policies are investigated through the entire product life cycle of a remanufacturable product. Benefiting from the MDP analysis, the optimal or

Transcription

1 ABSTRACT AHISKA, SEMRA SEBNEM. Invenory Opimizaion in a One Produc Recoverable Manufacuring Sysem. (Under he direcion of Dr. Russell E. King and Dr. Thom J. Hodgson.) Environmenal regulaions or he necessiy for a green image due o growing environmenal concerns as well as he poenial economical benefis of produc recovery have pushed manufacurers o inegrae produc recovery managemen wih heir manufacuring process. Consequenly, producion planning and invenory conrol of recoverable manufacuring sysems has gained significan ineres among researchers who aim o conribue o indusrial pracice. This disseraion considers invenory opimizaion of a single produc recoverable manufacuring sysem where sochasic demand is me by eiher newly manufacured iems or remanufacured iems. Lead imes and se up coss for manufacuring and remanufacuring are considered. The invenory opimizaion problem for his sysem is formulaed as a Markov decision process (MDP) and hrough an empirical sudy, opimal or near-opimal policy characerizaions under several cos configuraions and several lead ime cases for manufacuring and remanufacuring are deermined. The effecs of a change in cos parameers of he sysem on he opimal policy srucure as well as policy parameer values are invesigaed. Resuls indicae ha he exisence of se up cos for eiher manufacuring or remanufacuring has a significan effec on policy srucure. Consequenly, an MDPbased search procedure is inroduced o deermine he invenory policy characerizaions given ha appropriae policy srucures under cerain cos configuraions are known. Furher, a neural nework analysis is performed o deermine he funcional relaionships beween cos parameers of he sysem and he invenory policy parameer values. Resuls indicae ha he policy characerizaions found by eiher MDP-based search mehodology or he formulae provided by neural nework are opimal or near-opimal wih small deviaions (usually, less han 1%) from opimal cos. Finally, he opimal invenory

2 policies are invesigaed hrough he enire produc life cycle of a remanufacurable produc. Benefiing from he MDP analysis, he opimal or near-opimal policy characerizaions wih only a few parameers are deermined for every sage of he produc life cycle. The effecs of a change in he demand and reurn raes on he opimal invenory policies are invesigaed. Furher, he performance of hese long-run policy characerizaions is evaluaed in a finie-horizon seing, and he imporance of frequenly revising he invenory policies over he produc life cycle is illusraed numerically.

3 Invenory Opimizaion in a One Produc Recoverable Manufacuring Sysem by Semra Sebnem Ahiska A disseraion submied o he Graduae Faculy of Norh Carolina Sae Universiy in parial fulfillmen of he requiremens for he Degree of Docor of Philosophy Indusrial and Sysems Engineering Raleigh, Norh Carolina 2008 APPROVED BY: Dr. Russell E. King Chair of Advisory Commiee Dr. Thom J. Hodgson Co-chair of Advisory Commiee Dr. Krisin A. Thoney-Barlea Dr. Jeffrey A. Joines

4 DEDICATION To my parens and my auns, who have always encouraged me o complee my sudies no maer how ough hey are. ii

5 BIOGRAPHY S. Sebnem Ahiska was born in Isanbul, Turkey on January 30, For her secondary educaion, she aended Saine Pulchérie French middle school and Sain Joseph French High school. She received her Bachelor of Science in Indusrial Engineering from Isanbul Technical Universiy in Upon graduaion, she aended Galaasaray Universiy where she received her Maser s degree in Indusrial Engineering in She received a scholarship from he Council of Higher Educaion of Turkey for her PhD educaion in USA, and she began her PhD educaion in Indusrial and Sysems Engineering a Norh Carolina Sae Universiy in Her research ineress include he applicaions of Operaions Research, Markov Decision Processes, mahemaical modeling, decision making. iii

6 ACKNOWLEDGMENTS I would like o especially hank Dr. Russell E. King for all his help, suppor, guidance, and valuable advices he has given me during my PhD educaion. I would also like o hank Dr. Thom Hodgson, Dr. Krisin Thoney and Dr. Jeffrey Joines for heir valuable commens regarding my disseraion. Finally, I would like o hank my parens, Ayla and Tufan Ahiska, my auns, Oya and Mine Dogru, and my broher, Tolga Ahiska, for all heir encouragemen, suppor and love. iv

7 TABLE OF CONTENTS Lis of Tables.vii Lis of Figures xii 1. Inroducion Lieraure survey on invenory conrol in recoverable manufacuring sysems Deerminisic recoverable invenory models Sochasic recoverable invenory models Periodic-review models Coninuous-review models Invenory opimizaion for a periodically-reviewed recoverable manufacuring sysem Problem descripion and MDP formulaions Experimens seing and he analysis Resuls regarding policy srucures under differen lead ime cases Resuls for uniy lead ime case for boh manufacuring and remanufacuring Resuls for he case where remanufacuring lead ime is wo periods and manufacuring lead ime is one period Resuls for he case where remanufacuring lead ime is one period and manufacuring lead ime is wo periods Sensiiviy analysis regarding he effec of changing he coefficien of variaion of demand disribuion on he opimal policy Discussion Invenory policy characerizaion mehods for a single-produc recoverable sysem The recoverable sysem An MDP-based search mehodology and a facorial analysis A neural nework analysis Discussion v

8 5. Invenory policy characerizaions for a single-produc recoverable sysem over differen sages of he produc life cycle Problem descripion Produc life cycle analysis Resuls The policy characerizaions over differen sages of produc life cycle under differen se up cos configuraions The performance comparison of he PULL policy wih he MDPbased policy characerizaions The performance evaluaion of long run opimal or near-opimal policy characerizaions in a finie horizon seing Frequen policy revision vs. no revision or parial revision in finiehorizon seing Discussion Conclusion and furher work. 79 References.. 82 Appendices vi

