. Romeijn and Romero Morales [ 43 42 ] subsequen tly generalized this heuristic and the analysis to the basic MPSSP as w ell as some extensions thereof. In this section w e will describ e a further generalization of the greedy heuristic to the MPSSP with the additional capacit y constrain ts describ ed ab o v e. In addition, w e will generalize results b y Romero Morales [ 44 ] and pro vide an asymptotic p erformance guaran tee for particular sto c hastic mo dels for the problem data. Using the assignmen t form ulation (AP 1 ) of the MPSSP the idea of the heuristic is to ev aluate eac h p ossible assignmen t using some pseudocost function f ( i; j ), whic h should measure the actual assignmen t costs, as w ell as the cost of using the limited capacities. F or eac h assignmen t to b e made, the dierence b et w een the second smallest and the smallest v alues of f ( i; j ) (called the desir ability of making the c heap est assignmen t with resp ect to the pseudocost) is computed, and assignmen ts are made in decreasing order of this dierence. Along the w a y the v alues of the desirabilities are up dated to tak e in to accoun t the fact that the reduction in capacities caused b y earlier assignmen ts mak es certain other assignmen ts infeasible. Greedy Heuristic Step 0. Set L = f 1 ; : : : ; N g and x G = 0. PAGE 59 59 Step 1. F or all j 2 L let F j = f i : retailer j can feasibly b e assigned to facilit y i giv en x G g : If F j = for some j 2 L : let L = L nf j g and rep eat Step 1. Otherwise, let i j 2 arg min i 2F j f ( i; j ) for j 2 L j = min s 2 F j s 6 = i j f ( s; j ) ¡ f ( i j ; j ) for j 2 L: Step 2. Let ^  2 arg max j 2 L j and set x Gi ^  ^  = 1 L = L n f ^  g : Step 3. If L = : STOP x G is a (partial) solution to (AP 1 ). Otherwise, go to Step 1. The output of this greedy heuristic is a v ector of assignmen ts x G whic h is either a full or a partial solution of the reform ulated problem (AP1). In the follo wing section, w e generalize the pseudocost functions prop osed earlier for the basic v ersion of the MPSSP to the case with additional capacit y constrain ts. 3.4.2 PseudoCost F unctions F ollo wing Romeijn and Romero Morales [ 43 42 ] and Romero Morales [ 44 ], w e emplo y the follo wing pseudocost function for (P): f ( i; j ) = 8><>: c ij + P Tt =1 ¤it + ¤ it ¡ P min( k ;t ¡ 1) ` =1 ¤ i;t ¡ ` d j t for the acyclic case c ij + P Tt =1 ¤it + ¤ it ¡ P k` =1 ¤ i [ t ¡ ` ] d j t for the cyclic case where ¤ ¤ and ¤ are, resp ectiv ely the v ectors of optimal dual m ultipliers for the pro duction capacit y constrain ts (equation ( 3.1 )), the throughput capacit y constrain ts (equation ( 3.6 )), and the p erishabilit y constrain ts (equation ( 3.8 ) or PAGE 60 60 ( 3.8 0 )) in the LPrelaxation of (P). Note that all capacit y constrain ts ha v e b een rewritten as \ "constrain ts, so that the m ultipliers are all nonnegativ e. Romeijn and Romero Morales [ 43 42 ] analyzed the greedy heuristic with the pseudocost function giv en ab o v e in the presence of pro duction capacit y constrain ts only Under a v ery general sto c hastic mo del for the data that yields feasible problem instances with probabilit y one, the greedy heuristic has b een sho wn to pro duce solutions that are asymptotically feasible and optimal with probabilit y one for the cyclic case. The same p erformance guaran tee holds in the acyclic case if the retailer demands share a common seasonalit y pattern. This result can b e extended to the MPSSP with throughput and ph ysical in v en tory capacities and p erishabilit y constrain ts, again when the retailer demands share a common seasonalit y pattern. Next w e will deriv e a suitable sto c hastic mo del for this case, that is then also used to generate problem instances for testing purp oses. 3.4.3 Sto c hastic Mo del for the Problem Data W e will follo w Romeijn and Romero Morales [ 43 42 41 ] and Romero Morales [ 44 ], and discuss a sto c hastic mo del for the problem data of the capacitated MPSSP 1 F or eac h j = 1 ; : : : ; N let ( D j ¢ ; C ¢ j ) b e i.i.d. random v ectors in [ D ; D ] T £ [ C ; C ] M (with D > 0), where D j ¢ = ( D j t ) t =1 ;::: ;T and C ¢ j = ( C ij ) i =1 ;::: ;M W e assume that the v ectors ( D j ¢ ; C ¢ j ) ( j = 1 ; : : : ; N ) are i.i.d. according to an absolutely con tin uous probabilit y distribution for eac h j = 1 ; : : : ; N Note that the demands and costs for a particular retailer are allo w ed to b e dep enden t. W e will often assume that the demands of all retailers share some common seasonalit y pattern, that is, D j t = t D j for a xed v ector of seasonalit y factors suc h that t 0 1 In the remainder of this dissertation, random v ariables will b e denoted b y capital letters, and their realizations b y the corresp onding lo w ercase letters. In addition, the sym b ol E will b e used to denote exp e ctation PAGE 61 61 for all t = 1 ; : : : ; T and P Tt =1 t = 1. The pro duction and in v en tory holding costs p it and h it are assumed to b e xed nonnegativ e constan ts. F or con v enience, let p = min i;t p it p = max i;t p it h = min i;t h it and h = max i;t h it T o allo w for sucien t capacit y as the n um b er of retailers gro ws, w e let all capacities b it ; r it and I it dep end linearly on N : b it = it N r it = it N I it = it N where it ; it and it are p ositiv e constan ts. This w a y of mo delling the capacities is customary in probabilistic mo dels for assignmen t problems [ 21 51 39 ]. Observ e that M and T are xed, th us the size of (P) only dep ends on the n um b er of retailers N 3.4.4 F easibilit y Condition In the presence of pro duction capacities only Romeijn and Romero Morales [ 43 ] sho w that acyclic instances of (P) generated using this mo del are feasible with probabilit y one if X t =1 E ( D 1 t ) < X t =1 M X i =1 it for = 1 ; : : : ; T and infeasible with probabilit y one if at least one of the inequalities is rev ersed. Similarly Romeijn and Romero Morales [ 42 ] sho w that the corresp onding condition is T X t =1 E ( D 1 t ) < T X t =1 M X i =1 it for the cyclic case. These conditions are closely related to the inequalities c haracterizing the domain of H i In the presence of throughput, ph ysical in v en tory and p erishabilit y constrain ts, feasibilit y conditions can b e deriv ed for the case where the retailer PAGE 62 62 demands share a common seasonalit y pattern. In this case, the capacit y constrain ts ( 3.9 ) and the additional constrain ts ensuring that the ob jectiv e function is nite, or the domain of H i then reduce to N X j =1 d j x ij B i N i = 1 ; : : : ; M where B i = min ( min t =1 ;::: ;T it t ; min = 1 ; : : : ; T r = 0 ; : : : ; T ¡ P + r t = it + i; ¡ 1 P + r t = t ; min = 2 ; : : : ; T ¡ k r = k ; : : : ; T ¡ P + r t = it P + r t = t ) in the acyclic case, and B i = min ( P Tt =1 it P Tt =1 t ; min t =1 ;::: ;T it t ; min = 1 ; : : : ; T r = 0 ; : : : ; T ¡ 2 P + r t = i [ t ] + i [ ¡ 1] P + r t = [ t ] ; min = 1 ; : : : ; T r = k ; : : : ; T ¡ 2 P + r t = i [ t ] P + r t = [ t ] !) in the cyclic case. This means that the feasible region of (AP 1 ) is in fact the feasible region of a Generalized Assignmen t Problem with agen tindep enden t requiremen ts. The feasibilit y of this problem w as studied b y Romeijn and Piersma [ 39 ]. The follo wing assumption ensures that problem instances generated according to the probabilistic mo del giv en ab o v e are asymptotically (as N 1 ) feasible with probabilit y one. Assumption 7 The normalize d aggr e gate c ap acity exc e e ds the exp e cte d demand p er r etailer, that is, E ( D 1 ) < M X i =1 B i : In addition, Romeijn and Piersma [ 39 ] sho w that asymptotic infe asibility is guaran teed with probabilit y one if the inequalit y in the assumption is rev ersed. PAGE 63 63 My ers [ 37 ] analyzes the eect of p erishabilit y on the maxim um demand that a compan y can accommo date. He tak es in to accoun t pro duction capacit y and p erishabilit y constrain ts. The demands are aggregated o v er all retailers, and the capacities are aggregated o v er all facilities. F or eac h p erio d, a seasonalit y factor similar to our constan t t is kno wn, whic h represen ts the fraction of the total retailer demand in p erio d t He prop oses a linear programming mo del to forecast the maxim um total demand that the facilit y can satisfy Note that this mo del resem bles the onefacilit y acyclic MPSSP with pro duction capacit y and p erishabilit y constrain ts, where the retailer demands share a common seasonalit y pattern. In the MPSSP the demands are kno wn for eac h retailer and w e are in terested in conditions under whic h w e can satisfy this demand, while My ers kno ws the prop ortion of total demand that will b e consumed in eac h p erio d and is in terested in the total demand that can b e satised. The upp er b ound on the maximal demand deriv ed b y My ers coincides with B 1 N where 1 t = 1 t = 1 for all t = 1 ; : : : ; T and M = 1. 3.4.5 Asymptotic feasibilit y and optimalit y In this section w e pro v e asymptotic feasibilit y and optimalit y of the greedy heuristic with the pseudocost function prop osed in Section 3.4.2 when the retailer demands exhibit a common seasonalit y pattern. W e denote the optimal assignmen ts in the LPrelaxation of (P) b y x LP and its ob jectiv e v alue b y z LP F urthermore, let x G denote the (partial) solution to (AP 1 ) giv en b y the greedy heuristic, and z G b e its ob jectiv e v alue. Note that the in tegral assignmen ts in x LP as w ell as x G are (partial) solutions to (AP 1 ). Let A N b e the set of assignmen ts for whic h x G and x LP do not coincide. The follo wing result sho ws that the n um b er of dierences for whic h x LP and x G do not coincide can b e b ounded from ab o v e b y a constan t indep enden t of N PAGE 64 64 Theorem 8 (cf. Romero Morales [ 44 ]) Ther e exists some c onstant R indep endent of n such that j A N j R for al l instanc es of (LP) that ar e fe asible and nonde gener ate. The follo wing result ensures that under the sto c hastic mo del prop osed in Section 3.4.3 (LP) is nondegenerate with probabilit y one. Lemma 9 (cf. Romero Morales [ 44 ]) (LP) is nonde gener ate with pr ob ability one, under the sto chastic mo del pr op ose d. W e will use the follo wing lemma to pro v e the asymptotic feasibilit y result in Theorem 11 Lemma 10 (cf. Romeijn and Piersma [ 39 ]) Under Assumption 7 M X i =1 B i ¡ 1 N M X i =1 N X j =1 D j X LP ij > 0 with pr ob ability one when n go es to innity. W e are no w ready to pro v e that the greedy heuristic yields a solution that is asymptotically feasible and optimal with probabilit y one when the retailer demands follo w a common seasonalit y pattern. Theorem 11 Under Assumption 7 the gr e e dy heuristic is asymptotic al ly fe asible and optimal with pr ob ability one when the r etailer demands fol low a c ommon se asonality p attern. Pro of: (LP) is nondegenerate with probabilit y one (see Lemma 9 ) and feasible with probabilit y one when N 1 b y using Assumption 7 F rom Theorem 8 w e then kno w that the n um b er of assignmen ts that dier b et w een the optimal solution of the relaxation of (P) and the solution giv en b y the greedy heuristic is b ounded from ab o v e b y a constan t indep enden t of N Moreo v er, Lemma 10 ensures us that the remaining capacit y in the optimal solution for the relaxation of (AP 1 ) gro ws linearly with N Th us, when N gro ws to innit y the solution found b y the greedy heuristic is a feasible solution to (AP 1 ). PAGE 65 65 It remains to b e sho wn that the greedy solution is asymptotically optimal. It suces to sho w that j 1 N Z LP N ¡ 1 N Z G N j 0 with probabilit y one as n 1 It is easy to sho w that 1 N Z LP N ¡ 1 N Z G N ¡ C ¡ C + ¡ p ¡ p + T h ¢ D ¢ j A N j N : The desired result then follo ws directly from Theorem 8 The generalization of Theorem 11 to the case of general demands is an op en issue. It is fairly straigh tforw ard to generalize the result of Theorem 8 Ho w ev er, the main obstacle to pro ving asymptotic feasibilit y (and optimalit y) for the general case is the fact that the constrain ts can, unlik e in the seasonal demand case, not b e summarized in a single constrain t for eac h facilit y [ 43 ]. 3.5 Computational Results 3.5.1 Generation of Problem Instances W e ha v e tested our solution approac h on problem instances generated according to the sto c hastic mo del presen ted in Section 3.4.3 F or eac h problem instance, w e ha v e generated a set of facilities and a set of retailers uniformly in the square [0 ; 10] £ [0 ; 10]. F or the case where all retailers exhibit the same seasonalit y pattern, w e ha v e generated an aggregate demand D j from the uniform distribution on [5 ; 25] for eac h retailer j and set D j t = t D j ( t = 1 ; : : : ; T ). F or the more general case, w e ha v e generated a demand D j t from the uniform distribution on [5 t ; 25 t ] for eac h retailer in eac h time p erio d. W e ha v e xed the n um b er of time p erio ds T = 6, and, in most cases, c hosen the v ector of seasonal factors to b e = ( 1 9 ; 3 18 ; 2 9 ; 2 9 ; 3 18 ; 1 9 ) > Ho w ev er, in this case p erishabilit y constrain ts, ev en with k = 1 ; 2, often turn out to b e nonbinding. T o obtain more illustrativ e results, w e ha v e used a more extreme (alb eit less realistic from a practical p oin t of view) v ector of seasonal co efcien ts when p erishabilit y constrain ts are presen t; in particular: t = 6 t 2 T ( T +1)(2 T +1) PAGE 66 66 ( t = 1 ; : : : ; T ) for the acyclic case, and 1 = 6 T 2 T ( T +1)(2 T +1) and t = 6( t ¡ 1) 2 T ( T +1)(2 T +1) ( t = 2 ; : : : ; T ) for the cyclic case. The costs C ij are assumed to b e prop ortional to demand and distance, that is, C ij = dist ij P Tt =1 D j t where dist ij denotes the Euclidean distance b et w een facilit y i and retailer j Finally w e ha v e generated in v en tory holding costs h it uniformly in the in terv al [10 ; 30] and, without loss of generalit y due to the absence of sp eculativ e motiv es, c hosen the pro duction costs equal to zero. W e ha v e assumed that the capacities are equal for all facilities and all p erio ds, that is, b it = N r it = N and I it = N When all retailers demands ha v e the same seasonal b eha vior, w e can mak e the feasibilit y conditions giv en in Section 3.4.4 more explicit for the follo wing t yp es of problem instances: 1. The b asic MPSSP In this case, w e c ho ose = ¢ E ( D 1 ) M ¢ max t =1 ;::: ;T 1 t t X =1 in the acyclic case, and = ¢ E ( D 1 ) M ¢ 1 T T X t =1 t in the cyclic case. The feasibilit y condition for this mo del is then equiv alen t to > 1. In fact, it can b e sho wn that the same feasibilit y conditions are v alid for the general demand case discussed ab o v e. 2. The MPSSP with thr oughput c onstr aints In this case, w e c ho ose the same as in the basic case, and = 0 ¢ E ( D 1 ) M max t =1 ;::: ;T t for b oth the acyclic and the cyclic case. The feasibilit y condition for this mo del is then equiv alen t to > 1 and 0 > 1. PAGE 67 67 3. The MPSSP with physic al inventory c onstr aints In this case, w e ha v e the follo wing theorem: Theorem 12 If is the same as in the b asic c ase, and = 00 ¢ max r =0 ;::: ;T ¡ 2 E ( D 1 ) M ¢ max =2 ;::: ;T ¡ r + r X t = t ¡ (1 + r ) in the acyclic c ase, and = 00 ¢ max r =0 ;::: ;T ¡ 2 E ( D 1 ) M ¢ max =1 ;::: ;T + r X t = [ t ] ¡ (1 + r ) in the cyclic c ase, The fe asibility c ondition for this mo del is then e quivalent to > 1 and 00 > 1 Pro of: W e kno w that the this mo del is asymptotically feasible when Assumption 7 is satised. Since B i is the same for all i = 1 ; : : : ; M the feasibilit y conditions then can b e rewrite as B i > E ( D 1 ) M : Romero Morales [ 44 ] has sho wn that in this mo del w e ha v e B i = min = 1 ; : : : ; T r = 0 ; : : : ; T ¡ P + r t = it + i; ¡ 1 P + r t = t in acyclic case and B i = min ( min = 1 ; : : : ; T r = 0 ; : : : ; T ¡ 1 P + r t = +1 i [ t ] + i [ ¡ 1] P + r t = +1 [ t ] ; P Tt =1 it P Tt =1 t ) in cyclic case. It follo ws that in the acyclic case the mo del is feasible when ( r + 1) + P + r t = t > E ( D 1 ) M = 1 ; : : : ; T ; r = 0 ; : : : ; T ¡ or equiv alen tly > E ( D 1 ) M + r X t = t ¡ ( r + 1) = 1 ; : : : ; T ¡ r ; r = 0 ; : : : ; T ¡ 1 : PAGE 68 68 If w e c ho ose the same as in the basic case with > 1, It is easy to see that the ab o v e inequalit y is redundan t when r = T ¡ 1 and = 1. Th us w e ha v e > max r =0 ;::: ;T ¡ 2 E ( D 1 ) M ¢ max =2 ;::: ;T ¡ r + r X t = t ¡ (1 + r ) and this pro v e the conditions for acyclic case. The conditions for cyclic case can b e obtained in a similar manner. 4. The MPSSP with p erishability c onstr aints In this case, w e ha v e c hosen = ¢ E ( D 1 ) M ¢ max = 2 ; : : : ; T ¡ k r = k ; : : : ; T ¡ 1 r + 1 + r X t = t in the acyclic case, and = ¢ E ( D 1 ) M ¢ max = 1 ; : : : ; T r = k ; : : : ; T ¡ 2 1 r + 1 + r X t = [ t ] : The feasibilit y condition for this mo del is then equiv alen t to > 1. 