Study on Multi-objective Optimization Model of Inventory Control and Supplier Selection Problem Under Uncertainty

Transcription

1 Study on Mult-objectve Optmzaton Model of Inventory Control and Suppler Selecton Problem Under Uncertanty June, D O C T OR OF E N G INEERING D c k y F a t r a s T O Y O HASHI U N I VERSITY OF TECHNOLOGY

2 Study on Mult-objectve Optmzaton Model of Inventory Control and Suppler Selecton Problem Under Uncertanty Doctoral Thess Submtted as a partal fulfllment of Requrement for The Award of Doctor of Engneerng Dcky Fatras d Approved by Supervsor, Professor Yoshak Shmzu

3 year month day Dept. Electronc and Informaton Engneerng ID d Name Dcky Fatras Supervsor Professor Yoshak Shmzu Abstract Ttle Study on Mult-objectve Optmzaton Model of Inventory Control and Suppler Selecton Problem under Uncertanty (800 words) Most companes are now facng dynamc challenges that requre not only well-plannng capacty, but also robust supply chan networks (SCN) that allow the members nvolved to address and respond any changes n a short notce. In partcular, when nventory s stuck n the varous stages of the supply chan, the company may be forced to operate at crtcal cash flow levels. On the other hand, among the varous actvtes nvolved n SCN, purchasng s one of the most strategc functons because t provdes opportuntes to reduce costs across the entre supply chan. An essental task wthn the purchasng functon s suppler selecton. Ths comes from the fact that the cost of raw materals and component parts represents the largest percentage of the total product cost n most ndustres. From ths pont of vew, ths thess addresses ssues assocated wth nventory and suppler selecton problem. We study both ssues under uncertan envronment and consder such problem usng ether stochastc approach or fuzzy approach. At the frst part, we studes mult-objectve problem of perodc revew nventory n two-echelon supply chan system under uncertanty n demand and lead tme. We propose dfferent strateges to solve the stock-out problem n seral replenshment system whch requres a hgher level of coordnaton. Whle stochastc approach s utlzed to tackle the uncertanty, the mult-objectve Dfferental Evoluton (DE) s appled after gvng ts new algorthm to work wth the problem. We reveal that the coordnaton strategy becomes more effectve as the uncertanty ncreases n the system. Though retalers are requred to keep a bt hgh nventory level to mantan a hgh responsveness, ths stock level s more effectve to reduce the loss rate of supply chan. At the second part, we studed mult-objectve suppler selecton by consderng both qualtatve and quanttatve crtera. The fuzzy approach s appled due to the fact that most nformaton requred to assess suppler s not always avalable and/or usually not known precsely over the plannng horzon. Concernng such characterstcs, ths research develops an ntegrated methodology of fuzzy mult-objectve lnear programmng model for suppler selecton. To mprove the methodologcal process of dervng optmal soluton, the enhanced two-phase fuzzy programmng model has been proposed n ths study. Through numercal experment, we show some advantages of our proposed approach over the exstng methods n provdng a set of potental feasble solutons whch gude DMs to select the best soluton accordng to ther preference. At last part, we present mult-objectve possblstc mxed nteger programmng (MOPMIP) model of perodc revew nventory problem n mult-manufacturer mult-retaler SCN. We attempt to develop a mult-objectve model n a mxed mprecse and/or uncertan envronment by ncorporatng the fuzzness of demand, lead tme and cost parameters. A soluton procedure s developed usng the Torab and Hassn method to solve the model and to provde a systematc framework that facltates the fuzzy decson-makng process whle enablng the DM to adjust the decson and obtanng a more preferred satsfactory soluton. The proposed soluton procedures obtan a promsng result whch produces more balanced feasble solutons and provde decson support to dentfy crtcal objectve n both decentralzed and centralzed SCN. Moreover, as a supplementary consderaton, we consder daly plannng for three echelon SCN consderng nventory condtons. To cope wth the problem, we appld a practcal hybrd meta-heurstc method that supports decson makng at a tactcal and operatonal level. Its soluton procedure s developed by means of two modfed heurstc methods known as the savng method and tabu search together wth the graph algorthm of mnmum cost flow problem. Fnally, to enhance usablty of the method, vsualzaton of result s also realzed by vrtue of Google map API. Numercal experments are carred out to valdate effectveness of the proposed approach.

4 Thess Commttee Professor Yoshak Shmzu Department of Mechancal Engneerng Toyohash Unversty of Technology Professor Takao Fujwara Department of Archtecture and Cvl Engneerng Toyohash Unversty of Technology Assocate Professor Naok Uchyama Department of Mechancal Engneerng Toyohash Unversty of Technology Assocate Professor Rafael Batres Department of Mechancal Engneerng Toyohash Unversty of Technology

5 Acknowledgements My grattude s greatly expressed to the almghty Allah SWT for havng blessed my lfe. The prnted pages of ths thess hold far more than the culmnaton of three years of my study. I could not have come ths far wthout the assstance of many generous and nsprng people I have met snce begnnng my study and I want to express my deepest apprecaton to them. Frst of all, I would lke to express my deepest grateful to my supervsor, Professor Yoshak Shmzu, for hs excellent gudance, carng, patence, and provdng me wth an excellent atmosphere for dong research. Wthout hs constant trust and gentle proddng, my research would not have been completed. The good advce and support of Assocate Professor Rafael Batres has been nvaluable on both my academc and a personal level, for whch I am extremely grateful. I wsh to thank Dr. Tatsuhko Sakaguch and the secretary of ndustral systems engneerng laboratory, Ms. Nakao Yoshko. Ther gudance has served me well and I owe them my heartfelt apprecaton. I would also lke to thank my entre fellow n laboratory for ther help and cooperaton. Ther assstance has meant more to me than I could ever express. Specal acknowledgement to my parent and wfe, who have always beleved n me and helped me reach my goals. Ther support

6 forged my desre to acheve all that I could n lfe. I owe them everythng and wsh I could show them just how much I love and apprecate them. Lastly, I take ths opportunty to sncerely acknowledge the Mnstry of Educaton and Culture on behalf of the Indonesan government for provdng fnancal assstance of ths study. v

7 Contents Abstract... Thess Commttee... Acknowledgements... Contents... v Lst of Fgures... x Lst of Tables... x Chapter 1 : Introducton Background of Research Research Scope and Objectves Overvew of Thess... 5 Chapter 2 : Lterature Revew Supply Chan Concept Defnton of Supply chan Supply Chan Process Supply Chan System Uncertanty n Supply chan System Type of uncertanty Modelng uncertanty n Supply chan System Inventory n the Supply Chan System Inventory n the Sngle Dyadc Supply Chan Inventory n the Supply Chan network Supplers Selecton v

8 Evaluaton Crtera Evaluaton Model Mult-objectve Optmzaton Mult-objectve Optmzaton Problem Mult-objectve versus Sngle-objectve Optmzaton Search and Decson makng Concept of Domnaton Chapter 3 : Mult-objectve Analyss of Perodc Revew Inventory Problem wth Coordnated Replenshment n Twoechelon Supply Chan System through Dfferental Evoluton Introducton System Descrpton Replenshment Strateges Strategy Strategy Strategy Mult-objectve Dfferental Evoluton Numercal Experment Codng and Settng of Lower/upper Bound Result and Analyss Concluson Chapter 4 : An enhanced Two-phase Fuzzy Programmng Model for Mult-objectve Suppler Selecton Problem Introducton Problem Formulaton v

9 4.3. The Proposed Integrated Method Taguch Loss Functon Analytcal Herarchy Process Fuzzy Mult-objectve Lnear Programmng An Enhanced Two-phase Fuzzy Programmng method Soluton Procedures Numercal Example and Analyss Concluson Chapter 5 : Possblstc Programmng Model for Fuzzy Multobjectve Inventory problem n Two-echelon Supply Chan System Introducton Model Formulaton System Descrpton Mathematcal Formulaton The Objectve Functons Soluton Methodology The Equvalent Auxlary MOPMIP model The modfed S-curve Membershp Functon Torab and Hassn Fuzzy Aggregaton Functon The Integrated Soluton Procedures Computatonal Experment and Analyss Concluson Appendx Chapter 6 : Concluson Concludng Remark v

10 6.2. Future Research References Appendx : Daly plannng for three echelon logstcs Consderng Inventory Condton Lst of Publcaton v

11 Lst of Fgures Fgure 2. 1 : Supply Chan... 7 Fgure 2. 2 : Supply Chan Process Fgure 2. 3 : Level of Supply Chan Fgure 2. 4 : Optmal Soluton n Decson Space Fgure 2. 5 : Idea of Soluton Procedure n Objectve Space Fgure 2. 6 : Illustraton of Domnaton Concept Fgure 3. 1 : System Confguraton wth n-seral lnes Fgure 3. 2 : Replenshment Strategy Fgure 3. 3 : Replenshment Strategy Fgure 3. 4 : Pseudo code of DE (Strategy: DE/rand/1/bn) Fgure 3. 5 : Flowchart of the proposed MODE Fgure 3. 6 : Normalzed Pareto Optmal Soluton Fgure 3. 7 : Comparson of Backorder cost of Manufacturer Fgure 3. 8 : Comparson of Holdng cost of Manufacturer Fgure 3. 9 : Comparson of Holdng cost of Retaler Fgure : Comparson of Shortage cost of Retaler Fgure 5. 1 : Supply Chan network Fgure 5. 2 : Modfed S-Curve Membershp Functon Fgure 5. 3 : Flowchart of Soluton Procedures Fgure 5. 4 : Holdng cost of Manufacturers Fgure A. 1 : Global dea of DSS on logstcs plannng Fgure A. 2 : Flowchart od Soluton Procedures Fgure A. 3 : Example of MCF graph x

12 Fgure A. 4 : Varatons of demand Fgure A. 5 : Varatons of nventory Fgure A. 6 : Trend of Optmal Cost Fgure A. 7 : Varatons of number of workng depot Fgure A. 8 : Workng rate of each RS Fgure A. 9 : CPU tme vs. Problem sze Fgure A. 10 : Soluton Convergence Fgure A. 11 : Total cost wth demand devaton range and orderng pont Fgure A. 12 : Inventory cost wth demand devaton range and orderng pont Fgure A. 13 : A part of dsplay of result (Route from Depot 1 (Popup mark L)) x

13 Lst of Tables Table 2. 1 : Dckson 23 crtera for suppler selecton Table 3. 1 : Parameters value for Numercal Experment Table 3. 2 : Objectve Functon Values and Decson Varable of Selected Pareto Soluton Table 4. 1 : Supplers quanttatve nformaton Table 4. 2 : The specfcaton lmt and range of fve leadng crtera Table 4. 3 : Actual value of the four sub-crtera Table 4. 4 : Taguch Loss Table 4. 5 : Par-wse comparson matrx Table 4. 6 : Comparson of Max-mn operator and the proposed approach Table 5. 1 : Input parameters Table 5. 2 : Cost parameters Table 5. 3 : Summary of test result Table 5. 4 : Result obtaned for some weght combnatons Table A. 1 : Labelng on the edge for MCF graph Table A. 2 : Notes on parameter setup x

14 Chapter 1 INTRODUCTION 1.1. Background In today s compettve envronment characterzed by low proft margns and hgh responsveness to consumer demand for hgh qualty products and shorter lead-tmes, companes are forced to manage ther compettve advantage and opportunty to optmze ther busness processes. Manufacturng enterprses along wth ther dstrbuton systems face dynamc challenges that requre not only wellplannng capacty, but also robust SC networks wth coordnaton mechansms that allow the members nvolved n such networks to address and respond to any changes n a short notce. From radcal volume changes n customer demand, to varatons n prces of raw materal and fnshed products due to currency fluctuatons n the global marketplace, to ncreases n transportaton costs due to speculaton n the prce of crude ol, any number of factors can have a serous effect on corporate revenue projectons. In partcular, when nventory s stuck n the varous stages of the supply chan, the com- 1

15 pany may be forced to operate at crtcal cash flow levels. As known, hgh nventory levels ncrease the responsveness of the supply chan but decrease ts cost effcency because of the cost of holdng nventory. In ths stuaton, the ablty to ntellgently address crtcal ssues through effectve supply chan nventory and dstrbuton strateges not only keeps the wheels of busness turnng, but t also gves the company a relatve advantage over ts compettors n beng able to address suppler concerns and consumer needs n a way that slower, less agle manufacturers are unable to do. Hence, a relevant problem n SC network s to determne the approprate levels of nventory at the varous stages nvolved n a supply chan. Of the varous actvtes nvolved n SC network, purchasng s one of the most strategc functons because t provdes opportuntes to reduce costs across the entre supply chan. An essental task wthn the purchasng functon s suppler selecton, gven that the cost of raw materals and component parts represents the largest percentage of the total product cost n most ndustres. For nstance, n hgh technology frms, purchased materals and servces account for up to 80% of the total product cost [1]. In contemporary supply chan management, companes mantan long term partnershp wth supplers, and use fewer, but more relable supplers. The performance of potental supplers s evaluated aganst multple crtera rather than consderng a sngle factor [2]. Gven the prevalence of both nventory decsons and suppler selecton n a supply chan, ths study addresses both problems separately by studyng a mult-objectve nventory control problem and suppler selecton problem n a typcal two-echelon SC under uncertanty. 2

16 1.2. Research Scope and Objectves Investment n nventory should nether be excessve nor nadequate. It should just be optmum. Mantanng optmum level of nventory s the man am of nventory control system. Excessve nvestment n nventory slps the economcal effcency. At the same tme, nsuffcent nvestment n nventory creates stock-out problems, nterrupton n producton and sellng operaton. Therefore, the frm may loose the customers as they shft to the compettors. Whle nventory has always been mportant, establshng and mantanng long term partnershp wth supplers has also become mportant over the past several decades. Selectng the rght suppler not only gves the companes the opportunty to reduce materal cost, but also helps the companes to receve necessary products n a tmely and effectve manner to help mantan compettve advantage. Based on the above descrpton, ths research addresses nventory control and suppler selecton problem as follows: In the frst part, we develop mult-objectve nventory control problem n two echelon supply chan under uncertanty n demand and lead tme and proposes two nventory replenshment strateges, besde conventonal strategy, wth alternatve supply possbltes to fulfll the shortage whch nvolve dfferent level of coordnaton mechansm between manufacturers and retalers. A mult-objectve Dfferental Evoluton (MODE) s ntroduced to solve the problem and mult-objectve analyss s carred out to evaluate the performance of the system by smultaneously mnmzng total cost and loss rate of the supply chan. The am s to examne the stuaton when such coordnaton s proftable for all members n the system. 3

17 In the second part, we study a mult-objectve suppler selecton to determne the order quantty n a mult-sourcng envronment. Specfcally, ths research focuses on fuzzy mult-objectve lnear programmng (fuzzy MOLP) for solvng suppler selecton problem. Both quanttatve and qualtatve crtera are consdered durng the selecton process and the fuzzy approach s utlzed to cope wth uncertanty ssue. Ths research ntroduces an enhanced two-phase fuzzy MOLP to help obtan more reasonable compromse solutons. In the thrd part, a perodc revew nventory model n a typcal SC system wth multple-manufacturer, multple-retaler s consdered. The am of ths paper s to develop a mult-objectve perodc revew nventory model n uncertan envronment by smultaneously ncorporatng uncertanty n crtcal parameters. Unlke the research n the frst part, whch tackles the uncertanty usng a stochastc approach, ths research apples a fuzzy approach n codng the mprecse nature of demand, lead tme and cost parameters. The problem s to determne orderng polcy of raw materal and the safety stock level of each manufacturer, and the order allocaton and target stock level of each retaler that yeld satsfactory mnmum total cost whle mantanng low loss rates. Several alternatve solutons are obtaned by solvng a proposed mult-objectve possblstc mxed nteger programmng (MOPMIP) nventory model. Towards future real world applcatons, t s of specal mportance to concern operatonal problems besdes strategc ones consdered n the above. Notcng such crcumstance, and to enrch the prospects of ths thess, we also present a daly optmzaton of three echelon logstcs as a supplemental concern. Developng a hybrd approach for effcent soluton, we analyze an ssue of nventory assocates wth demand devatons. 4

18 In detal, the man objectves of the proposed research are summarzed as follow: 1. Develop a mult-objectve nventory control model for two echelon supply chan under uncertanty utlzng three dfferent scenaros of replenshment to fulfll the shortage and apply multobjectve Dfferental Evoluton to yeld several compromsed solutons. 2. Develop soluton procedures by enhancng a two-phase approach to solve the mult-objectve suppler selecton n a fuzzy envronment ncorporatng both quanttatve and qualtatve crtera smultaneously. 3. Propose an ntegrated soluton procedure of mult-objectve possblstc mxed nteger programmng (MOPMIP) nventory model to facltate the fuzzy decson-makng process n a multmanufacturer mult retaler supply chan Overvew of Thess Ths thess s composed of seven chapters. The frst chapter descrbes the ntroducton. It ncludes background, objectves of the thess and overvew of the thess. Chapter two s concerned wth lterature revew. Ths revew ncludes the supply chan concepts, uncertanty n supply chan, nventory n supply chan, suppler selecton, and the concept of mult-objectve optmzaton. Chapter three presents a mult-objectve Analyss of Perodc Revew Inventory Problem wth Coordnated Replenshment n Two-echelon Supply Chan System. Chapter four studes an enhanced two-phase fuzzy programmng model for mult-objectve suppler selecton problem. Chapter fve focuses on the applcaton of possblstc programmng 5

19 to solve fuzzy mult-objectve nventory problem n two-echelon supply chan system. The concluson and recommendaton for further study are brefly dscussed n Chapter sx. Lastly, we present a herarchcal approach to optmze a daly logstcs problem assocated wth nventory control aganst demand devatons n Appendx secton. 6

20 Chapter 2 LITERATURE REVIEW 2.1. Supply Chan The study about supply chans has grown very fast durng ths decade. Even so, there s no explct descrpton of supply chan management or ts actvtes n the lterature [3]. Each research defnes the term supply chan, ts processes and the complextes of the supply chan n dfferent ways. The lterature n ths secton starts by gvng the defnton of the supply chan and supply chan management and then descrbes the man process and the levels of the supply chan system Defnton of Supply chan Supply chans represent a coordnated network of frms nteractng to provde a product or servce to the end customer. They operate across functons wthn organzatons, company boundares, and natonal borders. A dagram llustratng the basc components of a supply chan s presented n Fgure 2.1. Most of the supply chans have three basc components: supplers, producers, and customer. 7

21 Suppler (source of supply) Producers Dstrbutor Retaler Customer (source of demand) Flow of products and servces Flow of demand and nformaton Flow of cash payment Fgure 2.1. Supply chan Supply chans often contan dstrbutors and retalers along wth servce and support functons. The components of the supply chan must nteract n a coordnated manner to acheve the ultmate goals: the delvery of goods and servces resultng n the creaton of customer satsfacton. Products and servces generally flow from sources of supply to sources of demand, whle nformaton and cash payments generally flow n the reverse drecton [4]. For many reasons, an nterest n logstc and supply chan management has grown explosvely n the last few years. Supply chan management s consdered to be an extenson of tradtonal logstcs. Whereas logstcs nvestgates the flow of nformaton, materals, captal and manpower n the nternal supply chan owned by a sngle frm; supply chan management deals wth the coordnaton of logstc processes wth n the external supply chan. The man goal of both tradtonal logstcs and supply chan management s to delver superor customer value at less cost to the supply chan as a whole. Unlke tradtonal logstcs, supply chan management nvolves the coordnaton of ndependently managed companes who seek to maxmze ther own profts. Although overall performance of the supply chan depends on the company s jont performance, the operatonal goals may conflct and result n neffcency for the entre chan. Therefore, one of the man ssues n supply chan management 8

22 s to fnd sutable mechansms for coordnatng the logstcal processes that are controlled by varous ndependent companes n order to acheve overall mnmal cost and maxmum proft. Other nterestng defntons of supply chan management are descrbed by the followng research. Smch-Lev et al. [5] defned supply chan management as a set of approaches utlzed to effcently ntegrate supplers, manufacturers, warehouses, and stores, so that merchandse s produced and dstrbuted at the rght quanttes, to the rght locatons, and at the rght tme, n order to mnmze system wde costs whle satsfyng servce level requrements. Scott and Westbrook [6] and New and Payne [7] descrbed supply chan management as a chan lnkng each element of the manufacturng and supply process from raw materals through to the end customer, encompassng several organzatonal boundares. Accordng to ths broad defnton, supply chan management encompasses the entre value of the chan and addresses materals and supply management from the extracton of raw materals untl the end of ts useful lfe. Supply chan management focuses on how frms utlze ther suppler s processes, technology, and capablty to enhance compettve advantage and the coordnaton of the manufacturng, logstcs, and materals management functons wthn and organzaton. When all strategc organzatons n the value chan ntegrate and act as a sngle unfed entty, performance s enhanced throughout the system of supplers Supply Chan Process As brefly descrbed earler, a supply chan s an ntegrated manufacturng process wheren raw materals are converted nto fnal products and then delvered to the customer. A supply chan s 9

23 comprsed of two basc, ntegrated processes: 1) the producton plannng and nventory control process, and 2) the dstrbuton and logstcs process. 1. The producton plannng and control process encompasses manufacturng and storage sub-processes, and ther nterface. In detal, producton plannng descrbes the desgn and management of the entre manufacturng process, ncludng raw materal schedulng and acquston, manufacturng process desgn and schedulng and materal handlng desgn and control. Inventory control descrbes the desgn and management of storage polces and procedures for raw materals, work-n-process and fnal products nventory. 2. The dstrbuton and logstcs process determnes how products are retreved and transported from the warehouse to the retaler. These products may be transported to retalers drectly, or may frst be moved to dstrbuton facltes, whch, n turn, transport products to the retalers. Ths process ncludes the management of nventory retreval, transportaton, and fnal product delvery. Both processes nteract wth one another to produce an ntegrated supply chan [8]. These processes are llustrated n Fgure 2.2, provdng the basc framework for the converson and movement of raw materals nto fnal products. Suppler Dstrbuton Center Manufacturng Faclty Storage Faclty Transp. Vehcles Retaler Producton Plannng & Inventory Control Dstrbuton and logstcs Fgure 2.2. Supply chan process 10