9 LIST OF TABLES Table 3.1 The cos parameers and he corresponding values considered in he experimens 26 Table 3.2 The effec of changing holding cos for serviceable iems, C HS, when l p =l m =1 28 Table 3.3 The effec of changing holding cos for recoverable iems, C HR, when l p =l m =1 29 Table 3.4 The effec of changing backordering cos, C BO, when l p =l m = Table 3.5 The effec of changing disposal cos for recoverable iems, C DISP, when l p =l m =1 29 Table 3.6 The effec of changing los sales cos, C LS, when l p =l m =1. 30 Table 3.7 The effec of changing se up cos for regular producion, S P, when l p =l m = Table 3.8 The effec of changing se up cos for remanufacuring, S M, when l p =l m = Table 3.9 Resuls in he exisence of non-zero manufacuring and remanufacuring se up coss, S P and S M, when l p =l m =1 33 Table 3.10 The summary of policy srucures found in he experimens for case l p =1 & l m =2 36 Table 3.11 The summary of policy srucures found in he experimens for case l p =2 & l m =1 37 Table 3.12 Summary of he resuls of experimens on CV of demand disribuion 40 Table 4.1 Relevan cos parameers and heir respecive values considered in he facorial analysis 46 Table 4.2 Sysem parameers ha are kep unchanged during he facorial analysis Table 4.3 Appropriae policy srucures wih respec o se up cos srucure of he sysem 47 vii

10 Table 4.4 Facorial analysis resuls for lead ime case 1: l p =l m =1. 51 Table 4.5 Facorial analysis resuls for lead ime case 2: l p =2; l m =1. 52 Table 4.6 Facorial analysis resuls for lead ime case 3: l p =1; l m =2. 52 Table 4.7 Resuls regarding he performance of he neural nework for each group of scenarios considered in each lead ime case. 56 Table 5.1 Demand and reurn means considered hroughou he produc life cycle Table 5.2 The se up cos values considered in he produc life cycle analysis. 63 Table 5.3 Sysem parameers ha are kep unchanged during he produc life cycle analysis.. 64 Table 5.4 Policy characerizaions and he corresponding cos informaion for he case where S P =S M =$ Table 5.5 Policy characerizaions and he corresponding cos informaion for he case where S P =$25, S M =$0.67 Table 5.6 Policy characerizaions and he corresponding cos informaion for he case where S P =$0, S M =$25.69 Table 5.7 Policy characerizaions and he corresponding cos informaion for he case where S P =$25, S M =$ Table 5.8 The performance comparison of he PULL policy wih he MDP-based policy characerizaions for he case where S p =S m =$0 73 Table 5.9 The performance comparison of he PULL policy wih he MDP-based policy characerizaions for he case where S p =$25 & S m =$ Table 5.10 The performance comparison of he PULL policy wih he MDP-based policy characerizaions for he case where S p =$0 & S m =$ Table 5.11 The performance comparison of he PULL policy wih he MDP-based policy characerizaions for he case where S p =$25 & S m =$ Table 5.12 The performance of he long-run policy characerizaions over PLCs wih several lenghs 76 Table 5.13 The percenage deviaions from opimal PLC coss for no policy revision, parial revision and frequen revision cases viii

11 Table A.1 The effec of changing several cos parameers when l p =1 & l m =2. 88 Table A.2 Resuls in he exisence of non-zero manufacuring and remanufacuring se up coss, S P and S M, when l p =1 & l m =2 89 Table A.3 The effec of changing several cos parameers when l p =2 & l m = Table A.4 Resuls in he exisence of non-zero manufacuring and remanufacuring se up coss, S P and S M, when l p =2 & l m =1. 91 Table A.5 Policy characerizaions under demand disribuions wih differen CVs for he case where l l = 1 and S S = p = m P = M Table A.6 Table A.7 Table A.8 Table A.9 Table A.10 Policy characerizaions under demand disribuions wih differen CVs for he case where l l = 1 and S = 10 & S = p = m Policy characerizaions under demand disribuions wih differen CVs P for he case where l = 1 & l = 2 and S S = p m P M = M Policy characerizaions under demand disribuions wih differen CVs for he case where l = 1 & l = 2 and S = 10 & = p m Policy characerizaions under demand disribuions wih differen CVs for he case where l = 2 & l = 1 and S S = p m P P = M Policy characerizaions under demand disribuions wih differen CVs for he case where l = 2 & l = 1 and S = 10 & = p m P S M S M Table B.1 The descripions of he policy srucures Table B.2 The relevan invenory posiion definiions for manufacuring and remanufacuring decisions under differen lead ime cases Table B.3 The policy characerizaions for he scenarios where S p =S m =0 and l p =l m =1 96 Table B.4 The policy characerizaions for he scenarios where S p >0 & S m =0 and l p =l m =1 97 Table B.5 The policy characerizaions for he scenarios where S p =0 & S m >0 and l p =l m =1 98 ix

12 Table B.6 The policy characerizaions for he scenarios where S p >0 & S m >0 and l p =l m = Table B.7 The policy characerizaions for he scenarios where S p =S m =0 and l p =1 & l m =2 103 Table B.8 The policy characerizaions for he scenarios where S p >0 & S m =0 and l p =1 & l m = Table B.9 The policy characerizaions for he scenarios where S p =0 & S m >0 and l p =1 & l m = Table B.10 The policy characerizaions for he scenarios where S p >0 & S m >0 and l p =1 & l m = Table B.11 The policy characerizaions for he scenarios where S p =S m =0 and l p =2 & l m = Table B.12 The policy characerizaions for he scenarios where S p >0 & S m =0 and l p =2 & l m = Table B.13 The policy characerizaions for he scenarios where S p =0 & S m >0 and l p =2 & l m = Table B.14 The policy characerizaions for he scenarios where S p >0 & S m >0 and l p =2 & l m = Table B.15 The raining daa for group 1 for he case where l p =l m =1 117 Table B.16 The esing daa for group 1 for he case where l p =l m = Table B.17 The prediced policies for he esing scenarios for group 1 for he case where l p =l m = Table B.18 The raining daa for group 2 for he case where l p =l m =1 121 Table B.19 The esing daa for group 2 for he case where l p =l m = Table B.20 The prediced policies for he esing scenarios for group 2 for he case where l p =l m = Table B.21 The raining daa for group 1 for he case where l p =1, l m = Table B.22 The esing daa for group 1 for he case where l p =1, l m = Table B.23 The prediced policies for he esing scenarios for group 1 for he case where l p =1, l m =2 130 x