3.5.2 Results W e ha v e mainly considered cases with M = 5 facilities, and N = 15, 25, 50, 100, 150, 200, 250, and 300 retailers. F or the pro duction capacities, w e ha v e mainly used a m ultiplier of = 1 : 1. Note that > 1 ensures asymptotic feasibilit y Our v alue of pro vides relativ ely tigh t (th us relativ ely hard) problem instances, while still allo wing for a reasonable fraction of feasible instances for small n um b ers of retailers. F or the throughput and ph ysical in v en tory constrain ts, c ho osing the tigh tness parameter to o small w ould yield instances (and solutions) that are impractical, since they w ould allo w for to o little v ariet y in the t yp es of feasible solutions. Therefore, for the throughput constrain ts, w e ha v e used 0 = 1 : 3, and for the ph ysical in v en tory constrain ts w e ha v e used 00 = 1 : 1 ; 1 : 5 ; 2. Finally w e ha v e considered p erishabilit y constrain ts with k = 1 ; 2. PAGE 69 69 F or most exp erimen ts w e ha v e used demand data that exhibits the same seasonalit y pattern for all retailers, since the feasibilit y analysis for the data mo del as w ell as the asymptotic p erformance guaran tee are v alid in this case. T o ev aluate the qualit y of the greedy heuristic and VLSN, w e compare the solution v alue to the LPlo w er b ound for instances with M 50, and with the exact in teger solution v alue for M = 15 ; 25. When the greedy heuristic do es not nd a feasible solution, w e emplo y the p enalized VLSN to nd a feasible solution (see Section 2.3 ). W e measure the infeasibilit y of the problem instance (H Ai ) b y com bining the maximal violation of constrain ts as follo ws: ¢ Ai ( x i ¢ ) = T X t =1 max ( 0 ; N X j =1 d j t x ij ¡ r it ) + T X t =0 max 8<: 0 ; I it ¡ min 0@ I it ; min ( T ;t + k ) X = t +1 N X j =1 d j x ij 1A 9=; and of the problem instance (H Ci ) as follo ws: ¢ Ci ( x i ¢ ) = T X t =1 max ( 0 ; N X j =1 d j t x ij ¡ r it ) + min =1 ;::: ;T T X t =0 max ( 0 ; I it ¡ min I it ; t + k X = t +1 N X j =1 d j [ ] x ij !) where the v alues of I it and I it are computed using the recursion ( 3.28 ), in the latter case treating as the last p erio d b y setting I i = 0 and p erforming the bac kw ards recursion in a cyclic manner from there. Using these measures of infeasibilit y w e then add the term M X i =1 L ¢ i ( x i ¢ ) to the ob jectiv e function of (AP 1 ). W e initially set L = 1000, and m ultiply it b y a factor of 10 when VLSN do es not nd an impro ving or feasible solution. As so on as VLSN do es nd a feasible solution, w e disregard the p enalt y term in the ob jectiv e PAGE 70 70 function, and retain feasibilit y for the remainder of the algorithm. If no feasible solution is obtained with L = 10 11 w e stop the pro cedure without a solution to the problem. F or eac h class of problem instances w e ha v e generated and solv ed 25 instances. T ables 3{1 3{22 rep ort the test results. The default n um b er of facilities is M = 5 in the tables. The tables sho w, for b oth the greedy heuristic and VLSN, the n um b er of instances for whic h the heuristic found a feasible solution, the computation time, and the error (a v eraged o v er all instances for whic h a feasible solution w as found). F or VLSN the a v erage n um b er of iterations, that is, the n um b er of impro ving solutions, is giv en. VLSN w as able to nd a feasible solution to almost all instances that w ere in fact feasible. A sup erscript on the n um b er of instances for whic h VLSN found a feasible solution indicates the n um b er of instances that w ere feasible, but for whic h VLSN w as not successful. In most cases, the error sho wn is an upp er b ound on the actual error, computed using the optimal solution v alue of the LPrelaxation. In T ables 3{1 and 3{2 as w ell as for N = 15 ; 25 in all other tables, the error w as computed using the optimal solution, or the b est lo w er b ound obtained b y CPLEX within 15 min utes of CPU time. Finally in T ables 3{1 and 3{2 the a v erage solution times using CPLEX are sho wn. The upp er b ound on the solution time w as 15 min utes, and a sup erscript indicates the n um b er of instances for whic h this limit w as reac hed. All tests w ere p erformed on a PC with a 667 MHz P en tium I I I pro cessor with 128 MB RAM. All LPproblems and MIP problems w ere solv ed using CPLEX 7.0. Main results. T ables 3{1 3{14 sho w results using the parameter c hoices indicated ab o v e. In most cases, VLSN w as able to nd a feasible solution whenev er the problem instance w as feasible, so the corresp onding column in the tables actually indicates the n um b er of feasible instances (out of the 25 generated). Only in 4.3% of the PAGE 71 71 instances with N = 15 and 0.3% of the instances with N = 25 w as VLSN unsuccessful in nding a feasible solution to the problem. Most tables clearly illustrate the asymptotic feasibilit y and optimalit y of the greedy heuristic. In T ables 3{5 and 3{8 w e see signican tly few er feasible instances than for all other cases. Additional exp erimen ts ha v e sho wn that the asymptotic nature of the feasibilit y result are more prominen t here, and only for N 2000 do w e stop seeing an y infeasible instances. Ho w ev er, the ratio N = M b et w een the n um b er of retailers and facilities often needs to b e quite large to obtain acceptable solution qualit y VLSN, ho w ev er, is able to nd go o d solutions to the problem in little time ev en for these apparen tly harder instances where the ratio N = M is small. The greedy heuristic and VLSN seem to b e complemen tary in the sense that the problem instances for whic h the time required to obtain a go o d solution using VLSN gets large, often allo w for a go o d solution using the greedy heuristic only in far less time. In comparison, the time used b y CPLEX to solv e the problems to optimalit y is v ery problem dep enden t, and increases quite rapidly in the size of the problem. F or N = 15, CPLEX tak es, on a v erage, 4230 times as m uc h time as VLSN. F or N = 25, this increases to 8270. The time used b y CPLEX for problem instances with N = 15 are an ywhere from less than a second to ab out half a min ute. This indicates that for the smallest problems, CPLEX could b e a viable alternativ e to VLSN. Com bining these observ ations, it seems that CPLEX should b e used for the smallest instances, up to N = 15, VLSN for the larger instances, up to sa y N = 300, and the greedy heuristic for the largest instances. The tables sho w that the computation times and solution qualit y for VLSN are remark ably insensitiv e to the t yp es of capacit y constrain ts presen t, as w ell as the particular tigh tness parameters c hosen. Another in teresting result is that apparen tly v ery few iterations of VLSN yield a dramatic impro v emen t in solution qualit y As for the computation time in v olv ed, PAGE 72 72 the op eration that curren tly tak es the most time is nding the negativ e cycle. Generally around 50% of the time of VLSN is sp en t there. General demands in basic case and with ph ysical in v en tory capacities T ables 3{15 3{18 sho w results obtained using the general demand mo del, for the basic MPSSP as w ell as the case with ph ysical in v en tory capacities. In the latter case the feasibilit y condition for general demands is an op en issue, but in the former the feasibilit y condition for seasonal and general demands coincide. W e ha v e therefore decided to use the same capacities as in the seasonal case for the mo del including in v en tory capacities as w ell. Apart from somewhat larger errors and a lo w er success rate for instances with a small ratio b et w een the n um b er of retailers and suppliers, the similarit y of the results for b oth demand mo dels sho ws that the results in Section 3.5.2 do not dep end signican tly on that particular c hoice of demand mo del. The dierence for small N is lik ely to b e caused b y apparen tly tigh ter capacities in case of general demands. More facilities in basic case and with ph ysical in v en tory capacities T o study the impact of the n um b er of facilities M w e ha v e tested the p erformance of the algorithms for the basic case, as w ell as the case with ph ysical in v en tory capacities when M = 10. T o a v oid memory problems in the negativ e cycle detection pro cedure of VLSN, w e ha v e here limited the cycle length to 3 retailers. The results in T ables 3{19 3{22 indicate that, with this setting, the main dierence b et w een M = 5 and M = 10 is the error exp erienced b y VLSN. Recall that the errors are with resp ect to the b est b ound found b y CPLEX within at most 15 min utes for N = 25, and with resp ect to the LP b ound for all other instances, explaining the jump in error from N = 25 to N = 50 for T ables 3{19 and 3{21 3{22 In T able 3{20 CPLEX w as rarely able to nd an optimal solution ev en within a 30 min ute time limit, and the errors are with resp ect to a probably w eak PAGE 73 73 T able 3{1: Basic case, acyclic, seasonal demands Greedy heuristic Including VLSN searc h CPLEX N F eas. Time Error feas. Time Error Iter. Time (sec) (%) (sec) (%) (sec) 15 18 0.001 11.29 21 (1) 0.09 2.07 2.1 4.09 25 18 0.006 13.62 21 0.15 1.56 2.8 21.22 50 24 0.018 6.23 24 0.41 0.58 3.4 45.59 (1) 100 25 0.061 3.87 25 1.93 0.24 4.7 16.69 150 25 0.132 3.03 25 5.22 0.12 5.1 131.02 (1) 200 25 0.244 2.31 25 10.03 0.09 4.7 224.38 (3) 250 25 0.374 1.55 25 20.20 0.08 5.2 171.68 (1) 300 25 0.546 1.31 25 28.66 0.06 5.3 378.22 (8) lo w er b ound. W e conclude that problems with v ery small ratio b et w een the n um b er of retailers and the n um b er of facilities are particularly hard problems. 3.6 Summary In this c hapter w e ha v e considered the MPSSP for ev aluating a logistics net w ork design in a dynamic en vironmen t in the presence of pro duction, in v en tory and throughput capacities and p erishabilit y constrain ts. W e ha v e dev elop ed ecien tly implemen table greedy heuristic and VLSN algorithm. In particular, w e ha v e prop osed a greedy heuristic approac h whic h enjo ys attractiv e theoretical prop erties, a VLSN algorithm with an ecien t subproblems solving approac hes, and sho wn that these algorithms are v ery successful at obtaining high qualit y solutions to the problem in limited time. W e are curren tly in v estigating extensions of the approac h to nonlinear pro duction cost structures, as w ell as cases that allo w for in v en tories at the retailer lev el. PAGE 74 74 T able 3{2: Basic case, cyclic, seasonal demands Greedy heuristic Including VLSN searc h CPLEX N F eas. Time Error feas. Time Error Iter. Time (sec) (%) (sec) (%) (sec) 15 19 0.001 10.61 22 0.08 1.69 2.1 3.99 25 16 0.006 16.49 21 0.16 0.95 3.0 30.93 50 21 0.012 8.73 24 0.41 0.89 3.5 12.85 100 25 0.046 3.06 25 1.79 0.37 4.0 26.00 150 25 0.098 2.27 25 5.55 0.21 4.7 115.95 200 25 0.171 1.63 25 10.78 0.15 4.3 329.62 (5) 250 25 0.268 1.45 25 22.80 0.12 5.1 280.56 (4) 300 25 0.389 1.17 25 34.50 0.08 4.7 325.43 (5) T able 3{3: Throughput constrain ts, acyclic, seasonal demands, 0 = 1 : 3 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 13 0.005 21.69 16 (2) 0.10 3.42 2.2 25 20 0.001 13.33 24 0.17 1.20 3.2 50 22 0.016 9.92 22 0.44 1.27 3.9 100 25 0.060 3.11 25 1.67 0.40 4.0 150 25 0.134 2.65 25 4.92 0.29 4.6 200 25 0.239 2.25 25 9.96 0.19 4.4 250 25 0.374 1.45 25 18.70 0.15 4.8 300 25 0.532 1.57 25 36.95 0.13 6.0 T able 3{4: Throughput constrain ts, cyclic, seasonal demands, 0 = 1 : 3 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 13 0.005 12.35 16 (2) 0.10 1.70 2.4 25 21 0.001 8.74 24 0.15 1.34 2.9 50 22 0.010 6.17 22 0.42 0.89 3.7 100 25 0.048 3.19 25 1.72 0.32 3.9 150 25 0.101 2.27 25 5.32 0.20 4.6 200 25 0.177 1.50 25 12.48 0.12 5.2 250 25 0.275 1.27 25 20.89 0.10 4.9 300 25 0.394 1.08 25 39.23 0.09 5.7 PAGE 75 75 T able 3{5: In v en tory capacities, acyclic, seasonal demands, 00 = 1 : 1 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 9 0.000 11.45 13 (1) 0.09 2.40 2.1 25 10 0.007 7.97 15 0.15 2.51 2.4 50 10 0.036 3.58 12 0.33 1.47 2.7 100 10 0.175 2.68 12 1.66 0.57 3.3 150 8 0.495 1.36 10 4.21 0.29 3.6 200 9 1.052 0.77 12 9.00 0.17 4.1 250 13 1.935 0.63 14 18.00 0.15 4.6 300 11 3.214 0.56 14 32.77 0.10 4.8 T able 3{6: In v en tory capacities, acyclic, seasonal demands, 00 = 1 : 5 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 12 0.007 12.27 19 (1) 0.12 3.88 2.4 25 11 0.008 9.89 15 0.19 2.14 3.0 50 18 0.034 4.74 20 0.37 1.43 3.2 100 24 0.176 2.87 25 1.63 0.54 3.6 150 24 0.494 2.44 24 4.50 0.27 4.2 200 24 1.054 1.93 24 10.20 0.20 4.9 250 25 1.928 1.50 25 18.71 0.15 4.8 300 25 3.195 1.08 25 28.52 0.10 4.3 T able 3{7: In v en tory capacities, acyclic, seasonal demands, 00 = 2 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 15 0.005 16.46 18 (2) 0.10 2.96 2.3 25 22 0.005 14.11 25 0.15 0.94 3.0 50 24 0.033 7.07 24 0.36 1.37 3.4 100 25 0.176 4.15 25 1.84 0.51 4.7 150 25 0.493 3.16 25 4.99 0.25 5.1 200 25 1.054 2.40 25 9.61 0.17 4.7 250 25 1.931 1.61 25 19.43 0.13 5.2 300 25 3.202 1.35 25 27.76 0.10 4.3 PAGE 76 76 T able 3{8: In v en tory capacities, cyclic, seasonal demands, 00 = 1 : 1 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 12 0.006 8.51 17 (2) 0.11 2.41 2.5 25 9 0.009 12.62 11 0.19 2.79 3.0 50 15 0.033 3.99 18 0.48 1.27 3.5 100 13 0.176 1.70 16 1.76 0.48 3.8 150 20 0.498 1.43 20 4.53 0.21 4.3 200 22 1.056 0.70 23 10.07 0.17 4.5 250 19 1.929 0.70 20 18.13 0.11 4.3 300 16 3.205 0.50 20 31.30 0.08 4.3 T able 3{9: In v en tory capacities, cyclic, seasonal demands, 00 = 1 : 5 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 18 0.003 11.44 20 (1) 0.09 2.85 2.3 25 17 0.008 9.77 19 0.14 1.51 2.7 50 21 0.034 5.88 22 0.42 0.89 3.6 100 24 0.178 3.40 24 1.84 0.33 4.2 150 25 0.497 2.17 25 5.37 0.22 4.8 200 25 1.060 1.59 25 12.51 0.12 5.3 250 25 1.929 1.23 25 21.00 0.09 5.1 300 25 3.190 1.06 25 35.73 0.09 5.3 T able 3{10: In v en tory capacities, cyclic, seasonal demands, 00 = 2 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 15 0.003 12.61 20 (1) 0.10 2.61 2.0 25 19 0.003 13.81 21 0.15 0.62 2.8 50 22 0.033 6.17 22 0.43 0.87 3.7 100 25 0.175 3.19 25 1.76 0.32 3.9 150 25 0.498 2.27 25 5.33 0.20 4.6 200 25 1.058 1.50 25 12.42 0.12 5.2 250 25 1.930 1.27 25 20.80 0.10 4.9 300 25 3.213 1.08 25 40.10 0.08 5.8 PAGE 77 77 T able 3{11: P erishabilit y constrain ts, acyclic, seasonal demands, k = 1 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 16 0.002 9.88 23 0.11 1.47 2.4 25 18 0.008 4.06 22 0.17 0.91 2.7 50 24 0.039 2.96 24 0.46 0.58 3.9 100 25 0.198 2.90 25 2.11 0.24 4.6 150 25 0.544 1.47 25 5.75 0.13 4.9 200 25 1.147 1.20 25 12.93 0.10 5.3 250 25 2.070 1.15 25 22.45 0.07 5.0 300 25 3.385 0.72 25 39.43 0.05 5.5 T able 3{12: P erishabilit y constrain ts, acyclic, seasonal demands, k = 2 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 13 0.002 6.18 17 (1) 0.10 0.70 2.1 25 19 0.009 4.38 23 0.19 0.44 3.5 50 24 0.041 2.31 24 0.50 0.31 4.3 100 25 0.201 1.41 25 2.00 0.18 4.4 150 25 0.549 0.66 25 5.53 0.08 4.6 200 25 1.153 0.60 25 12.03 0.06 4.9 250 25 2.090 0.54 25 23.39 0.04 5.3 300 25 3.393 0.41 25 37.55 0.03 5.1 T able 3{13: P erishabilit y constrain ts, cyclic, seasonal demands, k = 1 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 15 0.001 4.64 18 (1) 0.10 1.26 2.2 25 21 0.008 6.54 23 0.18 0.67 3.3 50 21 0.038 4.09 24 0.41 0.75 4.7 100 25 0.194 2.12 25 1.38 0.31 5.5 150 25 0.531 1.59 25 2.96 0.15 5.0 200 25 1.118 1.42 25 6.67 0.11 6.3 250 25 2.024 1.10 25 11.43 0.08 6.5 300 25 3.331 0.89 25 15.43 0.07 5.4 PAGE 78 78 T able 3{14: P erishabilit y Constrain ts, cyclic, seasonal demands, k = 2 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 18 0.003 8.94 22 0.11 0.66 2.6 25 19 0.008 3.72 21 (1) 0.18 0.29 3.5 50 21 0.032 2.71 24 0.41 0.44 4.7 100 25 0.194 1.13 25 1.28 0.18 5.2 150 25 0.533 0.77 25 2.98 0.09 5.3 200 25 1.122 0.50 25 5.94 0.06 5.8 250 25 2.033 0.46 25 9.41 0.04 5.2 300 25 3.346 0.37 25 16.14 0.04 6.0 T able 3{15: Basic case, acyclic, general demands Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 15 0.005 11.70 23 0.12 2.40 2.5 25 23 0.001 13.20 25 0.18 1.38 3.5 50 25 0.015 11.80 25 0.57 0.72 4.4 100 25 0.058 7.26 25 2.11 0.87 5.1 150 25 0.131 4.80 25 5.15 0.47 5.3 200 25 0.230 3.13 25 10.30 0.30 5.2 250 25 0.358 2.46 25 18.48 0.21 5.7 300 25 0.518 1.61 25 26.97 0.15 5.0 T able 3{16: Basic case, cyclic, general demands Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 15 0.005 17.09 21 (4) 0.13 2.41 3.2 25 25 0.000 12.62 25 0.16 0.98 3.4 50 25 0.012 7.33 25 0.42 0.45 4.0 100 25 0.044 5.19 25 1.94 0.43 5.1 150 25 0.097 3.99 25 5.23 0.29 5.5 200 25 0.172 2.59 25 10.37 0.19 5.3 250 25 0.266 2.25 25 19.97 0.15 5.7 300 25 0.383 1.50 25 35.31 0.11 6.0 PAGE 79 79 T able 3{17: In v en tory capacities, acyclic, general demands, = 1 : 1, 00 = 1 : 5 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 8 0.003 8.81 13 (2) 0.11 5.52 2.1 25 9 0.010 8.92 21 0.19 3.65 2.6 50 20 0.036 10.48 24 0.44 2.71 3.8 100 24 0.192 5.41 25 1.79 0.99 4.7 150 25 0.528 3.89 25 4.97 0.48 5.4 200 24 1.115 2.80 25 9.89 0.34 5.2 250 25 2.019 1.96 25 15.97 0.24 5.0 300 25 3.306 1.46 25 30.97 0.19 5.8 T able 3{18: In v en tory capacities, cyclic, general demands, = 1 : 1, 00 = 1 : 5 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 15 6 0.003 15.70 16 (3) 0.16 3.64 2.8 25 12 0.010 7.69 24 0.19 2.97 2.7 50 22 0.038 9.01 25 0.45 2.17 3.9 100 25 0.190 5.57 25 2.17 0.69 5.5 150 25 0.526 3.45 25 5.41 0.31 5.4 200 25 1.114 2.88 25 12.55 0.20 6.1 250 25 2.015 2.03 25 20.72 0.18 5.9 300 25 3.302 1.81 25 35.04 0.11 6.2 T able 3{19: Basic case, acyclic, general demands, m = 10 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 25 4 0.010 24.53 12 0.63 5.27 5.3 50 22 0.024 24.23 25 0.57 9.15 8.8 100 25 0.091 19.90 25 1.86 4.84 12.3 150 25 0.205 13.80 25 4.52 2.83 14.6 200 25 0.364 9.38 25 7.93 1.46 14.0 250 25 0.580 7.78 25 13.50 1.16 14.5 300 25 0.861 5.61 25 20.01 0.80 13.9 PAGE 80 80 T able 3{20: Basic case, cyclic, general demands, m = 10 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 25 5 0.020 47.28 11 (2) 0.61 28.69 6.1 50 24 0.019 18.92 25 1.89 4.77 8.0 100 25 0.082 11.45 25 2.68 1.71 10.6 150 25 0.171 10.59 25 4.61 1.64 14.8 200 25 0.301 7.41 25 8.11 1.18 13.8 250 25 0.478 5.59 25 14.38 0.78 15.0 300 25 0.683 5.14 25 22.82 0.62 16.0 T able 3{21: In v en tory capacities, acyclic, seasonal demands, m = 10, 00 = 1 : 1 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 25 4 0.015 10.56 11 0.30 5.98 3.8 50 10 0.046 14.13 16 0.46 7.13 5.1 100 13 0.218 7.43 19 1.59 2.12 8.3 150 10 0.579 3.93 15 3.61 1.33 8.5 200 16 1.193 3.12 19 5.90 0.73 9.5 250 12 2.138 2.64 18 11.21 0.69 8.7 300 17 3.469 2.27 19 16.47 0.44 10.5 T able 3{22: In v en tory capacities, cyclic, seasonal demands, m = 10, 00 = 1 : 1 Greedy heuristic Including VLSN searc h N F eas. Time (sec) Error (%) F eas. Time (sec) Error (%) Iter. 25 5 0.006 10.47 12 0.39 5.45 4.2 50 11 0.045 14.14 17 0.52 4.95 5.8 100 16 0.214 4.96 20 1.71 1.45 9.9 150 11 0.576 3.08 18 4.44 1.06 11.2 200 17 1.192 1.97 20 6.73 0.60 10.4 250 17 2.132 1.73 21 11.03 0.47 9.4 300 19 3.463 1.42 20 15.71 0.33 10.8 PAGE 81 CHAPTER 4 CONTINUOUSTIME SINGLESOUR CING PR OBLEM (CSSP) 4.1 In tro duction All v arian ts of the MPSSP studied to date, including the v arian ts studied in Chapter 3 of this dissertation, ha v e made the simplifying assumption that the pro duction and in v en tory holding costs are linear in pro duction v olume. In this c hapter, w e consider a con tin uoustime analog of the MPSSP whic h allo ws us to accoun t for the xedc harge structure of pro duction costs often found within pro duction facilities in practice. Th us, unlik e the MPSSP the mo del w e dev elop addresses the critical c hallenge of allo cating retailer demands across m ultiple facilities in order to optimally exploit pro duction economies of scale throughout the pro duction and distribution net w ork. While the MPSSP allo ws for timev arying demands and costs, to main tain analytical tractabilit y in the presence of xed setup costs our analysis requires a constan t demand rate with timein v arian t costs at eac h retailer. This allo ws us to dev elop eectiv e solution metho ds for these largescale and c hallenging problems, p ermitting application of these solution metho ds to con texts in whic h these assumptions hold appro ximately W e call this problem the c ontinuoustime singlesour cing pr oblem (CSSP) and study a basic uncapacitated v ersion of the problem, as w ell as extensions that deal with pro duction and in v en tory capacities. Finally w e will incorp orate opp ortunities for expanding these capacities. Similarly to the fact that the MPSSP can b e view ed as a generalization of the ELS problem, the CSSP can b e view ed as a generalization of the EOQ problem. 