24 Supply Chan System The supply chan system as descrbed by Harland [9] can be dvded nto 4 levels accordng to the scope of study of each researcher so that the term supply chan represents dfferent concepts relatng to dfferent spans of nfluence. The levels of supply chan system are llustrated n Fgure Internal chan: the frst level s the nternal chan, whch refers to the processes nsde a manufacturng organzaton startng from orderng and recevng materals through the transformaton processes of producton, to the dspatch and physcal dstrbuton of the product to the customers. 2. Dyadc Relatonshp: the second level of supply chan system s called dyadc relatonshp, and s concerned wth the relatonshp of two echelons n the supply chan, ncludng the flow of materal and nformaton, purchasng procurement, stock level etc. Most of the research n a supply chan system focuses on ths level. 3. External chan: the thrd level n a supply chan system s called an external chan. Ths vews supply chans more holstcally as the total chan of exchange from orgnal source of raw materal, Level 1 : Internal Chan Level 2 : Dyadc Relaton Shp Level 3 : External Chan Level 4 : Network Fgure 2.3. Levels of the supply chan 11

25 through the varous frms nvolved n extractng and processng raw materals, manufacturng, assemblng, dstrbutng and retalng to ultmate end customers. 4. Network level: ths level has undergone ncreased attenton on the total focal frm network, not only concerned n a sngle chan. For example, t s the network of relatonshps that a frm has wth ts supplers, ts suppler s suppler and so on n upstream, and ts customers and ts customer s customer and so on n downstream Uncertanty n the supply chan system Uncertanty rules the supply chan. Uncertantes n supply, process and demand are recognzed to have a major mpact on the supply chan. Uncertanty propagates throughout the network and leads to neffcent processng and non-value addng actvtes. Sales devate from forecast, components are damaged n transt, fabrcaton yelds fal to meet plan, shpments are held up n customs, are the most common events as drect results of uncertanty Types of uncertanty Supply chan uncertanty can be classfed nto four general types: process, supply, demand, and control uncertanty. Below s the descrpton of each type of uncertanty as cted from Geary et al. [10]. Process uncertanty. Process uncertanty affects a company s nternal ablty to meet a producton delvery target. The amount of process uncertanty can be establshed by understandng each work process s yeld ratos and lead tme estmates for operatons. Also, f the partcular product delvery process s competng aganst other 12

26 value streams for resources, then the nteracton between these must be studed and codfed. Supply uncertanty. Supply uncertanty results from poorly performng supplers due to ther nablty to meet company s requrements and thereby handcappng value-added processes. It can be evaluated by lookng at suppler delvery performance, tme seres of orders placed or call-offs and delveres from customers, actual lead tmes, suppler qualty reports, and raw materal stock tme seres. Demand uncertanty. Demand uncertanty can be thought of as the dfference between the actual end-marketplace demand and the orders placed wth a company by ts customers. Demand uncertanty can also be quantfed by measurng how well companes meet customer demand. For example, poor on-tme delvery or fll rates are often a result of demand uncertanty, though ths s not always the case. If a customer suddenly places a weekly order that s twce the typcal order sze, t may be the result of a shft n underlyng demand or t may just be that the customer has modfed safety stocks or orderng rules. Control uncertanty. Control uncertanty s assocated wth nformaton flow and the way an organzaton transforms customer orders nto producton targets and suppler raw materal requests. The level of control uncertanty can be determned by comparng customer requrements, suppler requests to delver, and producton targets over the same tme perods. Control uncertanty s drven by the algorthms an control systems that are used to transfer the customer orders nto producton targets and suppler raw materal requests. In a pure demand-pull envronment, the lnkage between supply and demand s clear and control uncertanty s elmnated. However, companes typcally use order batchng and lot szng, 13

27 whch obscures the lnkage between demands placed and true requrements. Each of these uncertantes creates a drag on operatonal performance. However, supply chan professonals are often so busy dealng wth the fallout from uncertanty (such as stock-outs, mssed shpments, and oversupply) that they do not have tme to attack the root cause of the problem. The ssue has been complcated even further over the course of the last decade by the movement away from the vertcally ntegrated supply chan. Now, rather than confrontng the uncertanty generated just by actvtes wthn the operatonal doman of a sngle organzaton, we must manage uncertanty across a host of supply chan partcpants. Outsourcng, the vrtual organzaton, and modular manufacturng all contrbute to supply chan uncertanty ssues. All of ths makes t more mportant to understand the relatonshp between supply chan performance and uncertanty Modelng uncertanty n the supply chan Wth the growng attenton pad n recent years to supply chan uncertanty, t s worth examnng n more detal dfferent models employed n the lteratures to cope wth uncertanty. There have been three broad phlosophes on whch several methods for optmzaton under uncertantes can be categorzed: stochastc programmng, fuzzy mathematcal programmng and chance constraned programmng [11]. In ths thess, only the frst two-approach wll be dscussed. 1. Stochastc programmng model Stochastc programmng s a typcal method of generatng an operatonal plan wthn an uncertan envronment when the precse 14

28 probablty dstrbuton of future uncertanty s known n advance. Stochastc programmng formulatons assume that the probablty dstrbutons governng the uncertan data are known or can be estmated [12]. Ths means that hstorcal record for the uncertan data s avalable, and ths data s analyzed usng statstcal approach to approxmate ts future trend. MrHassan et al. [13] consdered a two-stage model for multperod capacty plannng of supply chan networks. Here the frst stage decsons, comprsed of openngs and closngs of the plants and dstrbuton centers and settng ther capacty levels, are to be decded pror to the realzaton of future demands. Then, based upon the partcular demand scenaro realzed, the producton and dstrbuton decsons are to be decded optmally. The overall objectve s to mnmze the cost of the frst-stage strategc decsons and the expected producton and dstrbuton costs over the uncertan demand scenaros. The authors used Benders decomposton to solve the resultng stochastc nteger program, and presented computatonal results on supply chan networks nvolvng up to 8 plant stes, 15 dstrbuton centers, 30 customer locatons, and wth 100 scenaros. Georgads et al. [14] consdered a two-stage stochastc programmng model for supply chan network desgn under demand uncertanty. The authors developed a large-scale mxed-nteger lnear programmng model for ths problem, and presented a case study usng a European supply chan network nvolvng 14 products, 18 customer locatons, 6 dstrbuton center locatons, and 3 demand scenaros. Alonso-Ayuso et al. [15] proposed a branch-and-fx heurstc for solvng two-stage stochastc supply chan desgn problems. Com- 15

29 putatonal results on networks nvolvng 6 plants, 12 products, 24 markets, and 23 scenaros were presented. 2. Fuzzy programmng model Fuzzy mathematcal programmng s prolferated by Zmmermann [16, 17] by formulatng mathematcal programmng model that takes nto account (a) the decson maker's expectatons of a target range of objectve values and (b) soft constrants based on decson makng n a fuzzy envronment. In ths approach, uncertan parameters are treated as fuzzy numbers and constrants are treated as fuzzy sets. The degree of satsfacton of a constrant s defned n terms of a normalzed membershp functon of the constrant and a small extent of constrant volaton s allowed whle objectve functons may be ether a fuzzy goal or a crsp functon. The advantage of fuzzy approach over the other two approaches mentoned earler s that fuzzy approach nether assumes that the uncertan parameters have to follow any statstcal dstrbuton nor allows the fnal determnstc equvalent formulaton of the uncertan model to blow up n sze wth ncrease n number of uncertan parameters. The dsadvantage of the fuzzy approach les n ts nablty to represent the exact nature of the uncertanty and the results could depend on the fuzzfcaton approach. Mtra et al. [11] formulated the md-term plannng problem n a mult-ste, mult-product, mult-perod supply chan under uncertanty usng the fuzzy mathematcal programmng approach. They demonstrate that the fuzzy approach s qute generc, relatvely smple to use and can be adapted for bgger sze plannng problems, as the equvalent determnstc problem does not blow up n sze wth 16

30 ncrease n number of uncertan parameters whle usng fuzzy approach. Lu and Sahnds [18] solved a sngle objectve capacty plannng problem under uncertanty usng fuzzy and two stage stochastc programmng approaches and compared the performance of the two. Selm et al. [19] used fuzzy goal programmng approaches n handlng the mult-objectve collaboratve producton dstrbuton plannng problems n both centralzed and decentralzed supply chan desgn structures to make a comparatve analyss between them. Chen et al. [20] studed a mult-objectve supply chan desgn problem for locatng warehouses usng two-phase fuzzy approach where they have used scenaro-based approach to represent uncertanty n demand. Lang [21] developed an nteractve fuzzy multobjectve lnear programmng method to smultaneously mnmze the total dstrbuton costs and the total delvery tme wth reference to fuzzy avalable supply and total budget at each source, and fuzzy forecast demand and maxmum warehouse space at each destnaton Inventory n the Supply Chan System Inventory s the stock of any resource used n a company. An nventory system s the set of polces and controls that montor levels of nventory and determne what levels should be mantaned, when stock should be replenshed, and how large orders should be. By conventon, manufacturng nventory generally refers to tems that contrbute to or become part of a company s product output. Manufacturng nventory s typcally classfed nto raw materals, fnshed products, component parts, supples, and work-nprocess. In dstrbuton, nventory s classfed as n-transt, meanng 17

31 that t s beng moved n the system, and warehouse, whch s nventory n a warehouse or dstrbuton center. Retal stes carry nventory for mmedate sale to customers. In servces, nventory generally refers to the tangble goods to be sold and the supples necessary to admnster the servce. The basc purposes of nventory analyss are to specfy (1) when tems should be ordered and (2) how large the order should be. Many frms are tendng to enter nto longer-term relatonshps wth vendors to supply ther needs for perhaps the entre year. Ths changes the when and how many to order to when and how many to delver. Inventory decsons nvolve hgh rsk and hgh mpact for the supply chan management. Inventory commtted to support future sales, drves a number of antcpatory supply chan actvtes. Wthout a proper nventory assortment lost sales and customer dssatsfacton may occur. Lkewse, nventory plannng s crtcal to manufacturng. Materal or component shortages can shut down a manufacturng lne or force modfcaton of a producton schedule, whch creates added cost and potental fnshed goods shortage. Just as a shortage can dsrupt planned marketng and manufacturng operatons, nventory overstocks also create operatng problems. Overstocks ncrease cost and reduce proftablty as result of ncreasng warehousng, workng captal, nsurance, taxes, and obsolescence. Ths secton revews the nventory management accordng to dfferent confguraton structure of supply chan as follow: Inventory n the Dyadc Supply Chan Newhart et al. [22] desgned an optmal supply chan usng a two- phase approach. The frst phase combnes mathematcal pro- 18

32 gram and heurstc model wth the objectve of mnmzng the number of dstnct product types held n nventory throughout the supply chan. The second phase s a spreadsheet based nventory model, whch determnes the mnmum amount of safety stock requred to absorb demand and lead tme fluctuatons. Pyke and Cohen [23] developed a heurstc algorthm for determnng reorder at warehousng and orders up to levels S for retaler. The system s a three level supply chan, consstng of one product, one manufacturer, one warehousng and one retaler. Warehouse uses (Qn, R) system where retaler uses perodc revew wth order up to level (T, S). The model mnmzes total cost, subject to a servce level constrant, and holds the set up tmes, processng tmes, and replenshment lead tmes constant. Pyke and Cohen [24] furthered ther study on ntegrated producton and dstrbuton for mult-products n the three echelons supply chan system. The system conssts of a factory, a fnshed good stockple and a sngle retaler. The fnshed good stockple uses a (Q, R) system to control nventory, factory orders at Q when stock reach reorder pont R and a retaler uses a base stock nventory wth an order up to level S or (T, S) polcy. The system allows the retaler to make a second order f shortage has occurred. They developed an algorthm to determne order up to level at the retaler, normal and expedte replenshment reorder pont and normal and expedte replenshment batch sze for each product. Altok and Ranjan [25] consdered a producton and dstrbuton system. The system experences demand for fnshed products accordng to a compound Posson process. The nventory levels for nventores are controlled accordng to a contnuous revew (R, r) nventory polcy. Backorders are allowed n ther study. 19

33 Ish et al. [26] developed a determnstc model, known as a demand process, for determnng the base stock levels and leadtmes assocated wth the lowest cost soluton and ntegrated wth the lowest cost soluton for an ntegrated supply chan on a fnte horzon. The stock levels and lead-tmes are determned n such a way as to prevent stock out, and to mnmze the amount of obsolete nventory at each stock pont. Khouja et al. [27] tred to solve the economc lot szeschedulng problem (ELSP) for mult-products system by usng Genetc Algorthm (GA). They used a GA to solve the optmal floatng cycle tme and economc lot sze or nteger multplers of a basc perod for each product. They compared the result of usng GA wth the result from usng dynamc programmng. Dynamc Programmng (DP) allows for a tme-varyng lot sze approach wth a fxed cycle tme, but GA optmzes floatng cycle tme and fxed economc lot sze for the whole cycle tme. They concluded that the resultng solutons from GA were better than those obtaned usng DP Inventory n the Supply Chan network Ganeshan [28] presented a near-optmal (s, Q) type of nventory-logstcs cost mnmzng model for a producton/dstrbuton network wth multple supplers supplyng a dstrbuton center, whch n turn dstrbutes to a large number of dentcal retalers. The decsons n the model were made through a comprehensve dstrbuton-based cost framework that ncludes the nventory, transportaton, and transt components of the supply chan. Gjerdrum et al. [29] studed the replenshment control system n a supply chan network. The agent types n ths supply chan network conssts of two factores, two warehouses, external logstc, n- 20

34 ternal logstc, spot market and transportaton. The objectve of ths supply chan network system s to reduce operatng cost, whle mantanng a hgh level of customer order fulfllment. Mranda and Garrdo [30] studed three echelon supply chan network n whch a sngle plant suppled products to mult-regonal warehouses, and dstrbuted products to mult-retalers or customers. They proposed a smultaneous approach to ncorporate nventory control decsons whch are economc order quantty and safety stock. They presented a nonlnear mxed nteger model and a heurstc soluton approach, based on Lagrangan relaxaton and the subgradent method. They found that the potental cost reducton, compared to the tradtonal approach, ncreases when the holdng cost and/or the varablty of demand are hgher. Seferls and Gannelos [31] studed the four-echelon supply chan network that conssted of two producton modes, two warehouse nodes, four dstrbuton centers and sxteen retaler nodes. The optmzaton-based control scheme amed at adjustng the decson varables n the supply chan (e.g. transportaton load, producton nventory) to satsfy the customer orders wth the least operatng cost over a specfed rollng tme horzon usng a detaled dfferent model of the system. Smulaton results exhbted good dynamc performance under both stochastc and determnaton varaton Supplers Selecton Suppler selecton s the process by whch companes dentfy, evaluate, and contract wth supplers. The suppler selecton process deploys a tremendous amount of a company s fnancal resources. In 21

35 return, companes expect sgnfcant benefts from contractng wth supplers offerng hgh value. Several factors make new supplers mportant. Frst, there may exst new supplers that are superor n some way to a company s exstng supplers. For example, a new suppler may have developed a novel producton technology or streamlned process whch allows t to sgnfcantly reduce ts producton costs relatve to predomnate producton technology or processes. Or, a new suppler may have a structural cost advantage over exstng supplers, for example, due to low labor costs or favorable mport/export regulatons n ts home country. Second, exstng supplers may go out of busness, or ther costs may be ncreasng. Thrd, the buyer may need addtonal supplers smply to drve competton, reduce supply dsrupton rsks, or meet other busness objectves such as suppler dversty. In recognton of these reasons, buyers and ther nternal customers may be oblged by company polcy to locate a mnmum number of vable, potental supplers for every product or servce procured. Fndng a vable new suppler s challengng due to the need to verfy the suppler s ablty to meet the buyer s requrements [32]. Suppler non-performance on even the most basc level, and for the most-smple commodty, can have dre consequences for the buyer. Boeng s 787 Dreamlner producton schedule was sgnfcantly affected by shortages of fasteners, essentally bolts that secure sectons of the fuselage together [33]. In consumer products many product safety ssues have been traced back to supplers falng to meet a buyer s requrements, resultng n dangerous lead pant n toys [34], unsafe car tres [35], and pet food contanng posonous chemcals [36]. 22

36 Producton delays due to parts shortages and recalls of faulty products produced by noncomplant supplers have cost buyer frms mllons of dollars through recalls, warranty costs, and assocated nventory adjustments, and have nflcted untold damage on ther reputatons and future sales potental. New Jersey based tre mporter Foregn Tre Sales traced feld falures of ts tres to an unauthorzed desgn change made by ts suppler, whose desgn engneer decded to omt gum strps, apparently unaware of ther role n preventng tread separaton [35]. A surprsed Foregn Tre Sales was forced by U.S. government authortes to recall a quarter of a mllon tres, and rsked bankruptcy as a result [37] Suppler Selecton Crtera Suppler selecton decsons are complcated by the fact that varous crtera must be consdered n the decson makng process. The analyss of crtera for selecton and measurng the performance of supplers has been the focus of many scentsts and purchasng practtoners snce 1960 s. Suppler selecton crtera for a partcular product or servce category should be defned by a cross-functonal team of representatves from dfferent sectors wthn the company. In a manufacturng company, for example, members of the team typcally would nclude representatves from purchasng, qualty, engneerng and producton. Team members should nclude personnel wth techncal/applcatons knowledge of the product or servce to be purchased, as well as members of the department that uses the purchased tem. An nterestng work, whch has been adopted as reference for the majorty of papers dealng wth suppler problem, was presented 23

37 by Dckson [38]. Dckson s study was based on questonnares sent to 273 purchasng agents and managers selected form the membershp lst of the Natonal Assocaton of Purchasng managers. The lst ncluded purchasng agents and managers from the Unted States and Canada. From the returned 170 questonnares, the total of 23 crtera was regarded to be the most consdered crtera for suppler selecton. Indeed, the 23 crtera are ranked wth respect to ther mportance observed n the begnnng of the sxtes. At the tme (1966), the most sgnfcant crtera were qualty of the product, the ontme delvery, the performance hstory of the suppler and the warranty polcy used by the suppler. The complete lst of there crtera can be found n Table 2.1. In [39], the authors present a classfcaton of all the artcles publshed snce 1966 accordng to the treated crtera. Based on 74 Table 2.1. Dckson 23 crtera for suppler selecton Rank Crtera Ratng Evaluaton 1 Qualty Extreme mportance 2 Delvery Performance hstory Warranty and clam polces Producton faclty and capacty Consderable mportance 6 Prce Techncal capablty Fnancal poston Procedural complance Communcaton system Reputaton and poston n ndustry Desre of busness Management and organzaton Operatng controls Repar servce Average mportance 16 Atttude Impresson Packagng ablty Labor relatons record Geographcal locaton Amount of past busness Tranng ads Recprocal arrangement Slght mportance 24

38 papers, they observed that prce, delvery, qualty and producton capacty and locaton are the crtera most often treated n the lterature. Overall, the 23 crtera presented by Dckson stll cover the majorty of the crtera presented n the lterature untl today. On the other hand, the evoluton of the ndustral envronment modfed the degrees of the relatve mportance of these crtera. For example, Weber [39] nssts on the hgh mportance of the geographcal poston of the suppler n Just-n-Tme envronment, whereas ths crteron appeared n the 20 th poston n Also, the crterons n the 10 th, 12 th, 13 th poston (communcaton system, desre of busness, management and organzaton), of the Dckson s study, are very mportant for the actual ndustral envronment. Indeed, the actual stuaton requres a perfect coordnaton and a durable cooperaton between varous actors of the supply chan Suppler Selecton Methods Several well-known suppler selecton methods have been developed and classfed by numerous scholars over the years. Certan methods have been popular selecton choces for years, whle other methods have only emerged recently. As cted from Ho et al. [40], the classfcaton of suppler selecton methods are descrbed brefly n as follow. 1. Data envelopment analyss (DEA) Bragla and Petron [41] appled DEA to measure the effcences of alternatve supplers. Nne evaluatng factors were proposed to measure each suppler ratng. To avod selectng a sub-optmal or false postve suppler, both cross-effcency and Maverck ndex were measured. 25

39 Narasmhan et al. [42] appled DEA model to evaluate alternatve supplers for a multnatonal corporaton n the telecommuncatons ndustry. Eleven evaluatng factors were consdered n the model, n whch there are sx nputs related to the suppler capablty, and fve outputs related to the suppler performance. Based on the performance score, the supplers were classfed nto four categores: hgh performers and effcent, hgh performers and neffcent, low performers and effcent, and low performers and neffcent. Ross et al. [43] used DEA to evaluate the suppler performance wth respect to both buyer and suppler performance attrbutes. Three senstvty analyses were carred out. The frst analyss was to compute the suppler effcency scores wthout consderng the evaluaton team s weghts and bounds. The second analyss consdered the evaluaton team s preferences on the suppler performance attrbutes, whereas the thrd analyss consdered the buyer s preferences on the suppler performance attrbutes. Wu et al. [44] presented a so-called augmented mprecse DEA for suppler selecton. The proposed model was able to handle mprecse data (.e., to rank the effcent supplers) and allow for ncreased dscrmnatory power (.e., to dscrmnate effcent supplers from poor performng supplers). A web-based system was developed to allow potental buyers for suppler evaluaton and selecton. 2. Mathematcal programmng Hong et al. [45] presented a mxed-nteger lnear programmng model for the suppler selecton problem. The model was to determne the optmal number of supplers, and the optmal order quantty so that the revenue could be maxmzed. The change n supplers supply capabltes and customer needs over a perod of tme were consdered. 26

40 Ghodsypour and O Bren [46] formulated a mxed nteger nonlnear programmng model to solve the mult-crtera sourcng problem. The model was to determne the optmal allocaton of products to supplers so that the total annual purchasng cost could be mnmzed. Three constrants were consdered n the model. Karpak et al. [47] constructed a goal programmng (GP) model to evaluate and select the supplers. Three goals were consdered n the model, ncludng cost, qualty, and delvery relablty. The model was to determne the optmal amount of products ordered, whle subjectng to buyer s demand and suppler s capacty constrants. Narasmhan et al. [48] constructed a mult-objectve programmng model to select the optmal supplers and determne the optmal order quantty. Fve crtera were proposed to evaluate the performance of supplers. Before solvng the model to optmalty, the relatve mportance weghtngs of fve crtera were derved n advance. The authors suggested that AHP could be one of the possble ways for generatng the weghtngs. 3. Case-based reasonng Choy and Lee [49] presented a generc model usng the CBR technque for suppler selecton. Varous evaluatng crtera were grouped nto three categores: techncal capablty, qualty system, and organzatonal profle. The model was mplemented n a consumer products manufacturng company, whch had stored the performance of past supplers and ther attrbutes n a database system. The proposed model would then retreve or select a suppler who met the specfcaton predefned by the company most. Choy et al. [50] appled the CBR-based model to ad decson makers n the suppler selecton problem agan. The approach was very smlar to that proposed n Choy and Lee [51], ncludng the 27