13 Table B.24 The raining daa for group 2 for he case where l p =1, l m = Table B.25 The esing daa for group 2 for he case where l p =1, l m = Table B.26 The prediced policies for he esing scenarios for group 2 for he case where l p =1, l m = Table B.27 The raining daa for group 1 for he case where l p =2, l m = Table B.28 The esing daa for group 1 for he case where l p =2, l m = Table B.29 The prediced policies for he esing scenarios for group 1 for he case where l p =2, l m = Table B.30 The raining daa for group 2 for he case where l p =2, l m = Table B.31 The esing daa for group 2 for he case where l p =2, l m = Table B.32 The prediced policies for he esing scenarios for group 2 for he case where l p =2, l m = Table C.1 The descripions of he invenory policies Table C.2 The cos sensiiviy of he invenory policies for he case where S P =S M = Table C.3 The cos sensiiviy of he invenory policies for he case where S P =25, S M = Table C.4 The cos sensiiviy of he invenory policies for he case where S P =0, S M = Table C.5 The cos sensiiviy of he invenory policies for he case where S P =25, S M = xi

14 LIST OF FIGURES Figure 3.1 The recoverable manufacuring sysem 18 Figure 3.2 Demand disribuions wih μ = 5 and coefficien of variaions Figure 5.1 Demand and reurn raes over a ypical life cycle of a recoverable produc Figure 5.2 Demand disribuions used over he produc life cycle..63 Figure 5.3 Reurn disribuions used over he produc life cycle.63 xii

15 1. Inroducion In recen years, manufacurers have paid growing aenion o reuse aciviies ha provide maerial wase reducion via he recovery of some conen of used producs. Moivaion behind hese produc recovery aciviies is wo-fold: growing environmenal concerns and poenial economical benefis. In several counries environmenal regulaions are in place, which make manufacurers responsible for he whole life cycle of he produc hey produce. A common example of hese regulaions is ake-back obligaions afer usage (Fleischmann e al., 1997). Bu even in he absence of such regulaions, he expecaions of environmenally conscious consumers pu pressure on companies o consider environmenal issues in heir manufacuring process. Nowadays, a green image, which can be obained by implemening recoverable manufacuring sysems, has become a powerful markeing ool and provides a significan compeiive advanage o companies who seek o have a place in he global marke. Reuse of producs or maerials can be economically aracive as well in addiion o conribuing o susainable developmen. Disposal coss have increased significanly in recen years due o depleion of incineraion and land filling capaciies. Companies are considering produc recovery o avoid disposal cos. Furher, wih produc recovery, no only savings in disposal cos are obained bu also he value incorporaed in he used produc is regained, which provides energy, maerial and labor savings. Remanufacuring is a ypical example for economically aracive reuse aciviies. Remanufacuring ransforms used producs ino like new producs. Afer disassembly, modules and pars are exensively inspeced and problemaic pars are repaired, or if no possible, replaced wih new pars. These operaions allow a considerable amoun of value incorporaed in he used produc o be regained. Remanufacured producs have usually he same qualiy as he new producs and are sold for he same price, bu hey are less cosly. Examples of remanufacurable producs include mosly high-value componens such as aircraf or auomobile engines, aviaion equipmen, medical equipmen, office furniure, machine ools, copiers, compuers, elecronics equipmen, oner carridges, cellular elephones, single-use cameras ec. (Thierry e al., 1995; Fleischmann e al., 1997; US EPA, 1997; Guide e al., 1999; Tokay e al., 2000). 1

16 Recoverable manufacuring sysems can be defined as closed loop sysems wih discarded iems used in place of exernally supplied virgin maerials o he greaes exen possible in he fabricaion of new producs (Guide e al., 2000). These sysems are capable of dealing wih produc reurns via several produc recovery opions, which can be caegorized as direc reuse, repair, refurbishing, remanufacuring, cannibalizaion and recycling wih respec o increasing degree of required disassembly level (Thierry e al., 1995). There are several review papers ha emphasize he challenges of considering produc recovery. Thierry e al. (1995) describe sraegic issues ha manufacurers face in implemening produc recovery managemen policies. Fleischmann e al. (1997) provide a sysemaic review of reverse logisics issues and mahemaical models for dealing wih reurns in disribuion planning, producion planning and invenory conrol areas. Guide e al. (2000) discuss he complicaing characerisics of recoverable manufacuring sysems including uncerainies in iming, quaniy and qualiy of reurns, he need for balancing demands wih reurns, disassembly need for reurned producs, he requiremen of a reverse logisics nework, maerial maching resricions and sochasic rouings for maerials o be used in recovery operaions. Two main addiional sources of complexiy appear in invenory conrol of recoverable manufacuring sysems compared wih radiional invenory sysems wihou produc reurns. Firs, due o uncerainy of he produc reurns, an addiional sochasic impac needs o be considered. Second, he produc recovery opion (e.g. remanufacuring) mus be coordinaed wih regular procuremen (e.g. manufacuring), which complicaes he invenory conrol siuaion. In recen years, he challenges ha are faced when dealing wih reurns in he conex of producion planning and invenory managemen have gained considerable aenion of researchers. Consequenly, sudies regarding recoverable manufacuring sysems have found a place in he lieraure, which are menioned horoughly in he nex secion. In his disseraion, we consider he invenory opimizaion problem of a singleproduc sochasic recoverable manufacuring sysem where regular producion (manufacuring) as well as remanufacuring are available as modes of supply o mee cusomer demand. Lead imes as well as se up coss for manufacuring and 2