81 PAGE 82 82 A substan tial b o dy of past researc h has considered a n um b er of v arian ts of m ultistage EOQ mo dels with setup costs at eac h stage, applying socalled \p o w eroft w o" p olicies that pro vide go o d qualit y solutions in reasonable computing times [ 32 36 46 ]. Additional past w ork considers m ultilev el replenishmen t frequency and v ehicle routing (or direct deliv ery) transp ortation decisions under constan t and deterministic retailer demand rates [ 8 9 7 26 52 17 19 ]. In eac h of these pap ers, the assignmen t of retailers to w arehouses is predetermined and the fo cus is on the op erationallev el replenishmen t frequency and deliv ery sc heduling problems. In con trast, the CSSP considers the tacticallev el problem of optimally assigning retailer demands to w arehouses, assuming direct deliv ery from w arehouse to retailer and a singlesourcing strategy The remainder of this c hapter is organized as follo ws. In Section 4.2.1 w e in tro duce the basic uncapacitated mo del. W e the sp ecialize the pricing problem as dev elop ed for the general class (AP) to the CSSP in Section 4.2.2 whic h then completely sp ecies the column generation and corresp onding branc handprice algorithm for solving the CSSP In Section 4.3 w e then sho w ho w this metho dology can b e applied to sev eral v arian ts of the CSSP Since the branc handprice pro cedure b ecomes v ery timeconsuming as problem sizes b ecome v ery large, Section 4.4 presen ts t w o greedy heuristic algorithms, as w ell as an implemen tation of the VLSN searc h impro v emen t algorithm (see Section 2.3 ). Finally w e presen t extensiv e computational results in Section 4.5 and conclude the c hapter in Section 4.6 b y discussing topics for future researc h. 4.2 Con tin uousTime SingleSourcing Problem In this section w e study solution approac hes for a general class of assignmen t problems that encompasses all v arian ts of the CSSP that w e will discuss in this c hapter. Ho w ev er, to motiv ate studying this class of problems, w e will start b y in tro ducing our basic, uncapacitated v arian t of the CSSP PAGE 83 83 4.2.1 F orm ulation W e consider a distribution net w ork with N retailers and M pro duction facilities. F ollo wing the EOQ mo del, w e assume that the demand rate of retailer j is constan t and deterministic and is denoted b y d j ( j = 1 ; : : : ; N ), and a xed c harge of f i is asso ciated with eac h pro duction batc h at facilit y i ( i = 1 ; : : : ; M ). Let a ij denote the total cost (p er unit time) asso ciated with satisfying retailer j demand using facilit y i including, but not necessarily limited to, a measure of the transp ortation cost, as w ell as a facilit ydep enden t v ariable pro duction cost. The p erunit holding cost (p er unit time) for an item pro duced at facilit y i is denoted b y h i W e dene decision v ariables x ij ha ving the v alue 1 if retailer j is serv ed b y facilit y i and 0 otherwise. In addition, w e let T i denote the time b et w een t w o consecutiv e pro duction setups at facilit y i In our basic mo del w e assume that the pro duction rate is innite, that is, pro ducts are a v ailable immediately Geunes et al. [ 29 ] studied a related mo del, in whic h a single facilit y faces the problem of c ho osing the set of retailers whose demands should b e satised to maximize prot. This problem, called the EOQ problem with mark et c hoice (EOQMC), as w ell as sev eral v arian ts thereof, will app ear in this c hapter as subproblems in solving retailer assignmen t problems. In a similar w a y as w as done for the EOQMC, w e can nd the optimal pro duction frequencies at the facilities easily from a basic EOQ analysis if the assignmen t v ariables are xed. In that case, the a v erage ann ual setup and holding costs at facilit y i are equal to C i ( x i ¢ ; T i ) = f i T i + 1 2 h i T i N X j =1 d j x ij : Clearly the cost function C i is con v ex in T i for an y xed x i ¢ W e obtain the optimal v alue of T i (i.e., the v alue minimizing the cost function C i ( x i ¢ ; T i ) for xed PAGE 84 84 x i ¢ ) b y setting @ C i ( x i ¢ ; T i ) =@ T i = 0: T ¤ i ( x i ¢ ) = s 2 f i h i P Nj =1 d j x ij and the corresp onding minimal a v erage ann ual setup and holding costs are then equal to: C ¤ i ( x i ¢ ) = C i ( x i ¢ ; T ¤ i ( x i ¢ )) = vuut 2 f i h i N X j =1 d j x ij : The uncapacitated CSSP or CSSPU, is the problem of nding an assignmen t of retailers to facilities that minimizes the total costs: minimize M X i =1 0@ N X j =1 a ij x ij + vuut 2 f i h i N X j =1 d j x ij 1A sub ject to M X i =1 x ij = 1 j = 1 ; : : : ; N x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N : Since the ob jectiv e function of CSSPU mo del is conca v e, according to Lemma 1 w e can obtain an equiv alen t problem b y relaxing the binary constrain ts on x ij to nonnegativ e constrain ts. Ho w ev er b ecause the ob jectiv e function is a nonlinear function, the problem is still hard to solv e ev en without binary constrain ts. Although whether the problem is NPHard is an op en issue, w e b eliev e that it is so. Observ e that our cost mo del lends itself to at least three distinct in terpretations with resp ect to applications con texts. In the rst in terpretation, pro duction batc hes are pro duced instan taneously at eac h setup o ccurrence at the facilities (observing the innite pro duction rate EOQ assumption), and items are depleted from facility i in v en tory at a con tin uous rate equal to the sum of the demand rates of retailers assigned to facilit y i This in terpretation corresp onds to applications con texts in whic h do wnstream retailers (or customers), for example, receiv e direct PAGE 85 85 pac k age deliv eries (rather than batc h deliv eries) from a facilit y in resp onse to individual orders (i.e., retailer or customer demands are \shipp ed to order" at a con tin uous rate and in v en tory is held only at the facilities). In the second in terpretation, facilit y pro duction batc hes are b oth pro duced and shipp ed instan taneously to retailers with iden tical holding costs, and in v en tory is depleted con tin uously in time at the retailers, that is, the retailers hold in v en tory and the facilities do not. In this case w e replace h i with h R in the ob jectiv e function (for all i = 1 ; : : : ; M ), where the single parameter h R denotes the holding cost at ev ery (iden tical) retailer. Iden tical retailer holding costs ma y apply for example, when the retailers are actually dieren t retail stores from the same retail c hain. In our third and nal in terpretation of the mo del, in v en tory is held b oth at the facilities and at retail lo cations, and the pro duction rate at eac h facilit y is exactly matc hed to the demand rate assigned to the facilit y That is, eac h facilit y pro duces and builds in v en tory con tin uously in time in ev ery pro duction cycle (at a rate equal to the total demand rate assigned to the facilit y 1 ), and ev ery T i time units, the facilit y ships in v en tory in batc hes to retail sites. In this con text, since pro duction is con tin uous in time at eac h facilit y the parameter f i no w denotes a xed dispatc h cost for deliv eries at the end of ev ery dispatc h cycle (an y v ariable dispatc h costs ma y b e absorb ed b y the a ij term). Moreo v er, the parameter h i is set equal to h 0i + h R where h R is the holding cost at ev ery (iden tical) retailer, and h 0i is the lo cal holding cost at facilit y i Our cost mo del is insensitiv e to these three p oten tial applications con text in terpretations, and the mo del therefore pro vides a fair appro ximation to a v ariet y of p oten tial practical settings. 1 Observ e that in steady state, on a v erage, a facilit y's pro duction rate m ust equal its assigned demand rate to ensure b oth that con tin uous in v en tory accum ulation do es not o ccur, and that shortages do not o ccur. PAGE 86 86 In the CSPPU, w e can ev en generalize our mo del further under the second and third in terpretations ab o v e, to accoun t for retailerdep enden t holding costs (see Section 4.2.3 for details). Ho w ev er, w e will mainly restrict ourselv es in this c hapter to the case of retailerindep enden t holding costs, whic h will allo w us to extend our algorithmic approac h to sev eral generalizations of the CSSPU. If w e dene the function H i to represen t the total costs incurred b y facilit y i and represen t the sets of retailers that ma y b e assigned to facilit y i b y X i w e obtain that the CSSP is a sp ecial case of the (AP). In particular, if w e c ho ose H i ( x i ¢ ) = N X j =1 a ij x ij + vuut 2 f i h i N X j =1 d j x ij w e obtain the CSSPU discussed in the previous section. As w e will sho w in Section 4.3 man y v arian ts of the CSSP share a common structure for the ob jectiv e functions H i or, equiv alen tly the pricing problem. This structure allo ws for the dev elopmen t of an ecien t p olynomialtime algorithm for solving the pricing problem, or for the dev elopmen t of an ecien t branc handb ound algorithm. In the next subsection, w e will study pricing problems ha ving this structure in detail. 4.2.2 Pricing Problem for LP(SP) In our column generation implemen tation, w e solv e a pricing problem for eac h of the facilities at eac h iteration, and add at least one column for eac h facilit y when one that prices out exists. The pricing problem for LP(SP) and facilit y i reads: minimize i N X j =1 d j z j ¡ N X j =1 ( u ¤j ¡ a ij ) z j + v ¤ i sub ject to (PP i ) z j 2 f 0 ; 1 g j = 1 ; : : : ; N where u ¤j and v ¤ i are optimal dual v ectors corresp onding to the constrain ts in LP(SP). PAGE 87 87 The general pricing problem that w e consider in this section is of the form maximize f ( z ) N X j =1 j z j ¡ N X j =1 d j z j sub ject to (Q) z j 2 f 0 ; 1 g j = 1 ; : : : ; N where : R + R [ f + 1g is an extended realv alued function, and w e will call j the rev en ue asso ciated with retailer j F or con v enience, w e will denote the ob jectiv e function of (Q) b y f Note that an y additional constrain ts that only dep end on z = ( z 1 ; : : : ; z N ) > 2 [0 ; 1] N through P Nj =1 d j z j can b e incorp orated b y mo difying the function to ha v e v alue + 1 for all infeasible solutions. Without loss of generalit y w e will assume that the retailers j are ordered in nonincreasing order of the ratio j =d j that is, 1 =d 1 2 =d 2 ¢ ¢ ¢ N =d N The follo wing lemma establishes a prop ert y of optimal solutions to the r elaxation of (Q), denoted R(Q), obtained b y replacing the binary constrain ts on the v ariables z j b y 0 z j 1 j = 1 ; : : : ; N : Lemma 13 If R(Q) has an optimal solution, it wil l have an optimal solution z ¤ with the pr op erty such that if z ¤ k > 0 for any k = 1 ; : : : ; N then z ¤ ` = 1 for ` = 1 ; : : : ; k ¡ 1 Pro of: Supp ose suc h an optimal solution do es not exist. Then there will b e in tegers k and ` suc h that z ¤ k > 0, z ¤ ` < 1, and ` < k Let 0 < min z ¤ k ; d ` d k (1 ¡ z ¤ ` ) : PAGE 88 88 Then consider the follo wing solution to R(Q): z 0 j = 8>>>><>>>>: z ¤ k ¡ j = k z ¤ ` + d k d ` j = ` z ¤ j otherwise : Since N X j =1 d j z ¤ j = N X j =1 d j z 0 j w e can see that f ( z ¤ ) ¡ f ( z 0 ) = N X j =1 j z ¤ j ¡ N X j =1 d j z ¤ j # ¡ N X j =1 j z 0 j ¡ N X j =1 d j z 0 j !# = k ( z ¤ k ¡ z 0 k ) + ` ( z ¤ ` ¡ z 0 ` ) = ¡ k + ` d k d ` = d k ` d ` ¡ k d k : Since d k > 0, > 0, and ` =d ` k =d k w e ha v e f ( z ¤ ) f ( z 0 ) : This means w e can increase the v alue of x ¤` and decrease the v alue of x ¤k un til the former is equal to 1 or the latter is equal to 0 without making the solution v alue an y w orse. Rep eating the argumen t as necessary no w pro v es the result. W e can use this prop ert y to dev elop a branc handb ound algorithm for solving the pricing problem, assuming that w e can ecien tly nd the b est fraction of retailer k + 1 to include in a solution, giv en that retailers 1 ; : : : ; k ha v e b een included. F or example, if the function is piecewise conca v e, all candidate fractional v alues are determined b y the breakp oin ts of the piecewise conca v e function. This result forms the basis of the algorithms to most of the (sub)problems discussed in this c hapter. The follo wing theorem, generalizing a result in Geunes et PAGE 89 89 al. [ 29 ], sho ws that Lemma 13 implies that R(Q) has an in tegral optimal solution if the function is conca v e. This solution is then clearly also an optimal solution for (Q).Theorem 14 F or any c onc ave function the pr oblem R(Q) wil l have at le ast one optimal solution z ¤ satisfying the pr op erty that x ¤` = 1 for ` = 1 ; : : : ; k and z ¤ ` = 0 for ` = k + 1 ; : : : ; N for some k = 1 ; : : : ; N Mor e over, if is strictly c onc ave, al l optimal solutions to R(Q) ar e of that form. Pro of: By Lemma 13 the problem (Q) will ha v e an optimal solution z ¤ of the follo wing form: z ¤ j = 8>>>><>>>>: 1 j = 1 ; : : : ; k ¡ 1 j = k 0 j = k + 1 ; : : : ; N where 0 < 1. The result immediately follo ws if = 1, so in the remainder of the pro of w e will assume < 1. It is ob vious that the follo wing t w o solutions are feasible solutions to the problem (Q): z 0 j = 8><>: 1 j = 1 ; : : : ; k ¡ 1 0 j = k ; : : : ; N z 00 j = 8><>: 1 j = 1 ; : : : ; k 0 j = k + 1 ; : : : ; N and f ( z ¤ ) f ( z 0 ) and f ( z ¤ ) f ( z 00 ) b y the optimalit y of z ¤ Since 0 < 1 w e ha v e: f ( z ¤ ) (1 ¡ ) f ( z 0 ) + f ( z 00 ) : (4.1) Since the function is conca v e w e ha v e: (1 ¡ ) f ( z 0 ) + f ( z 00 ) = (1 ¡ ) N X j =1 j z 0 j ¡ (1 ¡ ) N X j =1 d j z 0 j + N X j =1 j x 00j ¡ N X j =1 d j z 00 j PAGE 90 90 k ¡ 1 X j =1 j + k ¡ k ¡ 1 X j =1 d j + d k = f ( z ¤ ) : It follo ws that f ( z ¤ ) = f ( z 0 ) = f ( z 00 ), whic h means that there exists an in tegral optimal solution z ¤ to (Q) with the prop ert y that z ¤ j = 1 for j = 1 ; : : : ; k and z ¤ j = 0 for j = k + 1 ; : : : ; N If is strictly conca v e, w e will ha v e strict inequalit y in ( 4.1 ), con tradicting that < 1. This implies that an y optimal solution is of the desired form. The follo wing Decreasing Net Rev en ue Selection (DNRS) algorithm can no w b e used to ecien tly solv e (Q) to optimalit y in O ( N log N ) time for the case where is conca v e. DNRS algorithm Step 0. Sort the retailers in decreasing order of j =d j Step 1. Let k = 1, j ¤ = 0, and f = f (0). Let z 1 = 1 and z j = 0 for j = 2 ; : : : ; N Step 2. If f > f ( z ), set f = f ( z ) and let j ¤ = k Step 3. If k = N stop with optimal solution v alue f Otherwise, let k = k + 1, set z k = 1 and return to Step 2. The optimal solution to the problem has z ¤ j = 1 for j = 1 ; : : : ; j ¤ and z j = 0 otherwise. 4.2.3 The CSSPU revisited W e no w apply the results obtained so far to the CSSPU. Recall that, in the CSSPU, w e ha v e H i ( z ) = N X j =1 a ij z j + vuut 2 f i h i N X j =1 d j z j : Since these functions are conca v e, w e ma y conclude from the fact that R(AP) con tains all in tegral extreme p oin ts, that the singlesourcing constrain ts are not PAGE 91 91 necessary in this v arian t of the CSSP In other w ords, a singlesourcing optimal solution automatically exists, ev en if the singlesourcing constrain ts are not imp osed explicitly This c hoice of cost function leads to a pricing problem of the form (Q), with (for xed facilit y i ) j = u ¤j ¡ a ij ( d ) = p 2 f i h i d : Since is a conca v e function, the optimal solution of (Q) can b e found ecien tly b y the DNRS algorithm. Due to the absence of capacit y constrain ts in the CSSPU, it is p ossible in this mo del to accoun t for holding costs that dep end on b oth the facilit y and the retailer, denoted b y sa y h ij In terms of the last t w o of the three in terpretations of our mo del describ ed in Section 4.2.1 this means that the holding costs either dep end on the retailer only or are equal to the sum of the facilit y and retailer holding costs. In this case, w e obtain H i ( z ) = N X j =1 a ij z j + vuut 2 f i N X j =1 h ij d j z j : Dening j = u ¤j ¡ a ij ( d ) = p 2 f i d w e then obtain a pricing problem of the form maximize f ( z ) N X j =1 j z j ¡ N X j =1 h ij d j z j sub ject to z j 2 f 0 ; 1 g j = 1 ; : : : ; N PAGE 92 92 whic h can b e solv ed to optimalit y using the DNRS algorithm, with h ij d j replacing d j 4.3 Extensions of the CSSPU This section discusses sev eral generalizations of the basic CSSP In our uncapacitated CSSP w e ha v e implicitly assumed that an y pro duction facilit y ma y b e assigned an y subset of the retailers, including, for example, the en tire set of retailers. It seems unlik ely that this is indeed p ossible in man y practical settings. Instead, there ma y b e a limit to the supply rate of ra w materials that precludes a giv en facilit y from pro ducing extremely large quan tities of the pro duct. F urthermore, the a v ailabilit y of pro duction equipmen t or lab or ma y dictate a nite throughput rate. Our rst extension (Section 4.3.1 ) deals with these situations b y allo wing for a limit on the total demand rate of all retailers assigned to a giv en facilit y In some situations, suc h capacit y constrain ts ma y exist in principle, but w e ha v e the option to expand the capacit y at a certain cost. Our second extension (Section 4.3.2 ) incorp orates suc h expansion opp ortunities in the problem. If the pro duction facilit y has limited storage capacit y it ma y then face a b ound on the pro duction batc h size. This means that, for a giv en assignmen t of retailers to a facilit y that facilit y ma y not b e able to c ho ose the EOQ batc h size if it exceeds the storage capacit y W e address this issue in Section 4.3.3 If w e are able to expand the storage capacit y b y ren ting additional storage space, w e can apply the mo del discussed Section 4.3.4 This mo del extends the previous mo del b y allo wing for in v en tory capacit y expansion opp ortunities at a cost. W e will deriv e the cost functions for all of these extensions, and sho w that the corresp onding v arian ts of the CSSP fall in the class of assignmen t problems (AP). This means that their asso ciated pricing problem is of the form (Q). In addition, w e sho w that the structure of the cost functions is suc h that either (Q) or its PAGE 93 93 relaxation can b e solv ed ecien tly Lemma 13 sa ys that, in general, the relaxation of (Q) has an optimal solution in whic h only a single v ariable has a fractional v alue. Similar solution metho ds as ha v e b een deriv ed for the knapsac k problem [ 35 ] can then b e applied to solv e (Q) to optimalit y relativ ely ecien tly ev en if its relaxation do es not p ossess an in tegral optimal solution. Note that in all cases that w e will discuss b elo w w e will ha v e j = u ¤j ¡ a ij in the pricing problem. Ho w ev er, the function will b e dieren t for eac h v arian t. 4.3.1 Pro duction Capacit y Constrain ts In this section w e extend the CSSPU to include pro duction capacit y limits, and denote the corresp onding problem b y CSSPP The capacit y constrain ts w e consider limit the total demand rate of all retailers that can b e assigned to facilit y i This leads to the follo wing additional constrain ts in the CSSP: N X j =1 d j x ij b i i = 1 ; : : : ; M : Alternativ ely w e ma y incorp orate the constrain t in the ob jectiv e function as follo ws: H i ( z ) = 8><>: P Nj =1 a ij z j + q 2 f i h i P Nj =1 d j z j if P Nj =1 d j z j b i + 1 otherwise whic h leads to a pricing problem of the form (Q) with (for xed facilit y i ) ( d ) = 8><>: p 2 f i h i d if d b i + 1 otherwise : Note that adding the capacit y constrain t destro ys the conca vit y of the function H i and therefore that of the functions in the pricing problems. Ho w ev er, w e still kno w b y Lemma 13 that the optimal solution to the relaxation of the pricing PAGE 94 94 problem con tains at most one fractional v alue. Moreo v er, w e only ha v e a fractional v alue if the capacit y constrain t is binding. This observ ation means that the relaxation of the pricing problem can b e solv ed ecien tly using a mo dication of the DNRS algorithm, where w e consider retailers in order of decreasing net rev en ue (ratio j =d j ) un til the capacit y constrain t is binding. The last considered solution will then, in general, con tain a single fractional v alue. The pricing problem in this case is in fact the Economic Order Quan tit y Problem with Mark et Choice and total pro duction capacit y constrain t (EOQMC 2 ), studied b y Geunes et al. [ 29 ]. 