41 suppler selecton workflow. In addton, the model was deployed to the same company. 4. Analytcal herarchy process (AHP) Muraldharan et al. [52] proposed a fve-step AHP-based model to ad decson makers n ratng and selectng supplers wth respect to nne evaluatng crtera. People from dfferent functons of the company, such as purchasng, stores, and qualty control, were nvolved n the selecton process. Akarte et al. [53] developed a web-based AHP system to evaluate the castng supplers wth respect to 18 crtera. In the system, supplers had to regster, and then nput ther castng specfcatons. To evaluate the supplers, buyers had to determne the relatve mportance weghtngs for the crtera based on the castng specfcatons, and then assgned the performance ratng for each crteron usng a parwse comparson. Chan [54] developed an nteractve selecton model wth AHP to facltate decson makers n selectng supplers. The model was so-called because t ncorporated a method called chan of nteracton, whch was deployed to determne the relatve mportance of evaluatng crtera wthout subjectve human judgment. AHP was only appled to generate the overall score for alternatve supplers based on the relatve mportance ratngs. 5. Fuzzy set theory Chen et al. [55] presented a herarchy model based on fuzzysets theory to deal wth the suppler selecton problem. The lngustc values were used to assess the ratngs and weghts for the suppler evaluatng factors. These lngustc ratngs could be expressed n 28

42 trapezodal or trangular fuzzy numbers. The proposed model was capable of dealng wth both quanttatve and qualtatve crtera. Sarkar and Mohapatra [56] suggested that performance and capablty were two major measures n the suppler evaluaton and selecton problem. The authors used the fuzzy set approach to account for the mprecson nvolved n numerous subjectve characterstcs of supplers. A hypothetcal case was adopted to llustrate how the two best supplers were selected wth respect to four performance-based and ten capablty-based factors. Florez-Lopez [57] pcked up 14 most mportant evaluatng factors from 84 potental added-value attrbutes, whch were based on the questonnare response from US purchasng managers. To obtan a better representaton of supplers ablty to create value for the customers, a two-tuple fuzzy lngustc model was llustrated to combne both numercal and lngustc nformaton. Besdes, the proposed model could generate a graphcal vew showng the relatve sutablty of supplers and dentfyng strategc groups of supplers. 6. Integrated approaches Sevkl et al. [58] appled an ntegrated AHP DEA approach for suppler selecton. In the approach, AHP was used to derve local weghts from a gven parwse comparson matrx, and aggregate local weghts to yeld overall weghts. Each row and column of the matrx was assumed as a decson makng unt (DMU) and an output, respectvely. A dummy nput that had a value of one for all DMUs was deployed n DEA to calculate the effcency scores of all supplers. However, the authors ponted out that the approach was relatvely more cumbersome to apply than the ndvdual AHP. Kull and Tallur [59] utlzed an ntegrated AHP Goal Programmng approach to evaluate and select supplers wth respect to 29

43 rsk factors and product lfe cycle consderatons. In the proposed model, AHP was used to assess supplers along the rsk crtera, and to derve rsk scores. The GP model was then constructed to evaluate alternatve supplers based on multple rsk goals and varous hard constrants. Xa and Wu [60] ncorporated AHP nto the mult-objectve mxed nteger programmng model for suppler selecton. The model appled AHP to calculate the performance scores of potental supplers frst. The scores were then used as coeffcents of one of the four objectve functons. The model was to determne the optmal number of supplers, select the best set of supplers, and to determne the optmal order quantty. Chan and Kumar [61] also used a fuzzy AHP for suppler selecton as the case wth Kahraman et al. [50]. In the proposed approach, trangular fuzzy numbers and fuzzy synthetc extent analyss method were used to represent decson makers comparson judgment and decde the fnal prorty of dfferent crtera. Amd et al. [62] developed a fuzzy mult-objectve lnear programmng model for suppler selecton. The model could handle the vagueness and mprecson of nput data, and help the decson makers to fnd out the optmal order quantty from each suppler. Three objectve functons wth dfferent weghts were ncluded n the model. An algorthm was developed to solve the model Mult-objectve Optmzaton A mult-objectve problem (MOP) deals wth more than one objectve functon. Most real world problems nvolve the smultaneous optmzaton of two or more (often conflctng) objectves. The solu- 30

44 ton of such problems (called mult-objectve ) s dfferent from that of a sngle-objectve optmzaton problem. The man dfference s that mult-objectve optmzaton problems normally have not one but a set of solutons whch are all equally good Mult-objectve Optmzaton Problem As cted from Shmzu et al. [63], a MOP can be descrbed as a trplet lke (x, f, x), smlar to the usual sngle-objectve optmzaton. However, t should be notced that the objectve functon n ths case s not a scalar but a vector. Consequently, the MOP s wrtten, n general, by [Problem] mn f(x) = {f1(x), f2(x),..., fn(x)} subject to x X, where x denotes an n-dmensonal decson varable vector, X a feasble regon defned by a set of constrants, and f an N-dmensonal objectve functon vector, some elements of whch conflct and are ncommensurable wth each other. The conflcts occur when one tres to mprove a certan objectve functon, at least one of the other objectve functons deterorates. As a typcal example, f one weghs on the economy, the envronment wll deterorate, and vce versa. On the other hand, the term ncommensurable means that the objectve functons lack a common scale to evaluate them under the same standard, and hence t s mpossble to ncorporate all objectve functons nto a sngle objectve functon. For example, envronmental mpact cannot be measured n terms of money, but money s usually used to account economc affars. 31

45 Fgure 2.4. Optmal soluton n decson space (Source: Shmzu et al.[63]) To grasp the entre dea, let us llustrate the feature of MOP schematcally. Fgure 2.4 descrbes the contours of two objectve functons f1 and f2 n a two-dmensonal decson varable space. There, t should be noted that t s mpossble to reach the mnmum ponts of the two objectve functons p and q smultaneously. Here, let us make a comparson between three solutons, A, B and C. It s apparent that A and B are superor to C because f1(a) < f1(c), and f2(a) = f2(c), and f1(b) = f1(c), and f2(b) < f2(c). We call A and B as non-domnated soluton. Thus we can rank the solutons from these comparsons. However, t s not true for the comparson between A and B. We cannot rank these as just the magntudes of the objectve values because f1(a) < f1(b), and f2(a) > f2(b). Lkewse, a comparson between any solutons on the curve, p q, whch s a trajectory of the tangent of both contour curves s mpossble. These solutons are known as Pareto optmal solutons. Such a Pareto optmal soluton (POS) becomes a ratonal bass for MOP snce any other solutons are nferor to every POS. It should be also recalled, however, that there exst nfnte POSs that are mpossble to rank. Hence the fnal decson s left unsolved. To understand ntutvely the POS as a key ssue of MOP, t s depcted agan n Fgure 2.5 n the objectve functon space when N = 32

46 + Fgure 2.5. Idea of soluton procedure n objectve space (Source: Shmzu et al.[63]) 2. From ths, we also know that there exst no solutons that can completely outperform any soluton on the POS set (also called Pareto front). For any soluton belongng to the POS set, f we try to mprove one objectve, the rest of the objectves are urged to degrade. It s also apparent that t never provdes a unque or fnal soluton for the problem under consderaton. For the fnal decson under mult-objectves, therefore, we have to decde a partcular one among an nfnte number of POSs. For ths purpose, t s necessary to reveal a certan value functon of decson maker (DM) ether explctly or mplctly. Ths means that the fnal soluton wll be derved through the tradeoff analyss among the conflctng objectves by the DM. In other words, the soluton process needs a certan subjectve judgment to reflect the DM s preference n addton to the mathematcal procedures [63] Mult-objectve versus sngle-objectve Optmzaton Besdes havng multple objectves, there are a number of fundamental dfferences between sngle-objectve and mult-objectve optmzaton. 33

47 In sngle objectve optmzaton, there s one goal the search for the optmum soluton. Although the search space may have a number of local optmal solutons, the goal s always to fnd the global optmal soluton. However, n mult-objectve optmzaton, there are clearly two goals. Progressng toward the optmal front s certanly an mportant goal. However, mantanng a dverse set of solutons n the Pareto front s also essental [64]. An algorthm that fnds a closely packed set of soluton n Pareto optmal front satsfes the frst goal of convergence of the solutons, but does not satsfy mantenance of a dverse set of solutons. Snce all objectve are mportant, the dverse set of the obtaned solutons close to the Pareto front provde a varety of optmal solutons, tradng objectve dfferently. Snce both goals are mportant, an effcent mult-objectve optmzaton algorthm must work on satsfyng both of them. It s mportant to realze that both of these tasks are somewhat orthogonal to each other. The achevement of one goal does not necessarly acheve the other goal. Explct or mplct mechansms to emphasze convergence near the optmal Pareto front and the mantenance of a dverse set of solutons must be ntroduced n an algorthm. Because of these dual tasks, mult-objectve optmzaton s more dffcult than sngle objectve optmzaton. Another dffculty s that mult-objectve optmzaton nvolves two search space, nstead of one. In sngle objectves optmzaton, there s only one search space the decson varable space. An algorthm works n ths space by acceptng and rejectng solutons based on ther objectve functon values. Here n addton to the decson varable space, there also exsts the objectve or crteron space. Although these two spaces are related by a unque mappng between 34

48 them, often the mappng s non-lnear and the propertes of the two search spaces are not smlar. For example, proxmty of two solutons n one space does not mean proxmty n the other space. Thus, whle achevng the second task of mantanng dversty n the obtaned set of solutons, t s mportant to decde the space n whch the dversty must be acheved. In any optmzaton algorthm, the search s performed n the decson varable space. However the proceedng of an algorthm n the decson varable space can be traced n the objectve space. In some algorthms, the resultng proceedngs n the objectve space are used to steer the search n the decson varable space. When ths happens, the proceedngs n both spaces must be coordnated n such away that the creaton of new soluton s complmentary to the dversty needed n the objectve space. Ths, by no means, s an easy task and depends on the mappng between the decson varables and the objectve functon values Search and decson makng In solvng an MOP, two conceptually dstnct types of problem dffculty can be dentfed [65]: search and decson makng. The former refers to the optmzaton process n whch the feasble set s sampled for Pareto optmal solutons. Even n sngle-objectve optmzaton, large and complex search spaces can make search dffcult and preclude the use of exact optmzaton methods lke lnear programmng [66]. The latter addresses the problem of selectng a sutable compromse soluton from the Pareto-optmal set. The decson maker s necessary to make the often dffcult trade-offs between conflctng objectves. 35

49 Dependng on how optmzaton and the decson process are combned, mult-objectve optmzaton methods can be broadly classfed nto three categores [67, 68]: Decson makng before search: The objectves of the MOP are aggregated nto a sngle objectve whch mplctly ncludes preference nformaton gven by the decson maker. Search before decson makng: Optmzaton s performed wthout any preference nformaton gven. The result of the search process s a set of (deally Pareto-optmal) canddate solutons from whch the fnal choce s made by the decson maker. Decson makng durng search: The decson maker can artculate preferences durng the nteractve optmzaton process. After each optmzaton step, a number of alternatve trade-offs s presented on the bass of whch the decson maker specfes further preference nformaton, respectvely gudes the search. The aggregaton of multple objectves nto one optmzaton crteron has the advantage that the classcal sngle-objectve optmzaton strateges can be appled wthout further modfcatons. However, t requres profound doman knowledge whch s usually not avalable. For example, n computer engneerng desgn space exploraton specfcally ams at ganng deeper knowledge about the problem and the alternatve solutons. Performng the search before decson makng overcomes ths drawback, but excludes DM s preference artculaton whch mght reduce the search space complexty. Another problem wth ths and also the thrd algorthm category mght be the vsualzaton and the presentaton of non-domnated sets for hgher dmensonal MOPs. Fnally, the ntegraton of search and decson makng s a promsng 36

50 way to combne the other two approaches, untng the advantages of both [68] Concept of domnaton Most mult-objectve optmzaton algorthms use the concept of domnaton. In these algorthms, two solutons are compared on the bass of whether one domnates the other soluton or not. We assume that there are M objectve functons. In order to cover both mnmzaton and maxmzaton of objectve functons, we use the operator between soluton and j as j to denote that soluton s better than soluton j on a partcular objectve. Smlarly, j for a partcular objectve mples that soluton s worse than soluton j on ths objectve. For example, f an objectve functon s to be mnmzed, the operator would be mean the < operator, whereas f the objectve functon s to be maxmzed, the operator would mean the > operator. The followng defnton covers mxed problems wth the mnmzaton of some objectve functons and maxmzaton of the rest of them. A soluton x (1) s sad to domnate the other soluton x (2), f both condton 1 and 2 are true: 1. The soluton x (1) s no worse than x (2) n all objectves, or fj(x (1) ) fj(x (2) ) for all j = 1,2,,M. 2. The soluton x (1) s strctly better than x (2) n at least one objectve, or fj(x (1) ) fj(x (2) ) for at least one j {1,2,,M}. If any of the above condton s volated, the soluton x (1) does not domnate the soluton x (2). If x (1) domnates the soluton x (2), t s also customary to wrte any of the followng: x (2) s domnated by x (1) ; 37

51 f 2 (mnmze) f 1 (maxmze) Fgure 2.6. Illustraton of domnaton concept x (1) s non-domnated by x (2), or; x (1) s non-nferor to x (2). Let us consder a two-objectve optmzaton wth fve dfferent solutons shown n the objectve space, as llustrated n Fgure 2.6. Let us also assume that the objectve functon 1 s needs to be maxmzed whle the objectve functon 2 needs to be mnmzed. Fve solutons wth dfferent objectve functon values are shown n ths fgure. Snce both objectve functons are of mportant to us, t s usually dffcult to fnd one soluton whch s the best wth respect wth both objectves. However, we can use the above defnton of domnaton to decde whch soluton s better among any two gven solutons n term of both objectves. For example, f soluton 1 and 2 are to be compared, we observe that soluton 1 s better than soluton 2 n objectve functon 1 and soluton 1 s also better than soluton 2 n objectve functon 2. Thus, both the above condtons are satsfed and we may wrte the soluton 1 domnates soluton 2. We take another nstance of comparng soluton 1 and soluton 5. Here, soluton 5 s better than soluton 1 n the frst objectve and soluton one s no worse (n fact they are equal) than soluton 1 n the second objectve. Thus, both the above condtons for domnaton are also satsfed and 38

52 we may wrte that soluton 5 domnates soluton 1. Moreover, f we compare soluton 1 and soluton 4, we can see that soluton one s better than soluton 4 n objectve functon 1, but soluton 1 s worse than soluton 4 n objectve functon 2. Thus, the above condtons are volated. In ths case, we say that soluton 1 and soluton 4 are non-domnated solutons. 39

53 Chapter 3 MULTI-OBJECTIVE PERIODIC REVIEW INVENTORY WITH COORDINATED REPLE- NISHMENT IN TWO-STAGE SUPPLY CHAIN TROUGH DIFFERENTIAL EVOLUTION 3.1. Introducton In today s globalzaton, every supply chan s expected to mnmze the system wde cost ncludng nventory cost along the supply chan whle mnmzng loss rate to meet customer demand as much as possble. Ths s because recent nnovatve technologes have shortened the product lfe cycle and ncreased the demand varablty. The excess nventory n the supply chan block the cash flow and ndeed gves an adversely effect on the enterprse. It s dffcult to determne an optmal nventory polcy for a mult-echelon supply chan system due to the nteracton among the dfferent levels wth dfferent goals. Moreover, such polcy depends on the structure of the system and should be based on the states of 41

54 the whole system. Even f such a polcy can be dentfed, t usually has a very complex structure and s not sutable for mplementaton. Therefore t s reasonable to consder smple, cost-effectve heurstc polces, whch can be readly applcable n practce [69]. Among nventory control polces, perodc revew nventory system s commonly mplemented n practce. In the survey of Smch-Lev et al. [70], materal managers ndcate the effectveness of perodc revew systems for reducng nventory levels n the supply chan. However, just as same as the others, one of the dsadvantages of the perodc revew s that the stock out s normally occurs before recevng the new replenshment due to the demand and lead tme varatons n the system. To overcome ths dsadvantage, an dea of alternatve supply n nventory system has been studed extensvely n the lterature. The majorty of the papers n ths area deal wth ether the concept of emergency supply modes [71, 72], or the concept of expedted supply modes [73]. Along wth ths dea, coordnaton and nformaton sharng between members n the supply chan have recently become other key ssues. Many companes undertake ntatves drected to re-engneerng ther supply chan to reduce costs whle beng more responsve to customer demand. Recent research carred out n ths drecton nclude Snha and Sarmah [74], Ouyang [75], Zhou and Benton [76], Sarmah et al [77], and Gupta and Weerawat [78]. In ths study, we note two nventory replenshment strateges n two-level supply chan wth alternatve supply possbltes whch nvolve dfferent level of coordnaton mechansm between manufacturers and retalers. Then, we compare the performance of these strateges wth that of a tradtonal one where orderng from only one of manufacturers s allowed for each retaler. 42

55 Under normal crcumstances, orders from a retaler are fulflled by only one manufacturer whch s desgnated to serve that retaler. However, n the case where the correspondng manufacturer cannot fully satsfy the order, a centralzed decson can be made to assgn another manufacturer n the chan to cover the shortage. Ths research also proposes another strategy whch mples a hgher level of coordnaton between manufacturer and retaler. In the case where another manufacturer cannot cover the shortage due to nsuffcent stock, the shortage s consdered as backorder to the manufacturer whch can serve faster. Under these crcumstances, n ths study, a mult-objectve nventory analyss s proposed to evaluate the performance of the system by smultaneously mnmzng total cost and loss rate of the supply chan. The am s to examne the stuaton when such coordnaton s proftable for all members n the system System descrpton The supply chan operates under the make-to-stock envronment, n whch stochastc demand and lead-tme are consdered. The Central company Suppler Manufacturer Retaler customer 1 1 shpment demand n n Fgure 3.1. System Confguraton wth n-seral lnes 43

56 system s controlled by a sngle major company whch operates n- seral sub systems, as depcted n Fg Each seral subsystem conssts of one manufacturer whch serves one retaler. In what follows, characterstc of the supply chan systems concerned here wll be descrbed brefly. For each member n the system, the followng assumptons are ntroduced: 1. The manufacturer uses perodc revew system wth safety stock and lot szng polcy to control ts nventory. 2. The retaler uses perodc revew system wth target stock level (T, S) to control ts nventory. 3. Only a sngle product s consdered n the model. Wthout loss of the generalty, the manufacturer uses one unt of raw materal to produce one unt of fnshed product. 4. End customer s demand and delvery lead-tme are randomly generated based on the normal dstrbuton. 5. Producton rate of the manufacturer s assumed to be fxed and hgher than the mean demands. 6. Unfulflled demand at manufacturer s consdered as backorder whle unfulflled demand at retaler s consdered as loss. The system model s descrbed based on the followng notaton lsted for major parameters. Index T N t j Number of plannng horzon. Number of seral lnes. Perod (t = 1,2,,T). Manufacturer ( N) Retalers ( j N). 44

57 Parameters of Manufacturer FD tp PR lm lr Qp Qpr Bs Sr Es BSS ESS Qm Qf Qa Qb Qsh Qsl BI EI Forecast demand of manufacturer at perod t Number of days n each perod Producton rate of Manufacturer Mean lead tme of raw materal delvery to manufacturer Actual lead tme of raw materal delvery to manufacturer at perod t Mean delvery lead tme from manufacturer Actual delvery lead tme from manufacturer at perod t Producton quantty of manufacturer durng suppler s late delvery at perod t Producton quantty at manufacturer at perod t Begnnng stock level of raw materals of manufacturer at perod t Amount of raw materal left at manufacturer after producton durng suppler s late delvery at perod t Endng stock level of raw materals of manufacturer at perod t Begnnng safety stock level of manufacturer at perod t Endng safety stock level of manufacturer at perod t Orderng quantty of manufacturer at perod t Quantty to fll back the safety stock of manufacturer at perod t Avalable quantty of manufacturer at perod t Backorder quantty of manufacturer at perod t shortage quantty of manufacturer at perod t Sales volume of manufacturer at perod t Begnnng nventory level of manufacturer at perod t Endng nventory level of manufacturer at perod t 45

58 Parameters of Retaler Dj,t dj,t Brj,t Ioj,t Qsj,t Qrej,t Irj,t Qshj,t Qorj,t Qsej,t Total customer demand of retaler j at perod t Customer demand at retaler j before recevng replenshment at perod t Begnnng nventory level of retaler j at perod t Inventory level left at retaler j durng manufacturer s lead tme at perod t Shortage quantty at retaler j durng manufacturer s lead tme at perod t Replenshment quantty receved by retaler j at perod t Endng nventory level of retaler j at perod t Shortage quantty of retaler j at perod t Orderng quantty of retaler j at perod t Sales volume of retaler j at perod t Cost Parameters Com Orderng cost of manufacturer Cp Unt purchasng cost of manufacturer Cpr Unt producton cost of manufacturer Chr Unt holdng cost of raw materal of manufacturer Chf Unt holdng cost of fnshed product of manufacturer Cb Unt backorderng cost of manufacturer Ctr Unt transportaton cost of manufacturer Cpuj Unt purchasng cost of retaler j Choj Unt holdng cost of fnshed product of retaler j Csj Unt shortage cost of fnshed product of retaler j Decson Varbales LS Lot szng polcy of manufacturer SS Safety stock level of manufacturer 46

59 Rj Target stock level of retaler j Manufacturer The nventory level of the manufacturer s revewed at every perod t, over totally T perods (plannng horzon). Each perod conssts of tp days. The manufacturer receves raw materals from an outsde suppler whch has unlmted capacty, transforms t to the fnshed product and then dstrbutes the product to correspondng retaler. However, under uncertanty n the delvery lead-tme, the suppler may delay the supply of raw materals to the manufacturer. Therefore, the manufacturer has to select the approprate materal orderng polcy and hold some safety stock of fnsh product. The orderng quantty of the manufacturer s drectly nfluenced by the lot szng polcy (LS) whch s adopted for orderng raw materal. For nstance, the frst polcy s to place an order n every perod. The second polcy s to place an order at the frst perod and combne the orders of remanng perods together, and so on. After the best pattern of LS s selected, the manufacturer checks current nventory level at the begnnng of the perod. If the nventory level s less than the sum of the forecasted demand, the quantty to fll back the safety stock and the backorder quantty, then the manufacturer wll place an order. Otherwse no order wll be ssued (Eq. 3.1). Qm, t 0, FD Qf 1 Qb BI Bs max (3.1) It s assumes that we always starts revewng the nventory at tme t. Then the manufacturer start producng the product at the lead-tme contract t +. When late delvery occurs (lm > ), the manufacturer stll can start the producton at tme t + f there s 47