17 remanufacuring are aken ino consideraion. Sochasic recoverable sysems wih similar characerisics have been widely invesigaed in he las decade; however, no work exiss in he lieraure which conducs an analysis o find he opimal invenory policy srucure in he exisence of fixed cos for regular producion and/or remanufacuring. Furher, o our knowledge, none of he work ha uses a pre-deermined policy srucure for invenory opimizaion indicaes how well he policy srucure hey consider characerizes he opimal invenory conrol policy. This disseraion aims o find he srucure of opimal invenory conrol policies for he recoverable manufacuring sysem under several cos configuraions including nonzero se up coss for manufacuring and remanufacuring operaions. The opimal invenory policies are found by solving he Markov Decision Process (MDP) model of he recoverable sysem, and an empirical sudy is conduced o deermine policy characerizaions under several cos configuraions and lead ime relaions. The performance of he policy characerizaions provided is evaluaed numerically considering he percenage deviaion of heir coss from he opimal cos. Based on he policy characerizaions, i is invesigaed how he policy srucure as well as policy parameer values change as cos parameers of he sysem change. Consequenly, an MDP-based search procedure is inroduced o deermine invenory policy characerizaions given ha appropriae policy srucures under cerain cos configuraions are known. Furher, a neural nework analysis is performed o deermine he funcional relaionships beween cos parameers of he sysem and he invenory policy parameer values. Finally, he opimal invenory policies are invesigaed hrough he enire produc life cycle of a remanufacurable produc. Benefiing from he long-run opimal policies found hrough MDP analysis, he opimal or near-opimal policy characerizaions are deermined for every sage of he produc life cycle. The effecs of a change in he demand and reurn raes on he opimal invenory policies are invesigaed. Furher, he performance of hese long-run policy characerizaions is evaluaed in a finie-horizon seing, and he imporance of frequenly revising he invenory policies over he produc life cycle is illusraed numerically. 3

18 The res of his disseraion is organized as follows: in chaper 2, a lieraure survey on he invenory managemen of sochasic recoverable manufacuring sysems is provided. In chaper 3, a periodically reviewed recoverable manufacuring sysem is presened and formulaed as a discree-ime Markov Decision Process under differen lead ime cases for manufacuring and remanufacuring. Through an empirical sudy, how he policy srucure as well as policy parameer values change as cos parameers of he sysem change is invesigaed. Furher, a sensiiviy analysis is provided regarding he effec of changing coefficien of variaion of demand disribuion on he opimal policy srucure. In chaper 4, an MDP-based characerizaion process is inroduced and illusraed hrough a facorial analysis considering he mos relevan cos parameers of he sysem. Furher, wheher a neural nework analysis can be used o successfully deermine he funcional relaionship beween he cos parameers of he sysem and he policy parameers values is invesigaed. Chaper 5 considers invenory opimizaion over he enire life cycle of a remanufacurable produc. The effecs of a change in demand and reurn raes on he opimal invenory policies are invesigaed via a numerical sudy, and he imporance of frequen policy revision over he life cycle is emphasized. 4

19 2. Lieraure survey on invenory conrol in recoverable manufacuring sysems Tradiional invenory models do no consider ha manufacured iems ha are sold o he marke may be reurned o he manufacurer by he end users afer a cerain ime and may need o be recovered by he manufacurer. Nowadays, many manufacurers ake back heir producs from cusomers afer heir consumpion, eiher due o responsibiliies resuling from legislaion or due o economic profis of produc recovery (Inderfurh and Van der Laan, 2001). Two main addiional sources of complexiy appear in invenory conrol of recoverable manufacuring sysems compared wih radiional invenory sysems wihou produc reurns. Firs, due o uncerainy of he produc reurns, an addiional sochasic impac needs o be considered. Second, produc recovery opions (e.g. remanufacuring) mus be coordinaed wih regular procuremen opions (e.g. manufacuring), which complicaes he invenory conrol siuaion. The challenges ha are faced when dealing wih reurns in he conex of producion planning and invenory managemen have gained considerable aenion of researchers who aim o conribue o indusrial pracice. Consequenly, opimizaion of recoverable manufacuring sysems has found a place in he lieraure. Invenory conrol of single produc-hybrid manufacuring/remanufacuring sysems has been invesigaed by several auhors. A general framework for hese sysems can be briefly described as follows: used producs ha are no longer needed by cusomers are reurned back o he manufacurer and are considered as an inpu resource in addiion o exernally supplied maerials and/or pars for producion of new iems. In oher words, here are wo supply modes o fulfill cusomer demand in hese sysems; one being regular producion (manufacuring) of new iems using exernally supplied resources, he oher being remanufacuring of reurned producs. Models found in he lieraure wihin his framework differ wih respec o he assumpions made regarding several model characerisics including he demand and reurn processes, he exisence of a disposal opion for reurned iems, relevan cos parameers, lead imes of producion and remanufacuring aciviies, and he lengh of he planning horizon. Such assumpions allow several classificaions such as deerminisic versus sochasic models wih respec o cusomer demand and reurns, finie horizon 5