4.3.2 The CSSP with Pro duction Capacit y Expansion In this section w e consider the case where a particular pro duction capacit y is installed, but w e ha v e the opp ortunit y to expand the capacit y at some cost. This v ersion of the problem is called CSSPPE. F or facilit y i the cost function asso ciated with capacit y expansion is denoted b y g i W e assume that the functions g i are monotone increasing on R + and without loss of generalit y that g i (0) = 0. Clearly the capacit y will then only b e expanded b y as m uc h as is necessary for a giv en retailer assignmen t, so that H i ( z ) = N X j =1 a ij z j + vuut 2 f i h i N X j =1 d j z j + g i max ( 0 ; N X j =1 d j z j ¡ b i )! : This leads to a pricing problem of the form (Q) with (for xed facilit y i ) ( d ) = p 2 f i h i d + g i (max f 0 ; d ¡ b i g ) : If the function g i is conca v e, the function will b e piecewise conca v e. In addition, the relaxation of the corresp onding pricing problem can then b e solv ed ecien tly b y noting that this problem has an in tegral optimal solution that can b e found using the DNRS algorithm unless the original capacit y constrain t is binding. If the original capacit y constrain t is binding, the optimal solution to the relaxation of PAGE 95 95 the pricing problem ma y ha v e a single fractional v ariable, and this solution can b e found in the same w a y as the solution to the CSSPP studied in Section 4.3.1 4.3.3 The CSSP with In v en tory Capacit y So far, w e ha v e assumed that the facilities ha v e unlimited storage capacit y If facilit y i faces an in v en tory capacit y of I i units (yielding the CSSPI), this capacit y limits the batc h size at that facilit y In particular, the optimal batc h size is the smaller of the optimal unconstrained EOQ pro duction quan tit y and the in v en tory capacit y This means that the optimal batc h size and corresp onding time b et w een setups are equal to: Q ¤i = min 0@ I i ; s 2 f i P Nj =1 d j x ij h i 1A T ¤ i = min I i P nj =1 d j x ij ; s 2 f i h i P Nj =1 d j x ij : When the assignmen t v ariables x i ¢ are xed, the optimal costs for facilit y i are giv en b y: H i ( x i ¢ ) = N X j =1 a ij z j + f i min I i P Nj =1 d j x ij ; q 2 f i h i P Nj =1 d j x ij + 1 2 h i N X j =1 d j x ij min I i P Nj =1 d j x ij ; s 2 f i h i P Nj =1 d j x ij = N X j =1 a ij z j + 1 2 max 0@ 2 f i N X j =1 d j x ij = I i ; vuut 2 f i h i N X j =1 d j x ij 1A + 1 2 min 0@ h i I i ; vuut 2 f i h i N X j =1 d j x ij 1A : Alternativ ely this cost function can b e written as: H i ( x i ¢ ) = 8><>: P Nj =1 a ij z j + 1 2 h i I i + f i P Nj =1 d j x ij = I i if P Nj =1 d j x ij > 1 2 h i I 2 i =f i P Nj =1 a ij z j + q 2 f i h i P Nj =1 d j x ij otherwise : PAGE 96 96 This leads to a pricing problem of the form (Q) with (for xed facilit y i ) ( d ) = 8><>: 1 2 h i I i + d f i = I i if d > 1 2 h i I 2 i =f i p 2 f i h i d otherwise : The function is clearly piecewise conca v e, since the t w o segmen ts are conca v e functions of d Ho w ev er, also note that the left and righ t deriv ativ es of at d = 1 2 h i I 2 i =f i are b oth equal to f i = I i whic h means that is a conca v e function of d Th us w e kno w that there exists an optimal solution to the pricing problem satisfying the DNRS prop ert y 4.3.4 The CSSP with In v en tory Capacit y Expansion Finally w e will consider the case in whic h a giv en in v en tory capacit y can b e expanded at some cost. As ab o v e, let I i b e the in v en tory capacit y at facilit y i W e denote the cost function asso ciated with expanding the capacit y at facilit y i b y g i As in the case of pro duction capacit y expansion, w e again assume that the functions g i are monotone increasing and conca v e functions on R + and g i ( y ) = 0 if y < 0. If the in v en tory capacities after expansion are kno wn, it is easy to see that the problem reduces to the CSSPI, and the pricing problem has an optimal DNRS solution. Clearly this also holds for the optimal capacit y expansion v alues, whic h implies that the pricing problems for CSSPIE ha v e optimal DNRS solutions as w ell. If w e can nd optimal in v en tory capacit y expansion v alues for an y giv en assignmen t of retailers to a facilit y w e can solv e the pricing problem again b y simply considering eac h DNRS solution. So supp ose w e consider assignmen ts where x ij = 1 for j = 1 ; : : : ; k and x ij = 0 otherwise (for k = 0 ; : : : ; N ). The optimal in v en tory capacit y expansion for facilit y i and a giv en k is then giv en b y PAGE 97 97 the solution to the follo wing optimization problem: min y 0 8<: max 0@ f i P kj =1 d j I i + y ; 1 2 vuut 2 f i h i k X j =1 d j 1A + 1 2 min 0@ h i ( I i + y ) ; vuut 2 f i h i k X j =1 d j 1A + g i ( y ) 9=; : If q 2 f i P kj =1 d j =h i I i that is, the EOQ solution satises the original capacit y constrain t, w e ha v e that the optimal capacit y expansion is y ¤ = 0. Otherwise, w e need to solv e the problem min y 0 i ( y ) where i ( y ) = 8><>: 1 2 i ( I i + y ) + f i I i + y P kj =1 d j + g i ( y ) if 0 y < q 2 f i P kj =1 d j =h i ¡ I i q 2 f i h i P kj =1 d j + g i ( y ) if y q 2 f i P kj =1 d j =h i ¡ I i : Note that the fact that g i is monotone increasing and that i is con tin uous implies that w e only need to consider v alues 0 y q 2 f i P kj =1 d j =h i ¡ I i so that w e need to solv e the problem minimize 1 2 i ( I i + y ) + f i I i + y k X j =1 d j + g i ( y ) sub ject to 0 y vuut 2 f i k X j =1 d j =h i ¡ I i : The complicating factor here is that p i ( y ) 1 2 i ( I i + y ) + f i I i + y k X j =1 d j is a con v ex function of y whereas g i ( y ) is a conca v e function of y whic h in general leads to a global optimization problem. PAGE 98 98 In the sp ecial case where g i is linear, the problem is a con v ex optimization problem, and is easily solv ed. In the more general case where g i is a piecewise linear conca v e function with a nite n um b er of segmen ts (whic h commonly o ccurs in practice), w e can solv e the problem b y solving a n um b er of easy con v ex optimization problems. Note that, in this case, w e ma y write g i ( y ) = min s q ( s ) i ( y ) where q ( s ) i are all socalled xedc harge cost functions, that is, functions that are linear apart from a xed cost that is only c harged when y > 0. Then let y ¤ s = arg min 8<: p i ( y ) + q ( s ) i ( y ) : 0 y vuut 2 f i k X j =1 d j =h i ¡ I i 9=; s ¤ = arg min s p i ( y ¤ s ) + q ( s ) i ( y ¤ s ) : W e then ha v e p i ( y ¤ s ¤ ) + q s ¤ i ( y ¤ s ¤ ) p i ( y ¤ s ) + q s i ( y ¤ s ) p i ( y ¤ s ¤ ) + q s i ( y ¤ s ¤ ) for s 6 = s ¤ ) q s ¤ i ( y ¤ s ¤ ) q s i ( y ¤ s ¤ ) for s 6 = s ¤ ) q s ¤ i ( y ¤ s ¤ ) = min s f q s i ( y ¤ s ¤ ) g = g i ( y ¤ s ¤ ) : Th us the optimal solution of the problem is y ¤ = y ¤ s ¤ F or a general conca v e cost function g i the problem is harder. In that case, w e iterativ ely appro ximate the function g i from b elo w b y a piecewise linear conca v e cost function, where g i and its appro ximation coincide at the breakp oin ts of the piecewise linear appro ximations. The algorithm is based on the observ ation that, if the solution obtained using the appro ximating function is attained at a breakp oin t, that solution is optimal for the original problem as w ell. If the solution obtained using the appro ximating function is not attained at a breakp oin t, w e impro v e PAGE 99 99 the appro ximating function b y creating a new breakp oin t at that solution. The follo wing describ es this sc heme. Step 0. Set S = 1. Let 0 = y (0) < y ( S ) = q 2 f i P kj =1 d j =h i ¡ I i Step 1. Construct an appro ximating function g ( S ) i b y connecting p oin ts ( y ( s ) ; g i ( y ( s ) )) and ( y ( s +1) ; g i ( y ( s +1) )) for s = 0 ; : : : ; S ¡ 1. Step 2. Solv e the problem with resp ect to the cost function g ( S ) i Denote the solution b y ^ y ( S ) and let f S = p i ( ^ y ( S ) ) + g ( S ) i ( ^ y ( S ) ). Step 3. If ^ y ( S ) = y ( s ) for some s = 0 ; : : : ; S stop and ^ y ( S ) is the optimal solution to the problem with resp ect to the original cost function g i Otherwise there exists an in teger 0 ` S ¡ 1 suc h that ^ y ( S ) 2 ( y ( ` ) ; y ( ` +1) ). Let y ( s +1) = y ( s ) for s = S ; : : : ; ` + 1 and set y ( ` +1) = ^ y ( S ) Set S = S + 1 and and return to Step 1. Theorem 15 Supp ose y ¤ is the optimal solution to the pr oblem (CP) with gener al c onc ave c ost function g i The se quenc e f f s g gener ate d by the ab ove algorithm wil l c onver ge to f ¤ = p i ( y ¤ ) + g i ( y ¤ ) that is, lim s !1 f s = f ¤ : Pro of: If the algorithm terminates in a nite n um b er of iterations, w e immediately see that the optimal solution y ¤ is obtained. Th us w e only need to consider the case in whic h the algorithm do es not terminate nitely and th us generates an innite sequence of solutions. Since g i is conca v e, w e kno w that g i ( y ) g ( s ) i ( y ) and g ( s +1) i ( y ) g ( s ) i ( y ) for an y feasible y and in teger k 0. It follo ws that f ¤ = p i ( y ¤ ) + g i ( y ¤ ) p i ( y ¤ ) + g ( s ) i ( y ¤ ) p i ( ^ y ( s ) ) + g ( s ) i ( ^ y ( s ) ) = f s andf s +1 = p i ( ^ y ( s +1) ) + g ( s +1) i ( ^ y ( s +1) ) p i ( ^ y ( s +1) ) + g ( s ) i ( ^ y ( s +1) ) p i ( ^ y ( s ) ) + g ( s ) i ( ^ y ( s ) ) = f s : PAGE 100 100 This implies that the sequence f f s g is nondecreasing and b ounded from ab o v e. Th us it m ust con v erge to a limit, sa y F and f k F f ¤ It no w remains to b e sho wn that F = f ¤ W e will pro v e this b y con tradiction. Supp ose F 6 = f ¤ W e kno w then that F < f ¤ and g i ( ^ y ( s ) ) ¡ g ( s ) i ( ^ y ( s ) ) = p i ( ^ y ( s ) ) + g i ( ^ y ( s ) ) ¡ p i ( ^ y ( s ) ) ¡ g ( s ) i ( ^ y ( s ) ) f ¤ ¡ f ( s ) f ¤ ¡ F > 0 : a ( ) y ( 1 ) s y + ( 1 ) y + B C A E D i g ( ) s i g ( 1 ) s ig + Figure 4{1: Appro ximation algorithm As sho wn in Figure 4{1 in iteration s w e ha v e j AD j and line segmen t B C is replaced b y B A and AC Let ( y 0 ; g i ( y 0 )) b e a p oin t on function g i suc h that the line segmen t from it to (0 ; g i (0) has length that is, p y 2 0 + ( g i ( y 0 ) ¡ g i (0)) 2 = Let ¡ b e the slop e of this segmen t. Since the function g i is a nondecreasing conca v e function, it follo ws that tan ¡ < 1 for an y c hoice of ^ y ( s +1) and j B C j j AB j j AD j Supp ose AE is the heigh t of the triangle AB C Then the area of the triangle AB C is then B C ¢ AE 2 = B C ¢ AD ¢ cos 2 2 2 p 1 + tan 2 2 1 + ¡ 2 ¢ > 0 : PAGE 101 101 This means that at eac h iteration the algorithm cuts an area of at least ¢. Let V k b e the area b et w een curv es g i and g ( s ) i for an y in teger s 1. Then 0 V s +1 V s ¡ ¢ V s ¡ 1 ¡ 2¢ ¢ ¢ ¢ V 1 ¡ s ¢ : It follo ws that V 1 s ¢. When s go es to innit y w e ha v e S 1 lim s !1 s ¢ = 1 Th us w e ha v e a con tradiction and w e can conclude that F = f ¤ 4.4 Greedy Heuristic In this section w e prop ose t w o greedy heuristic algorithms that can b e used to obtain an initial solution. The idea of our rst greedy heuristic is to iterativ ely add retailers to facilities based on minimizing the incremen tal cost asso ciated with doing so. In particular, this heuristic uses a \b est rst" strategy for assigning retailers to facilities, and will therefore b e referred to as the b est rst (BF) gr e e dy heuristic F or eac h unassigned retailer j and facilit y i the b est rst strategy determines the increase in total cost incurred if the asso ciated assignmen t w ere to b e made. The least costly assignmen t is then made, and new incremen tal costs are determined for all remaining retailers un til all retailers are assigned. Note that the cost of the solution th us obtained ma y b e + 1 whic h usually corresp onds to a hard constrain t that is violated. This heuristic has a running time of O ( M N 2 ). The second greedy heuristic, whic h w e will call the DNRS gr e e dy heuristic is inspired b y the DNRS prop ert y Note that the pricing problem can b e view ed as an instance of the CSSP with t w o facilities, one of whic h is a virtual facilit y represen ting the fact that w e are selecting only a subset of the retailers (i.e., the retailers not selected are assigned to the virtual facilit y). Using this in terpretation, ¡ a ij represen ts the (negativ e) rev en ue asso ciated with assigning retailer j to facilit y i whereas the costs are a function of P Nj =1 d j x ij In the spirit of the DNRS prop ert y w e no w determine, for eac h of the facilities, an ordering of the retailers PAGE 102 102 according to decreasing desirabilit y that is, according to decreasing v alue of ¡ a ij =d j or, equiv alen tly according to increasing v alue of a ij =d j F or eac h facilit y i w e then denote the set con taining the ` most desirable retailers b y R i ( ` ). The a v erage cost asso ciated with assigning the retailers in R i ( ` ) to facilit y i is equal to K i ( R i ( ` )) = P j 2 R i ( ` ) a ij + i P j 2 R i ( ` ) d j j R i ( ` ) j or K i ( R i ( ` )) = P j 2 R i ( ` ) a ij + i P j 2 R i ( ` ) d j P j 2 R i ( ` ) d j : The heuristic no w pro ceeds b y nding the pair ( i; ` ) with smallest a v erage cost, and assigns all retailers in set R i ( ` ) to facilit y i The a v erage costs are then recalculated after discarding these retailers and this facilit y and the pro cedure is rep eated un til all facilities are considered. Note that this pro cedure ma y only yield a partial solution, in whic h a subset of retailers is iden tied for eac h facilit y but not all retailers are assigned to a facilit y An y retailers that are not assigned b y the DNRS greedy heuristic are assigned according to the BF greedy heuristic describ ed earlier in this section. DNRS greedy heuristic Step 0. Let F = f 1 ; : : : ; m g R = f 1 ; : : : ; N g and x ij = 0 for all i 2 P and j 2 R Step 1. F or all i 2 F and ` = 1 ; : : : ; N determine K i ( R i ( ` ) \ R ). F or all i 2 F let ` ¤i = arg min ` =1 ;::: ;N f K i ( R i ( ` ) \ R ) g Let i ¤ = arg min i 2 F K i ( R i ( ` ¤i ) \ R ). Step 2. Set x i ¤ j = 8><>: 1 for j 2 R i ¤ ( ` ¤i ¤ ) 0 otherwise : Set F = F nf i ¤ g and R = R n R i ¤ ( ` ¤i ¤ ). Step 3. If R = then stop. Otherwise, if P 6 = then return to Step 1. PAGE 103 103 Step 4. If R 6 = x the assignmen ts in x and apply the BF heuristic to assign the remaining retailers in R Denote the resulting solution b y x This heuristic has the same running time as the BF heuristic, that is, it will run in O( M N 2 ) time. But since this algorithm exploits the DNRS prop ert y w e exp ect that this algorithm will generally pro vide a higher qualit y solution to the CSSP 4.5 Computational Results In this section, w e presen t the results of computational tests of our exact branc handprice algorithms as w ell as our heuristics on the basic uncapacitated case (CSSPU), the case with pro duction capacities (CSSPP), and the case with pro duction capacit y expansion (CSSPPE). In order to create a manageable scop e of test problems, w e fo cused on testing uncapacitated instances as w ell as instances with pro duction capacit y limits. Because of the similar problem structures, w e exp ect our algorithms to p erform similarly on the problem classes with in v en tory capacities and in v en tory capacit y expansion. W e tested the algorithms on a randomly generated set of problem instances that are represen tativ e of a wide range of practical scenarios. In particular, w e considered problem instances with M = 5 and M = 10 facilities, and mainly fo cused on problems with N = 15, 25, and 50 retailers. These problem sizes are sucien tly large, but can still b e solv ed in reasonable time using branc handprice. Our exp erimen ts therefore enables us to eectiv ely gauge the p erformance of our heuristics against the true optimal v alues. In addition, w e p erformed limited tests on instances with N = 100 retailers, to illustrate that the computational time required for the branc handprice algorithm increases dramatically when the n um b er of retailers is increased. The heuristics, in con trast, are still able to nd a v ery high qualit y solution in v ery limited time for these large instances. F or eac h problem size, w e generated facilit y and retailer lo cations uniformly in the square [0 ; 50] 2 with retailer demands d j generated indep enden tly from a uniform distribution on the in terv al [500 ; 2500]. The unit PAGE 104 104 holding costs w ere generated i.i.d. uniformly in the in terv al [1 ; 4]. The assignmen t costs a ij w ere c hosen to include v ariable pro duction costs (generated i.i.d. uniformly in [5 ; 20]), as w ell as linear transp ortation costs (with the unit transp ortation costs equal to the Euclidean distance b et w een a facilit y and a retailer). Finally the setup costs w ere generated i.i.d. uniformly as w ell. T o study the impact of v arying the magnitude of the setup costs, w e considered 3 dieren t in terv als: [20 ; 100], [100 ; 500], and [500 ; 2500]. F or the CSSPP w e generated the pro duction capacit y limits b i uniformly in the in terv al [ D ; 1 : 5 D ], where D is the exp ected total demand faced p er facilit y when the retailers are assigned randomly: D = ( d + d ) N 2 M = 15 M where d = 5 and d = 25 are the lo w er and upp er b ound of the in terv al used to generate the retailer demands. Finally for the CSSPPE w e c hose tigh ter pro duction capacities: uniform in the in terv al [0 : 9 D ; 1 : 1 D ]. The capacit y expansion costs w ere c hosen to b e the conca v e function g i ( y ) = 30 p y : The results of the tests are sho wn in T ables 4{1 4{2 and 4{3 F or all scenarios, w e generated 25 random instances for eac h c hoice of parameter settings, and pro vide the a v erage results o v er these instances. F or the branc handprice algorithm, w e rep ort (i) the a v erage computation time for nding the optimal solution to the ro ot problem as w ell as the n um b er of columns generated (rep orted in the column lab elled \B&P ro ot" under the subheadings \Time" and \Cols."); and (ii) the a v erage computation time for