60 begnnng raw materal on hand. Otherwse, the manufacturer has to wat untl t receves replenshment from the suppler at tme t + lm. The producton quanttes durng late delvery of suppler and amount of raw materal left after producton durng suppler s late delvery are calculated usng equatons (3.2) and (3.3), respectvely. Consequently, we can determne the total producton quantty (Eq. (3.4)) and endng stock level of raw materal as shown n equaton Eq. (3.5). Endng nventory level (EI) s the amount of fnshed product left after fulfllng retaler s demand and the use of safety stock. The safety stock of the manufacturer s used only when the order quantty of the retalers s greater than the sum of the producton quantty (Qpr) and amount of begnnng nventory of fnshed products (BI). The EI and ESS are calculated usng Eq. (3.6) and Eq. (3.7), respectvely. Qp lm t lm~ lm t lm~ lm t lm~, PR f, and Bs, Bs f lm lm~ Bs lm lm~ 0, t, t and f lm, t lm~, t, t PR PR (3.2) Sr Bs 0 Bs, t lm lm~ PR Bs lm lm~, t f f f lm lm, t, t lm Bs, t lm~ lm~ lm~ lm and, t and, t lm~ PR PR (3.3) 48

61 49 PR lm Tp Qm Bs PR lm tp Qm Bs PR lm tp Qm Sr Bs PR lm tp Qm Sr Bs Qm PR lm tp Qm Sr Qp PR lm tp Qp Qpr and 0 and 0 and 0 and 0 f f f f ) ( (3.4) otherwse and 0 and 0 f f f 0 PR lts Tp Qm Bs PR lts Tp Qm Sr Bs PR lts Tp Qm PR lts Tp Qm Sr Es (3.5) otherwse ) ( ) ( and and f and f 0 ) ( ) ( ) ( j,t 1 j,t 1 j,t 1 j,t 1 j,t 1 BSS SS Qb Qor BI Qpr Qb Qor BI Qpr SS BSS Qb Qor BI Qpr SS BSS BSS SS Qb Qor BI Qpr Qb Qor BI Qpr EI (3.6)

62 ESS 0 Qa Qor j,t BSS SS BSS Qpr BI Qor 1 j,t f Qor f Qor f BSS f BSS j,t and j,t otherwse Qa SS Qpr Qpr 1 SS 1 BI BI and Qor and Qor and j,t j,t Qor Qpr Qpr j,t Qor j,t 1 1 Qa BI BI SS BSS (3.7) Qsl Qor j,t Qb f Qa Qor j, t (3.8) Qa Qb otherwse As shown n Eq. (3.6) and Eq. (3.7), the endng safety stock level (ESS) and the endng nventory level (EI) have dfferent meanngs. The ESS s the amount of safety stock left at the end of each perod whle EI s the number of fnshed products that s produced beyond the retaler s demand and s left from fulfllng the safety stock. The sales volume (Qsl) of the manufacturer s determned based on total order quantty of retaler (Eq. (3.8)). When the total order quantty of the retaler exceeds the avalable quantty on hand, the shortage quantty (Qsh) wll be backordered the next perod (Qb). Retaler The retaler perodcally places the order to a correspondng manufacturer to rase up ts nventory to the target stock level. The order quantty (Qorj,t) s determned by comparng the endng stock level (Irj,t) at perod t wth the desred target stock level (Rj), whch s equal to (Rj - Irj,t). The target stock level s not only for coverng the end customer s demand but also to cover the effect of end cus- 50

63 tomer demand s fluctuaton as well as the late delvery and unfulflled quantty of products from the manufacturer. After placng the order to the correspondng manufacturer, the retaler receves replenshment after some lead tme. At the begnnng of the perod, each retaler uses begnnng nventory of fnshed product (Brj,t) to fulfll end customer demand durng replenshment lead tme. Due to uncertanty n the system, sometmes the retaler may not receve a full replenshment from correspondng retaler. When ths stuaton occurs, the manufacturer promses to delver the remanng quantty the next perod. The nventory that s left and the amount of shortage durng ths tme are shown n equaton (3.9) and (3.10). Consequently, endng nventory, total shortage quantty and sales volume of retaler after recevng replenshment can be calculated usng equaton (3.11), (3.12) and (3.13), respectvely. Io Qs max Br d, 0 (3.9) j, t j,t j, t max d Br, 0 (3.10) j, t j,t j, t Io j,t Qre j,t ( D j,t d j, t ) f Io j,t Qre j,t D j,t d j, t Ir j,t (3.11) 0 otherwse Qs j,t ( D j,t d j, t ) f Io j,t Qre j,t D j,t d j, t Qsh j,t ( Io j,t Qre j,t ) (3.12) Qs j,t otherwse D j,t f Qsh j,t 0 Qse j,t (3.13) D j,t Qsh j,t otherwse 51

64 Any stock-out that the retaler faces s consdered as a total lost due to the fact that n today hgh compettve market, the customers have a plenty of choces to acqure products, especally for retalers such as department stores, supermarkets or conventonal stores. Consequently, the stuaton n whch the customers buy the same product from a second shop when there s the stock-out occurs at the frst shop s qute common n the real world busness. Ths study consders two objectve functons to evaluate the system performance. The frst objectve mnmzes system total cost (ZTCS) consstng of total cost of manufacturers (TCM) and total cost of retalers (TCRj). The second objectve functon mnmzes loss rate of supply chan (ZLRS) calculated from total shortage of retalers. These two objectves requre a trade-off n soluton as they are conflctng wth each other. As the system faces uncertanty, one way to mantan the responsveness (mnmum loss rate) to the customer demand s holdng nventory n hgher volume. However, ths causes an ncrease n system total cost as an excess nventory blockng system cash flow. On the contrary, holdng lower nventores gve the opposte consequence Replenshment Strategy The detals of each replenshment strategy wll be descrbed n ths secton Replenshment Strategy 1 (S1) The nventory flow of strategy 1 s shown n Fg.1. It s assumed that manufacturer 1 s the only source of supply for retaler 1, manufacturer 2 s the only source of supply of retaler 2 and so forth. If a 52

65 retaler places an order to ts correspondng manufacturer and that manufacturer holds nsuffcent quantty, the retaler s order s partally backordered wth the correspondng manufacturer. Then, the unfulflled quantty s delvered by the correspondng manufacturer the next perod. In the rest of ths paper, ths strategy s referred to as a tradtonal strategy whch s common for the nventory systems wth sngle manufacturer and sngle retaler. In general, the objectve functon of the mult-objectve optmzaton problem for p objectves can be descrbed as mnmzng F = {F1, F2, F3,, Fp}. In the present problem, we gve the objectve functon as follows: Mnmze, F Z Z F (3.14) 1 2 TCS, LRS where ZTCS s gven by J Z TCM (S1) TCR (S1) (3.15) I TCS 1 j1 j T T T Com Cp Qm Cpr Qpr t 1 t1 t 1 T T TCM (S1) Chr Es Chf ESS EI (3.16) t 1 t1 T T Cb Qb tp Ctr Qsl t 1 t1 T t1 T TCR (S1) Cpu j Qre Cho j Irj,t Cs j Qsh j,t (3.17) t1 T t1 and ZLRS s gven by 53

66 J T Qsh j,t Z LRS (3.18) j1 t1 D j, t Replenshment Strategy 2 (S2) Under crcumstance of strategy 1, manufacturers only fulfll the demands of ther correspondng retalers. However, under strategy 2, a retaler may receve an alternatve supply from another manufacturer n the chan f the correspondng manufacturer fals to meet the demand. Ths alternatve supply s llustrated n Fg When a certan retaler n does not receve full replenshment from manufacturer n as ts correspondng manufacturer, the central decson maker wll check the remanng nventory of the fnshed product of other manufacturers (say manufacturer k) after supplyng ther correspondng retaler and then assgn one of them to serve retaler n. The requred quantty s backordered to manufacturer n f no other manufacturers can supply all quantty of shortage. Central Company Manufacturer n Retaler n Assgnment Assgnment Manufacturer k (k n) Suffcent nventory? No Informaton Order Yes Full shpment Partal shpment Start Suffcent nventory? No Yes Add. delvery Hgher transp. cost Informaton Backordered at perod t+1 End Fgure 3.2. Replenshment Strategy 2 54

67 Snce each retaler can possbly receve a product from more than one manufacturer, a new transportaton cost from manufacturer to the retaler should be added to the frst objectves functon. Thus for each manufacturer, there are two types of transportaton cost that are ncurred whch correspond to the transportaton cost to regular retaler and to other retaler, where the latter s more costly due to non-regular delvery. On the other hand, the other objectve functon remans the same. T TCM (S2) TCM (1) Cta Qsa (3.19) 1 T 1 l,t TCR (S2) TCR (1) Cpu Qsa (3.20) j j j k j,t where Qsa l,t Sales volume of manufacturer to retaler l at perod t, l j where l, j ϵ J (unt). Cta Unt transportaton cost of manufacturer to other retaler ($) Replenshment Strategy 3 (S3) Ths strategy works n the same way as strategy 2 except for the stuaton where one manufacturer experences a shortage whle the other manufacturers hold nadequate nventory to fulfll ths shortage. Unlke strategy 2, n whch the shortage s backordered partally to the correspondng manufacturer when the stock-out occurs, ths strategy proposes dfferent coordnaton mechansm. The central decson maker examnes the suppler s lead tme for the upcomng de- 55

68 Central Company Manufacturer n Retaler n Assgnment Assgnment Manufacturer l (faster response) Manufacturer k (k n) Suffcent nventory? No Informaton Order Yes Full shpment Partal shpment Start Suffcent nventory? No Yes Addtonal delvery Hgher transp. cost Informaton Addtonal delvery Hghest transp. cost End Fgure 3.3. Replenshment Strategy 3 lvery and the current nventory status of fnshed product of all manufacturers. Ths nformaton s used to determne the tme requred to produce the products to fulfll the shortage. Suppler s lead tme affects the startng tme of producton and the nventory status determnes the quantty of product to be produced. Lastly, a manufacturer whch has a shorter lead tme s assgned to fulfll the shortage and the backordered quantty s delvered as soon as producton s fnshed. Accordngly, the new transportaton cost for addtonal replenshment s added to the cost of manufacturers whle others reman the same as strategy 2 after adjustng the backorder cost at manufacturer (Refer to Fg.3.3). T T TCM (S2) Cpr Qpx Ctx Qpx 1 1 TCM (S3) T (3.21) Cb Qb tf 1 56

69 Qpx Ctx tf Non-regular producton quantty of manufacturer at perod t to fulfll shortage (unt). Unt transportaton cost of manufacturer to fulfll shortage quantty of manufacturer, where ctr< cta< ctx ($). Total tme to produce backorder at manufacturer at perod t (days). It should be noted n Eq that for a manufacturer whch s suffered from stock-out, there s no cost component of transportaton ( because the non-regular producton quantty s desgnated to fulfll ts own requrement. On the other hand, the cost component of backorder s not appled ( for manufacturers whch are assgned as backup resource Mult-objectve Dfferental Evoluton (MODE) Dfferental Evoluton (DE) s a recent optmzaton technque n the famly of evolutonal algorthms. It s proposed as a varant of genetc algorthms to acheve the goal of robustness n optmzaton Fgure 3.4. Pseudo code of DE (Strategy: DE/rand/1/bn) 57

70 and faster convergence [79, 80]. DE uses a self-organzng scheme to take the dfferent vector of two or more vector to create a mutant vectors. So that a few nput s requred from the user and t eases to mplement DE to solve the problem. The pseudo code of DE can be seen n Fgure 3.4. One of the most commonly appled strateges of DE to solve the problem s DE/rand/1/bn. Notaton rand/1 means one dfferent set of vector (from 2 vectors) s randomly chosen n the populatons to be mutated. Then bn means the ndependent bnomal experment s used for crossover schemes. The detaled of DE s algorthm s summarzed as follow: 1. Randomly generate the ntal populaton to yeld target vector (x,g) by usng Eq. (3.22). x, G 2 1,,..., M (3.22) where M = populaton sze 2. Apply mutaton to generate the mutant vector by addng the weght dfference between target vectors to the thrd target vector as show n Eq. (3.23). v, G 1 x3,g F x 2,G x1,g (3.23) where F s a scalng factor whch control controls the amplfcaton of the dfferental varaton (x2,g x1,g) 3. Apply the crossover operaton to generate the tral vector by mxng some elements of the target vector wth the mutant vector through comparson between random value and crossover rate. 58

71 u j,g 1 v x j,g 1 j,g f f rand rand j j CR CR or or j rnbr j rbnr a a (3.24) where rand(j) s a random number, CR s crossover rate [0, 1] and rnbr() s a randomly chosen ndex (1,2,,n) whch ensure that the tral vector gets at least one element from the mutant vector. 4. Perform the selecton operaton by comparng the target vector and the tral vector. If the tral vector s better than the target vector, the tral vector replaces the target vector at the next generaton. Otherwse, the target vector s remaned. Then check pre-specfed stoppng crtera. If t s satsfed, stop and return the overall best vector as a fnal soluton. Otherwse, go back to step 2 by ncrementng the generaton number by 1. In the sngle-objectve DE, the selecton process s straghtforward. The vector whch has optmal soluton s chosen. In the multobjectve case, however, the selecton process becomes more complcated. Ths s because the need of Pareto optmal solutons. Prevous research attempted to deploy DE to a mult-objectve problem and showed that DE can be attractve alternatve for mult-objectve optmzaton [81, 82, 83, 84]. However, handlng mult-objectve DE poses certan dffcultes n ts mplementaton. Besdes preservng a unformly spread front of non-domnated solutons, t s also necessary to decde when to replace the parent (target vector) wth the canddate soluton (tral vector). Reddy and Kumar [85] proposed another methodology (MODE) to resolve these dffcultes by combnng Pareto Domnance prncples wth DE and usng eltsm n ts 59

72 S t a r t G e n e r a t e n t a l t a r g e t v e c t o r s ( T V ) A r c h v e n o n - d o m n a t e d T V = 1 C h o o s e o n e T V T V T V S R D T V b e t G e n e r a t e m u t a n t v e c t o r ( M V ) C r e a t e t r a l v e c t o r ( T V ) N o n - d o m n a t e d s e l e c t o n T V < T V T V < T V T V = T V N o A r c h e v e d? Y e s S R D T V * w n n e r N o A r c h e v e d? b o t h Y e s l o s e r S R D T V * A b a n d o n S e l e c t A b a n d o n = + 1 G = G + 1 = N p? G = G m a x? R e t r e v e a r c h e v e * S e l e c t r a n d o m l y d o m n a t e d t a r g e t v e c t o r A < B : A s d o m n a t e d b y B A = B : A a n d B a r e n o n - d o m n a t e d m u t u a l l y E n d Fgure 3.5. Flowchart of the proposed MODE evoluton. The man algorthm conssts of ntalzaton of populaton, evaluaton, Pareto domnance selecton, performng DE operatons and repeatng the search on populaton to reach the Pareto optmal solutons. One of the crucal ponts of ths method s that the use of external archve to store the non-domnated solutons found so far over the generaton. In ths study, we propose a new procedure wth a dfferent selecton mechansm to dscover the set of Pareto optmal solutons. 60

73 Snce ths procedure uses a sngle and fxed sze populaton durng the entre process, the algorthm becomes much smpler than the conventonal one. All vectors are frst evaluated and checked usng domnaton relaton to dstngush those nto non-domnated and domnated classes. Thereafter, n order to select the new populaton for the next generaton, a unque selecton mechansm s appled between the target vector and the tral vector. The proposed MODE methodology s descrbed as follows (See also Fg.3.5): Step 1. Input the necessary DE parameters. Generate n- dmensonal ntal vectors (M) n the populaton randomly wthn the bounds of specfed decson varables. Clear the archves. Step 2. Evaluate each vector n the populaton. Step 3. Perform non-domnated sortng. Apply the concept of domnance to dentfy vectors that become non-domnated vector n the current populaton, and archve them as nondomnated vectors. Step 4. Perform mutaton and crossover operatons for every target vectors x,g n the populaton as same as the sngleobjectve DE,.e., a. Select three dfferent vectors randomly from the current populaton other than the target vector. b. Calculate a mutant vector usng Eq. (3.23). c. Create a tral vector by mxng some element from target vector and mutant vector usng crossover rate (eq.3.24). d. Restrct the varables to ts boundares f any varable s outsde the lower or upper bound. Step 5. Evaluate each target vector 61

74 Step 6. Perform the selecton operaton based on the domnance concept accordng to the followng cases: Case 1 : The tral vector domnates the target vector - Select tral vector as canddate parent for the next generaton. - Check the status of the target vector. If the target vector s prevously non-domnated, keep t n the populaton by randomly removng one domnated target vector. Otherwse, remove the target vector from populaton. Case 2 : The target vector domnates the tral vector - Select the target vector as a canddate parent for the next generaton. - Remove the tral vector from the populaton. Case 3 : Nether vector are domnated by the other - Check the status of the target vector. If the target vector was prevously non-domnated, keep t n the populaton by removng one domnated target vector and select a tral vector as the canddate parent. - Otherwse, select randomly avector (target vector or tral vector) to be the canddate parent. Step 7. Increment the generaton counter, G to G + 1. Check the stoppng crtera. If t s not satsfed, then go to step 3. Otherwse, return the non-domnated solutons Numercal Experment In practce, a sngle decson could hardly be formulated to handle problems characterzed by dfferent stuatons. The nature of the stuaton typcally changes the characterstc of the problem 62

75 causng the decson needs to be adjusted. The problem becomes more complcated when the decson maker faces several objectves, whch s normally conflctng and need to be optmzed smultaneously. Ths research attempts to address such stuaton mentoned above. Ths numercal experment evaluates the three replenshment strateges by solvng a stock out problem for a gven supply chan system and decdng the optmal nventory control polcy for each strategy. Then we compare the performance of the system for three dfferent uncertan stuatons and analyzed how the decson s adjusted for each of them. Snce we approach the problem usng multobjectve decson makng, the comparson s carred out based on the Pareto soluton set. A numercal experment was carred out for a system consstng of two-seral lnes usng the nput parameters shown n Table 3.1. The experment was performed accordng to the followng steps: 1. Apply the proposed MODE to fnd a set of Pareto optmal soluton of each proposed strategy by determnng approprate orderng polcy and nventory level of each member. 2. Apply senstvty analyss to Pareto front to see the effectveness of the proposed strategy under dfferent uncertan stuatons gven by standard devatons of demand. Here the demand s selected as an uncertan parameter because varablty affects greatly the whole system. 3. Examne the effect of each strategy for the approprately selected compromsed soluton regardng the holdng cost and backorder cost of manufacturers, and holdng cost and shortage cost of retalers. 63

76 Codng and settng of lower/upper Bound In ths experment, the decson varables consst of orderng polcy of raw materal (LS) and safety stock level (ss) of each manufacturer, and target stock level (S) of each retaler. In order to generate ntal populaton, lower and upper bound Table 3.1. Parameters values for Numercal Experment Input parameters M1 M2 T 6 perod tp 7 days FD 250 unt / day 25, 50, , 50, days Normal(3,1) PR 360 unt 400 unt Input parameters R1 R2 Dj,t 250 unt / day 25, 50, , 50, days Normal(3,1) Cost Parameters M1 M2 Com $200 / order Cp $10 / unt Cpr $20 / unt Chr 0.15 * (Cp) Chf 0.20 * (Cp + Cpr) Cb $15 / unt $17 / unt Ctr $13 / unt $15 / unt Cta $20 / unt Ctx $30 / unt Cost Parameters R1 R2 Cpuj $77 / unt Choj 0.15 * (Cpu) Csj $20 / unt DE Parameters Np 300 CR 0.5 F 0.8 G

77 of each decson varable should be set as follows: For LS, the manufacturer has to decde whether to make the order at the begnnng of every perod or combne the order n a bg batch. Therefore, the bnary codng s appled to represent the value of LS. For example, wth 6 perods plannng horzons, means lot-for-lot polcy s used for perod 1, 2 and 3. For perod 4 to 6, the order s combned and t s placed at perod 4. For ss and S, These decson varables are consdered as the amount of safety stock at the manufacturer and the amount of target stock retaler, respectvely. Thefore, the nteger codng s used to represent these values. Let,, and are the mean and standard devaton of the end customer demand and forecast demand, respectvely, and, and are standard devaton of the delvery lead tme of raw materal and product, respectvely. - The lower bound of ss s set to 0 or no safety stock s hold at the manufacturer. - The upper bound of ss s calculated by the usng Eq. (3.25). In order to mantan the mnmum loss rate (hgher servce level) and to make sure that all searchng space s bounded, the upper bound of ss s set at the value correspondng z = 4. (see [86] for further nformaton). ( ) ( ) (3.25) - The lower bound of S should be at least or equal to the expected demand durng the revew tme plus delvery lead tme contract, whch s equal to ( ). 65

78 Normalzed F2 - The upper bound of S can be determned n the same way as done by [4], whch s equal to: ( ) ( ) ( ) (3.26) Results and analyss a. Pareto Optmal Solutons Fgure 3.6 shows Pareto optmal soluton sets for each strategy under varous standard devatons of demand. Each plot s normalzed between 0.0 and 1.0 usng the maxmum and/or mnmum values all over the computatons. In every case, we can obtan favorable features of Pareto front usng the developed MODE method. Under the small standard devatons of demand (σ = 25), strateges 2 and 3 have less nfluence to the system performance snce only slght dfference can be seen at Pareto front as shown n Fg. 3.6 (a). Ths s because when the demand s relatvely steady, a shortage Normalzed Pareto Optmal Soluton (Std. Devaton of Demand = 25) Normalzed F1 Strategy 1 Model 1 Strategy Model 2 Model 2 3 Strategy 3 (a). Std. Devaton of Demand = 25 66

79 Normalzed F2 Normalzed F2 Normalzed Pareto Optmal Soluton (Std. Devaton of Demand = 50) Normalzed F1 Strategy 1 Strategy 2 Strategy 3 Model 1 Model 2 Model 3 (b). Std. Devaton of Demand = 50 Normalzed Pareto Optmal Soluton (Std. Devaton of Demand = 100) Normalzed F1 Strategy 1 Strategy 2 Strategy 3 Model 1 Model 2 Model 3 (c). Std. Devaton of Demand = 100 Fg.3.6. Normalzed Pareto Optmal Soluton such that the proposed coordnaton can be effectvely appled s rarely occurred. In other words, each seral lne can cope wth ts own market wthout cooperatng wth each other. In ths case, the decson can be made not to ntroduce such coordnaton strategy n the system. However, as varatons of demand ncrease (σ = 50, 100), the dfferent strateges produce remarkably dstnct Pareto optmal solutons (Fgures 3.6(b) and (c)). Ths ndcates that strateges 2 and 3 67