20 versus infinie horizon models, periodic review versus coninuous review models, models wih/wihou disposal opion, ec. A major classificaion observed in he lieraure is deerminisic versus sochasic recoverable sysems. Furhermore, sochasic produc recovery models are classified ino periodic versus coninuous review models. In periodic review models, decisions regarding invenory replenishmen are made on a periodic basis while in coninuous review models invenory levels are moniored coninuously and decisions can be made a any ime. The following wo secions review several works on deerminisic and sochasic invenory sysems wih produc recovery opion, respecively Deerminisic recoverable invenory models In deerminisic recoverable invenory models, all he componens of he sysem (e.g., cos informaion, lead imes, cusomer demand and produc reurns over he enire planning horizon) are assumed o be known wih cerainy (Fleischmann, 2001). The deerminisic recoverable models can primarily be subdivided ino saic and dynamic models. In he former, demands and reurns are saionary. In oher words, demand and reurn raes are consan over he planning horizon. While in he laer, demand and reurn raes are dynamic, i.e. hey vary hrough ime. For saic recoverable models, he exensions o Economic Order Quaniy (EOQ) model ha ake ino accoun reurn flows have been proposed by several auhors. These models seek o find an opimal rade-off beween fixed ordering/se up coss and variable holding coss (Fleischmann, 2001). The firs saic model in he reverse logisics conex was proposed by Schrady (1967). Schrady (1967) considers a deerminisic recoverable sysem wih wo socking poins, serviceable iems and recoverable iems, wihou a disposal opion for reurned iems. Demand and reurn raes as well as lead imes for exernal order and recovery operaions are deerminisic. Sock-ous are no allowed. The cos funcion o minimize consiss of fixed coss for exernal order and recovery se up as well as linear holding coss for recoverable and serviceable iems. Schrady proposes a hree-parameer conrol policy ha consiss of an order quaniy for exernal procuremen, a bach size for recovery opion and he number of idenical baches ha should follow 6

21 every exernal procuremen order. The expressions for he opimal policy parameers are derived in an analogous way o EOQ model. Mabini e al. (1992) provide EOQ-ype formulaions for conrolling invenory in a single iem repairable sysem where sock ous are allowed up o a cerain level. Their aim is o find he purchase and repair quaniies ha would minimize oal cos while saisfying a cerain service level. Furhermore, hey exend heir model o a muli-iem siuaion where muliple iems share a common limied repair capaciy. Richer (1996) presens a deerminisic wo-sage EOQ model ha describes he producion of new iems and he repair of used iems in a firs shop and he employmen of he new or repaired producs in a second shop a a consan demand rae. A he end of a collecion period, he used producs in he second shop are eiher disposed of or brough back o he firs shop for repair according o a cerain repair rae. In addiion o he fixed se up coss and he holding coss, he sysem cos includes he variable producion, repair and disposal coss as well. More han one se ups for producion and repair wihin he collecion inerval are allowed. One shall noe ha all he above menioned works are common in he following sense: Firs, hey all consider a predeermined invenory holding policy wihou invesigaing is opimaliy. Second, a remanufacured or repaired iem is assumed o be as-good-as a newly produced iem in erms of qualiy. Third, he qualiy of he reurned iems is assumed o be perfec for reuse. More recenly, Dobos and Richer (2006) invesigae a producion/recycling model wih qualiy consideraions. They show ha i is beer o ousource he qualiy conrol and repurchase only reusable producs in order o minimize he oal EOQ relaed (i.e. he fixed se up/order coss and he holding coss) and non-eoq relaed (i.e. he linear wase disposal, recycling, producion and repurchasing coss) coss. Beside he saic recoverable models, he dynamic recoverable models as well have received he aenion of researchers. The varians of Wagner/Whiin algorihm (Wagner and Whiin, 1958) have been developed in he conex of Reverse Logisics o solve dynamic recoverable models. For insance, Richer and Sombruzki (2000) sudy he pure reverse Wagner/Whiin model and some of is exensions, where only he remanufacuring opion is available o mee demand. The oal cos consiss of he se up cos for remanufacuring, he holding cos for reurned iems and he holding cos for 7

22 remanufacured iems. Furhermore, hey analyze he model of alernae applicaion of remanufacuring and manufacuring processes where boh newly produced iems and remanufacured iems have he same value and can be used o saisfy cusomer demand. In a laer work, Richer and Weber (2001) exend he reverse Wagner/Whiin model by incorporaing he variable manufacuring and remanufacuring coss ino he model. This model can be viewed as a combinaion of he classical Wagner/Whiin model and a pure reverse Wagner/Whiin model. They also invesigae how he exisence of a disposal opion for used producs would affec he soluion Sochasic recoverable invenory models Regarding sochasic invenory conrol problems in he exisence of remanufacuring opions, wo sreams of conribuions can be found in he lieraure: one includes periodic-review models, he oher includes coninuous-review models. Two main approaches are observed in he lieraure regarding invenory managemen of sochasic recovery models. One approach, which has been rarely used, is o invesigae he srucure of he opimal conrol policy using dynamic programming approaches (e.g., Simpson, 1978; Inderfurh, 1997). A second approach is o find opimal or near-opimal values for he parameers of a predeermined reasonable, bu no necessarily opimal, conrol policy srucures, benefiing from he resuls of queuing heory, Markov decision processes, enumeraion echniques, simulaion or heurisics (e.g. Kiesmuller, 2003; Kiesmuller and Scherer, 2003; Kiesmuller and Minner, 2003; Mahadevan e al., 2003; Van der Laan e al. 1995, 1996; Van der Laan and Salomon, 1997; Van der Laan e al. 1999a, 1999b; Van der Laan and Teuner, 2006). The laer approach has been widely used for boh periodically and coninuously reviewed recoverable invenory sysems due o is simpliciy and is applicabiliy o more comprehensive, larger-scale models wih relaxed assumpions compared o models o which he former approach has been applied. However, i has he major drawback of considering a pre-deermined policy which is no guaraneed o be opimal. In hese sudies, i is no indicaed how good he pre-deermined policy is compared wih he opimal policy. 8