nding the optimal solution to the assignmen t problem, the a v erage n um b er of columns generated, and the a v erage n um b er of no des in the branc handprice tree that w ere examined; note that this latter metric is only rep orted (in the column lab elled simply \B&P" under the PAGE 105 105T able 4{1: Results for M = 5 facilities B&P ro ot B&P DNRS VLSN Mo del fi N Time Cols. Time Cols. No des F eas. Opt. Time Error F eas. Opt. Time Error (sec) (sec) (sec) (%) (sec) (%) 15 1.3 172 3 0.001 11.4 25 0.165 0.0 [20 ; 100] 25 4.0 462 7 0.001 0.8 25 0.104 0.0 50 139.7 5169 9 0.003 0.7 25 0.212 0.0 15 1.2 158 3 0.001 4.7 25 0.138 0.0 CSSPU [100 ; 500] 25 4.2 515 6 0.001 1.7 25 0.137 0.0 50 81.5 4543 12 0.002 0.2 24 0.151 0.0 15 1.3 171 1 0.001 8.4 25 0.181 0.0 [500 ; 2500] 25 4.0 465 6 0.002 3.4 25 0.153 0.0 50 73.7 4322 11 0.003 0.6 25 0.198 0.0 15 1.1 117 1.5 158 1.3 18 1 0.000 9.0 ¤24 10 0.188 1.8 [20 ; 100] 25 4.6 375 6.2 499 1.4 23 0 0.001 7.8 25 8 0.220 1.6 50 194.3 4946 223.7 5895 1.2 25 0 0.002 5.0 25 5 0.433 1.1 15 1.1 122 1.3 139 1.1 16 0 0.000 9.2 ¤24 8 0.179 2.7 CSSPP [100 ; 500] 25 4.4 375 5.0 426 1.2 24 2 0.001 5.4 25 9 0.192 1.0 50 309.3 4971 344.0 6017 1.4 25 1 0.002 6.3 25 6 0.476 1.7 15 1.0 117 4.0 460 4.3 21 2 0.000 10.1 25 8 0.164 1.7 [500 ; 2500] 25 4.6 390 5.8 485 1.3 24 1 0.001 7.4 25 8 0.212 1.4 50 148.1 4066 207.9 6221 2.6 25 0 0.003 5.5 25 3 0.441 1.1 15 1.2 138 7 0.000 2.7 25 0.112 0.0 [20 ; 100] 25 5.8 481 10 0.002 2.3 25 0.144 0.0 50 196.9 6360 7 0.003 0.4 24 0.199 0.0 15 1.3 147 3 0.001 5.9 25 0.138 0.0 CSSPPE [100 ; 500] 25 6.0 502 3 0.002 1.6 25 0.160 0.0 50 194.3 6592 10 0.004 0.8 25 0.213 0.0 15 1.1 146 5 0.001 4.4 25 0.130 0.0 [500 ; 2500] 25 4.9 467 4 0.001 1.7 24 0.167 0.0 50 267.1 7626 7 0.007 0.6 23 0.234 0.0 ¤The remaining instance w as infeasible. PAGE 106 106T able 4{2: Results for M = 10 facilities B&P ro ot B&P DNRS VLSN Mo del fi N Time Cols time Cols No des F eas. Opt. Time Error F eas. Opt. Time Error (sec) (sec) (sec) (%) (sec) (%) 15 1.2 164 0 0.002 29.6 25 0.270 0.0 [20 ; 100] 25 3.8 456 0 0.002 11.4 25 0.274 0.0 50 32.5 2076 1 0.006 2.9 25 0.301 0.0 15 1.1 170 0 0.001 34.7 25 0.260 0.0 CSSPU [100 ; 500] 25 3.8 493 0 0.000 15.1 25 0.375 0.0 50 28.4 1948 0 0.006 3.3 25 0.349 0.0 15 1.1 177 0 0.000 20.3 25 0.255 0.0 [500 ; 2500] 25 3.6 498 0 0.000 13.8 24 0.327 0.0 50 30.7 2094 0 0.003 3.6 25 0.388 0.0 15 2.6 91 2.6 319 3.0 8 0 0.000 15.5 ¤24 3 0.293 3.9 [20 ; 100] 25 4.0 260 4.0 433 1.6 19 0 0.001 12.1 25 5 0.395 2.1 50 81.5 1545 81.5 4174 3.4 25 0 0.004 9.0 25 4 0.790 1.1 15 2.0 93 2.0 172 1.7 9 0 0.000 12.1 ¤¤22 4 0.285 3.2 CSSPP [100 ; 500] 25 8.3 256 8.3 911 4.0 16 0 0.002 13.3 ¤24 7 0.373 1.5 50 58.6 1438 58.6 2830 2.2 24 0 0.003 9.8 ¤24 3 0.804 2.0 15 1.7 104 1.7 218 2.2 11 0 0.001 16.9 ¤¤¤22 2 0.315 3.7 [500 ; 2500] 25 9.7 269 9.7 1051 4.6 14 0 0.001 12.3 25 4 0.356 2.4 50 67.7 1473 67.7 3288 2.6 24 0 0.002 8.4 25 2 0.819 1.6 15 1.1 138 0 0.001 34.0 25 0.274 0.0 [20 ; 100] 25 3.3 355 0 0.003 10.3 25 0.286 0.0 50 52.6 2372 0 0.009 3.8 25 0.405 0.0 15 1.1 147 0 0.001 22.5 25 0.272 0.0 CSSPPE [100 ; 500] 25 3.3 365 0 0.002 13.3 25 0.332 0.0 50 61.7 2716 0 0.009 4.1 24 0.336 0.0 15 1.1 142 0 0.002 28.5 25 0.281 0.0 [500 ; 2500] 25 3.1 342 0 0.004 13.1 24 0.264 0.0 50 42.7 2090 0 0.008 3.0 25 0.370 0.0 ¤The remaining instance w as infeasible.¤¤2 of the 3 remaining instances w ere infeasible.¤¤¤1 of the 3 remaining instances w as infeasible. PAGE 107 107T able 4{3: Results for n = 100 retailers, with fi2 [100 ; 500] B&P ro ot B&P DNRS VLSN Mo del M Time Cols. Time cols No des F eas. Opt. Time Error F eas. Opt. Time Error (sec) (sec) (sec) (%) (sec) (%) CSSPU 5 7,510 25,337 11 0.011 0.1 25 0.196 0.0 CSSPU 10 707 11,043 2 0.020 1.0 25 0.714 0.0 CSSPP 10 5,433 17,086 10,557 47,117 6.0 25 0 0.018 6.4 25 5 2.363 1.3 CSSPPE 10 11,865 17,978 0 0.037 0.6 25 24 0.748 0.0 PAGE 108 108 subheadings \Time", \Cols.", and \No des") for the problem classes for whic h the optimal in tegral solution w as not found at the ro ot no de for an y of the 25 instances. F or the DNRS and VLSN heuristics, w e considered three primary p erformance metrics: (i) the n um b er of problem instances (out of 25) in eac h category in whic h our heuristic metho ds pro vided a pro v ably optimal solution (rep orted under the subheading \Opt."); (ii) a v erage computation time for obtaining the heuristic solution (rep orted under the subheading \Time"); and (iii) p ercen t optimalit y gap (rep orted under the subheading \Error"), computed as the relativ e amoun t b y whic h the heuristic solution v alue exceeds the optimal solution v alue found using the branc handprice algorithm. In addition to these metrics, for the CSSPP w e also considered the n um b er of problem instances within eac h problem class for whic h the heuristic solution metho ds generated capacit y feasible solutions (rep orted under the subheading \F eas."). All tests w ere p erformed on a PC with a 650 MHz P en tium I I I pro cessor and 512 MB RAM. The results sho w that the branc handprice algorithm is able to nd the optimal solution to problems with up to 25 retailers in seconds of computation time. Ho w ev er, when the n um b er of retailers gro ws to 50, the computation times increase to 14 min utes. While this increase is fairly dramatic, these problems are still solv ed in acceptable computing time. As T able 4{3 indicates, branc handprice solution time can extend to sev eral hours with 100 retailers (in fact, the column generation algorithm w as not able to nd the solution at the ro ot no de within da ys of computation time for CSSPP and CSSPPE for an y of the instances tested). F or all cases, ho w ev er, the DNRS and VLSN heuristic solutions are obtained in seconds. The DNRS heuristic exhibits relativ ely large errors when the n um b er of retailers is small or in the presence of capacities. Ho w ev er, applying the VLSN lo cal searc h heuristic alw a ys yields the optimal solution for the cases without capacities PAGE 109 109 and with capacit y expansion, and v ery highqualit y solutions in the presence of capacities. Th us the VLSN heuristic has pro v ed extremely eectiv e in quic kly nding high qualit y solutions, while the DNRS heuristic can quic kly nd a feasible solution when the n um b er of retailers is large. It is in teresting to note that, for the uncapacitated case as w ell as the case allo wing for capacit y expansion, the optimal in tegral solution is alw a ys found at the ro ot of the branc handprice tree, indicating that the set partitioning form ulation generally pro vides excellen t b ounds. Note also that the n um b er of no des required in the branc handprice algorithm (for those problems not solv ed at the ro ot no de) is apparen tly insensitiv e to the n um b er of retailers. The tables also indicate that the solution time, n um b er of columns, and n um b er of no des required are apparen tly insensitiv e to the magnitude of the xed costs (across the ranges w e tested), although the solution time and n um b er of columns are of course strongly correlated with the n um b er of retailers. 4.6 Summary In this c hapter w e studied a logistics net w ork design problem in v olving the allo cation of the demands of eac h of a set of retailers to a n um b er of serv er facilities. As is often the case in practice, w e apply a singlesour cing restriction, whic h ensures that eac h retailer will order and receiv e shipmen ts from a unique source facilit y Suc h arrangemen ts reduce crossco ordination planning complexit y and facilitate impro v ed ordering, shipping, and receiving co ordination b et w een eac h retailer and its source facilit y Our mo del allo ws us to accoun t for pro duction economies of scale in a m ultifacilit y con text, and leads to a largescale nonlinear, binary in teger programming problem. The sp ecial assignmen t structure of this problem allo w ed us to cater exact branc handprice algorithms to this problem class. T o enable the successful application of these algorithms, w e dev elop ed ecien t algorithms for solving the asso ciated pricing problems. W e also dev elop ed PAGE 110 110 a greedy heuristic algorithm, whic h w as then augmen ted with v ery largescale neigh b orho o d searc h metho ds that seek further solution impro v emen ts. In addition to the basic uncapacitated v ersion of the mo del, w e also considered sev eral applicable t yp es of capacit y constrain ts, along with the p ossibilit y of capacit y expansion under general capacit yexpansion cost functions. Our exp erimen tal results sho w ed that the solutions pro vided b y our sp ecialized exact algorithms are obtained in reasonable computation time when the n um b er of retailers is no more than ab out 50. Ho w ev er, our heuristic and neigh b orho o d searc h metho ds are able to nd v ery high qualit y solutions in negligible computation time. In our analysis, w e ha v e emplo y ed an assumption of constan t deterministic demand rates at eac h retailer, while this is clearly an appro ximation for practical con texts. F uture researc h migh t therefore fo cus on a dynamic discretetime v ersion of the mo del, whic h also generalizes the MPSSP to allo w for pro duction setup costs at facilities. F uture w ork on the sp ecial assignmen t structure of the mo del migh t also consider more general cost function structures, as w ell as accoun ting for dieren t t yp es of pro duction and assignmen t constrain ts in conjunction with these cost functions. PAGE 111 CHAPTER 5 MUL TIPERIOD FLEXIBLE DEMAND ASSIGNMENT PR OBLEM (MPFD A) 5.1 In tro duction In this c hapter, w e will study a class of exible pro duction planning problems o v er a discrete time horizon, whic h w e call the multip erio d exible demand assignment (MPFD A) problem. This class of problems is inspired b y a pro duction planning problem faced b y an U.S. man ufacturer in the steel industry whic h pro duces mak etoorder sp ecialt y steel plates from slabs. T o pro duce plates, the man ufacturer will rst cut eac h slab in to smaller pieces. Next, eac h piece is rolled in to a steel plate. While eac h slab can b e used to pro duce m ultiple plates of dieren t dimensions, the material required for pro ducing an individual plate should come from a single slab. It is p ossible that this assignmen t of plates to slabs do es not fully utilize eac h slab. The un used part of a slab, called a surplus, can b e recycled, yielding a rev en ue. W e assume that demand sp ecications for all customer orders are kno wn in adv ance, and that the plates are built to order. The k ey prop ert y of our problem that distinguishes it from more traditional pro duction planning problems is the presence of demand exibility That is, eac h customer order is for a steel plate with dimension (w eigh t) sp ecied b y a range, and customers will accept an y steel plate whose dimension is within that range. T o plan pro duction, the man ufacturer m ust not only decide on the timing of pro duction of eac h plate and the slab that is used to pro duce eac h plate, but also the size of eac h plate. Since plate rev en ues are t ypically prop ortional to their size, and the slab surplus can b e recycled, the c hoice of plate sizes will not only aect the rev en ue, but also the utilization of resources and thereb y the cost. 111 PAGE 112 112 Balakrishnan and Geunes [ 11 ] studied a singlep erio d v arian t of this problem. In particular, they ha v e dev elop ed a solution approac h that com bines mo del enhancemen t using v arious strong v alid inequalities, Lagrangian relaxation, and heuristic algorithms. Although the singlep erio d problem ma y b e sucien t for op erational lev el pro duction planning decisions, it is p ossible that signican t cost sa vings can b e ac hiev ed if the man ufacturer emplo ys a tactical lev el mo del to mak e longer term pro duction planning decisions. In this c hapter, w e will extend the singlep erio d mo del to a m ultip erio d setting and form ulate it as a problem in the class (AP). Although our problem is motiv ated b y the steel industry the mo del can b e applied to man y other con texts when customer demands are exible in terms of pro duct quan tities or size. F or example, customers ma y b e willing to accept lot sizes within a certain range, or man ufacturers ma y ha v e the exibilit y to decide the p ercen tage of demands to b e satised in eac h of the mark ets that it serv es. In our study w e will discuss three dieren t kinds of pro duction and in v en tory strategies, where w e can sto c k only end pro ducts but acquire ra w materials justintime, sto c k ra w materials only but pro duce justintime, or sto c k b oth. In the steel industry and man y other industries, the ra w materials suc h as steel slabs are v ery exp ensiv e and require a v ery large storage place. Th us it is often not economic to sto c k the ra w material. In this case the b est c hoice will b e rst strategy that is, to sto c k only the end pro ducts but acquire ra w material justintime. In some other industries, w e ma y face a situation where the ra w materials, for example b er rolls, are v ery easy to sto c k and cut. If, in addition, the demands for the end pro ducts v ary dramatically o v er time and therefore will require high safet y so c k lev els, w e will apply the second strategy and sto c k ra w materials only but pro duce plates justintime. The third strategy will b e suitable when the in v en tory cost for slab and plates are basically the same. It also can b e applied to cases where the end PAGE 113 113 pro ducts are fairly standard but the demands are large or v ary dramatically A example of this case is the w o o d building parts industry In this c hapter, w e will dev elop MPFD A mo dels for all three in v en tory strategies and fo cus our study on ecien t algorithms to solv e the subproblem of the assignmen t form ulation as w ell as the pricing problem in the branc handprice algorithm, whic h is a problem called the knapsack pr oblem with exp andable items (KPEI). The KPEI w as studied previously b y Balakrishnan and Geunes [ 11 ]. W e will lo ok further in to this problem and its algorithms, and prop ose a greedy heuristic based on structural prop erties of KPEI. W e will sho w that this heuristic is asymptotically optimal with probabilit y one under a v ery general mo del for the problem data. In addition, the prop erties of the KPEI also suggest that a branc handb ound algorithm will b e v ery eectiv e in solving this problem to optimalit y Finally w e dev elop greedy heuristics for the MPFD A mo dels based on insigh ts obtained b y studying the structure of optimal solutions to appropriate subproblems of these mo dels. 5.2 The MPFD A with Plate In v en tory 5.2.1 F orm ulation In the rst MPFD A mo del w e will assume an in v en tory p olicy in whic h only end pro ducts (plates) can b e stored and ra w materials (slabs) are obtained just in time and denote this problem b y MPFD AP W e assume that M slabs are a v ailable at the b eginning of eac h of T p erio ds. In particular, let W it and C it b e the w eigh t and pro duction cost of slab i ( i = 1 ; : : : ; M ) in p erio d t ( t = 1 ; : : : ; T ), and denote the total n um b er of plate orders o v er the time horizon b y N F or eac h order j ( j = 1 ; : : : ; N ), let j b e the p erio d in whic h order j is demanded, and let S t denote the set of orders whose demand p erio d is no earlier than t : S t = f j : j t g In addition, let R j l j and u j b e the unit rev en ue, minim um w eigh t, and maxim um w eigh t for order j The unit holding cost for order j in p erio d t is equal to h j t PAGE 114 114 Finally at the end of eac h p erio d, slab surplus of partially used slabs m ust b e recycled, yielding a surplus rev en ue r p er unit w eigh t. W e require that eac h plate is pro duced from exactly one slab. Without loss of generalit y w e assume that all orders are protable, and that w e cannot obtain a prot from simply recycling un used slabs, that is, w e assume that R j r for j = 1 ; : : : ; N C it r W it for i = 1 ; : : : ; M ; t = 1 ; : : : ; T : The ob jectiv e is to maximize the total prot while satisfying all customer demands. The MPFD AP problem can b e form ulated as an assignmen t problem in the form of (AP) as follo ws: maximize M X i =1 T X t =1 H it ( x i ¢ t ) sub ject to (AP P3 ) M X i =1 j X t =1 x ij t = 1 j = 1 ; : : : ; N (5.1) X j 2 S t l j x ij t W it i = 1 ; : : : ; M ; t = 1 ; : : : ; T (5.2) x ij t 2 f 0 ; 1 g i = 1 ; : : : ; M ; j 2 S t ; t = 1 ; : : : ; T where x ij t = 1 if order j is pro duced from slab i in p erio d t and 0 otherwise, and x i ¢ t = ( x ij t : j 2 S t ). Constrain ts ( 5.1 ) ensure that eac h plate is assigned to exactly one slab in one p erio d, and constrain ts ( 5.2 ) ensure feasibilit y of the assignmen t of plates to slabs. The function H it : f 0 ; 1 g j S t j R represen ts the maximal prot that can b e obtained from slab i in p erio d t as a function of the assignmen t v ector x 2 f 0 ; 1 g j S t j of plates to slab i in p erio d t : maximize X j 2 S t R j v j + r W it z ¡ X j 2 S t v j ¡ C it z ¡ X j 2 S t j ¡ 1 X k = t h j k v j PAGE 115 115 = X j 2 S t R j ¡ r ¡ j ¡ 1 X k = t h j k v j ¡ ( C it ¡ r W it ) z sub ject to (SP Pit ) l j x j v j u j x j for j 2 S t (5.3) X j 2 S t v j W it z (5.4) v j 0 for j 2 S t z 2 f 0 ; 1 g : Here z = 1 if slab i is used in p erio d t and 0 otherwise, and v j is the w eigh t of plate j pro duced from slab i in p erio d t The ob jectiv e function of the subproblem includes four parts: rev en ue from orders, rev en ue from recycling slabs, cost of slabs and in v en tory holding cost. In the subproblem (SP Pit ), constrain ts ( 5.3 ) mo del the demand exibilit y for plate j and constrain t ( 5.4 ) is the slab capacit y constrain t. The problem MPFD AP is NPHard since the knapsac k problem is a sp ecial case of this problem, obtained b y c ho osing M = T = 1 and l j = u j for j = 1 ; : : : ; N Since it has b een form ulated as an instance of the class of the assignmen t problems (AP), w e can apply all metho dologies dev elop ed for this class in Chapter 2 to the MPFD AP As in Chapter 4 w e will fo cus our study on the subproblem (Section 5.2.2 ) and the pricing problem (Section 5.2.3 ). In next section w e will analyze the subproblem and discuss its solution approac hes, as w ell as the greedy heuristic. 