80 Table 3.2. Objectve Functon Values and Decson Varable of Selected Pareto Soluton Std. Dev. demand = 25 Strategy 1 Strategy 2 Strategy 3 TCSC ($-thousand) 2,372 2,363 2,368 LRSC (%) DLS ss (unt) S (unt) Std. Dev. demand = 50 Strategy 1 Strategy 2 Strategy 3 TCSC ($-thousand) 2,403 2,381 2,359 LRSC (%) DLS ss (unt) S (unt) Std. Dev. demand = 100 Strategy 1 Strategy 2 Strategy 3 TCSC ($-thousand) 2,404 2,373 2,254 LRSC (%) DLS ss (unt) S (unt) are more sutable compared to strategy 1 snce the Pareto optmal soluton of both strateges move down toward the lower regon and spread wdely over the decson space. Eventually, ths means strategy 3 s the one. b. Uncertanty Effect on Each Objectve Functon The fnal goal of mult-objectve decson-makng should be to present decson maker a unque soluton known as preferentally optmal soluton that wll be derved through mult-objectve optmzaton. 68

81 H oldng C os t ($) B ac korder C os t ($) Durng the mult-objectve optmzaton, we assume to take a posteror approach that s search and then proceed to the actual decson-makng n whch the decson s made wthout any pror preference nformaton. At the search stage, the result s a set of Pareto optmal solutons. The fnal choce wll be made at the decsonmakng stage. To show how the uncertanty analyss s avalable for the preferental optmal soluton, t s necessary to have a reference or a sample from the Pareto optmal front. Presently, assumng that both goals are of equal mportance (w1 = w2 = 0.5, for nstance), we derve a reference soluton from the Average Backorder Cost of Manufacturers S td. D evaton of D emand Strategy Model 1 1 Strategy Model 22 Model Strategy 3 3 Fgure 3.7. Comparson of Backorder cost of Manufacturer Average Holdng Cost of Manufacturers S td. D evaton of D emand Strategy Model 1 1 Strategy Model 22 Model Strategy 3 3 Fgure 3.8. Comparson of Holdng cost of Manufacturer 69

82 S hortag ec os t ($) H oldng C os t ($) followng objectve functons: Mnmze 0.5Z TCSC 0.5Z LRSC As suggested by the strateges n Table 3.2, as the demand varaton ncreases, decson can be made for both manufacturers to order raw materal n a bg batch and to hold moderate amount of safety stock rather than to order usng lot for lot polcy and to hold hgh amount safety stock. Ths decson avod both manufacturers from experencng frequent backorder. Fgure 3.7 proves that strateges 2 and 3 produce lower backorder cost compared to strategy 1 whle Average Holdng Cost of Retalers S td. D evaton of D emand Strategy Model 1 Strategy Model 2 2 Model Strategy 3 3 Fgure 3.9. Comparson of Holdng cost of Retaler Average Shortage Cost of Retalers S td. D evaton of D emand Strategy 1 Strategy 2 Strategy 3 Model 1 Model 2 Model 3 Fgure Comparson of Shortage cost of Retaler 70

83 strategy 3 outperforms strategy 2 n two cases (σ = 50, 100). It can also be notced from Table 3.2 that the optmal quantty of safety stock under strategy 3 s always the lowest n every cases. It s reflected n Fgure. 3.8 where strategy 3 s superor n reducng the holdng cost n most cases. Moreover, reducton n holdng cost s drven by the fact that fulfllng the shortage quantty experenced by one manufacturer leads to reducng excess nventory of the donor manufacturer and hence reducng nventory n the system. However, retalers requre settng slghtly hgh target stock level to mantan the responsveness to the customer demand. Even though ths decson ncreases the average holdng and shortage costs as shown n Fgure. 3.9 and 3.10, respectvely, ths stock level s more effectve to reduce the loss rate of the supply chan Concluson A mult-objectve analyss of a perodc revew nventory problem for two-echelon supply chan system was nvestgated. Three strateges of coordnated replenshment are proposed to manage nventory n effcent way whle tryng to mprove the system performance by smultaneously mnmzng total cost and loss rate of the supply chan. For ths purpose, a mult-objectve DE was adopted to the problem n queston and appled for each of the three strateges. Fnally, the most proftable stuaton was examned n whch all members n the chan can acheve proportonal satsfacton under these conflctng goals. The result shows that the coordnaton strategy becomes more effectve as the uncertanty ncreases n the system. By cooperatng, manufacturers can avod frequent backorder and reduce excess n- 71

84 ventory n the system. Even though retalers are requred to keep a bt hgh nventory level to mantan the good responsveness to the customer demand, ths stock level s more effectve to reduce the loss rate of supply chan. Future studes should be devoted to more general and complcated confguratons of the supply chan system n real world. Relyng on the ablty of the proposed MODE revealed n ths study, ths seems to be straghtforward. It s also meanngful to move on the mult-objectve optmzaton to derve the preferentally optmal soluton by revealng value system of the central decson maker. 72

85 Chapter 4 AN ENHANCED TWO-PHASE FUZZY PRO- GRAMMING MODEL FOR MULTI OBJECTIVES SUPPLIER SELECTION PROBLEM 4.1. Introducton In global and compettve market, the need for establshng a longer-term relatonshp that fosters cooperaton among supplers and ther customers has been hghlghted. However, many purchasers fnd t dffcult to determne whch supplers should be targeted as each of them has varyng strengths and weaknesses n performance. Moreover, the mportance of each crteron tends to vary from one purchaser to others. Ths problem becomes more complcated as the smultaneous evaluaton s requred n terms of qualtatve and quanttatve crtera. So, every decson must be ntegrated by tradng off performances of dfferent supplers at each supply chan stage. One of the man characterstc of suppler selecton s that ths task s characterzed by an mprecson and ncomplete of data whch results n vagueness of nformaton related to decson crtera. Sto- 73

86 chastc models are usually based on representaton of exstng uncertanty by probablty concepts and are, consequently, lmted to tacklng the uncertantes captured [87]. Moreover, the estmaton of probablty dstrbuton s dffcult to carry out n a fuzzy envronment because of the mprecson of the data. Ths s why, Fuzzy set theory (FST) s suggested as an approprate tool to handle ths problem effectvely. A number of studes have been devoted to examnng suppler selecton methods. Quanttatve technques have become ncreasngly appled recently. A comprehensve revew of numerous quanttatve technques used for suppler selecton has been done by Weber et al. [1]. They found that lnear weghtng models, mathematcal programmng models and statstcal/probablstc approaches have been most common approaches. Some researches used a sngle objectve, such as cost, to evaluate supplers. Kaslngam & Lee [88] developed an nteger programmng model to select supplers and to determne order quanttes wth the objectve of mnmzng total supplyng costs whch nclude purchasng and transportaton costs. Caudhry at al. [89] used lnear and mxed bnary nteger programmng to mnmze aggregate prce consderng both all unt and ncremental quantty dscount. As an extenson of sngle objectve technques, mult-objectve mathematcal programmng has been proposed to solve a more complex suppler selecton problem. Weber et al. [90] combned multobjectve programmng (MOP) and Data Envelope Analyss (DEA) to deal wth non-cooperatve suppler negotaton strateges where the selecton of one suppler results n another beng left out of the soluton. Dahel [91] studed a mult-objectve mxed nteger programmng model to select supplers and allocate product to them n mult- 74

87 product envronment. Xa & Wu [92] mproved the Analytcal Herarchy Process (AHP) usng rough set theory and mult-objectve mxed nteger programmng to determne the best supplers and optmal quantty allocated to each of them n the case of multple sourcng, multple product wth multple crtera. Kokangul & Susuz [93] proposed an ntegraton of analytcal herarchy process (AHP) and nonlnear nteger MOP to determne the best suppler and optmal order quantty among them that smultaneously maxmze total value of purchase and mnmze total cost of purchase. Chamodrakaz et al. [94] provded new approach of two-stage suppler selecton problem. At the frst stage, an ntal screenng s performed through the enforcement of hard constrant on the selecton crtera, and n the second stage, fnal selecton s performed usng a modfed varant of fuzzy preference programmng (FPP). Eroll & Farell [95] used qualtatve and quanttatve factors n the suppler selecton. A fuzzy QFD (Qualty functon Deployment) s used to translate lngustc nput nto qualtatve data and then combne t wth other quanttatve data to develop a mult-objectve mathematcal programmng model. Ths research focuses on fuzzy mult-objectve lnear programmng (fuzzy MOLP) to deal wth the suppler selecton problem. Kumar et al. [96] developed a fuzzy mult-objectve nteger programmng approach for vendor selecton problem subject to constrants ncludng buyer s demand, vendors capacty, and derved an optmal soluton usng max-mn operator (Zmmermann s approach). To evaluate the performance of the model, they perform senstvty analyss on the order allocaton and objectve functon by changng the degree of uncertanty n vendor capacty. Amd et al. [97] solved fuzzy MOLP suppler selecton problem by applyng weghted addtve method to facltate an asymmetrc fuzzy decson makng tech- 75

88 nque. Snce they found the performance of such a method to be nadequate to support decson makng process, α-cut approach s then proposed to mprove the resulted achevement level. Later on, Amd et al. [98] appled weghted max-mn approach n suppler selecton problem and compared the performance of the proposed approach wth max-mn operator and weghted addtve model. They found that the rato of achevement level of objectves matches the rato of the objectves weght. Although there are a number of publcatons adoptng fuzzy programmng models n suppler selecton problem n the lterature, most of them rely on the applcaton of the exstng methods and very few papers are concerned wth the mprovement n the methodologcal process of dervng the optmal soluton. Kagncoglu [99] proposed super-transtve approxmaton to determne the weghts of objectves and constrant n formulatng fuzzy MOLP model n suppler selecton and solved the model usng max-mn operator and weghted addtve model. Yucel & Guner [100] proposed a new method of weghts calculaton n fuzzy MOLP suppler selecton. Both researches mentoned above only focus on the process for weghts calculaton for fuzzy objectve and constrants. It has been verfed that solvng fuzzy MOLP usng max-mn operator may not result n an optmal soluton (Tzeng & Tsaur [101]; Tzeng & Chen [102]; L, [103]). Such a lmtaton has been resolved by L et al. [103] who proposed two-phase approach to compute effcent solutons of fuzzy MOLP problems as the mprovement of compromse approach of Wu & Guu [104]. They found that the performance of compromse approach decreases when the DM prefers to choose the mnmum acceptable achevement level closer or equal to the most optmstc value. In ther proposed method, the mnmum 76

89 acceptable achevement level s set to the soluton of max-mn operator. In ths sense, the performance of compromse approach can be mproved and, on the other hand, the dsadvantage of max-mn operator can be overcome. However, the two-phase approach wll face the same obstacle f max-mn operator outputs the result closer or equal to the most optmstc value, and hence, cannot provde the mprovement. To tackle the above-mentoned shortcomngs and to help obtan a more reasonable compromse soluton, ths paper proposes an enhanced two-phase approach of fuzzy MOLP by ntroducng addtonal varables whch control the relaxaton of resulted overall achevement level and apply t to solve suppler selecton problem. For that purpose, a suppler selecton model s proposed n whch net cost mnmzaton, servce level maxmzaton and purchasng rsk mnmzaton are ncorporated as fuzzy goals. The frst two crtera are cted most often n orderng decson [105]. Purchasng rsk s ncluded as one objectve to measure the rsk of potental loss ncurred f purchaser allocates a certan amount of product to purchase to a certan suppler. To ths end, the Taguch loss functon (TLF) s used to quantfy ths rsk. AHP s employed to determne the relatve mportant between fuzzy goals and constrants Problem Formulaton In ths secton, we formulate a mathematcal model of fuzzy MOLP suppler selecton. The followng notatons are defned n order to descrbe the model. = ndex for suppler ( = 1, 2,.., N) D = demand of buyer (unt) B = total budget of buyer to purchase product ($) 77

90 x = order quantty to suppler (unt) p = unt prce of suppler ($) f r C = servce level of suppler (% fulfllment) = purchasng rsks of suppler (% rsk) = capacty of suppler (unt) The MOLP model for suppler selecton s as follow: Mn Z (4.35) n 1 p x 1 Max Z (4.36) n 2 f x 1 Mn Z (4.37) subject to: n 1 n 1 n 3 r x 1 x D (4.38) p x B (4.39) x C (4.40) x 0 (4.41) Eq. (4.19) mnmzes the net cost for orderng product to satsfy demand. Eq. (4.20) maxmzes the servce level of supplers. Eq. (4.21) mnmzes the purchasng rsk when the frm allocates a certan amount of product to purchase to a certan suppler. Eq. (4.22) puts restrcton that order quantty assgned to supplers must satsfy the total demand. Eq. (4.23) ensures that the total cost of purchasng does not exceed the amount of budget allocated by the frm. Eq. (4.24) guarantees that the order quantty assgned to each suppler wll not exceed suppler capacty lmt. Eq. (4.25) s non-negatvty constrant. 78

91 4.3. The proposed Integrated Method Ths secton presents all methods nvolved n the proposed fuzzy MOLP model. Frst, Taguch loss functon s descrbed to quantfy the rsk assocated wth purchasng decson, followed by AHP to calculate a relatve mportance of sub-crtera used to measure rsk as well as the relatve mportance between objectves and constrants n the fnal formulaton. Next, fuzzy MOLP suppler selecton model and an enhanced two-phase approach are presented Taguch loss functon In a tradtonal system, the product s accepted f the qualty measurement falls wthn the specfcaton lmt. Otherwse, the product s rejected. The qualty losses occur only when the product devates beyond the specfcaton lmts, therefore becomng unacceptable [106]. Taguch suggests a narrower vew of qualty acceptablty by ndcatng that any devaton from the qualty target value results n a loss. If the qualty measurement s the same as the target value, the loss s zero. Otherwse, the loss can be measured usng a quadratc functon [107]. There are three types of Taguch loss functons: target s best (two-sded equal specfcaton or two-sded unequal specfcaton), smaller s better and larger s better. If L(y) s the loss assocated wth a partcular value of qualty y, m s the target value of the specfcaton, and k s the loss coeffcent whose value s constant dependng on the cost at the specfcaton lmts and the wdth of the specfcaton, then for target s best two sded equal specfcaton type, target s best two sded unequal specfcaton type, smaller s better type, and larger s better type, the formulaton of L(y) are 79

92 gven s Eq.(4.1)-(4.4), respectvely. L(y) 2 k(y m) (4.1) L(y) 2 2 k1(y m) or L(y) k2(y m) (4.2) 2 L(y) k(y) (4.3) 2 L(y) k/y (4.4) Analytcal herarchy process The analytc herarchy process (AHP) was developed to provde a smple but theoretcally multple-crtera methodology for evaluatng alternatves [108]. The major reasons for applyng AHP are because t can handle both qualtatve and quanttatve crtera and because t can be easly understood and appled by the DMs. AHP nvolves decomposton, par-wse comparsons, and prorty vector generaton and synthess. The procedures of AHP to solve a complex problem nvolve sx essental steps [109]: 1. Defne the unstructured problem and state clearly the objectves and outcomes; 2. Decompose the problem nto a herarchcal structure wth decson elements (e.g., crtera and alternatves); 3. Employ par-wse comparsons among decson elements and form comparson matrces; 4. Use the egen value method to estmate the relatve weghts of the decson elements; 5. Check the consstency property of matrces to ensure that the judgments of decson makers are consstent; and 80

93 6. Aggregate the relatve weghts of decson elements to obtan an overall ratng for the alternatves Fuzzy mult-objectve lnear programmng A lnear mult-objectve problem can be stated as: fnd vector x n the transformed form X T = x 1, x 2,, X n whch mnmze objectve functon Z k and maxmze objectve functon Z l wth Z c x, k 1,2,..., p (4.5) n k 1 n l 1 k Z c x, l p 1, p 2,..., q (4.6) l subject to: x X, X x g(x) b, r 1, 2,..., m (4.7) d d r where constrants. X s the set of feasble soluton that satsfy the set of system d Zmmermann [110] frst adopted the fuzzy programmng model proposed by Bellman and Zadeh [111] nto conventonal LP problems. The fuzzy formulaton for (4.5)-(4.7) can be stated as 0 Z c x ~ Z, k 1,2,..., p (4.8) n k 1 n l 1 k k 0 Z c x ~ Z, l p 1, p 2,..., q (4.9) l subject to: ~ g r x n 1 a r x l ~ b r, r 1,2,..., h (4.10) 81

94 x g a x b, p h 1,..., m (4.11) p n 1 p p x 0, 1, 2,..., n (4.12) The above fuzzy MOLP s characterzed by a lnear membershp functon whose value changes between 0 and 1. The membershp functon for fuzzy objectves are gven as μz k 1 Z k (x) Z 0 max max k Z k (x) mn Z k f f f Z (x) Z Z k mn k k Z k Z (x) Z mn k (x) Z max k max k (4.13) μz l 0 Z l (x) Z l (x) max Z l Z l 1 mn mn f f f Z (x) Z Z l mn l Z (x) Z Z (x) Z l l mn l max l max l (4.14) and the lnear membershp functon for fuzzy constrants s gven as μ gr 0 1 (x) 1 gr (x) b d r r f f f g (x) b b r r r g r r g (x) b d (x) b r r, r 1, 2,..., h (4.15) where d r s subjectvely chosen tolerance nterval expressng the lmt of the volaton of the r-th nequaltes constrants. In the above formulaton, Z max k, Z max mn mn l, Z k and Z l means the maxmum value (worst soluton) and the mnmum value (best soluton) of Z k and Z l, respectvely. They are obtaned by solvng a sngle objectve optm- 82

95 zaton problem respectvely under each objectve functon [112]. Zmmermann [110] proposed a max-mn operator approach to solve the above fuzzy MOLP. The Eq. (4.5)-(4.7) can be transformed nto the followng crsp formulaton by ntroducng addtonal varable λ whch represent an overall achevement level for both fuzzy objectves and constrants. Max λ (4.16) Subject to: λ μzj( x), j 1,2,..., q for fuzzy objectves (4.17) λ μgr( x), r 1,2,..., h for fuzzy constrants (4.18) g ( x) b, p h1,...,m for crsp constrants (4.19) p p x 0, 1, 2,.., n and μzj, μgr, λ [0,1] (4.20) An enhanced two-phase fuzzy programmng L et al. [103] proposed a two-phase approach to compute effcent solutons of fuzzy MOLP as the mprovement of compromse approach of Wu et al. [104]. The steps of two-phase approach are as follow: Step 1: Solve the max-mn operator problem and output the optmal value, say x 0. Step 2: Set the lower bound λ l = x 0 ) for objectve functon and r l = r x 0 ) for fuzzy constrants and solve the followng model to get a fnal soluton x. q j1 j h j r1 Max ω λ β γ (4.21) r r subject to: 83

96 l λ λ μ (x), j 1, 2,..., q (4.22) j j zj γ l r γ μ (x), r 1, 2,..., h (4.23) r gr g (x) b, p h 1,...,m (4.24) p p x 0, 1, 2,.., n (4.25) λ j, γr [0,1] (4.26) q j1 ω j h r1 β r 1, ω, β j r 0 (4.27) It should be noted that the value of mnmum acceptable achevement level s a compromsed preference value of decson maker. However, ths method may not necessarly yeld a feasble soluton when the mnmum acceptable achevement level s closer or equal to the most optmstc value. Moreover, due to the problem structure of suppler selecton under consderaton, formulatng lnear programmng model requres a careful parameter settng because selecton crtera are quantfed usng wde range of numercal nput. Inapproprate parameter settng may also result n nfeasble soluton. To release the above-mentoned shortcomngs and to help obtan a more reasonable compromse soluton, therefore, ths research proposes an enhanced two-phase approach of fuzzy MOLP. Therefore, we propose to solve the followng model to get the fnal soluton x: q h q h Max (1 p) ω jλ j βrγr pε j δr (4.28) j1 r1 j1 r1 Subject to: 84

97 l λ ε λ μ (x), j 1, 2,..., q (4.29) γ j l r j j zj δ γ μ (x), r 1, 2,..., h (4.30) r r gr g (x) b, p h 1,...,m (4.31) p p x 0, 1, 2,.., n and λ, γ [0,1] (4.32) j r q j1 ω j h r1 β r 1; ω j,β r 0 (4.33) 0 ε λ ; 0 δ γ (4.34) j l j r l r where ε and δ r are augmented varables to relax the overall achevement level resulted from the foregong max-mn operator problem, respectvely, and p s a weghtng factor whch control the orgnal objectve functon value and the relaxaton value. Apparently, t s desrable such relaxaton s as small as possble as long as the feasblty s hold Soluton procedures The proposed fuzzy MOLP suppler selecton problem s constructed through the followng steps: Step 1: Defne the crtera for suppler selecton problem Step 2: Construct the MOLP suppler selecton problem accordng to defned crtera (mnmze purchasng cost, maxmze servce level, and mnmze purchasng rsk) and constrant of the buyer and supplers. The purchasng rsk s quantfed as followed: a. Defne sub-crtera b. Measure the relatve mportant of each sub-crtera usng AHP c. For each sub-crteron, defne a target value, calculate loss coeffcent and Taguch loss 85

98 d. Fnd the weghted Taguch loss by employng the output of AHP. Ths value s used n MOLP model as the coeffcent of objectve of mnmzng purchasng rsk Step 3: Fnd membershp functon for each selecton crtera and constrant. a. Determne a lower bound of each objectve by solvng MOLP as a sngle objectve suppler selecton problem usng each tme only one objectve. b. As n a), determne an upper bound of each objectve by solvng MOLP as a sngle objectve suppler selecton problem usng each tme only one objectve. Step 4: Calculate the relatve mportance of crtera and constrants usng AHP. Step 5: Reformulate the MOLP suppler selecton nto equvalent crsp model usng the enhanced two-phase fuzzy MOLP and fnd the set of feasble soluton Numercal Experment and Analyss Suppose that one frm should manage three supplers for one product. Management wants to mprove the effcency of the purchasng process by evaluatng ther supplers. The management consders three objectve functons.e. mnmzng net cost, maxmzng servce level and mnmzng purchasng rsk, subject to constrants regardng the demand of product, suppler capacty lmtaton, frm s budget allocaton, etc. The estmated value of supplers net prce, servce level and supplers capacty are gven n Table 4.1. An allocated budget of the frm to purchase the product s $20,000. The demand s a fuzzy number and s predcted to be about 1400 unt wth refrac- 86