23 Periodic-review models Periodic review models describe pracical siuaions in a more suiable way. However, only a few papers can be found in he lieraure which considered periodic review models in he conex of produc recovery. Furhermore, he majoriy of hese invesigae he opimal parameer values for a pre-deermined policy raher han finding he opimal policy srucure iself for he sysem under consideraion. Kiesmuller (2003) considers a pre-deermined periodic review PULL policy, named as (S, M) policy, for a sochasic hybrid manufacuring/remanufacuring sysem wih wo socking poins wihou a disposal opion and wih differen lead ime cases for manufacuring and remanufacuring operaions. The key poin in heir approach is defining differen invenory posiions on which producion and remanufacuring decisions will be based. They provide a comparaive analysis o jusify he use of separae invenory posiion definiions for producion and remanufacuring decisions raher han using a single invenory posiion definiion for boh decisions. They calculae opimal parameer values of he pre-deermined policy and he opimal cos using grid search and simulaion. Kiesmuller and Minner (2003) provide simple Newsvendor ype heurisic formulae o calculae he parameer values of he (S, M) policy. Their numerical sudy shows ha he proposed formulae lead o near opimal parameer values in mos cases. Mahadevan e al. (2003) employ a periodic review push policy, named as (R, S) policy, o analyze a similar recoverable invenory sysem. They develop several heurisics based on radiional models o find he values of his pre-deermined policy s parameers. To our knowledge, only a couple of papers in he lieraure emphasize he generaion of opimal conrol policy srucures for a one produc recoverable sysem. Simpson (1978) generaes he opimal policy srucure for a finie-horizon repairable invenory problem wih wo socking poins using a backward dynamic programming echnique. The opimal policy srucure, which is defined by hree period dependen parameers, namely repair-up-o level, purchase-up-o level, and scrap-down o level, is valid under a 0-lead ime assumpion for repairing and purchasing aciviies. A more comprehensive work has been provided by Inderfurh (1997), who addresses he problem of invenory opimizaion for a recoverable sysem wih and wihou sock keeping of reurned iems 9

24 for which he uses he mehod of sochasic dynamic programming o derive opimal decision rules for procuremen, remanufacuring and disposal. He explores he impac of procuremen and remanufacuring lead imes o he complexiy of he opimal decision rules and shows ha up o a cerain exen, simple rules wih only a few parameers can be proved o be opimal for he sochasic recovery problem. For he case wihou sock keeping of recoverable iems, he formulaes opimal policy srucures for differen lead ime cases. In addiion, for he sock keeping of recoverable iems case, he provides he opimal policy srucure only for he case of equal lead imes and no se up cos for producion and remanufacuring. Inderfurh indicaes ha if here exised fixed coss of remanufacuring or normal producion, he policies may no longer be opimal. Kiesmuller and Scherer (2003) consider he opimal policy srucures provided by Inderfurh (1997) for one produc recovery sysem wih/wihou sock keeping of reurns under equal lead imes for manufacuring and remanufacuring, and provide a mehod for exac compuaion of he policy parameers and a couple of heurisic mehods Coninuous-review models The firs coninuous review model for he manufacuring sysems wih produc reurns was proposed by Heyman (1977). Heyman considers a single iem coninuousreview invenory model wih sochasic uncorrelaed demand and reurns. A reurned iem is eiher disposed of or repaired upon arrival. He disregards he fixed cos of ordering (implying no lo size reordering) and assumes insananeous procuremen and repair (implying no backorders or los sales). Under hese assumpions, he opimizes a single parameer policy ha consiss of only a disposal level, which is defined as he serviceable invenory a which reurned producs are disposed of. Mucksad and Isaac (1981) consider a single iem, single locaion invenory sysem wih produc reurns ha differs from he one considered by Heyman (1977) in he sense ha hey disregard he disposal opion for reurned iems, and hey incorporae ino he model he fixed ouside procuremen coss and non-zero lead imes for boh procuremen and repair. Under he assumpion ha demand and reurn are Poisson disribued, hey propose he use of an (r,q) order policy, where r is he serviceable invenory posiion a 10

25 which an ouside procuremen order of Q unis is placed. Closed form expressions for calculaing reorder level, r, and order quaniy, Q, are provided. Furher, he resuls of he single echelon case are exended o a specific wo echelon case where a warehouse (upper echelon) suppors several reailers (lower echelon). Vander Laan e al. (1995) propose wo conrol policies, namely (s m, Q m, Q r ) PUSH and (s m, Q m, s r, S r ) PULL sraegies for invenory conrol of hybrid manufacuring and remanufacuring sysems wihou a disposal opion for reurned iems. Alhough hese sraegies are non-opimal, hey have he advanage ha hey are easy o implemen and hey are acually used in pracice. The wo sraegies are he same in erms of manufacuring decision. According o boh sraegies, when serviceable invenory posiion is below s m, a manufacuring order of size Q m is placed. The main difference beween PUSH and PULL sraegies is ha wih PUSH conrol, reurned producs are remanufacured as early as possible while wih he oher, hey are remanufacured as lae as possible. Under PUSH conrol, reurned producs are pushed ino he remanufacuring process every ime recoverable invenory reaches a cerain level (i.e. Q r ) while under PULL conrol, remanufacuring operaions do no sar unless he serviceable invenory posiion is sufficienly low (i.e. below s r ) and here are enough reurned iems in recoverable invenory o increase he serviceable invenory posiion up o a cerain level, S r. Van der Laan e al. (1996) consider a single produc single echelon remanufacuring sysem wih produc disposals where ouside procuremen is considered as a second supply mode in addiion o remanufacuring o mee demand. Lead ime for ouside procuremen is deerminisic while remanufacuring lead ime is sochasic. There is a fixed ouside procuremen cos per order and variable ouside procuremen cos per ordered uni of produc. No fixed cos is considered for remanufacuring. There are c parallel machines for remanufacuring, and once one become available, a reurned iem in queue is pushed ino remanufacuring. When here is no room in queue for remanufacuring or invenory posiion reaches a cerain level, incoming reurns are disposed of. For his sysem, Van der Laan e al. (1996) consider hree conrol policies, namely (s p,q p,s d,n), (s p,q p,s d ) and (s p,q p,n), which differ form each oher only wih 11