5.2.2 Subproblem and Greedy Heuristic In this section w e will study the subproblem (SP Pit ) and dev elop an ecien t algorithm for solving it to optimalit y Then, based on the insigh ts obtained from the structure of the optimal solution to this problem, w e dev elop a greedy heuristic for MPFD AP Clearly if the assignmen t v ector x = 0 in the subproblem, the optimal solution v alue is equal to 0. Alternativ ely if x 6 = 0, w e set z = 1 and obtain PAGE 116 116 a linear programming problem. Ho w ev er, the optimal solution to this problem can b e found m uc h more ecien tly as follo ws. First, w e sort all plates in the set N x = f j : x j = 1 g in decreasing order of their net rev en ue co ecien t R j = R j ¡ r ¡ P j ¡ 1 s = t h j s and ren um b er them as j = 1 ; : : : ; j N x j Then, let m b e the n um b er of plates in N x that ha v e a p ositiv e net rev en ue co ecien t, that is, are protable to pro duce, and let n b e the leading n um b er of plates from the list 1 ; : : : ; j N x j that can b e pro duced at their upp er b ound due to the capacit y constrain t. More formally w e ha v e m = max j 2 N x : R j > 0 n = max 8<: j 2 N x : W it ¡ j X k =1 u k ¡ j N x j X k = j +1 l k > 0 9=; : Note that, capacit y p ermitting, w e w ould lik e the rst m plates to b e pro duced at their upp erb ound, and the remaining ones at their lo w erb ound. On the other hand, the slab capacit y allo ws the rst n plates to b e pro duced at their upp erb ound. Therefore, if m n the optimal solution of (SP pit ) is giv en b y: v j = 8><>: u j for j = 1 ; : : : ; m l j for j = m + 1 ; : : : ; j N x j : Alternativ ely if m > n the optimal solution of (SP pit ) is giv en b y: v j = 8>>>><>>>>: u j for j = 1 ; : : : ; n W it ¡ P jk =1 u k ¡ P j N x j k = j +1 l k for j = n + 1 l j for j = n + 2 ; : : : ; j N x j : The running time of this algorithm is dominated b y the time needed to sort the plates, that is, w e can solv e the subproblem in O ( N log N ) time, since m and n can b e computed in O ( N ) time and the solution can b e obtained in O ( N ) time once m and n are kno wn. PAGE 117 117 W e can emplo y insigh ts from the ab o v e algorithm to dev elop a greedy heuristic for MPFD AP The general idea of this greedy heuristic is to pro duce the most protable plates using the c heap est slabs, where the protabilit y of pro ducing plate j in p erio d t is measured b y its net rev en ue co ecien t R j t = R j ¡ r ¡ P j ¡ 1 s = t h j s and the cost of a slab is dened as its pro duction cost p er unit w eigh t, that is, the ratio b et w een C it and W it W e start assigning plates to slabs in the last p erio d and w ork bac kw ards in time. Then, at an y p oin t in time, w e assign the most protable unassigned plate to the least costly a v ailable slab at its upp er b ound whenev er the slab can accommo date it. More formally the greedy heuristic can b e describ ed as follo ws: Greedy Heuristic Step 0. Set x ij t = v ij t = 0 for i = 1 ; : : : ; M j 2 S t and t = 1 ; : : : ; T Set curren t p erio d s = T Step 1. If curren t p erio d s = 0, stop and a solution is found if all plates are assigned. Otherwise, let P = n j : P Mi =1 P Tt = s +1 x ij t = 0 ; j s o denote the set of plates due in or after p erio d s that ha v e not b een assigned y et and reindex all plates in P in decreasing order of the net rev en ue co ecien ts R j s Sort all slabs a v ailable in p erio d s in increasing order of their cost p er unit w eigh t C is =W is and denote the sorted set b y Q Set the curren t plate p = 1, the curren t slab q = 1, and the remaining capacit y of the curren t slab w = W q s Step 2. If curren t plate p > j P j or curren t slab q > j Q j set s = s ¡ 1 and return to step 1. Otherwise con tin ue to step 3. Step 3. If u p w set x q ps = 1, v q ps = u p w = w ¡ u p p = p + 1 and go to step 2. Step 4. If l p w set x q ps = 1, v q ps = w w = 0, p = p + 1 and return to step 2. Otherwise con tin ue to step 5. PAGE 118 118 Step 5. Set q = q + 1, w = W q s to select next slab in set Q as curren t slab and return to step 2. Note that if an y plates remain unassigned at the end of the pro cedure, the greedy heuristic has failed to nd a feasible solution, resulting in only a partial solution to the assignmen t problem. W e will use the p enalt y VLSN algorithm to try to transfer this partial solution to a feasible solution. 5.2.3 Knapsac k Problem with Expandable Items W e kno w that the eciency of the branc handprice algorithm critically dep ends on the abilit y to ecien tly solv e the pricing problem. W e will form ulate the pricing problem and deriv e some attractiv e prop erties that will help us to dev elop eectiv e algorithms for solving it. The pricing problem can b e form ulated as: maximize X j 2 S t ¡ u ¤j ( K ) x j + X j 2 S t R j ¡ r ¡ j ¡ 1 X k = t h j k v j ¡ ( C it ¡ r W it ) z ¡ v ¤ it ( K ) sub ject to (PP Pit ) l j x j v j u j x j for j 2 S t X j 2 S t v j W it z v j 0 for j 2 S t x j 2 f 0 ; 1 g for j 2 S t z 2 f 0 ; 1 g : In this pricing problem, w e can only ha v e z = 0 if x = 0. In this case, the solution v alue of the pricing problem is trivially seen to b e ¡ v ¤ it ( K ). If this v alue is p ositiv e, the v ector x = 0 yields a protable column. Therefore, in the remainder w e can restrict ourselv es in the pricing problem to solutions with z = 1. By setting z = 1 and eliminating the last t w o constan t terms in the ob jectiv e function, the pricing PAGE 119 119 problem b ecomes the socalled knapsack pr oblem with exp andable items (KPEI). This problem w as previously studied b y Balakrishnan and Geunes [ 11 ]. In this section, w e will study this problem further and prop ose alternativ e algorithms for solving it. In the general KPEI w e need to assign items from a set S to a knapsac k with capacit y W Eac h item j 2 S has a exible v olume b et w een l j and u j and has a xed rev en ue j and v ariable rev en ue j p er unit v olume. Our ob jectiv e is to maximize the total rev en ue of items in the knapsac k. Th us, the KPEI can b e form ulated as follo ws: maximize X j 2 S j x j + X j 2 S j v j sub ject to l j x j v j u j x j for j 2 S X j 2 S v j W v j 0 for j 2 S x j 2 f 0 ; 1 g for j 2 S : If b oth j and j are nonp ositiv e for an item j w e can set x j = 0 without loss of optimalit y Therefore, in the remainder w e will assume that either j > 0 or j > 0 (or b oth). The KPEI is clearly an NPHard problem since the knapsac k problem is a sp ecial case of the KPEI, obtained when l j = u j for all j It will b e con v enien t to dene the maximal p ossible unit prot for eac h item j 2 S as 0 j = j l j + j if j > 0 and 0 j = j u j + j if j 0 : Lemma 16 L et ( x ¤ v ¤ ) b e the optimal solution of the LP r elaxation of KPEI. Then 1. If 0 j 0 then x ¤ = 0 and v ¤ = 0 PAGE 120 120 2. When 0 < x ¤j < 1 we have 8>>>><>>>>: v ¤ j = l j x ¤j if j > 0 and v ¤ j = u j x ¤j if j 0 or j x ¤j + j v ¤ j = 0 j v ¤ j : 3. Ther e is at most one p 2 S such that 0 < x ¤p < 1 4. Ther e is at most one p 2 S such that x ¤p = 1 and l p < v ¤ p < u p 5. If 0 < x ¤p < 1 ther e is no q such that l q < v ¤ q < u q and x ¤q = 1 6. If P j 2 S v ¤ j < W we have x ¤j = 8><>: 0 if 0 j 0 1 if 0 j > 0 and v ¤ j = 8><>: l j x ¤j if j 0 u j x ¤j if j > 0 : 7. If 0 < x ¤p < 1 we have x ¤j = 8><>: 0 if 0 j < 0 p 1 if 0 j > 0 p and v ¤ j = 8><>: l j x ¤j if j < 0 p u j x ¤j if j > 0 p : 8. If x ¤p = 1 and l p < v ¤ p < u p we have x ¤j = 8><>: 0 if 0 j < p 1 if 0 j > p and v ¤ j = 8><>: l j x ¤j if j < p u j x ¤j if j > p : 9. Supp ose P j 2 S v ¤ j = W x ¤j 2 f 0 ; 1 g and v ¤ j 2 f l j x ¤j ; u j x ¤j g for al l j 2 S L et p = arg max j f 0 j : x ¤j = 0 g and p = arg max j f j : x ¤j = 1 ; v ¤ j = l j g We have x ¤j = 8><>: 0 if 0 j < 0 p 1 if 0 j > 0 p and v ¤ j = 8><>: l j x ¤j if j < q u j x ¤j if j > q : Pro of: 1. When j > 0 and 0 j 0, w e ha v e j < 0. It follo ws that j x ¤j + j v ¤ j j x ¤j + j l j x ¤j 0 j l j x ¤j 0 : PAGE 121 121 When j 0, w e ha v e j > 0 from the assumption j > 0 or j > 0. It follo ws that j x ¤j + j v ¤ j j x ¤j + j u j x ¤j 0 j u j x ¤j 0 : Th us w e ha v e x ¤ = 0 and v ¤ = 0 if 0 j 0. Notice that this prop ert y holds in KPEI. Without loss of generalit y w e can assume 0 j > 0 for all j 2 S 2. If it is not true, w e can increase (or k eep) the ob jectiv e function v alue b y increasing x ¤j when j > 0 and decreasing x ¤j when j 0. It follo ws that ( x ¤ v ¤ ) is not the optimal solution, whic h is a con tradiction, or an alternate optimal solution satisfying the prop ert y exists. 3. W e can pro v e it b y con tradiction. Supp ose 0 < x ¤p < 1, 0 < x ¤q < 1 and p 6 = q By decreasing v ¤ p (or v ¤ q ) and increasing v ¤ q (or v ¤ p ) b y the same amoun t ,sa y when 0 p < 0 q (or 0 p > 0 q ), W e can increase the ob jectiv e function v alue b y j 0 q ¡ 0 p j W e will set x ¤p and x ¤q according to the prop ert y 2 in this pro cess. It follo ws that ( x ¤ v ¤ ) is not the optimal solution, whic h is a con tradiction. When 0 p = 0 q w e will ha v e an alternate optimal solution satisfying the prop ert y 4. The pro of is similar to the pro of of the prop ert y 3 except that w e compare p and q 5. The pro of is similar to the pro of of the prop ert y 3 except that w e compare 0 p and q 6. F rom prop ert y 1, w e kno w x ¤j = 0 if 0 j 0. F rom prop ert y 2 and P j 2 S v ¤ j < W w e kno w that x ¤j = 1 if 0 j > 0. If the statemen t ab out the v alue of v ¤ is not true, w e can increase the ob jectiv e function v alue b y increasing v ¤ j when j > 0 and decreasing v ¤ j when j < 0. It follo ws that ( x ¤ v ¤ ) is not the optimal solution, whic h is a con tradiction. When j = 0 w e can reset v ¤ j = l j without c hanging an ything. PAGE 122 122 7. W e can pro v e this prop ert y b y con tradiction follo wing the pro of of the prop ert y 3 and 5. 8. W e can pro v e this prop ert y b y con tradiction follo wing the pro of of the prop ert y 4 and 5. 9. W e can pro v e this prop ert y b y con tradiction follo wing the pro of of the prop ert y 3 and 4. Th us without loss of generalit y w e can assume that 0 j > 0 for all j 2 S from prop ert y 1 in ab o v e Lemma 16 F urthermore from Lemma 16 w e will ha v e the follo wing theorem, whic h states a prop ert y of the optimal solution to the LPrelaxation of the KPEI: Theorem 17 Ther e exists an optimal solution ( x j ; v j ) of the fol lowing form to the LP r elaxation of the KPEI: x j = 8><>: 0 if 0 j < 1 if 0 j > and v j = 8><>: l j x j if j < u j x j if j > wher e ar e two c onstants. Pro of: This theorem is a direct results of the prop erties 6, 7, 8 and 9 in Lemma 16 Ab o v e Theorem leads to the follo wing p olynomial time algorithm for solving the LPrelaxation of the KPEI: Step 0. Compute 0 j for all j 2 S F or all j suc h that 0 j 0, set x j = 0 and v j = 0 and remo v e j from S Set = j S j and add a dumm y item + 1 to S with +1 = +1 = 0. Sort all items in S in decreasing order of 0 j Step 1. Set x j = 1 and v j = u j if j > 0, and x j = 1 and v j = l j if j 0 for all j 2 S If W P j 2 S v j w e ha v e the optimal solution and stop. Otherwise con tin ue to step 2. PAGE 123 123 Step 2. Set x j = 0 and v j = 0 for all j 2 S Initialize the curren t plate index p = 1. Step 3. If p > stop. If p > 0 set x p = 1 and v p = l p Otherwise set x p = 1 and v p = u p Step 4. If w = W ¡ P j 2 S v j > 0, con tin ue to step 5. Otherwise, the curren t item can only b e pro duced partially If p > 0 set x p = w =l p and if p 0 set x p = w =u p F urthermore, set v p = w and stop. Step 5. Let q = arg max j 2 S f j : v j = l j g If 0 p +1 > q set p = p + 1 and return to step 3. Otherwise con tin ue to step 6. Step 6. If w u q ¡ l q set v q = v q + w and stop. Otherwise set v q = u q w = w ¡ u q + l q and return to step 5. By solving the relaxed KPEI with ab o v e algorithm, w e will ha v e a solution with at most one fractional item if the algorithm stop in Step 4. If there is no fractional item, the solution is also the optimal solution of the KPEI. Otherwise w e can remo v e the fractional item, and add or expand some (most) protable items to obtain a go o d feasible solution for KPEI. Th us w e prop ose a heuristic for the KPEI b y mo difying step 4 of the algorithm for the relaxed KPEI as follo ws: Step 4. If w = W ¡ P j 2 S v j > 0, con tin ue to step 5. Otherwise, the curren t item can only b e pro duced partially If p > 0 set x p = w =l p and if p 0 set x p = w =u p F urthermore, set v p = w and return to step 3. The follo wing Theorem sho ws that our heuristic for KPEI is asymptotically optimal when the data are randomly generated with nite exp ectation. Theorem 18 L et z I n and z LP n b e the optimal obje ctive function value of the KPEI and the r elaxe d KPEI, and let z H n b e the solution value of the heuristic. If al l data ar e r andomly gener ate d with nite exp e ctation, we have lim n !1 z H n ¡ z I n n w :p: 1 : PAGE 124 124 Pro of: Dene n = j S j Let max = max j 2 S f j g max = max j 2 S f j g and u max = max j 2 S f u j g Since z H n is at least as go o d as the solution b y remo ving the fractional item p if exists, from Z LP n w e ha v e z H n z LP n ¡ p x p ¡ p v p z LP n ¡ max ¡ max u max W e also ha v e z H n z I n z LP n since z H n is feasible to KPEI and z I n is feasible to relaxed KPEI. Then z I n n ¡ max + max u max n z H n n z I n n : Since all data ha v e nite exp ectation, from the follo wing Lemma 19 w e ha v e lim n !1 max + max u max n = 0 w :p: 1 : Th us z H n =n and z I n =n are asymptotic equiv alen t w.p. 1. Lemma 19 L et X 1 ; X 2 ; : : : ; X n b e i.i.d. r andom variables with nite exp e ctation. Dene X ( n ) = max i f X i g Then we have l im n !1 X ( n ) =n = 0 w.p.1 Pro of: Let F n and F b e the c.d.f. of X ( n ) and X i It is ob viously F n ( x ) = F ( x ) n F ( x ). Let a b b e t w o constan ts and a < b W e consider the follo wing probabilit y P ( a X ( n ) =n b ) = P ( na X ( n ) nb ) = F n ( nb ) ¡ F n ( na ) : When a < b < 0, w e ha v e 0 lim n !1 P ( a X ( n ) =n b ) = lim n !1 [ F n ( nb ) ¡ F n ( na )] lim n !1 F n ( nb ) lim n !1 F ( nb ) = 0 : Since this is true for an y constan ts a < b < 0 w e ha v e lim n !1 P ( X ( n ) =n < 0) = 0 : When 0 < a < b w e ha v e 0 lim n !1 P ( a X ( n ) =n b ) PAGE 125 125 = lim n !1 [ F n ( nb ) ¡ F n ( na )] = lim n !1 [ F ( nb ) n ¡ F ( na ) n ] = lim n !1 [ F ( nb ) ¡ F ( na )] £ F ( nb ) n ¡ 1 + F ( nb ) n ¡ 2 F ( na ) + : : : + F ( na ) n ¡ 1 ¤ lim n !1 n [ F ( nb ) ¡ F ( na )] lim n !1 n [ 1 ¡ F ( na )] = lim n !1 na [1 ¡ F ( na )] a = 0 : The last equalit y holds since the existence of E ( X ) or E ( X ) < 1 implies lim x !1 x (1 ¡ F ( x )) = 0. 1 Similarly w e ha v e lim n !1 P ( X ( n ) =n > 0) = 0. No w w e can conclude that the probabilit y of random v ariable X ( n ) =n in an y in terv al excluding 0 is 0. Th us lim n !1 X ( n ) =n = 0 w.p.1 Balakrishnan and Geunes [ 11 ] dev elop ed a pseudop olynomial time dynamic programming algorithm for solving the KPEI to optimalit y when all problem data are in tegral. Let f j ( U ) denote the optimal v alue of KPEI for a knapsac k with remaining (in teger) capacit y U ( U = 0 ; : : : ; W ) and allo wing items 1 ; : : : ; j ( j = 1 ; : : : ; j S j ). W e set f j ( U ) = 0 if j = 0 or 0 U < min f l k : k = 1 ; : : : ; j g and f j ( U ) = ¡1 if U < 0. W e will ha v e follo wing recurrence relationship in the dynamic programming: f j ( U ) = max f j ¡ 1 ( U ); max v j = l j ;::: ;u j f j + j v j + f j ¡ 1 ( U ¡ v j ) g for U = min j 2 S f l j g ; : : : ; W j = 1 ; : : : ; j S j 1 0 = lim a !1 Z 1 a xdF ( x ) lim a !1 a Z 1 a dF ( x ) = lim a !1 a (1 ¡ F ( a )) 0 PAGE 126 126 As an alternativ e to this approac h, whic h can also b e used when not all problem data are in tegral, w e can dev elop a branc handb ound algorithm to solv e the KPEI exactly b y emplo ying the algorithm for solving its LPrelaxation giv en ab o v e. A t eac h no de of the branc handb ound tree, w e rst solv e the relaxed problem, whic h usually giv es a go o d upp er b ound, and then branc h on the fractional item (if one exists). W e can also use additional tec hniques to impro v e the p erformance of this algorithm [ 35 ]. 5.3 The MPFD A with Slab In v en tory 5.3.1 Basic F orm ulation In the second mo del w e assume that in v en tory can only consist of slabs and plates are pro duced just in time, and denote this problem b y MPFD AS. Unless men tioned b elo w, w e will use the same notation as in the rst mo del. Supp ose there are at most M slabs during the whole time horizon. Let W i and C i b e the initial w eigh t and cost of slab i ( i = 1 ; : : : ; M ), and h it b e the holding cost of slab i ( i = 1 ; : : : ; M ) in p erio d t ( t = 1 ; : : : ; T ). Similar to the rst mo del, without loss of generalit y w e can assume R j r j = 1 ; : : : ; N C i r W i i = 1 ; : : : ; M : W e rst restrict ourself to a basic mo del in whic h the slabs are purc hased at the b eginning of the planning horizon and the surplus can only b e recycled at the end of the planning horizon. Later, w e will expand this basic mo del to eliminate these restrictions. As b efore, w e can form ulate the problem MPFD AS in the form of (AP): maximize M X i =1 H i ( x i ¢ ) PAGE 127 127 sub ject to (AP S3 ) M X i =1 x ij = 1 j = 1 ; : : : ; N N X j =1 l j x ij W i i = 1 ; : : : ; M x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N where x ij = 1 if plate j is pro duced from slab i and 0 otherwise. Here the function H i : f 0 ; 1 g N R represen ts the maxim um prot that can b e obtained from slab i as a function of the assignmen t v ector x 2 f 0 ; 1 g N of plates to slab i : maximize N X j =1 R j v j + r W i z ¡ N X j =1 v j ¡ C i z ¡ T X t =1 h it 0@ W i z ¡ X j 2 S t v j 1A = N X j =1 0@ R j ¡ r + T X t = j h it 1A v j ¡ C i ¡ r W i + T X t =1 h it W i z sub ject to (SP Si ) l j x j v j u j x j j = 1 ; : : : ; N N X j =1 v j W i z v j 0 j = 1 ; : : : ; N z 2 f 0 ; 1 g where S t = f j : j t g is the set of plates that m ust b e pro duced b efore p erio d t Here z = 1 if slab i is used and 0 otherwise, and v j is the w eigh t of order j pro duced from slab i The ob jectiv e function of the subproblem (SP Si ) includes four parts: rev en ue from orders, rev en ue from recycling slabs, cost of slabs and in v en tory holding cost. The constrain ts include demand exibilit y constrain ts and slab capacit y constrain ts. When the assignmen ts x are giv en, w e can use a similar PAGE 128 128 algorithm as the one dev elop ed for (SP Pit ) to solv e the subproblems b y c hanging the denition of the net rev en ue co ecien t to R j = R j ¡ r + P Tt = j h it In the greedy heuristic, the net rev en ue co ecien t R ij = R j ¡ r + P Tt = j h it will b e the protabilit y measuremen t of plate j when w e consider the assignmen t of slab i Since slabs are a v ailable for the en tire time horizon there is no need to go bac k in time in the greedy heuristic in this mo del, that is, w e can treat it in a similar w a y as a v ersion of (AP P3 ) with only one p erio d. This leads to the follo wing greedy heuristic for (AP S3 ): Greedy Heuristic Step 0. Set x ij = v ij = 0 for i = 1 ; : : : ; M and j = 1 ; : : : ; N Let Q = f 1 ; : : : ; M g b e the set of slabs sorted in increasing order of cost p er unit w eigh t C i =W i and P = f 1 ; : : : ; N g b e the set of plates. Set curren t slab q = 1, the remaining capacit y of the curren t slab w = W q Step 1. If curren t slab q > M stop and greedy heuristic can not nd a feasible solution. Sort and reindex all plates in P in decreasing order of the net rev en ue co ecien ts R q j Set curren t plate p = 1. Step 2. If curren t plate p > j P j stop and a feasible solution is found. Otherwise con tin