99 Table 4.1. Supplers quanttatve nformaton Net Cost/unt ($) Servce Level (% fll rate) Capacty (unt) Suppler Suppler Suppler Table 4.2. The specfcaton lmt and range of fve leadng crtera Crtera Target Value Range Specfcaton Lmt Qualty (% defect rate) 0% 0-3% 3% Order fulfllment 100% 80%-100% 80% On-tme delvery (days) 0-10 to 0 and 0 to 5 10 days earler, 5 days delay Dstance/Proxmty (mles) The closest 0-40% 40% Suppler Table 4.3. Actual value of the four sub-crtera Qualty (% defect) Order Fulfllment (% unt) On-tme delvery (days) Dstance/ Proxmty (mles) 1 1.0% 90% % 95% % 97% -1 9 ton of -100 and 150 unts. Purchasng rsk s measured from four sub-crtera: qualty, order fulfllment, on-tme delvery, and dstance/proxmty. Concernng product qualty, DM set the target value of defect parts at zero and the upper specfcaton lmt at 3% to ndcate the allowable devaton from the target value. Zero loss wll occur for 0% defectve parts and 100% loss wll occur at the specfcaton lmt of 3% defectve parts. For order fulfllment rate, the loss wll be zero for the suppler who fulflls all order quantty (100%) and the total loss wll occur f suppler can only satsfy 80% of total order. For on-tme delvery, the specfcaton lmt of delvery s 10 days and 5 days for early and de- 87

100 lay shpment, respectvely. The DM wll tolerate the shpment for a maxmum of 5 days delay and 10 days early. In ths case, the manufacturer wll ncur 100% loss f shpment s delayed for 5 days or earler for 10 days from scheduled shpment, and on contrary, no loss ncurred f the shpment s on tme. For dstance/proxmty, a zeroloss wll occur at the closest suppler and the specfcaton lmt s up to 40% of the closest suppler. It means that the manufacturer wll ncur 100% loss f there are other supplers n consderaton whose dstance reaches the specfcaton lmt. The specfcaton lmt and range value of each selecton crteron are presented n Table 4.2. Calculatng the value of k from Eq. (4.1)-(4.4) gves , 0.64, and 6.25 for qualty, order fulfllment, dstance/proxmty, respectvely. For on-tme delvery, k1 = 4 and k2 = 1 (snce an unequal two sde specfcaton s consdered for on-tme delvery, there exsts two loses coeffcents, k1 and k2). The actual values n Table 4.3, together wth the value of loss coeffcent k prevously calculated for these four sub-crtera, are used to calculate the ndvdual Taguch Loss for each suppler for each crteron usng Eq. (4.1)-(4.4). For example, the actual qualty value of suppler A s 1.0% defectve rate, whch means 1.0% devaton from the target value. Indvdual Taguch loss s then calculated by enterng ths value nto eq. (3.1)- (3.4). The result s shown n Table 4.4. Suppler Suppose the par-wse comparson matrx and local weght for Qualty Table 4.4. Taguch loss ( )) Order Fulfllment On-tme delvery Dstance/ Proxmty Weghted Taguch Loss Normalzed Taguch Loss Total

101 Table 4.5. Par-wse comparson matrx a) Sub-crtera Qualty Order Fulfllment delvery Proxmty Weght On-tme Dstance / Local Qualty Order Fulfllment 1/ On-tme delvery 1/2 1/ Dstance/Proxmty 1/5 1/5 1/ b) Crtera Net Cost Servce Level Purchasng rsk Demand Net prce 1 1/2 1/3 1/3 Servce Level 2 1 1/2 1/3 Purchasng Rsk /2 Demand Local Weght each of these four sub-crtera usng AHP s that shown n Table 4.5a. The consstency Rato (CR) of table 5 s (less than 0.1). The weghted Taguch loss s then calculated usng Taguch losses and the local weght of crtera (Table 4.5a). Table 4.4 shows the weghted Taguch loss and the normalzed Taguch loss for each suppler. The normalzed Taguch loss, whch represents the loss score, s then used as a coeffcent of purchasng rsk n fuzzy MOLP. Based on supplers data n Table 4.1 and the normalzed Taguch Loss n Table 4.4, the fuzzy MOLP suppler selecton of the presented problem s constructed accordng to Eq. (4.8)-(4.12) as follow: Mn Max Mn Z 1 10x1 12x 2 9x3 ~ Z Z Z x 1 0.9x x 3 ~ x x x 3 ~ Z 0 2 Z 0 3 Subject to: x x x ~ x 12x 2 9x3 1 x

102 x x x 0 The crtera and constrant can be consdered equally mportant and added together for comparson. However, such a comparson s generally unfar due to certan crtera that that may be more mportant than others. In ths model, the weght of cost, servce level, purchasng rsk and demand are derved from AHP. Table 4.5b shows the par-wse comparson matrx and local weghts for crtera and constrant. The consstency Rato (CR) s (less than 0.1). Calculatng the membershp functon usng max-mn operator (Eq. (4.16)) gves 0.566, and for z1 x 0 ), z2 x 0 ), z3 x 0 ), respectvely. The crsp formulaton of the above fuzzy MOLP usng the enhanced two-phase approach accordng to Eq. (4.28)- (4.34) s gven as Max (1 p p) 0.447λ λ λ γ 1 ε ε ε δ subject to: ε ε ε δ 1 λ 1 λ 2 λ 3 γ x x1 12x x1 12x x 9x 0.90x x 37 x1 x x

103 12x 16x2 12x3 1 x x x ω 1 ω2 ω3 β ε , ε , ε , δ λ, λ x, x, λ 2 3, x, γ 3 1, ω [0,1] 1, ω 2, ω 3, β 1 0 In ths problem, the orgnal two-phase approach fals to yeld a feasble soluton. The constrant assocated wth λ 1 cannot be satsfed because the value of λ 1 equal to whch s lower than the desgnated value of ts lower bound (λ l 1 =0.566). Table 4.6 provdes a set of the feasble solutons resulted by utlzng the proposed method whch ncludes the overall achevement level, ndvdual achevement level, orderng plan and the objectve value of the equvalent crsp model along wth the upper and lower bounds of fuzzy objectves and constrant. As shown n the Table 4.6, the overall achevement level of the proposed approach s known to be better than that of Max-mn operator (λ = 0.566) when value of p s lower than 0.5. When p s equal or greater than 0.5, the overall achevement level decreases. A lower p value ndcates the model attempts to fnd a soluton by relaxng more the crtcal objectve related to the correspondng constrant to acheve a better achevement level of the other objectve. In ths model, the servce level (Z 2 ) and the purchasng rsk (Z 3 ) are a crtcal objectves as the correspondng constrants are relaxed for almost any p value (crtcal constrant). Ths mples that the 91

104 92

105 model tends to sacrfce the performance of these objectves because t s at less of cost decreasng the performance of these objectves rather than decreasng other. The greatest relaxaton s occurred when p s The achevement level of Z 2 s totally relaxed ( z2 = 0) to acheve a better achevement level for Z 1 followed by Z 3. The achevement level of Z 2 reaches the best possble value for the entre value of p when Z 3 s relaxed for p s equal to 0.54 and 0.6. Moreover, Z 1 s free from relaxaton as t s the most mportant objectve, whose assgned weght s the hghest, accordng to the DM s preference (ω 1 ω 2 > ω 3 ). In ths fuzzy formulaton, all supplers are selected to supply the product to the frm. Moreover, upon more careful observaton, t s revealed that orderng to Suppler 1 and Suppler 3 s more preferable. It s nferred form the order quantty assgned to these supplers as they receve the bggest amount of order quantty whch s equal/closer to ther full capacty. In ths case, t s not proftable to order more quantty to Suppler 2 because t offers the most expensve prce and the hghest purchasng rsk among others. As mentoned above, the prce (net cost) s put as the man concern of the DM (the hghest weght). Thus, placng a smaller order quantty to suppler 2 s the best decson. Wthout loss of generalty, suppose that the DM wants to select p equals In ths soluton, z1 and z3 mprove to and 0.980, respectvely, whch results n the best value of Z 1 and Z 3. However, the DMs should carefully notce that the achevement level of servce level, the second most mportant crtera, declnes toward the worst performance ( z2 = 0). Eventually, fnal decson should be made by the DM to choose the most favorable decson among the feasble alternatve solutons accordng to hs/her preference. 93

106 4.5. Concluson Suppler selecton s an essental task wthn the purchasng functon that needs careful screenng under some qualtatve and quanttatve crtera. Moreover, most nformaton requred to assess suppler s usually not known precsely and typcally fuzzy n nature over the plannng horzon. Concernng such characterstcs, ths research proposes ntegrated methodology for FMOLP model for suppler selecton. In formulated problem, the most common fuzzy objectves and parameter n practcal orderng decson have been presented. AHP s used to facltate the subjectve judgment on qualtatve/quanttatve crtera and TLF s employed to quantfy the purchasng rsk. For the purpose of solvng the FMOLP problem, the enhanced two-phase fuzzy programmng model has been developed. Through numercal experment, we demonstrate the promsng advantage of our proposed approach over the max-mn operator (Zmmermann s approach). Fnally, ths ntegrated approach provdes a set of potental feasble solutons whch gude DMs to select the best soluton accordng to ther preference. Ths also refers to a mult-objectve optmzaton problem that should be concerned n future studes. 94

107 Chapter 5 POSSIBILISTIC PROGRAMMING MODEL FOR FUZZY MULTI-OBJECTIVE PERIODIC REVIEW INVENTORY IN TWO-STAGE SUP- PLY CHAIN 5.1. Introducton In today s global market, nnovatve technologes have shortened the product lfe cycle and ncreased the demand varablty along the supply chan system. Ths ultmately forces enterprses to ncreasngly focus on the role of nventory. The excess nventory n the supply chan blocks the cash flow and ndeed gves a tremendous mpact on the enterprse whle nsuffcent nventory results n poor responsveness to the customer demand. In the context of supply chan (SC) system, determnng nventory control polcy s qute challengng due to the nteracton among the dfferent levels wth dfferent goals. Even such a polcy can be dentfed, t usually has a very complex structure and s not sutable for mplementaton. Therefore, t s reasonable to consder smple, cost-effectve heurstc polces, 95

108 whch can be readly applcable n practce [113]. In ths regard, the perodc revew nventory system plays an mportant role because of ts smplcty and easness n mplementaton, and therefore, t s mostly employed at dfferent levels of the supply chan network [114]. The ssue of consderng uncertanty n nventory control has receved a great deal of concern n the feld of producton/nventory management. In the context of perodc revew nventory system, many research have focused on the use of stochastc approach under dfferent concerns. Whle the usefulness of stochastc approach has been documented, t s not always applcable n codng the nformaton regardng the mprecson of data and vagueness of goals. To avod ths drawback, the fuzzy approach s employed for modelng uncertan parameters and goals n nventory problem. The fuzzy approach copes wth the uncertanty related to mprecson due to unavalablty and ncompleteness of data as well as vagueness of goals n whch the use of conventonal probablty dstrbuton s mpossble n ths case. Moreover, t s frequently emphaszed n the lterature that fuzzy approach has had a great mpact n preference modelng and mult-objectve problem and has helped brng optmzaton technques closer to the users needs [115]. Recently, the fuzzy approach has also been employed extensvely for the modelng of parameters n perodc revew nventory system. Petrovc et al. [116] consdered fuzzy demand n a sngle product nventory control of dstrbuton supply chan of sngle-warehouse, multple-retaler n whch nventory of each member s replenshed by adoptng perodc revew system. They proposed nteractve method to determne the optmal revew perod and order-up-to level nventory that gve the lowest total cost n the fuzzy sense characterzed by lnear member- 96

109 shp functon. Vjayan and Kumaran [117] examned the mpact and senstveness of the mprecseness of fuzzy cost components to the orderng quantty n perodc revew and contnuous revew nventory models n whch a fracton of demand s backordered, and the remanng fracton s lost durng the stockout perod. Ln [118] provded a soluton procedure to fnd the optmal revew perod and optmal lead tme n perodc revew nventory model consderng the fuzzness of expected demand shortage and backorder rate and used the sgned dstance method to defuzzfy the total expected annual cost. Dey and Chakraborty [114] developed the perodc revew nventory model wth a constant lead-tme n a mxed fuzzy and stochastc envronment by ncorporatng the customer demand as a fuzzy random varable. The am s to determne the optmal nventory level and the optmal perod of revew such that the total fuzzy expected annual cost whch s defuzzfed by ts possblstc mean s mnmzed. Chang et al. [119] studes a mxtures model of perodc revew nventory for sngle-retaler sngle-suppler nvolvng varable lead tme wth backorders and lost sales by further consderng the fuzzness of lead-tme demand and annual average demand. Usng centrod method for defuzzfcaton, they found the optmal soluton for order quantty and lead tme n the fuzzy sense such that the total cost has a mnmum value. Earler research of fuzzy perodc revew system n the SC were lmted n mplementaton n that the nventory problem s solved for a smple SC structure to satsfy a sngle objectve problem. In current globalzaton, however, the SC network has become dynamc and complex n a structure whch mposes a hgh degree of uncertanty and ths sgnfcantly nfluences the overall performance of the SC network [120]. Moreover, t s mostly the case that each SC member 97

110 consders more than one factor n ther goals to sustan or mprove ther compettve poston. The nature of operatons makes one goal often conflcts wth each other and complexty n SC structure makes t s dffcult to algn all of these goals. Ths research proposes a perodc revew nventory model n a typcal SC system where multple manufacturers, multple retalers s consdered. The am of ths paper s to develop a mult-objectve perodc revew nventory model n an uncertan envronment by smultaneously ncorporatng the mprecse nature of some crtcal parameters such as demand, lead tme and cost parameters. N ths model, each retaler places orders perodcally to multple manufacturers and each manufacturer s replenshed perodcally from an external suppler. The problem s to determne the orderng polcy for raw materal and the safety stock level of each manufacturer, and the order allocaton and the target stock level of each retaler that can yeld satsfactory mnmum total cost whle mantanng low loss rates. Several alternatve solutons are obtaned by solvng a proposed mult-objectve possblstc mxed nteger programmng (MOPMIP) nventory model. Ths research contrbutes to the exstng lterature n the followng ways. Frst, t presents a comprehensve and practcal, but tractable, MOPMIP nventory model for two-stage supply chan system consderng the mprecson of some crtcal data. And second, t ntroduces an ntegrated soluton procedure whch provdes a systematc framework to facltate the fuzzy decson-makng process, enablng the DM to adjust the decson and to obtan a more preferred satsfactory soluton n both decentralzed and centralzed SC system. 98

111 5.2. Model Formulaton System descrpton The model has the followng assumptons. 1. Only a sngle product s consdered n the model. Wthout loss of the generalty, the manufacturer uses one unt of raw materal to produce one unt of product. 2. For both manufacturer and retaler, only one order s allowed to place at any perod. 3. The producton rate of the manufacturer s assumed fxed and hgher than the average demands. 4. Unfulflled demand at the manufacturer s consdered as backorder whle unfulflled demand at the retaler s consdered as shortages. 5. Due to ncompleteness and/or unavalablty of requred data over the specfed plannng horzon, uncertanty n demand, lead tme and cost parameter are assumed to be mprecse (fuzzy) n nature wth a trangular pattern. The supply chan n ths study operates under the make-tostock envronment, n whch mprecse parameters such as demand, Manufacturer Retaler customer Suppler j Fgure 5.1. Supply chan Network 99

112 lead-tme and cost parameters are consdered. The supply chan conssts of suppler, manufacturer, retaler and the end customers of whch each of these members s a representatve dfferent supply chan echelons. However, ths research focuses on a relatonshp between manufacturer and retaler (supplers and end customers are consdered as external members n the chan) as shown n Fgure Manufacturer The manufacturer uses the perodc revew system to control the nventory and orders raw materal based on the dscrete lot szng polcy. In addton, the manufacturer also holds some safety stock to cover the effect of uncertanty n demand and delvery lead-tme. Ths safety stock wll only be used when a normal nventory level cannot satsfy the retalers demand and t must be flled back as soon as possble after havng used. - Retaler The retaler uses the perodc revew wth order up to the target stock level (T, R) polcy. The retaler makes a sngle order n every cycle to the manufacturer to rase up ts nventory to the target stock level. In addton, snce the retaler may receve replenshment from more than one manufacturer, the retaler must determne the approprate strategy to allocate the order quantty to each manufacturer. Any unfulflled demand s consdered as lost sales due to the fact that n practce the customers have a plenty of choces to acqure the products and buy the same product from other shops when the stock out occurs at the frst shop, especally when the retaler s vewed as department stores, supermarkets or conventonal stores. 100

113 Formulaton of the model In ths secton, we develop a mult-objectve mx-nteger programmng nventory model stated n the prevous secton. The model s descrbed based on the followng notaton lsted for major parameters. Index T = Number of plannng horzon t = Perod (t = 1,2,,T). = Number of manufacturer j = Number of retaler tp = Number of days n each perod Parameters of Manufacturer Forecast demand of manufacturer at perod t. Q Order quantty of manufacturer at perod t. Qp Sr Producton quantty of manufacturer durng suppler s late delvery at perod t. Amount of raw materal left at manufacturer after producton durng suppler s late delvery at perod t. PR Producton rate of Manufacturer. lead tme of suppler to manufacturer at perod t. Qpr Producton quantty of manufacturer at perod t. Es Endng stock of raw materals of manufacturer at perod t. Ess Endng safety stock of manufacturer at perod t. Qm Orderng quantty of manufacturer at perod t. Qbj,t Backorder quantty of manufacturer for retaler j at perod t. Qslj,t Sales volume of manufacturer at perod t. Qa Avalable quantty of product at manufacturer at perod t. EI Endng nventory of manufacturer at perod t. 101

114 Parameters of Retaler Total end customer demand at perod t. dbj,t drj,t End customer demand at retaler j at perod t before recevng replenshment. End customer demand at retaler j at perod t after recevng replenshment. Lead tme of manufacturer to retaler j at perod t. Ir t Endng nventory of product of retaler j at perod t. Qsrj,t Shortage quantty of retaler j at perod t after recevng replenshment. Qorj Order quantty of retaler j to manufacturer at perod t. LRj Loss rate of retaler j. Cost Parameters Order cost of manufacturer at perod t ($). Unt purchasng cost of manufacturer ($). Unt producton cost of manufacturer ($). Unt holdng cost of raw materal of manufacturer ($). Unt holdng cost of product of manufacturer ($). Unt backorder cost of manufacturer ($). Unt transportaton cost of manufacturer ($). Unt purchasng cost of retaler j ($). Unt holdng cost of fnshed product of retaler j ($). Unt shortage cost of fnshed product of retaler j ($). TCM Total Cost of manufacturers ($). TCRj Total Cost of retalers ($). Decson Varables LS Lot szng polcy of manufacturer, LS = 1 f manufacturer 102

115 place the order, and LS = 0 f not place the order but combne t to foregong order placement. SS Safety stock level of manufacturer (unt). Rj Target stock level of retaler j (unt). OAj Order allocaton of retaler j to manufacturer. It should be noted that the notatons wth wave sgned on t ndcate parameters tanted wth uncertanty (mprecse parameters) The objectve Functons In today s dynamc globalzaton, competton among organzatons exhbts a dynamc nature and t s based on varous factors such as cost, tme, servce level, qualty, etc. In such envronment, the organzaton may formulate one or more factors n ther goals to sustan or mprove ther compettve poston. In ths sense, t s assumed for the case problem we deal wth that the manufacturers consder cost whle the retalers consder cost and loss rate as the core factors of ther compettveness. Followng are the objectve functons for the proposed nventory model. T T T r ~ Q m ~ ~ Qpr h Es t1 t1 t1 Mn TCM T T J (5.1) T J ~ ~ ~ f (Ess EI ) b Qbj,t t Qslj,t t1 t1 j1 t1 j1 T I T T Mn TCR p ~ Qsl c ~ Ir ~ s Qsr (5.2) j t1 1 j j,t t1 j j,t t1 j j,t Mn T Qsr j,t LR j ~ t1 d j (5.3) 103

116 104 where 1 t, Bs BI Qb f D, Q ~ 0 max (5.4) otherwse ) ( ) ( f ) ( PR ls tp Q Sr Q Sr Qp PR ls tp Qp Qpr ~ ~ (5.5) otherwse ) ( ) f ( 0 ) ( ) ( t, PR ls tp Q Sr PR ls tp Q Sr Es ~ ~ (5.6) otherwse and and f and f and f 0 j j 1 1 j j 1 j j j j 1 j j j j j j 1 j j BSS SS Qor BI Qpr BI Qpr Qor SS BSS BI Qpr Qor SS BSS Qa Qor BI Qpr Qor Qa Qor Qor BI Qpr BSS SS BSS Qor Qa ESS f (5.7) otherwse ) ( ) ( and and f and f 0 ) ( ) ( ) ( j j,t j j 1 j j,t j j 1 j j,t j j 1 j j,t j j 1 j j,t j j 1 BSS SS Qb Qor BI Qpr Qb Qor BI Qpr SS BSS Qb Qor BI Qpr SS BSS BSS SS Qb Qor BI Qpr Qb Qor BI Qpr EI (5.8)

117 Qor j Qbj,t f Qa Qor j Qbj,t Qsl j,t (5.9) Qa Qbj,t otherwse db j,t ~ lm ~ (d /tp) (5.10) j, t dr j, t d j,t db j,t ~ (5.11) The orderng quantty of manufacturer (Q) s drectly nfluenced by lot szng polcy (LS) whch s adopted for orderng raw materal. For example, f there are 6 plannng horzon, there are 2 T-1 or 32 possble orderng polces for manufacturer to choose. In ths case, the frst polcy may follow lot for lot or place order at every perod. The second possble polcy may combne order of perod 1, 2 and 3 and then use lot for lot the rest four perods. After the best pattern of LS s selected, the manufacturer wll check the amount of nventory on hand at the begnnng of the perod (Eq. 5.4). If the amount on hand s less than the sum of the demand and the amount to fll back the safety stock, then the manufacturer wll place the order to suppler. Otherwse no order wll be ssued. The uncertanty of the delvery lead-tme from the suppler has an nfluence on the producton quantty of manufacturer. Normally, producton tme at perod t s equal to the revew perod deducted by actual delvery lead tme of the suppler. However, when late delvery happens, the manufacturer can stll start producng the product (Qp) f raw materal s avalable on hand. Otherwse, the manufacturer has to wat for new replenshment. Consequently, we can determne total producton quantty and endng raw materal on hand wth Eq. (5.5) and Eq. (5.6), respectvely. As shown n Eq. (5.7) and (5.8), the endng safety stock (ESS) 105