26 respec o disposal conrol rule. Noe ha he las wo policies are a special case of (s p,q p,s d,n) policy. According o (s p,q p,s d,n) policy, he sraegy wih respec o ouside procuremen is: whenever invenory posiion, which consiss of ne serviceable invenory, reurned iems in he remanufacuring faciliy and ouside procuremen orders, drops o s p an ouside procuremen order of Q p unis of producs is given. The sraegy wih respec o remanufacuring or disposal is whenever he number of reurned iems in remanufacuring faciliy is N or he serviceable invenory equals s d, reurned unis are disposed of upon arrival. In order o calculae expeced cos for pre-deermined policies, Van der Laan e al. (1996) employ an analyical procedure ha requires formulaing and solving a coninuous-ime Markov chain model wih wo sae variables, invenory posiion and he reurned iems in he remanufacuring faciliy. They use an enumeraive search procedure o find he opimal parameer values for he policies under consideraion. Via a numerical sudy, hey show how he performance of each policy varies when he reurn rae or he remanufacuring rae changes. In a laer sudy, Van der Laan and Salomon (1997) consider a manufacuring/remanufacuring sysem wih planned disposal operaions where cusomer demands and reurns are correlaed. Inerarrival imes for demand and reurn are Coxian-2 disribued. Boh remanufacuring and producion have deerminisic non-zero lead imes. They exend PUSH and PULL sraegies ha Vander Laan e al. (1995) developed for a similar sysem wihou planned disposal opion o PUSH-disposal and PULL-disposal sraegies ha include one addiional parameer, i.e. s d, which defines disposal sraegy. Through numerical examples, hey provide a comparison beween sysems wih and wihou disposals, a comparison beween PUSH-disposal and PULL-disposal sraegies, and a sudy on he robusness of conrol parameers over he differen sages of produc life cycle. Laer, Van der Laan e al. (1999a) consider a single produc hybrid manufacuring/remanufacuring sysem wihou disposal opion where manufacuring and remanufacuring lead imes are modeled as deerminisic as well as a discreely disribued sochasic variable. They invesigae he influence of lead ime duraion (for deerminisic lead ime case) and lead ime variabiliy (for sochasic lead ime case) on oal expeced coss when PUSH and PULL sraegies are used o conrol he invenory 12

27 sysem. They model he sysem under consideraion as a coninuous-ime Markov chain model and use an enumeraive search procedure o find he opimal conrol parameers (or, opimal cos) for PUSH and PULL policy. Similarly, Van der Laan e al. (1999b) presen an exac mehodology ha requires an exensive enumeraive search o opimize a PUSH conrol sraegy and a PULL conrol sraegy. Through a numerical sudy, hey compare he cos of a radiional invenory sysem wihou remanufacuring operaions ha is conrolled by a simple (s, S) policy o he coss of hybrid manufacuring/remanufacuring sysems conrolled by a PULL or a PUSH policy. Furhermore, hey compare he cos of a PUSH conrolled sysem o a PULL conrolled sysem under several scenarios. Inderfurh and Van der Laan (2001) invesigae he informaion srucure (invenory informaion necessary for opimal sock replenishmen) of a predeermined conrol policy from a heoreical poin of view. They consider a simple 4-parameer conrol policy and show ha he performance of he policy can be improved considerably if he invenory informaion o consider when making a producion decision is defined appropriaely. Fleischmann e al. (2002) consider a basic single-echelon invenory model where here is no disincion beween new and reurned iems in erms of qualiy. Reurned iems upon arrival join direcly serviceable iem sock wihou requiring any recovery operaions. In addiion o reurned iems, new iems ha are obained by ouside procuremen are also used o mee cusomer demand. They show he opimaliy of he convenional (s, Q) policy for ordering new iems under he assumpion of sochasic demand and reurns, and hey provide a procedure o calculae opimal conrol parameers. Their analysis sars wih formulaing a coninuous-ime Markov chain wih a one-dimensional sae space, invenory posiion being he sae variable. They use seady sae probabiliies o calculae long-run average cos under he (s, Q) policy, and make furher analysis o prove opimaliy of he (s, Q) policy. Furher, hrough a numerical sudy, hey invesigae he effec of varying he raio of reurn o demand on he opimal coss. In a laer work, Fleishmann and Kuik (2003) again analyze a single socking poin sysem wih independen sochasic demand and reurns where reurned iems are in a direcly usable condiion. The sysem is considered as a varian of a radiional 13

28 sochasic invenory model where demand can ake boh posiive and negaive values, a negaive demand represening he arrival of a reurned iem. Benefiing from he general heory of Markov decision processes, hey show he average cos opimaliy of an (s, S) policy. Through a numerical sudy, he effec of reurn flow on he sysem coss (fixed cos, holding cos and backordering cos) is invesigaed. More recenly, Van der Laan and Teuner (2006) consider PUSH and PULL remanufacuring policies o conrol a hybrid invenory sysem wih uni produc reurns and demands where he remanufacuring opion is cheaper han he manufacuring opion. The sysem coss include se up coss, holding coss and backordering coss. They develop simple closed form expressions for calculaing near-opimal parameer values for hese policies, and in an exensive numerical sudy, hey evaluae he performance of he proposed heurisics by comparing heir coss o he coss associaed wih opimal parameer seings for policy srucures under consideraion. 14