ue to step 3. Step 3. If u p w set x q p = 1, v q p = u p w = w ¡ u p P = P ¡ f p g p = p + 1 and return to step 2. Step 4. If l p w set x q p = 1, v q p = w w = 0, P = P ¡ f p g p = p + 1 and return to step 2. Step 5. Set q = q + 1, w = W q to select next slab in set Q as curren t slab and return to step 1. Next w e consider the pricing problems for (AP S3 ), whic h can b e form ulated as: PAGE 129 129 maximize N X j =1 ¡ u ¤j ( K ) x j + N X j =1 0@ R j ¡ r + T X t = j h it 1A v j ¡ C i ¡ r W i + T X t =1 h it W i z ¡ v ¤ i ( K ) sub ject to (PP Si ) l j x j v j u j x j j = 1 ; : : : ; N N X j =1 v j W i z v j 0 j = 1 ; : : : ; N x j 2 f 0 ; 1 g j = 1 ; : : : ; N z 2 f 0 ; 1 g : As b efore, the only non trivial case is the case where x 6 = 0, that is, z = 1. When z = 1, it is easy to see that this pricing problem is a KPEI b y eliminating the last t w o constan t terms. Th us, the algorithms prop osed in Section 5.2.3 can b e used to solv e it. 5.3.2 Mo del Expansions If, in con trast to the assumptions in the basic mo del presen ted ab o v e, w e can recycle surpluses and/or purc hase slabs in an y p erio d, that is, recycle a surplus as so on as the corresp onding slab will not b e used an ymore and/or purc hase a slab just b efore it is used, w e can reduce the in v en tory costs and impro v e our prot. In this section, w e will consider these expansions of the basic mo del. Note that since an y solution to problem (AP S3 ) will also b e feasible to these expansions, w e can still apply the same greedy heuristic to obtain a feasible solution and c hange the recycle p erio ds or (and) the purc hase p erio ds of the slabs accordingly to impro v e the solution. Next w e will form ulate these mo died mo dels, and discuss ho w to solv e the subproblems as w ell as the pricing problems. PAGE 130 130 Recycle in an y p erio d First, w e consider the case where a surplus can b e recycled in an y p erio d. The form ulation of the problem will b e: maximize M X i =1 H i ( x i ¢ ) sub ject to (AP S R 3 ) M X i =1 x ij = 1 j = 1 ; : : : ; N N X j =1 l j x ij W i i = 1 ; : : : ; M x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N where the function H i : f 0 ; 1 g N R is the optimal ob jectiv e function v alue of the follo wing subproblem: maximize N X j =1 R j v j + r T X t =1 s t ¡ C i z ¡ T X t =1 h it 0@ W i z ¡ X j 2 S t v j ¡ t X k =1 s k 1A = N X j =1 0@ R j + T X t = j h it 1A v j + T X t =1 r + T X k = t h ik s t ¡ C i + T X t =1 h it W i z sub ject to (SP S R i ) l j x j v j u j x j j = 1 ; : : : ; N N X j =1 v j + T X t =1 s t W i z (5.5) s t W i (1 ¡ x j ) j 2 f k : k > t g ; t = 1 ; : : : ; T (5.6) v j 0 j = 1 ; : : : ; N s t 0 t = 1 ; : : : ; T z 2 f 0 ; 1 g where s t is the w eigh t of the surplus of slab i recycled in p erio d t Constrain t ( 5.5 ) is the slab capacit y constrain t, and constrain ts ( 5.6 ) ensure s t = 0 if in p erio d t a PAGE 131 131 order will b e pro duced from slab i that is, surplus of slab i will only b e recycled after all plates assigned to it ha v e b een pro duced. F or an y giv en assignmen t v ariable x let t = max j f j : x j = 1 g b e the last p erio d in whic h slab i is used to pro duce plates. T o maximize the prot, the surplus of slab i m ust b e recycled at the end of p erio d t since the unit surplus rev en ue r is constan t and th us an y dela y in recycling will decrease the total rev en ue b y increasing slab holding cost. It follo ws that s t = W i z ¡ P Nj =1 v j and s t = 0 for all t 6 = t in the optimal solution. Th us w e can reform ulate the subproblem as: maximize N X j =1 0@ R j ¡ r + t ¡ 1 X t = j h it 1A v j ¡ C i ¡ r W i + t ¡ 1 X t =1 h it W i z sub ject to l j x j v j u j x j j = 1 ; : : : ; N N X j =1 v j W i z v j 0 j = 1 ; : : : ; N z 2 f 0 ; 1 g : This problem is the same as (SP Pit ) and th us the subproblem (SP S R i ) can b e solv ed b y using the algorithm in Section 5.2.2 with R j = R j ¡ r + P t ¡ 1 t = j h it Notice that for an y subproblem the dierence b et w een these net rev en ue co ecien ts and those of the mo del (AP S3 ) is a constan t P Tt = t h it Th us, the ordering of the plates for an y slab will remain the same in the greedy heuristic no matter whic h set of co ecien ts is used. In fact, this prop ert y holds for all three expansions discussed in this section. Th us w e can alw a ys use the same greedy heuristic for (AP S3 ) to obtain a feasible solution for an y of the three expansions. Ho w ev er, only the v alues of x ij and v ij will remain the same in the feasible solution of the expansions. W e need to compute the v alue of the other v ariables, for example z and the ob jectiv e PAGE 132 132 function b y using a reform ulated subproblem suc h as the one ab o v e. Th us, in the follo wing discussion of the other t w o expansions, w e will omit the discussion ab out greedy heuristic and fo cus on the subproblems and pricing problems. But rst let us tak e a lo ok at the pricing problem of this expansion, whic h has the follo wing form ulation: maximize N X j =1 ¡ u ¤j ( K ) x j + N X j =1 0@ R j + T X t = j h it 1A v j + T X t =1 r + T X k = t h ik s t sub ject to (PP S R i ) l j x j v j u j x j j = 1 ; : : : ; N N X j =1 v j + T X t =1 s t W i z s t W i (1 ¡ x j ) j 2 f k : k > t g ; t = 1 ; : : : ; T v j 0 j = 1 ; : : : ; N s t 0 t = 1 ; : : : ; T x j 2 f 0 ; 1 g j = 1 ; : : : ; N z 2 f 0 ; 1 g : Supp ose that surplus is recycle in p erio d t 2 f j : j = 1 ; : : : ; N g Then w e only need to consider the plates in set S t Since only s t can b e p ositiv e and the capacit y constrain t will b e binding in the optimal solution, w e ha v e s t = W i ¡ X j 2 S t v j : No w for eac h xed t 2 f j : j = 1 ; : : : ; N g the pricing problem (PP S R i ) b ecomes the follo wing KPEI problem: maximize X j 2 S t ¡ u ¤j ( K ) x j + X j 2 S t 0@ R j ¡ r + t ¡ 1 X t = j h it 1A v j PAGE 133 133 sub ject to l j x j v j u j x j j 2 S t X j 2 S t v j W i v j 0 j 2 S t x j 2 f 0 ; 1 g j 2 S t : Let h ( t ) b e the optimal ob jectiv e function v alue of the ab o v e KPEI corresp onding to a giv en c hoice of t Then the optimal ob jectiv e function v alue of the pricing problem (PP S R i ) will b e: min t 2f j : j =1 ;::: ;N g ( h ( t ) ¡ C i ¡ r W i + t ¡ 1 X t =1 h it W i ¡ v ¤ i ( K ) ) : Th us w e need to solv e at most T KPEI problems to obtain the optimal solution of the pricing problem. Purc hase in an y p erio d. Next, w e consider the case that w e can obtain slabs in an y p erio d. In this case, w e obtain the follo wing assignmen t form ulation: maximize M X i =1 H i ( x i ¢ ) sub ject to (AP S P 3 ) M X i =1 x ij = 1 j = 1 ; : : : ; N N X j =1 l j x ij W i i = 1 ; : : : ; M x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N PAGE 134 134 where the function H i : f 0 ; 1 g N R represen ts the optimal ob jectiv e function v alue of the follo wing subproblem: maximize N X j =1 R j v j + r T X k =1 W i z k ¡ T X j =1 v j ¡ T X t =1 C i z t ¡ T X t =1 h it 0@ t X k =1 W i z k ¡ X j 2 S t v j 1A = N X j =1 0@ R j ¡ r + T X t = j h it 1A v j ¡ T X t =1 C i ¡ r W i + T X k = t h ik W i z t sub ject to (SP S P i ) l j x j v j u j x j j = 1 ; : : : ; N X j 2 S t v j W i t X k =1 z k t = 1 ; : : : ; T (5.7) T X t =1 z t 1 v j 0 j = 1 ; : : : ; N (5.8) z t 2 f 0 ; 1 g t = 1 ; : : : ; T where z t is 1 if slab i is purc hase in p erio d t and 0 otherwise. Constrain ts ( 5.7 ) ensure that the plates can only b e made after slab i is purc hased and the total w eigh t of plates made from slab i can not exceed the capacit y of slab i Constrain t ( 5.8 ) states that slab i can b y purc hased at most once during the time horizon. F or giv en assignmen ts x 6 = 0 let t = min j f j : x j = 1 g b e the rst p erio d during whic h w e use slab i to pro duce plates. Clearly due to the holding costs w e will ha v e z t = 1 for t = t and 0 otherwise in the optimal solution. And from feasibilit y considerations only plates in the set S t can b e pro duced. Th us w e can reduce the ab o v e subproblem to the follo wing problem and solv e it with net rev en ue co ecien ts R j = R j ¡ r + P Tt = j h it b y using the algorithm dev elop ed for the PAGE 135 135 subproblem (SP Pit ) as b efore: maximize X j 2 S t 0@ R j ¡ r + T X t = j h it 1A v j ¡ 0@ C i ¡ r W i + W i T X k = t h ik 1A z t sub ject to l j x j v j u j x j j 2 S t X j 2 S t v j W i z v j 0 j 2 S t z t 2 f 0 ; 1 g : Next, w e will consider the pricing problem in the branc handprice algorithm, whic h has the follo wing form ulation: maximize ¡ N X j =1 u ¤j ( K ) x j + N X j =1 0@ R j ¡ r + T X t = j h it 1A v j ¡ T X t =1 C i ¡ r W i + T X k = t h ik W i z t sub ject to (PP S P i ) l j x j v j u j x j j = 1 ; : : : ; N X j 2 S t v j W i t X k =1 z k t = 1 ; : : : ; T T X t =1 z t 1 v j 0 j = 1 ; : : : ; N x j 2 f 0 ; 1 g j = 1 ; : : : ; N z t 2 f 0 ; 1 g t = 1 ; : : : ; T : When x 6 = 0 w e kno w that exactly one z t is 1, that is, w e will purc hase slab i in the rst p erio d in whic h it is used in the optimal solution. Supp ose that the purc hase p erio d t 2 f j : j = 1 ; : : : ; N g is giv en. It is ob vious that only plates in the set S t PAGE 136 136 can b e pro duced. Th us, for xed t the pricing problem (PP S P i ) reduces to: maximize X j 2 S t ¡ u ¤j ( K ) x j + X j 2 S t 0@ R j ¡ r + T X t = j h it 1A v j sub ject to l j x j v j u j x j j 2 S t X j 2 S t v j W i v j 0 j 2 S t x j 2 f 0 ; 1 g j 2 S t : Let h ( t ) b e the optimal ob jectiv e function v alue of this problem. The optimal ob jectiv e function v alue of the pricing problem (PP S P i ) will then b e equal to: min t 2f j : j =1 ;::: ;N g 8<: h ( t ) ¡ 0@ C i ¡ r W i + T X k = t h ik W i 1A ¡ v ¤ i ( K ) 9=; : Ho w ev er, since the co ecien ts of the items in the ob jectiv e function are constan ts and not related to t if w e use dynamic programming to solv e the pricing problem w e do not need to solv e a set of KPEI problems. In order to obtain the optimal solution of the pricing problem, only one KPEI problem with t = 1 needs to b e solv ed. If w e order all items in set S 1 in decreasing order of j w e will ha v e follo wing relations b et w een h ( t ) and f j ( U ), whic h is dened in the dynamic programming algorithm in Section 5.2.3 : h ( t ) = f j ( t ) ( W i ) where j ( t ) = arg min j f j t g : Recycle and purc hase in an y p erio d. As the last expansion to the problem MPFD AS w e will com bine the previous t w o mo dels to allo w slabs to b e b oth recycled and purc hased in an y p erio d. The PAGE 137 137 com bined mo del reads: maximize M X i =1 H i ( x i ¢ ) sub ject to (AP S A 3 ) M X i =1 x ij = 1 j = 1 ; : : : ; N N X j =1 l j x ij W i i = 1 ; : : : ; M x ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N where function H t : f 0 ; 1 g N R is the optimal ob jectiv e function v alue of the follo wing subproblem: maximize N X j =1 R j v j + r T X t =1 s t ¡ T X t =1 C i z t ¡ T X t =1 h it 0@ t X k =1 W i z k ¡ X j 2 S t v j ¡ t X k =1 s k 1A = N X j =1 0@ R j + T X t = j h it 1A v j + T X t =1 r + T X k = t h ik s t ¡ T X t =1 C i + T X k = t h ik W i z t sub ject to (SP S A i ) l j x j v j u j x j j = 1 ; : : : ; N X j 2 S t v j + t X k =1 s k W i t X k =1 z k t = 1 ; : : : ; T s t W i (1 ¡ x j ) j 2 f k : k > t g ; t = 1 ; : : : ; T T X t =1 z t 1 v j 0 j = 1 ; : : : ; N s t 0 t = 1 ; : : : ; T z t 2 f 0 ; 1 g t = 1 ; : : : ; T : PAGE 138 138 F or giv en assignmen t x 6 = 0 of plates to slab i it is ob viously that the rst p erio d t = min f j : x j = 1 g and the last p erio d t = max f j : x j = 1 g during whic h slab i is used will b e the p erio d in whic h slab i is purc hased and recycled, resp ectiv ely in the optimal solution of (SP S A i ). It follo ws that z t = 1 for t = t and 0 otherwise, and s t = W i z t ¡ P j 2 S t \ S t v j for t = t and 0 otherwise in the optimal solution. Th us the subproblem (SP S A i ) can b e reduced to the follo wing problem: maximize X j 2 S t \ S t 0@ R j ¡ r + t ¡ 1 X t = j h it 1A v j ¡ 0@ C i ¡ r W i + t ¡ 1 X k = t h ik W i 1A z t sub ject to l j x j v j u j x j j 2 S t \ S t X j 2 S t \ S t v j W i z t v j 0 j 2 S t \ S t z t 2 f 0 ; 1 g : This problem is similar to (SP Pit ) and w e can solv e it in p olynomial time for xed x b y setting the net rev en ue co ecien ts to R j = R j ¡ r + P t ¡ 1 t = j h it A similar approac h can b e used to solv e the pricing problem in this case b y solving a set of problems with xed recycle p erio d t 2 f 1 ; : : : ; T g and purc hase p erio d t 2 f t ; : : : ; T g F or giv en t t w e then need to solv e the follo wing KPEI problem: maximize X j 2 S t \ S t ¡ u ¤j ( K ) x j + X j 2 S t \ S t 0@ R j ¡ r + t ¡ 1 X t = j h it 1A v j sub ject to l j x j v j u j x j j 2 S t \ S t PAGE 139 139 X j 2 S t \ S t v j W i v j 0 j 2 S t \ S t x j 2 f 0 ; 1 g j 2 S t \ S t : Let h ( t ; t ) b e the optimal ob jectiv e function v alue of this problem. The optimal ob jectiv e function v alue of the pricing problem will then b e: max t =1 ;::: ;T ; t = t ;::: ;T 8<: h ( t ; t ) ¡ 0@ C i ¡ r W i + t ¡ 1 X k = t h ik W i 1A ¡ v ¤ i ( K ) 9=; : T o solv e the pricing problem w e ma y need to obtain the optimal solution of at most T ( T + 1) = 2 KPEI problems. Notice that the co ecien ts of plates in ab o v e problem are constan t when t is xed. Then if all data are in tegral and the pseudop olynomial dynamic programming algorithm is applied, w e can apply an approac h similar to the one for (PP S P i ), and th us w e only need to solv e at most T KPEI problems, one for eac h t 2 f j : j = 1 ; : : : ; T g 5.4 The MPFD A with Plate and Slab In v en tory 5.4.1 Basic F orm ulation In this section w e consider the third in v en tory strategy that is, in v en tory can consist of b oth slabs and plates. W e denote this problem b y MPFD AA. Supp ose that there are at most M slabs are a v ailable during the time horizon. Let h 0it and h j t b e the holding cost of slab i ( i = 1 ; : : : ; M ) and plate j ( j = 1 ; : : : ; N ) in p erio d t ( t = 1 ; : : : ; T ). W e will ha v e the same assumption as in the second mo del: R j r j = 1 ; : : : ; N C i r W i i = 1 ; : : : ; M : F or ease of exp osition, w e assume that the slabs are purc hased at the b eginning of the time horizon and the surplus can only b e recycled at the end of the time horizon. In a similar w a y as w as done for the MPFD AS, w e can expand the PAGE 140 140 MPFD AA and eliminate these restrictions. Ho w ev er, w e will only discuss one expansion briey b ecause of these similarities. W e can form ulated MPFD AA in the form of (AP) as follo ws: maximize M X i =1 H i ( x i ¢¢ ) sub ject to (AP A3 ) M X i =1 j X t =1 x ij t = 1 j = 1 ; : : : ; N N X j =1 j X t =1 l j x ij t W i i = 1 ; : : : ; M x ij t 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N ; t = 1 ; : : : ; j where x ij t = 1 if order j is pro duced from slab i in p erio d t and 0 otherwise. Here the function H i : f 0 ; 1 g N £ R is the maximal prot that can b e obtained from slab i as a function of the assignmen t v ector x : maximize N X j =1 j X t =1 R j v j t + r W i z ¡ T X t =1 X j 2 S t v j t ¡ C i z ¡ T X t =1 h 0it W i z ¡ t X k =1 X j 2 S k v j k ¡ N X j =1 j ¡ 1 X t =1 h j t t X k =1 v j k = N X j =1 j X t =1 R j ¡ r + T X k = t h 0ik ¡ j ¡ 1 X k = t h j k v j t ¡ C i ¡ r W i + T X t =1 h 0it W i z sub ject to (SP Ai ) l j x j t v j t u j x j t j = 1 ; : : : ; N ; t = 1 ; : : : ; j N X j =1 j X t =1 v j t W i z v j t 0 j = 1 ; : : : ; N ; t = 1 ; : : : ; j z 2 f 0 ; 1 g PAGE 141 141 where z is 1 if slab i is used and 0 otherwise, and v j t is the w eigh t of plate j pro duced from slab i in p erio d t The ob jectiv e function consists of v e parts: rev en ue from orders, rev en ue from recycling slabs, cost of slabs, in v en tory holding cost of slabs and in v en tory holding cost of plates. Let t ¤j = arg max t =1 ;::: ; j ( R j ¡ r + T X k = t h 0ik ¡ j ¡ 1 X k = t h j k ) R j = R j ¡ r + T X k = t ¤j h 0ik ¡ j ¡ 1 X k = t ¤j h j k : Supp ose that plate j is pro duced from slab i in the optimal solution ( x ¤ v ¤ ). Then x ¤ij t = 1 and v ¤ ij t > 0 if and only if t = t ¤j that is, the plate will b e pro duced in the p erio d with maximal co ecien t. Otherwise, w e can alw a ys pro duce the plate in the p erio d with maximal co ecien t and increase the ob jectiv e function v alue. Based on this observ ation, w e can reform ulate (AP A3 ) as the follo wing problem: maximize M X i =1 H i ( X i ¢ ) sub ject to M X i =1 X ij = 1 j = 1 ; : : : ; N N X j =1 l j X ij W i i = 1 ; : : : ; M X ij 2 f 0 ; 1 g i = 1 ; : : : ; M ; j = 1 ; : : : ; N where X ij = 1 if order j is pro duced from slab i (in p erio d t ¤j ) and 0 otherwise. Here the function H i : f 0 ; 1 g N R is the maximal prot that can b e obtained from slab i as a function of the assignmen t v ector X : maximize N X j =1 R j V j ¡ C i ¡ r W i + T X t =1 h 0it W i z PAGE 142 142 sub ject to l j X j V j u j X j j = 1 ; : : : ; N N X j =1 V j W i z V j 0 j = 1 ; : : : ; N where V j is the w eigh t of plate j pro duced from slab i (in p erio d t ¤j ). Notice that this form ulation has the same form as (AP S3 ), whic h means that w e can transfer the basic mo del of MPFD AA to the basic mo del of MPFD AS. The reason b ehind this equiv alence is that giv en an y pair of slab and plate w e can immediately conclude the pro duction p erio d b y the ab o v e discussion. Th us all algorithms dev elop ed for solving (AP S3 ) can b e directly applied to the ab o v e simplied mo del of (AP A3 ). 