118 and the endng nventory of fnshed product (EI) have dfferent meanngs. Ess s the amount of safety stock left at the end of each perod whle EI s the number of fnshed products that s produced beyond the retaler s demand and s left from fulfllng the safety stock. The sales volume (Qsl) of the manufacturer s determned based on total order quantty of retaler (Eq. (5.9)). When total order quantty of retaler exceeds the avalable quantty on hand, the shortage quantty wll be backordered next perod (Qb). The retaler makes a regular order to the manufacturer perodcally to rase up the nventory to the target stock level. The order quantty (Qorj,t) s determned by comparng the endng nventory (Irj,t-1) wth the desred target stock level (Rj), whch s equal to (Rj- Irj,t-1). As stated n Eqs. (5.10) and (5.11), the end customer demand ( ) at retaler s dvded nto two types: the demand before recevng replenshment (dbj,t) and the demand after recevng replenshment (drj,t). The shortage occurred when end customer demand exceeds the nventory and the shortage quantty (Qsrj,t) s consdered as a full lost Soluton methodology Generally, fuzzy programmng can be classfed nto flexble programmng and possblstc programmng [121]. Flexble programmng deals wth flexblty n the gven target value of the objectve functon and the elastcty of constrants whle the possblstc programmng handle the uncertanty related to ll-known parameter due to the lack of hstorcal data. The possblstc programmng uses possblty dstrbuton to measure the occurrence of possble value 106

119 for each of uncertan parameter. Ths value s mostly determned based on avalable data as well as experts knowledge. Based on the above defnton and snce we are dealng wth mprecse parameters, possblstc programmng s used to handle mprecse parameters such as demand, lead tme and cost parameters n the proposed nventory model. Thus, our proposed nventory model s a multobjectve possblstc mxed nteger programmng (MOPMIP) nventory model. Subsequently, a soluton procedure s proposed to solve MOP- MIP nventory model. Frst, MOPMIP nventory model s transformed nto the auxlary crsp MOPMIP model. Ths step adopts modfed S- curve membershp functons to represent all objectve functons. The Torab and Hassn s aggregaton functon s then proposed to solve the auxlary crsp MOMIP model The equvalent auxlary crsp MOPMIP model Solvng optmzaton problem requres a condton n whch all varables and parameters nvolved must be defned n crsp form. Therefore, n the case where fuzzness s embedded to optmzaton problem, such fuzzness should be cleared up by transformng the problem nto equvalent crsp formulaton before movng to the soluton stage. In ths research, ths transformaton s carred out usng possblstc programmng. For ths purpose, one method of possblstc programmng named Jmenez method s appled due to ts computatonal effcency [122]. The Jmenez method s based on the defnton of the expected value and the expected nterval of a fuzzy number. Assume that s a trangular fuzzy number. The followng equaton can be defned as the possblty dstrbuton of. 107

120 pes x c pes mos fc x c x c ( ) f mos pes c c mos 1 f x c μc ~ ( x) (5.12) opt c x mos opt g c x c c( x) f opt mos c c pes opt f x c or x c 0 where, and are the three promnent ponts (the most lkely, the most pessmstc and the most optmstc value), respectvely. Eq. (5.13) and (5.14) defne the expected nterval (EI) and the expected value (EV) of trangular fuzzy number. EI(c) ~ 1 (c 2 c c 1 E1,E2 fc (x)dx, pes c 1 0 mos 1 ), (c 2 mos 1 0 g 1 c c (x)dx opt ) (5.13) E EV(c) ~ c 1 E 2 c 2 c pes 2c 4 mos c opt (4.14) Assumng all mprecse parameters n ths model follow the trangular pattern, accordng to Eq. (5.13) and (5.14) the auxlary crsp formulaton of the proposed nventory model (except for Eq. (5.7), (5.8) and (5.9)) can be reformulated as follows. Mn TCM T r t1 T h t1 T J t1 j pes 2r 4 r mos 2h h 4 pes mos b 2b 4 pes 1 mos opt T m Q t1 T f Es t1 opt b Qbj,t opt pes pes T 2m 4 t1 mos 2 f 4 J t j1 mos pes m f opt opt 2t 4 mos Qpr Ess EI opt t Qslj,t (5.15) 108

121 109 T 1 t j,t opt mos pes T 1 t j,t opt mos pes T 1 t I 1 j,t opt mos pes j Qsr s s s Ir c c c Qsl p p p TCR Mn (5.16) T 1 t opt j,t mos j,t pes j,t j,t j d d d Qsr LR 2 4 Mn (5.17) t, 1 1 opt mos pes t, Bs BI Qs f D D D, Q max (5.18) otherwse 4 2 f 4 2 opt mos pes opt mos pes PR ls ls ls tp ) Q (Sr Q Sr Qp PR ls ls ls tp Qp Qpr (5.19) otherwse 4 2 f opt mos pes opt mos pes PR ls ls ls tp ) Q (Sr PR ls ls ls tp ) Q (Sr Es (5.20) tp d d d lm lm lm db opt j,t mos j,t pes j,t opt j,t mos j,t pes j,t j,t (5.21) The modfed S-curve membershp functon Membershp functon s a functon whch represent of satsfacton degree (possblty degree) of a certan varables. There are many types of membershp functon such as lnear, exponental, etc.

122 In ths research, the modfed s-curve membershp functon (Fgure 5.2) s employed because t s consdered much more easly n handlng compared to other membershp functons [123]. The modfed s-curve membershp functon s a partcular case of the logstc functon wth specfed parameters and ts formulaton s defned as follow: μ B 1 Ce x x x x a a a b ( x) δ x x x x (5.22) x x b x b where μ s the degree of satsfacton. The term δ determnes the shape of membershp functon μ(x), where δ > 0. The larger the value of parameter δ, the greater the vagueness s. As can be seen n Fgure 5.2, the degree of satsfacton s redefned as μ(x) Ths range s chosen because we assume that the avalablty of supply of resources (raw materal and product) are not necessarly be always 100% of the requrement whch, at the same tme, mples that the total cost and the unsatsfed demand wll not be 0% at both manufacturers and retalers. In ths research ths type of member- μ ( x ) x 0 x a x b x c Fgure 5.2. Modfed S-Curve Membershp Functon 110

123 shp s appled to represent the fuzzy goals: mnmzng total cost of manufacturer, mnmzng the total cost of retaler, and mnmzng the loss rate of retalers Torab and Hassn aggregaton functon Torab and Hassn [124] proposed a new aggregaton functon of fuzzy approach (TH method). Ths method has been proven to yeld an effcent soluton [127]. Accordng to Torab and Hassn, a fuzzy mult-objectve model could be transformed n a sngle objectve model as follows: Max γλ 0 subject to (1 γ) : λ 0 μ λ, μ 0 Z k Z k k, γ w k μ Z k [0,1] (5.23) where and { } denote the satsfacton degree of k-th objectve (ndvdual satsfacton degree of each objectve) and the mnmum satsfacton degree of the objectves, respectvely. Moreover, wk and γ ndcate the relatve mportance of the k-th objectve functon and the coeffcent of compensaton, respectvely. The wk parameters are determned by the decson maker based on her/hs preferences so that wk = 1, wk > 0. Ths aggregaton functon results n a compromse value between the max-mn operator and weghted-sum operator based on the value of γ The ntegrated soluton procedures As menton earler, our proposed nventory model s a fuzzy mult-objectve model whch belongs to a possblstc mxed-nteger programmng (MOPMIP) problem. To solve the problem, we propose ntegrated soluton procedures whch provde a systematc frame- 111

124 work that facltates the fuzzy decson-makng process as shown n Fgure 5.3. Snce the proposed MOPMIP nventory model s typcally dffcult to solve optmally n most real-lfe cases, performng the above procedure to search satsfactory solutons n the proposed MOPMIP model requres a tough computatonal experence. Hence, n order to allevate such computatonal complexty, the model s solved wth a Dfferental Evoluton (DE). The DE s a knd of evolutonal search methods and known as a practcal and effectve method to solve such Formulate fuzzy MOMIP perodc nvetory model Convert the MOMIP nto an equvalent crsp MOPMIP model Fnd the range of each of objectve functon by calculatng ther mnmum and maxmum value Develop the modfed S-curve membershp functon for objectve functons Transform the equvalent crsp MOPMIP model nto a sngleobjectve PMIP Detemne the weght of the th objectve ( w ) and the value of coeffcent of compensaton ( ) Start Generate ntal populaton of target vector, G = 0 Compute and evaluate the ftness of each target vector Apply mutaton, crossover and selecton operator to generate new target vector Form new populaton of target vector Termnaton satsfed? End yes Generaton G+1 no Dfferental Evoluton no Decson maker satsfed? yes Gan the acceptable solutons Fgure 5.3. Flowchart of Soluton Procedures 112

125 problems [79, 80]. The steps of the proposed soluton procedures are summarzed as follows: Step 1: Formulate the fuzzy mult objectves mxed-nteger programmng (MOMIP) perodc revew nventory model (Eqs. (5.1) (5.11)). Step 2: Convert the MOMIP nventory model nto an equvalent crsp MOPMIP model. To ths end, all the mprecse cost parameters n the objectve functons as well as the demand and lead tme parameters are converted nto the crsp ones usng Jmenez method (see secton 5.3.1). Step 3: Determne the rage of each objectve functon by calculatng the upper bound (UB) and lower bound (LB) of each objectve functon. To do so, equvalent crsp MOPMIP model should be solved each tme only one objectve. I UB LB Z1 mn TCM, Z1 max 1 J UB LB Z2 mn TCR j, Z2 max j1 J UB LB Z3 mn LRj, Z3 max j1 I 1 J j1 J j1 TCM TCR LR j j Step 4: Develop the modfed S-curve membershp functon for the objectve functons usng Eq. (5.22). l 1 Z1 Z 1 1 l Z1 Z B B l u l u δz Z /Z Z Z1 Z1 Z μ 1 Z l u l δz2 Z2/Z2 Z 1 Ce 1 Ce 2 u Z1 Z l 0 Z1 Z 0 1 μ Z1 l 1 Z Z Z Z Z 2 2 l Z Z l 2 Z l 2 2 Z Z u 2 l 2 Z u 2 113

126 B δz3 1 Ce μ Z3 l u l Z3/Z3 Z 3 Z Z Z Z Z 3 3 l Z l 3 Z Z l 2 3 Z Z u 3 l 3 Z u 3 Step 5: Transform the equvalent crsp MOPMIP model nto a sngleobjectve model based on the TH method. Employng the TH method n Eq. (5.23), the sngle objectve formulaton of the nventory model can be formulated as follow. Max subject to : γλ 0 (1 γ) λ 0 0 μ λ, μ Z k k Z1, w λ, γ [0,1] Z2 Eqs.(4.19)-(4.23) 1 0 μ Z1 μ w μ 2, λ 0 Z2 w μ μ Z3 3 Z3 Step 6: Determne the weght of the k-th objectve (wk) and the value of coeffcent of compensaton ( ), and solve the correspondng sngle-objectve model usng the Dfferental Evoluton algorthm (See Chapter 3 Secton 2.4). Step 7: Present the set of compromse soluton to the DM. If the decson maker(s) s satsfed wth the current soluton, stop. Otherwse, search another soluton by repeatng step Computatonal Experment and Analyss In ths research, a hypothetcally constructed SC system consstng of mult-manufacturer and mult-retaler s developed for 114

127 Table 5.1. Input Parameters Input Parameters Value M1 M2 (unt) {3661, 3950, 4476} a, {3011, 3500, 4036}, {3611, 4450, 4808}, {3276, 3900, 4404}, {3105, 3700, 4202}, {3252, 4050, 4550} {3214, 3500, 4050}, {3246, 3900, 4544}, {3654, 4450, 4985}, {3545, 4100, 4665}, {3589, 4200, 4721}, {3108, 3700, 4280} (day) {3, 4, 6},{3, 4, 6},{1, 3, 7}, {3, 4, 6},{1, 4, 8},{3, 4, 6} {1, 3, 6},{2, 4, 8},{1, 3, 5}, {2, 4, 7},{1, 3, 6},{1, 3, 6} (day) to R1: {1, 3, 5},{3, 6, 8},{4, 6, 9}, {2, 4, 6},{2, 4, 7},{1, 3, 5}, to R2: {4, 6, 9},{1, 3, 5},{2, 4, 6}, {4, 6, 9},{3, 5, 8},{1, 3, 5} to R3: {2, 4, 6},{1, 3, 5},{2, 4, 7}, {1, 3, 5},{4, 6, 9},{4, 7, 9} to R1: {3, 5, 7},{1, 3, 5},{1, 3, 5}, {4, 6, 8},{5, 7, 9},{1, 3, 5}, to R2: {1, 3, 5},{5, 7, 9},{3, 5, 7}, {1, 3, 5},{2, 4, 6},{5, 7, 9} to R3: {5, 7, 9},{4, 6, 8},{1, 3, 5}, {3, 6, 8},{2, 4, 6},{1, 3, 5} R1 R2 R3 (unt) {1576, 2500, 2976} {1836, 2300, 2836} {1908, 2700, 3208} {1504, 2200, 2804} {1702, 2500, 3020} {1850, 2800, 2650} {2081, 2800, 3462} {2254, 3000, 3399} {1952, 2500, 2921} {2184, 2800, 3442} {1602, 2400, 2931} {2650, 3200, 3684} {2050, 2500, 2922} {1996, 2700, 3252} {2291, 3000, 3511} {1854, 2600, 3220} {1540, 2200, 2931} {2050, 2700, 3280} Table 5.2. Cost Parameters Cost Value Parameters M1 M2 ($) {80, 100, 130} {70, 110, 140} ($) {3, 5, 7} {3, 5, 7} ($) {3, 5, 8} {3, 6, 9} ($) {0.11, 0.15, 0.17} {0.11, 0.15, 0.17} ($) {0.3, 0.4, 0.5} {0.3, 0.4, 0.5} ($) {8, 10, 14} {8, 10, 14} ($) to R1: {3, 5, 7} to R2: {3, 6, 8} to R3: {4, 6, 8} to R1: {3, 5, 7} to R2: {3, 6, 8} to R3: {4, 6, 8} R1 R2 R3 ($) to M1: {15, 20, 25} to M2: {15, 20, 25} to M1: {15, 20, 25} to M2: {15, 20, 25} to M1: {15, 20, 25} to M2: {15, 20, 25} ($) {0.1, 0.3, 0.5} {0.1, 0.3, 0.5} {0.1, 0.3, 0.5} ($) {17, 20, 23} {17, 20, 23} {17, 20, 23} computatonal experment. To llustrate the usefulness of the fuzzy MOPMIP nventory model usng the proposed soluton procedure, the model s mplemented to solve the SC system consstng of two manufacturers and three retalers and the result s reported n ths secton utlzng parameters shown n Table 5.1 and 5.2. It s as- 115

128 sumed that all members belong to one central major enterprse. It s also assumed that the DM of each member determnes the estmaton of the possblty dstrbuton of the mprecse parameters by decdng three promnent values (.e., the most lkely, the most pessmstc and the most optmstc values) based on ther avalable data and knowledge. In ths SC confguraton, each manufacturer has a smlar characterstc except that they are dstngushed by ther producton capabltes. As a result, producton rate of manufacture 1 (450 unts) s slghtly hgher than that of manufacturer 2 (400 unts). Plannng horzon for 6 perods are provded n whch each perod conssts of 10 days (tp). Wthout loss of generalty, the SCs control s measured based on the aggregate performance.e., the measures s aggregated for all manufacturers and for all retalers. To provde the DMs wth the broad decson spectrum, several results wth dfferent value of varable of compensaton s (γ-level) are provded n performance testng, and for each γ value a soluton set s generated by the ad of the TH method (Table 5.3). As seen, the proposed fuzzy nventory model s referred to the mxed-nteger programmng problem whch s very dffcult to solve n real-lfe cases. Hence, some alternatve fuzzy soluton methods such as LH method [125], SO method [126], etc., are developed n lteratures. Among them, the SO method s employed n ths study by means that the effectveness of the proposed model usng TH method s compared wth that of usng the SO method (see Appendx). The modfed S-curve membershp functon for all fuzzy objectves has adopted the same values for B, C and δ parameters. These values are B = 1; C = and δ = [123]. Supposed the relatve mportance of objectves s provded from lngustc state- 116

129 ment that all are equvalent.e., ω1 = ω2 = ω3. Based on ths relatonshp the weght vector s set as: ω = (0.33, 0.33, 0.33) whch means that all the objectves are equally mportant as each other. Due to the nature of problem dependence, key parameters (M, CR and F) n DE should be chosen wsely and approprately to get fast convergence and avod the problem of premature convergence. Accordng to our prelmnary experments, an approprate value of M, CR, F and G are set as follows: M = 100, F = 0.8 and CR = 0.5 and G = Consderaton from the major nterest Accordng to Table 5.3, the TH method has a superorty over the SO method n terms of the value of mnmum satsfacton degree of objectves because the solutons derved from the TH method are more balanced compared to those of SO method. Partcularly, the solutons provded by the SO method are unbalanced n low γ-levels (.e., ), that s to say that the comprehensve satsfacton degree of objectves s small and the method pays more attenton to some specfc objectves rather than to comprehensve satsfacton degree. In ths model, the satsfacton degree (μz1) of the frst objectve (Z1) s always the lowest compared to (μz2) and (μz3) n all γ-levels. Ths ndcates that the frst objectve s a crtcal objectve as t bounds the mnmum satsfacton degree of objectves nvolved. Consequently, the method attempts to maxmze the satsfacton degree of the second and the thrd objectve whle gvng less attenton to the performance of the frst objectve. As a result, for the lower γ- levels the SO method provdes totally unsatsfactory soluton for the frst objectve whose value lkely move toward ts worst possble 117

130 118

131 value. However, n the hgh γ-levels, the performance of the two methods s qute smlar and both of them fnd the solutons wth nsgnfcant dfferences whle the TH method provdes more evenly acceptable results. Accordng to nformaton provded n Table 5.3, changng the γ- level does not gve sgnfcant nfluences on decson adjustment n the TH method. However, regardng the SO method, orderng raw materal n a bg batch, holdng hgher volume of safety stock and settng a hgher volume of target stock level were found to cause the poor performance of the frst objectve n low γ-levels due to the ncrease n of nventory holdng cost. As can be seen n Fgure 5.4, the aggregate holdng cost of manufacturers n SO method reported sgnfcantly hgher n the low γ-levels. On the contrary, the TH method provdes more equtable result by proposng to order raw materal n bg batches and small lots, holdng more moderate amount of safety stock and settng lower level of target stock. From the above llustratve case, decson makng support can be provded to mprove the performance of Z1 as a crtcal objectve. From strategc level, ths can be done by gvng the man concern to prortze more Z1 than to other objectves. In ths case, the DM n Holdng cost of Manufacturers 600,000 γ= ,000 γ= , ,000 0 γ=0.8 γ=0.2 γ=0.4 TH method SO method γ=0.6 Fgure 5.4. Holdng cost of Manufacturers 119

132 major enterprse plays an mportant role to formulate the relatve mportance of each objectve. On the other hand, from the tactcal level, the mprovement can also be acheved by adjustng the decson toward reducng the nventory level by orderng raw materal n a smaller lot and lowerng the amount of safety stock. - Consderaton from the extended nterest The mnmum satsfacton degree of objectve represents the comprehensve satsfacton degree by whch all objectves nvolved are guaranteed to have the same value of satsfacton degree. Ths value s controlled by γ to ensure yeldng both balanced and unbalanced compromse soluton. Hgher γ-levels mean more concern s gven to acheve a hgher mnmum satsfacton degree of objectves (λ0) and hence more balanced soluton s derved. On the contrary, low γ-levels ndcate that the method pays more attenton to maxmze the satsfacton degree of some specfc objectves than to comprehensve satsfacton degree and accordngly more unbalanced soluton. In ths case, we can dstngush the unbalanced solutons from the least unbalanced soluton (γ = 0.9) to the most unbalanced soluton (γ = 0.0). Wth the vewpont of decson makng, the γ-level controls the power of the DM to the soluton. Whle the balanced solutons ndcate no contrbuton of the DM to the solutons, the most unbalanced soluton (the lowest γ-level,.e., γ = 0.0) allows a complete authorzaton of the DM to make a decson accordng to hs preferences. Consequently, we thnk that all unbalanced solutons and the balanced soluton may represent a typcal decentralzed and centralzed SC system, respectvely. In the decentralzed system, varous operatonal decsons are made at dfferent companes or a certan group of com- 120

133 Table 5.4. Result obtaned for some weght combnatons Item Problem nstances ω ω ω μz μz μz Z Z Z % 30.73% 11.50% 17.16% 30.28% 21.25% pany n the same stage whch tres to optmze ther own objectves. More specfcally, the company such as regonal dstrbutor of automotve products performs collaboratve operatonal decson wth other regonal dstrbutors n order to mprove ther responsveness to the customer demand. To ths end, ths study assumes that the members n the same SC stage collaborate as the ndependent group of enterprse n whch all operatonal decson are made on ther own authorty whle the major enterprse focuses more n a strategc plannng. In most stuatons, however, the major enterprse may be urged to decde whch member(s) should be prortzed n order to mprove ther performance(s). Generally speakng, n the centralzed SC, the major enterprse attempts to make a centralzed decson for synchronzng producton and dstrbuton flows across the SC. For ths purpose, the DM of major enterprse determnes the relatve mportance of objectves and ths wll result n dfferent weght combnatons as presented n Table 4. From Table 5.4 we know that the loss rate of retalers (Z3) s almost nsenstve to the weght structures. Whle changng the weghts nfluences the loss rate of retalers slghtly, t causes consderable changes n the total cost of manufacturers (Z1) and the total 121