29 3. Invenory Opimizaion for a Periodically-Reviewed Recoverable Manufacuring Sysem Recoverable manufacuring sysems can be defined as closed loop sysems wih discarded iems used in place of exernally supplied virgin maerials o he exen possible in he fabricaion of new producs (Guide e al., 2000). These sysems are capable of dealing wih produc reurns via several produc recovery opions, e.g. direc reuse, repair, refurbishing, remanufacuring, cannibalizaion and recycling (Thierry e al., 1995). We consider here a periodically-reviewed recoverable sysem where remanufacuring is used as he recovery opion. In addiion o newly produced iems, remanufacured iems as well can be used o mee cusomer demands. Remanufacured iems are considered o have he same qualiy as he new producs and are sold for he same price, bu hey are less cosly. Two main approaches are observed in he lieraure regarding invenory managemen of sochasic recovery models. One approach, which has been rarely used, is o analyically invesigae he srucure of opimal conrol policies using dynamic programming approaches (e.g., Simpson, 1978; Inderfurh, 1997). A second approach is o find opimal or near-opimal values for he parameers of a predeermined reasonable, bu no necessarily opimal, conrol policy srucure, benefiing from he resuls of queuing heory, Markov decision processes, enumeraion echniques, simulaion or heurisics (e.g. Van der Laan e al. 1996; Van der Laan and Salomon, 1997; Van der Laan e al. 1999; Kiesmuller, 2003; Kiesmuller and Minner, 2003; Kiesmuller and Scherer, 2003; Mahadevan e al., 2003; Van der Laan and Teuner, 2006). The laer approach has been widely used for boh periodically and coninuously reviewed recoverable invenory sysems due o is simpliciy and is applicabiliy o more comprehensive, larger-scale models wih relaxed assumpions compared o models o which he former approach has been applied. However, i has he drawback of considering a pre-deermined policy ha is no guaraneed o be opimal. In hese works, i is no indicaed how far he predeermined policy is from he opimal policy. Periodic review models describe pracical siuaions in a more suiable way, however only a few papers can be found in he lieraure which considered periodic review models 15

30 in he conex of produc recovery. Furher, he majoriy of hese invesigae he opimal parameer values for a pre-deermined policy raher han finding he opimal policy srucure iself for he sysem under consideraion (Kiesmuller, 2003; Kiesmuller and Minner, 2003; Mahadevan e al., 2003) To our knowledge, only a couple of papers in he lieraure emphasize he generaion of opimal conrol policy srucures for a one produc recoverable sysem. Simpson (1978) generaes he opimal policy srucure for a finiehorizon repairable invenory problem under no se up coss and zero lead ime assumpion for repair and purchasing aciviies. He uses a backward dynamic programming echnique. Inderfurh (1997) addresses he problem of invenory opimizaion for a recoverable sysem wih and wihou sock keeping of reurned iems for which he uses he mehod of sochasic dynamic programming o derive opimal decision rules for procuremen, remanufacuring and disposal. For he case where recoverable iems are socked, he provides he opimal policy srucure only for he case of equal lead imes and no se up coss for manufacuring and remanufacuring. He indicaes ha if fixed coss exis, he policies provided may no longer be opimal. To our knowledge, no work exiss in he lieraure which conducs an analysis o find opimal policy srucure in he exisence of fixed coss for manufacuring and/or remanufacuring in he conex of periodic-review invenory conrol. Furher, none of he work ha uses a pre-deermined policy srucure for invenory opimizaion indicaes how well he policy srucure hey consider characerizes he opimal invenory conrol policy. I is very imporan o know abou he srucure of he opimal invenory policy, because mos real life invenory problems are large-scale problems for which i migh no be compuaionally possible o deermine he opimal invenory policy. However, if he srucure of he opimal policy is more or less known beforehand, hen he opimal or near-opimal values of is parameers can be found hrough several ools ha can rea large-scale problems, e.g. simulaion, heurisics, some enumeraive search echniques, ec. Anoher advanage of he characerizaion of he opimal policy (i.e. defining he opimal policy in a srucured way using a few conrol parameers) is ha i enables a clear inerpreaion of he opimal policy. For insance, he effecs of changes in sysem parameers on he opimal invenory policy can be clearly seen hrough characerizaion. 16

31 This chaper considers invenory opimizaion problem of a periodically reviewed single produc sochasic manufacuring/remanufacuring sysem wih wo socking poins: recoverable invenory and serviceable invenory. Lead imes and se up coss for manufacuring and remanufacuring are considered. The sysem is modeled using Markov Decision Process, and an empirical sudy is conduced o deermine opimal or near opimal policy characerizaions under several cos configuraions and differen lead ime cases for manufacuring and remanufacuring. Through hese policy characerizaions, how he opimal policy srucure and/or policy parameer values change as several cos parameers including se up coss change is invesigaed. The performance of he policy characerizaions provided under several cos configuraions is evaluaed numerically considering he percenage deviaion of heir cos from he opimal cos. Resuls indicae ha he exisence of se up cos for eiher manufacuring or remanufacuring has a significan effec on policy srucure, and he policy characerizaions we provide represen he opimal policies well wih a maximum deviaion of 1% from opimal cos in almos all cases. This chaper is organized as follows: In secion 3.1, he recoverable manufacuring sysem under consideraion is presened and he invenory opimizaion problem is formulaed as a discree Markov Decision Process under differen cases for manufacuring and remanufacuring lead imes. In secion 3.2, he experimens o be performed are described. In secion 3.3, he effecs of a change in cos parameers of he sysem on he opimal invenory policies are invesigaed hrough policy characerizaions. Secions 3.3.1, and include he opimal or near-opimal policy characerizaions deermined under several cos configuraions for he hree lead ime cases considered, i.e. he case where manufacuring and remanufacuring lead imes equal one period; he case where remanufacuring lead ime is wo periods and manufacuring lead ime is one period; and he case where manufacuring lead ime is wo periods and remanufacuring lead ime is one period, respecively. Furher, in secion 3.4, sensiiviy analysis is provided regarding he effec of changing he coefficien of variaion of he demand disribuion on he policy srucure or policy parameer values. Finally, concluding remarks and suggesions for furher research are given in secion