5.4.2 Expansion: Recycle and Purc hase in An y P erio d In this section w e will consider an expansion of the basic mo del of MPFD AA. W e will remo v e the unrealistic assumptions made in ab o v e mo del and consider the case in whic h slabs can b e purc hased and recycled in an y p erio d. W e will ha v e the follo wing subproblem in this expansion: maximize N X j =1 j X t =1 R j v j t + r T X t =1 s t ¡ T X t =1 C i z t ¡ T X t =1 h 0it t X k =1 W i z k ¡ t X k =1 X j 2 S k v j k ¡ t X k =1 s k ¡ N X j =1 j ¡ 1 X t =1 h j t t X k =1 v j k = N X j =1 j X t =1 R j + T X k = t h 0ik ¡ j ¡ 1 X k = t h j k v j t + T X t =1 r + T X k = t h 0ik s t ¡ T X t =1 C i + T X k = t h 0ik W i z t sub ject to (SP AAi ) l j x j t v j t u j x j t j = 1 ; : : : ; N ; t = 1 ; : : : ; j PAGE 143 143 t X k =1 X j 2 S k v j k + t X k =1 s k W i t X k =1 z k t = 1 ; : : : ; T s t W i 1 ¡ j X k = t x j k j 2 f k : k > t g ; t = 1 ; : : : ; T T X t =1 z t 1 v j t 0 j = 1 ; : : : ; N ; t = 1 ; : : : ; j z t 2 f 0 ; 1 g t = 1 ; : : : ; T where z t is 1 if slab i is purc hased in p erio d t and 0 otherwise, and s t is the w eigh t of the surplus of slab i recycled in p erio d t Let t = min f t : x j t = 1 g and t = max f t : x j t = 1 g Similar to the third expansion of MPFD AS, w e kno w that when slab i is used w e ha v e z t = 1 if and only if t = t and only plates in set S t can b e pro duced from slab i F urthermore w e ha v e s t = W i z t ¡ P tk = t P j 2 S k v j k if t = t and 0 otherwise. Th us the subproblem can b e simplied and is equiv alen t to the follo wing problem: maximize X j 2 S t min f t; j g X t = t R j ¡ r + t ¡ 1 X k = t h 0ik ¡ j ¡ 1 X k = t h j k v j t ¡ 0@ C i ¡ r W i + t ¡ 1 X k = t h 0ik W i 1A z t sub ject to (SP Ai ) l j x j t v j t u j x j t j 2 S t ; t = t ; : : : ; min f t ; j g X j 2 S t min f t; j g X t = t v j t W i z t v j t 0 j 2 S t ; t = t ; : : : ; min f t; j g z t 2 f 0 ; 1 g : PAGE 144 144 Similar to the basic mo del, w e can dene t ¤j = arg max t = t ;::: ; min f t; j g ( R j ¡ r + t ¡ 1 X k = t h 0ik ¡ j ¡ 1 X k = t h j k ) R j = R j ¡ r + t ¡ 1 X k = t ¤j h 0ik ¡ j ¡ 1 X k = t ¤j h j k and w e kno w that if plate j is pro duced from slab i in the optimal solution ( x ¤ ; v ¤ ) w e ha v e x ¤ij t = 1 and v ¤ ij t > 0 if and only if t = t ¤j Th us w e can further simplify the subproblem to the follo wing problem as in the basic mo del: maximize X j 2 S t R j V j ¡ 0@ C i ¡ r W i + t ¡ 1 X k = t h 0it W i 1A z sub ject to l j X j V j u j X j j 2 S t X j 2 S t V j W i z V j 0 j 2 S t z 2 f 0 ; 1 g whic h can th us b e solv ed in p olynomial time. As in the third expansion of MPFD AS, w e can solv e the pricing problem b y solving a set of KPEI problems, one for eac h pair of purc hase p erio d and recycle p erio d. 5.5 Computational Results 5.5.1 Generation of Problem Instances W e apply a similar metho d to Balakrishnan and Geunes [ 11 ] to generate random data and p erform the test based on t w o mo dels: the MPFD AP or (AP P3 ), and the third expansion of MPFD AS or (AP S3 A ). This random data generation metho d is based on real data pro vided b y a steel man ufacturer. In our test instances, slab cost C it is uniformly distributed on [0 : 45 ; 0 : 55] $/lb, and PAGE 145 145 plate rev en ue R j is uniformly distributed on [1 ; 1 : 5] $/lb. W e set slab recycling rev en ue to r = 0 : 25 $/lb and the rate of in v en tory holding cost usually at 10%. The minim um and maxim um plate w eigh ts l j and u j and the slab w eigh t W it are uniformly distributed on [5 ; 10], [10 ; 15], and [50 ; 60], resp ectiv ely W e consider cases with planning horizons T = 3 ; 4 ; 5 ; 6 and n um b er of orders N = 10 T ; 20 T ; 30 T F or eac h problem set w e generate 10 instances. The n um b er of orders placed in an y p erio d t is giv en b y N t where the v ector of seasonal factors is sho wn in the follo wing table: T able 5{1: Seasonal factor T 1 2 3 4 5 6 3 0.3 0.4 0.3 4 0.2 0.3 0.3 0.2 5 0.15 0.25 0.20 0.25 0.15 6 0.1667 0.1110 0.1667 0.2222 0.1667 0.1667 Denoting the upp er b ound on the plate w eigh ts b y U w e ensure that most instances generated are feasible b y setting M = N max t f t g U E ( W it ) in MPFD AP and M = N U E ( W it ) in MPFD AS. 5.5.2 Main Results W e p erformed our tests on a PC with a 650MHz P en tium I I I pro cessor and 512Mb memory and used the CPLEX 9.0 linear programming solv er to solv e the master problem in the branc handprice algorithm. Based on some preliminary tests, w e found that the running time of the branc handprice algorithm ev en for some small instances is on the order of hours. W e th us c hose to stop the branc handprice algorithm after nding the ro ot no de PAGE 146 146 of the branc handprice tree, and refer to the solution of the master problem as the B&P ro ot solution. F or small instances, that is, instances generated with N =T = 10, w e further solv ed the master problem obtained b y CPLEX with in tegralit y constrain ts enforced (see Section 2.4.4 ). W e will call the solution obtained b y this heuristic the ro ot MIP solution. Besides using the algorithms dev elop ed in this c hapter to solving the instances generated, w e also attempted to use the MIP solv er of CPLEX to solv e the instances to optimalit y in order to accurately ev aluate the qualit y of our heuristics. T o sp eed up CPLEX, w e alw a ys use the solution found b y VLSN as the starting solution for CPLEX. W e then tell CPLEX to stop when the MIP gap is less than 0 : 1% or the running time exceeds 12( T ¡ 3) N =T seconds. Using this metho d, w e obtain at least an upp er b ound on the optimal solution v alue, whic h w e will refer to as the CPLEX b ound. Results of MPFD AP. T able 5{2 sho ws the p erformance of the algorithms for (AP P3 ). W e rep ort the a v erage running time and error for eac h algorithm. All errors are computed based on the CPLEX b ound. W e found that B&P ro ot b ounds are alw a ys w orse than CPLEX b ound and errors rep orted for B&P are actually the gap b et w een these b ounds. the In addition, w e rep ort the n um b er of columns generated in the branc handprice algorithm. W e found that CPLEX usually stopp ed b ecause of the time limit and without a b etter in teger solution than the VLSN solution, except for some instances in the problem sets with T = 3, N = 30 and T = 4, N = 40, that is, the t w o smallest problem sets. Ev en after w e increased the time limit of CPLEX b y a factor of 5, the a v erage CPLEX b ound only impro v ed b y ab out 0 : 05% in the 60 instances generated for the 6 problem sets with ratios N =T = 10 and 20. Th us, CPLEX in general is not a go o d c hoice to solv e these problems except for v ery PAGE 147 147 small instances. The greedy heuristic usually giv es a solution with error b ounds b et w een 4% and 5%. But considering that the running time of the greedy heuristic is almost negligible, it can b e a go o d c hoice when w e need a go o d solution for v ery large instances in realtime. The VLSN heuristic generally yields a go o d solution in short time, ab out only 1 = 5 to 1 = 10 of the time used b y CPLEX or B&P ro ot. Th us it can b e used to solv e up to reasonably large size instances to obtain a satisfactory solution in limited time. Since the ro ot MIP solutions are alw a ys at least as go o d as the VLSN solutions, in cases that solution qualit y is extremely imp ortan t the ro ot MIP solution can b e obtained in a reasonable time for small to in termediate size instances. Results for MPFD AS. Next w e obtained results for (AP S3 A ) in a similar w a y as for (AP P3 ), and all results are sho wn in T able 5{3 F rom our theoretical analysis, w e exp ect that in general MPFD AS will b e harder to solv e than MPFD AP The results illustrate this conclusion: CPLEX alw a ys stopp ed b ecause of the time limit, ev en for the smallest instances, and qualit y of the solutions obtained b y the greedy heuristic is m uc h lo w er, with errors larger than 10%. Mean while, the B&P b ounds are m uc h w orse than the CPLEX b ounds, and the branc handprice algorithm tak es hours to solv e ev en for medium size instances. Th us w e only obtained 6 sets of results for B&P ro ot. Ho w ev er, the VLSN heuristic can still pro duce solutions with only 1% to 2% error in limited time for all instances tested. Since the errors of the greedy solutions for MPFD AS are usually more than 10% while the running time of the greedy heuristic is negligible ev en for large instances of MPFD AS, w e ha v e tried to use a Greedy Randomized Adaptiv e Searc h Pro cedure (GRASP) to impro v e the solution of greedy heuristic. Instead of assigning the most protable plate to the c heap est slab according to their measuremen t, w e randomly c ho ose one plate from the three most protable plates. W e assign a probabilit y 0.6 to the most protable plate, 0.3 to the second most PAGE 148 148T able 5{2: Results for MPFD AP B&P ro ot Ro ot MIP Greedy heuristic VLSN CPLEX T N M Time Error Cols Time Error Time Error Time Error Time (sec) (%) (%) (sec) (%) (sec) (%) (sec) (%) (sec) 30 4 4.634 2.583 315 0.341 0.519 0.000 3.484 0.371 0.616 28.55 3 60 7 44.241 1.269 1186 0.000 4.789 15.985 1.236 240.00 90 10 358.931 0.454 2792 0.000 5.029 67.197 1.264 360.00 40 4 10.387 1.917 473 2.614 1.288 0.000 4.637 0.854 1.663 214.31 4 80 7 88.234 0.874 1694 0.001 4.991 25.965 0.926 480.00 120 10 724.498 0.267 3854 0.001 4.791 119.384 1.739 720.00 50 4 26.308 1.265 626 14.286 1.721 0.000 4.700 3.061 1.279 360.00 5 100 7 176.928 0.611 2303 0.000 5.347 43.757 1.270 720.00 150 11 1907.464 0.217 5726 0.000 4.786 226.321 1.484 1080.00 60 4 78.081 1.436 850 56.533 1.016 0.000 4.710 7.650 1.572 480.00 6 120 8 339.81 0.440 3173 0.001 4.968 71.536 1.288 960.00 180 11 3746.689 0.242 7513 0.003 4.560 409.900 1.284 1440.00 PAGE 149 149T able 5{3: Results for MPFD AS B&P ro ot Ro ot MIP Greedy heuristic VLSN CPLEX T N M Time Error Cols Time Error Time Error Time Error Time (sec) (%) (%) (sec) (%) (sec) (%) (sec) (%) (sec) 30 9 136.664 3.075 1750 93.400 2.831 0.003 10.692 0.792 2.879 120.00 3 60 17 672.039 3.585 6053 0.000 9.499 27.714 1.968 240.00 90 25 0.007 9.016 25.263 1.478 360.00 40 11 284.851 2.717 3328 138.400 2.337 0.000 12.316 1.874 2.382 240.00 4 80 22 2438.303 4.745 12268 0.000 11.189 22.894 1.530 480.00 120 33 0.008 10.041 55.365 1.846 720.00 50 14 772.566 3.884 5488 333.500 2.621 0.001 12.816 7.018 2.621 360.00 5 100 28 7754.420 6.410 24873 0.004 12.561 30.898 1.713 720.00 150 41 0.012 11.865 103.252 1.662 1080.00 60 17 1729.703 6.348 8691 480.000 1.290 0.002 15.134 15.963 2.815 480.00 6 120 33 14620.143 7.844 34071 0.007 15.020 50.936 2.009 960.00 180 50 0.016 13.786 181.873 1.730 1440.00 PAGE 150 150 protable plate, and 0.1 to the third. W e randomly c ho ose a slab in a same w a y W e then run the GRASP heuristic 100 times and c ho ose the b est solution obtained in the 100 runs as the GRASP solution. W e then use the VLSN heuristic to impro v e the GRASP solution as b efore, whic h w e will call the VLSN with GRASP heuristic. Finally w e run the original greedy heuristic and the VLSN heuristic on the same instances. W e rep ort the running times and errors for all four heuristic in T able 5{4 Notice that the errors are again computed based on the CPLEX b ounds. W e found that the GRASP solutions are usually b etter than the greedy solutions with a reduction in error of ab out 1% to 3%. Ho w ev er, the solutions obtained b y VLSN with GRASP do not alw a ys sho w an impro v emen t. In fact, in the ma jorit y of cases the original VLSN heuristic outp erforms VLSN with GRASP Th us, w e conclude that there is no b enet to using the GRASP heuristic if w e will use VLSN heuristic to impro v e the initial solution. Ho w ev er, in case that solution time is critical and the VLSN heuristic can not b e used, the GRASP heuristic will b e a b etter c hoice than the greedy heuristic since the running time of GRASP heuristic is still only on the order of seconds. 5.6 Summary In this c hapter w e studied a dynamic pro duction planning mo del under pro duct sp ecication exibilit y W e in v estigated three dieren t in v en tory strategies, and dev elop ed p olynomialtime algorithms to solv e the subproblem for eac h of the cases to optimalit y Based on our observ ations ab out the structure of the optimal solutions to the subproblems, w e prop osed greedy heuristics for the en tire problems. F urthermore, w e studied the pricing problems that are encoun tered in a branc handprice solution metho dology and iden tied it as the KPEI problem and extensions thereof. W e pro v ed some prop erties of KPEI and dev elop ecien t algorithms for KPEI based on these prop erties. By solving the subproblem and PAGE 151 151 T able 5{4: Results of GRASP for MPFD AS With GRASP Without GRASP Greedy VLSN Greedy VLSN T N M Time Error Time Error Time Error Time Error (sec) (%) (sec) (%) (sec) (%) (sec) (%) 30 9 0.086 7.168 0.805 2.926 0.003 10.692 0.792 2.879 3 60 17 0.365 7.226 20.926 1.611 0.000 9.499 27.714 1.968 90 25 0.880 7.955 24.404 1.484 0.007 9.016 25.263 1.478 40 11 0.159 9.795 2.108 2.626 0.000 12.316 1.874 2.382 4 80 22 0.679 9.470 22.110 1.807 0.000 11.189 22.894 1.530 120 33 1.640 9.340 55.142 1.706 0.008 10.041 55.365 1.846 50 14 0.252 10.985 8.616 2.896 0.001 12.816 7.018 2.621 5 100 28 1.104 11.189 28.876 1.538 0.004 12.561 30.898 1.713 150 41 2.645 10.789 99.441 1.731 0.012 11.865 103.252 1.662 60 17 0.374 12.756 17.289 3.021 0.002 15.134 15.963 2.815 6 120 33 1.636 12.937 52.221 1.931 0.007 15.020 50.936 2.009 180 50 4.307 12.565 191.455 1.822 0.016 13.786 181.873 1.730 pricing problem ecien tly w e can implemen t ecien t VLSN algorithm and branc handprice metho d. The computational results illustrated the eciency of these algorithms. The logistics net w ork setting in the rst t w o applications is v ery similar. In b oth cases, w e consider a logistics net w ork with a set of facilities and a set of retailers and a single pro duct. The main dierence b et w een the t w o applications is that in the rst application, the MultiP erio d SingleSourcing Problem (MPSSP), w e consider a discrete time horizon and assume linear pro duction and holding costs, while in the second application, the Con tin uousTime SingleSourcing Problem (CSSP), the problem will b e set in con tin uoustime horizon and w e consider xed setup costs in pro duction. In the absence of capacities, the MPSSP can b e view ed as a co op erativ e m ultisupplier and m ultiretailer v ersion of the classical ELS mo del. In this dissertation, w e will study an extended MPSSP mo del whic h will include v arious t yp es of constrain ts, in particular, pro duction capacit y throughput PAGE 152 152 capacit y in v en tory capacit y and p erish bilit y constrain ts. Similarly the uncapacitated CSSP can b e view ed as a co op erativ e m ultisupplier and m ultiretailer v ersion of the classical EOQ mo del. W e also extend the uncapacitated CSSP mo del and consider cases with pro duction and in v en tory capacit y and capacit y expansion opp ortunities. In the last application, the MultiP erio d Flexible Demand Assignmen t Problem (MPFD A), w e go bac k to a discrete time horizon but w e allo w for exible demands, that is, retailers will accept orders within some sp ecied range, giving the facilities some freedom to c ho ose the amoun t supplied. F or the MPFD A w e will in v estigate the impact of three dieren t in v en tory strategies. In particular, w e consider cases in whic h w e can sto c k ra w materials only but pro duce justintime, sto c k only end pro ducts but acquire ra w materials justintime, or sto c k b oth ra w materials and end pro ducts. Our goal is that b y studying these applications w e can pro vide insigh ts and dev elop ecien t algorithms to solv e a large set of real life problems arising in logistics net w ork design and co ordination. In the MPFD A mo del, w e assume that there is no setup up cost when w e use a slab to pro duce plates in one p erio d. Ho w ev er, in some situations, the cost to setup pro duction in one p erio d can not b e omitted. The presence of setup costs will c hange the linear cost structure in curren t mo del, and will as suc h b e an in teresting topic for future researc h. In this c hapter, w e ha v e assumed that there is sucien t pro duction and in v en tory capacit y so that w e can pro duce an y amoun t of plates or store an y amoun t of plates or slabs in in v en tory as needed. Ev en when in man y cases this ma y b e a reasonable assumption, it will b e in teresting to explore extensions of the MPFD A in whic h resource constrain ts are tak en in to accoun t. PAGE 153 CHAPTER 6 CONCLUSIONS AND FUTURE RESEAR CH DIRECTIONS In this dissertation, w e ha v e studied sev eral complex optimization problems arising in the area of supply c hain managemen t. The solution to these problems ha v e the p oten tial to signican tly reduce the costs of man ufacturers and logistics service pro viders b y appropriately redesigning the logistics net w ork as w ell as impro ving the planning of pro duction and con trol of in v en tories. W e ha v e dev elop ed b oth heuristics and exact algorithms to solv e suc h problems. W e ha v e studied the eciency and ecacy of these appro ximate and exact algorithms b y p erforming extensiv e tests on large sets of randomly generated problem instances. All our algorithms are based on a v ery general mo del structure. Th us, the general solution metho dologies ha v e the p oten tial to b e adapted to solv e man y other reallife problems faced b y companies, not only in man ufacturing but also in the areas of service and nance. And an ecien t implemen tation can usually b e obtained b y a careful study of the structure of the subproblem and the pricing problem. In all three applications studied in this dissertation, there is a v ery in teresting common structure among the optimal solutions of the linear relaxation of the pricing problems that generalizes a w ellkno wn prop ert y of the basic linear knapsac k problem. That is, it can b e sho wn that there exists at most one fractional v ariable in an optimal solution to the relaxed problem. Naturally some op en questions remain in this area. Can w e generalize this structure to a more wide class of problems? Can w e obtain an y insigh ts from this structure to build more ecien t algorithms to solv e these pricing problems to optimalit y or more generally an y problem with the same structure? The answ ers to these questions will b e an v ery in teresting future researc h topic. 153 PAGE 154 154 F or eac h application, there also exist some unexplored areas. F or the MPSSP Romero Morales [ 44 ] studied a threelev el mo del with facilities, w arehouses and retailers, and considered the assignmen t of b oth w arehouses to facilities and retailers to w arehouses under linear cost structures. This mo del can b e form ulated in the form of (AP) in the absence of pro duction capacities. Th us another future researc h direction will b e to consider this mo del in presence of v arious constrain ts similar to MPSSP or ev en with capacit y constrain t. In b oth MPSSP and MPFD A, w e only consider a linear pro duction and in v en tory holding cost structure. In the future, w e ma y study v arian ts of these problems with xedc harge or more general pro duction cost structures to capture economies of scale that are often presen t. If w e incorp orate, for example, a xedc harge pro duction cost structure to these problems, this results in a nonlinearit y in the ob jectiv e function of the problem. Ho w ev er, as w e ha v e seen for the CSSP if w e can mo v e this nonlinear term to the subproblem, w e can transfer the problem to a linear problem and solv e it exactly using the branc handprice algorithm. F urthermore, if w e can iden tify prop erties of the asso ciated subproblem that help us to solv e this subproblem ecien tly w e can use the VLSN heuristic and exp ect high qualit y solutions in limited time. In the CSSP w e no w only considered a constan t demand rate. 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PAGE 160 BIOGRAPHICAL SKETCH W ei Huang w as b orn on Septem b er 21, 1975, in Tianmen, a small cit y in the cen ter of China. He studied in the eld of pro cess con trol at the Departmen t of Automation, Tsingh ua Univ ersit y from 1992 to 2000, and obtained his master's degree in con trol theory and con trol engineering. He started his Ph.D. studies at the Departmen t of Industrial and Systems Engineering, Univ ersit y of Florida, in August 2002 in the area of Op erations Researc h. 