134 cost of retalers (Z2) especally n low weght level. Ths can be seen from Table 5.4 that whle changng the weght from 0.2 to 0.5 causes μz3 to slghtly change from to 0.998, t generates sgnfcant change of μz1 from to and μz3 from to Therefore, the major enterprse as a central DM should carefully consder these behavors when choosng the weght combnaton to make the rght decsons Concluson Ths study has proposed MOPMIP model of perodc revew nventory problem n mult-manufacturer mult-retaler SC system. A soluton procedure s developed to solve the model and to provde a systematc framework that facltates the fuzzy decson-makng process, enablng the DM to adjust the decson and to obtan a more preferred satsfactory soluton. The computatonal experment ndcates that the proposed soluton procedures obtan a promsng result whch produces more balanced solutons set based on the DM s preferences. It also provdes decson support to dentfy crtcal objectve and provdes a realstc recommendaton for mprovement from the perspectve of strategc and operatonal aspect by offerng the flexblty to adjust the decson consderng the DM s preference. Moreover, notcng that the proposed soluton method has a specfc feature to control the authorty of the DM, t s also becomes feasble to apply the result n both decentralzed and centralzed SC system. Ths research s an antcpatng works n the desgn of perodc revew nventory control under uncertanty n the context of SC system usng possblstc programmng approach. Therefore many possble researches can be developed n ths scope. For example, ad- 122

135 dressng collaboratve plannng n SC system such as suppler selecton-nventory control and nventory control-transportaton plannng wll becomes an attractve research avenue wth sgnfcance practcal relevance. Appendx 5.1 Selm and Ozkarahan Method: Max γλ 0 subject to (1 γ) λ In ths model, : 0 λ λ, μ 0 Z k k k μ, γ w k Z λ k k [0,1] denotes the dfference between the satsfacton degree of k-th objectve (ndvdual satsfacton degree of each objectve) wth the mnmum satsfacton degree of the objectves ( ). 123

136 Chapter 6 CONCLUSION 6.1. Concludng remark Most companes are now facng dynamc challenges that requre not only well-plannng capacty, but also robust SC networks that allow the members nvolved to address and respond any changes n a short notce. In partcular, when nventory s stuck n the varous stages of the supply chan, the company may be forced to operate at crtcal cash flow levels. On the other hand, of the varous actvtes nvolved n SC network, purchasng s one of the most strategc functons because t provdes opportuntes to reduce costs across the entre supply chan. An essental task wthn the purchasng functon s suppler selecton, gven that the cost of raw materals and component parts represents the largest percentage of the total product cost n most ndustres. From ths pont of vew, ths thess addresses some ssues n nventory and suppler selecton problem. To be more practcally relevant, we study both ssues under uncertan envronment and ap- 125

137 proach such problem usng ether stochastc approach or fuzzy approach. Chapter 3 nvestgated a mult-objectve problem of perodc revew nventory n two-echelon supply chan system under uncertanty n demand and lead tme. In ths study, we propose dfferent strateges to solve the stock-out problem - besde the tradtonal mechansm - n seral replenshment system whch requre a hgher level of coordnaton. Whle stochastc approach s utlzed to tackle the uncertanty, the mult-objectve Dfferental Evoluton (DE) s appled after gvng ts modfed algorthm to work wth the problem. It was found that the coordnaton strategy s become more effectve as the uncertanty ncreases n the system. By cooperatng, manufacturers can avod frequent backorder and reduce excess nventory as a whole. Though retalers are requred to keep a bt hgh nventory level to mantan the good responsveness to the customer demand, ths stock level s more effectve to reduce the loss rate of supply chan. Chapter 4 studed mult-objectve suppler selecton problem by consderng both qualtatve and quanttatve crtera under uncertanty. Unlke Chapter 1 whch appled stochastc approach, ths research ncorporated fuzzy approach s proposed due to the fact that most nformaton requred to assess suppler s not always avalable and/or usually not known precsely over the plannng horzon. Concernng such characterstcs, ths research proposes ntegrated methodology for fuzzy mult-objectve lnear programmng model for suppler selecton. To mprove n the methodologcal process of dervng optmal soluton, the enhanced two-phase fuzzy programmng model has been developed n ths study. Through numercal experment, we show some promsng advantage of our proposed 126

138 approach over the exstng methods n provdng a set of potental feasble solutons whch gude DMs to select the best soluton accordng to ther preference. Chapter 5 presented a mult-objectve possblstc mxed nteger programmng (MOPMIP) model of perodc revew nventory problem n mult-manufacturer mult-retaler Supply Chan system. Possblstc programmng s one method under Fuzzy Set Theory (FST) whch s desgned to handle the uncertanty related to llknown parameters due to the lack of precse data. Specfcally, we attempt to develop a mult-objectve perodc revew nventory model n a mxed mprecse and/or uncertan envronment by ncorporatng the fuzzness of demand, lead tme and cost parameters. A soluton procedure s developed usng the Torab and Hassn (TH) method to solve the model and to provde a systematc framework that facltates the fuzzy decson-makng process, enablng the DM to adjust the decson and to obtan a more preferred satsfactory soluton. Then the solutons are derved by the ad of Dfferental soluton. The proposed soluton procedures obtan a promsng result whch produces more balanced solutons and provde decson support to dentfy crtcal objectve. It s also becomes feasble to apply the result n both decentralzed and centralzed SC system Future research Several possble research can be developed to extend the current research. Snce the proposed research problem s carred out n small scale problem, t s more appealng to devote the future research plan to more general and large scale confguratons of the supply chan system. Moreover, combnng two or more dea of the 127

139 current researches nto collaboratve plannng n SC wll becomes an attractve research avenue wth sgnfcance practcal relevance. 128

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159 Appendx A

160 DAILY PLANNING FOR THREE ECHELON LOGISTICS CONSIDERING INVENTORY CONDITIONS A.1. Introducton Due to aglty, greenness and servce nnovaton, daly logstcs optmzaton s becomng extremely mportant especally for small busnesses lke convenence stores or super markets n Japan. Recently, a revew of artcles publshed n the last decade wthn the context of supply chan management has been consderably emerged [128]. Thereat, they reveal a scarcty of models that capture dynamc aspects relevant to real-world applcatons, and emphasze an ncreasng need for extensve studes on such topc. Notcng such crcumstance, and to enrch the prospects of ths thess, we present a daly logstcs optmzaton at a tactcal and operatonal level. Specfcally, we assocate wth vehcle routng problem (VRP) consderng a substantal nventory ssue. 151

161 Logstc sde Super **LOS Oracle DB Structured by NET Framework Inform status on nventory of products and resources Plannng secton Producton sde DC-1 Indcate DC-3 Delvery secton CS-1 RS-1 CS-5 Inventory CS-20 CS-6 Inform status on Inform status on nventory of nventory of products and products and resources resources Domestc ponts Oversea ponts Logstcs nformaton system Update nformaton Real tme montorng on nventory RS-J CS-3 DC-I Output optmal plan Inventory CS-K Desgn system for optmal logstcs plannng Fgure A.1. Global dea of DSS on logstcs plannng Moreover, takng nto account dynamc demand and nventory of warehouse, we try to gve an operatonal and practcal approach amenable to nnovatve resoluton to the daly logstcs optmzaton. The fnal scope of ths study refers to an ntegrated decson support system wth formaton system that can dynamcally manage approprate data based on the nventory of resources and the demand of products (See Fgure A.1). A.2. Problem statement A.2.2. Revew of related studes Regardng the transportaton among the depots and customers, each vehcle must take a crcular route from ts depot as a startng pont and a destnaton at the same tme. Ths generc problem has been studed popularly as VRP [129]. The VRP s a well-known NPhard combnatoral optmzaton problem, whch mnmzes the total dstance traveled by a fleet of vehcles under varous constrants. Recent studes on VRP applcaton can be roughly classfed nto the followng four knds. 152

162 One of them s an extenson from the generc customer demand satsfacton and vehcle payload lmt. For examples, practcal condtons such as customer avalablty or tme wndow [130, 131], pck up [132], splt and mxed delveres [133] are concerned not only separately but n a combned manner [134]. The second s known as the mult-depot problem that tres to delver from multple depots [135, 136, and 137]. The thrds are concerned n the mult-objectve formulaton for the sngle depot and mult-depot problems [ ]. Though these three classes mght belong to an operatonal level, the last one [144, 146, 147 and 147] corresponds to a tactcal concern. That s, the decson on the allocatons of depot s nvolved besdes VRP. Though many of those studes are solved by usng a certan meta-heurstc method [148, 149], a certan local search s appled n the lterature [150]. To effectvely reduce the computatonal dffculty, a hybrd algorthm of Benders decomposton wth genetc algorthm s also proposed [151]. Due to the dffculty of soluton, however, only small problems wth no less than a hundred customers are solved to valdate the effectveness except for the lterature [150]. Moreover, though actual transportaton cost depends not only on the dstance but also load (Ton-Klo bass), those studes consder only dstance (Klo bass) to derve the route. Hence, the tactcal concerns mentoned above are unfavorable to make a generc and consstent dealng over the multple decson levels,.e., allocaton problem and VRP. A.2.2. Problem formulatons Takng a global logstcs network composed of dstrbuton center (DC), depots, and customers, we try to decde the avalable depots, paths from major DCs to sub-dcs or depots (RS) and crcular routes 153

163 from every depot to ts clent customers (See Delvery secton n Fgure A.1). Index set I J K V P T ndex for DC ndex for depot ndex for customer ndex for vehcle = J K ndex for plannng horzon Varables fj(t) load from DC to depot j at perod t gpp v(t) load of vehcle v on the path from p P to p P at perod t rj(t) take over nventory at depot j at perod t sj(t) consume from nventory at depot j at perod t xj(t) = 1 f depot j s open; otherwse 0 at perod t yv(t) = 1 f vehcle v s used; otherwse 0 at perod t zpp v(t) = 1 f vehcle v travels on the path from p P to p P; otherwse 0 at perod t Parameters Cj cv Dk(t) Dpp Fv transportaton cost per unt load per unt dstance from DC to depot j transportaton cost per unt load per unt dstance of vehcle v demand of customer k at perod t path dstance between p P and p P fxed charge to operatng vehcle v 154

164 Haj Hoj Hp Hsj M Qv Qj Sj Wv handlng cost per unt load at depot j holdng cost per unt load at depot j shppng cost per unt load from DC shppng cost per unt load from depot j auxlary constant (Large real number) maxmum load avalable at DC mnmum load requred to shp from DC own weght of vehcle v maxmum capacty at depot j maxmum nventory at depot j maxmum capacty of vehcle v The goal of ths problem s to mnmze total cost for daly logstcs over plannng horzon T. Ths problem s formulated as the followng mxed-nteger programmng problem under mld assumptons, e.g., round-trp transport between DC and depot; un-modal transport; averaged tme nvarant unt costs and system parameters except for demand and nventory, ndependency or decson per each plannng perod, etc. Mnmze for every T I jj jj C d1 Hp f ( t) c d2 g ( t) q j j Ho jrj ( t) Ha j s j j v vv pp p' P ( t) Hs j s j ( t) pp I ppy fj ( t) v vv z ppv ( t) Fv yv ( t) (A.1) subject to pp z kpv ( t) 1, k K; v V, t T (A.2) 155

165 156 P p P p p pv p v p T t V, v P; p, t z t z 0 ) ( ) ( (A.3) T t V, v J, j, t z J j v j j 0 ) ( (A.4) T t, J j, t g t f t s I V v K k jkv j j ) ( ) ( ) ( (A.5) T t, J j, t x Q t f t s t r I j j j j j ) ( ) ( ) ( ) ( (A.6) T t V, v P; p P; p, t W z t g v pp v p v p ) ( ) ( (A.7) T t V, v, t My t z P p v P p p v p ) ( ) ( (A.8) T t V, v J; j, t g K k kjv 0 ) ( (A.9) T t K, k, t D t g t g V v k V v P p kpv P p pkv ) ( ) ( ) ( (A.10) T t V, v K, k, t g t z t D t g P p P p kpv pkv k pkv ) ( ) ( ) ( ) ( (A.11) T t V, v, t y t z J j v K k jkv ) ( ) ( (A.12) T t V, v, t y t z J j v K k kjv ) ( ) ( (A.13) T t I,, P t f P J j max j mn ) ( (A.14)

166 r ( t) s ( t) S x ( t), j J, t T (A.15) j j j j x j ( t) {0, 1}, j J, t T ; ( t) {0, 1}, v V, t T y v s j ( t) 0, j J, t T ; ( t) 0, j J, t T r j z ( t) {0, 1}, p P; p P; v V, t T pp v f j ( t) 0, I; j J, t T ; g ( t), p P; p P; v V, t T pp v 0 In (A.1), the objectve functon s composed of round-trp transportaton costs between every DC and depot, crcular transportaton costs for travelng to every customer, shppng costs at DC, holdng, handlng and shppng costs at depot, and fxed operatonal charge of vehcles. Several constrants are appled: vehcles cannot vst a customer twce (Eq.(A.2)); vehcles enterng a locaton must leave t (Eq.(A.3)); no travel between dstrbuton centers (DCs) (Eq.(A.4)); materal balance (Eq.(A.5)); upper bound of the capacty at depot (Eq.(A.6)); upper bound of the load capacty for vehcle (Eq.(A.7)); each vehcle must travel on a certan path (Eq.(A.8)); vehcles return to the depot empty (Eq.(A.9)); customer demand s satsfed (Eq.(A.10)); sum of nlet good must be greater than that of outlet by ts demand (Eq.(A.11)); each vehcle leaves only one depot and returns back there (Eq.(A.12) and (A.13)); and the amounts of good avalable from DC are bounded (Eq.(A.14)); and the amounts of nventory are upper bounded (Eq.(A.15)). We also assume the followng nventory control polcy. 157

167 (1 ζ)r j (t 1) ) rj (t) S j (1 ζ)r j (t 1) f f (1 ζ)r (t 1) R (1 ζ)r (t 1) R j j j j, j J, t T (A.16) where ζ (<1) and Rj are a foulng rate of unsold goods and an orderng pont at depot j, respectvely. We know that t s almost mpossble to solve the above problem wth practcal sze usng any currently avalable commercal software. Aganst ths, we have successfully solved varous logstcs optmzaton problems under complcated stuatons resultng from a varety of real-lfe condtons, by usng a method called hybrd Tabu search (HybTS) [152, 153]. Ths s a two-level soluton method n whch the upper level sub-problem optmzes the selecton of avalable depots whle the lower level sub-problem optmzes the paths from DCs to customers va depots so as to mnmze the total cost. Snce HybTS s not only a practcal and powerful method but flexble and sutable for a varety of extensons, we wll deploy the smlar dea to solve the problem under consderaton. Start ( Frst level) Intal locaton New locaton (Modfed tabu t = 1 search) Decde (Second level) assgnment (Graph algorthm) t := t + 1 Intal route (Modfed savng method) (Thrd level) New route (Modfed tabu search) No No End Yes t = T? No Yes Converged? Evaluate total cost Yes Converged? Fgure A.2. Flowchart od Soluton Procedures 158

168 A.3. Daly decson assocated wth nventory condton A.3.1. Mult-level approach ncorporatng vehcle routng problem A For the daly logstcs optmzaton, t s meanngful to take nto account the use of nventory control at every depot. To make the foregong herarchcal approach avalable for the present case, we have majorly nvented two new deas and ntegrated them nto the smlar framework of our hybrd method. In our best knowledge, such global approach has not been reported anywhere. In ts frst level, we choose the avalable depots usng the modfed Tabu search. Then, n the second level, we tentatvely obtan round trp paths from DCs to customers va depots usng a graph algorthm for the mnmum cost flow (MCF) problem. Assgnng the customers thus allocated as the clents for each depot, we derve eve- 2S1 11 S1 P1 max [D1+D2+S1+S2] 1 P1 max -P1 mn P1 mn P1 max 5 6 Handlng Transportaton cost + Shppng costs S1 Q1 Q2 Shppng 7 cost 8 Transp. cost Holdng cost D1 D2 P1 mn +P2 mn D D2 S2 DC 12 RS RE 2S2 S2 D1 13 D2 [D1+D2+S1+S2] P max :Upper supply P mn :Lower supply D:Demand S:Inventory Q:Upper capacty Fgure A.3. Example of MCF graph 159

169 Table A.1. Labelng on the edge for MCF graph Edge (from to) Cost Capacty Case n Fg.3 Source - (Dummy) -M I P mn #1 - #2 Source - DC 0 P max -P mn #1 - #3, #1 - #4 - DC 0 P mn #2 - #3, #2 - #4 DC - RS j C jd1 j +Hp P max #3 - #5, #3 - #6, etc. Between double nodes of RS j Hs j Q j #5 - #7, #6 - #8 Stock - RS j Ha j S j #11 - #5, #12 - #6 Source - Stock j 0 2S j #1 - #11, #1 - #12 Stock j - Snk Ho j S j #11 - #13, #12 - #13 RS j - Customer k c vd2 jk D j #7 - #9, #7 - #10, etc. Customer k - Snk 0 D k #9 - #13, #10 - #13 ry vehcle route of depot usng the modfed savng method and modfed Tabu search. The result thus obtaned s fed back to the frst level to evaluate another set of avalable depots. Ths procedure wll be repeated untl a gven convergence condton has been satsfed. The procedure of ths algorthm s llustrated n Fgure A.2. In developng the above algorthm, we need to obtan the MCF graph that consders the nventory at each the depot. For example, the case where I = J = K =2 s llustrated n Fgure A.3. In Table A.1, we summarzed nformaton requred to put on the edges and nodes n the graph. In terms of the MCF graph thus derved, we can solve the orgnal allocaton problem extremely fast through a graph algorthm lke RELAX4 [154] together wth ts senstvty analyss. The senstvty analyss s amenable to repeatedly solve the problem wth slghtly dfferent parameters one after another. After all, we can effcently allocate each depot to ts clent customers on the Ton-Klo bass. Then, to solve the VRP n terms of Ton-Klo bass, we appled the hybrd approach composed of the modfed savng method and the modfed Tabu search n a hybrd manner [155, 156]. Thereat, 160

170 we noted that the fxed operatonal cost for the workng vehcle should be nvolved n the economc evaluaton. After all, the algorthm of the modfed savng method s outlned as follows. Step 1: Create round trp routes from the depot to all customers. Compute the savng value by s,j = (d0,j-d0,-d,j)dj + (d0,j+d,0-d,j)w, where Dj, q and dj denotes the demand at locaton j, weght of vehcle tself and dstance between locatons and j, respectvely. Step2: Order these pars n descendng order of savngs. Step3: Merge the path followng the order as long as the feasblty s satsfed and the savng s greater than -Ctf/cv, where Ctf denotes the fxed operatonal cost of the vehcle. However, snce the modfed savng method derves only an approxmated soluton, we move on the modfed Tabu search to update such ntal soluton. The modfed Tabu search s a varant that probablstcally accepts the degraded canddate lke smulated annealng n ts local search. Here, we can emphasze such an advantage that transportaton costs are able to be accounted on the same Ton-klo bass at the upper (frst and second) and lower (thrd) level procedures n Fgure A.2. A.3.2. Analyss of nventory level on demand varaton It s commonly known holdng too much nventory slps the economcal effcency whle the stock-out or opportunty loss wll happen n the opposte case. For the daly logstcs, therefore, t s of specal mportance to correctly estmate the demand and properly manage the nventory. Generally speakng, however, correctly est- 161

171 mate the demand s almost mpossble n many cases whle t s possble to roughly estmate the extent of devaton from experence. Under such crcumstance, t s relevant and practcal to try to reveal the relaton between the extent of demand devaton and the nventory level through parametrc approach. Through such analyses, we can setup a relable nventory level to mantan the economcally effcent logstcs whle preventng from the state of stock-out. Though such consderaton s able to present many prospects for the robust and relable logstc systems, t has not been almost concerned n the network optmzaton of logstcs so far due to computatonal dffcultes. A.4. Numercal experment A.4.1. Setup of test problem To examne some performance of the proposed method, we provded several benchmark problems wth dfferent problem szes,.e., { I, J, K }. Every system parameter are set randomly wthn the respectve prescrbed nterval as summarzed n Table A.2. Locaton of every member s also generated randomly, and dstances between Table A.2. Notes on parameter setup Member Item Range Remarks Hp 100 [0.2, 0.8] <3> DC P max 1000 [0, 1]+ <5>, Total P max >Total capacty P mn of RS P mn 1000 [0.2, 0.8] <5>, Total P mn > Total demand Hs 100 [0.2, 0.8] <3> Ha 50 [0.2, 0.8] <3> RS Ho 100 [0.2, 0.8] <5> Q p [0.2, 0.8] <5>, p=100 K / J S x [0.5, 0.7] <3>, Varyng at each tme RE D 100 [0.2, 0.8] <3>, Total demand < Total capacty of RS Cj = 3, cv = 1, Wv = 500, Fv = 50000, qv = 10; <n> multple of n 162

172 them are calculated as the Eucldan dstance. A.4.2. Result for leadng condton Regardng each demand, we randomly changed the amount daly wthn 100(1±α)% from the foregong day. On the other hand, the unsold goods at each depot are remaned as the nventory and t s possble to use them n the followng days. However, t s supposed to be spoled randomly wthn ζ and the goods are suppled to the upper lmt when the nventory level becomes below the prescrbed safety level (Rj=βSj). Frst, we solved the smaller sze problems lke I =3, J =10 and K =100 over 30 days and I =5, J =20 and K =200 over 10 days. Parameters α, β and ζ are set at 0.3, 0.5 and 0.1, respectvely. Fgures A.4 and A.5 llustrate the changes of demands and nventory durng the plannng horzon. Under these condtons, we derve the optmal cost that broadly changes wth the demand fluctuaton as shown n I =3, J =10, K =100 I =5, J =20, K =200 Fgure A.4. Varatons of demand I =3, J =10, K =100 I =5, J =20, K =200 Fgure A.5. Varatons of nventory 163

173 Fgure A.6. In Fgure A.7, we can see that the change of workng numbers of depot s moderated and kept nearly constant (around 60%). However, workng rate of each depot dffers greatly as shown n Fgure A.8. Ths observaton s avalable for consderng the restructurng of logstcs at the next stage. That s, the depots of low workng rate may be ntegrated nto the other hgher ones. Moreover, we solved the larger problems to examne the necessary computaton tme. Fxng the plannng horzon at 1, I =10, and J =30, we solved the problems lke K ={250, 500, 1000, 1500, 2000, 2500}. As expected a pror, the requred CPU tme ncreases exponentally wth the problem sze as shown n Fgure A.9. Even for these larger sze problems, however, we can obtan the result wthn a reasonable tme or around several hours. Fgure A.10 shows the convergence profle for the largest problem. We can confrm the suffcent convergence. From all of these I =3, J =10, K =100 I =5, J =20, K =200 Fgure A.6. Trend of optmal cost I =3, J =10, K =100 I =5, J =20, K =200 Fgure A.7. Varatons of number of workng depot 164

174 CPU tme [sec] I =3, J =10, K =100 I =5, J =20, K =200 Fgure A.8. Workng rate of each RS Problem sze ( K ) [-] Fgure A.9. CPU tme vs. Problem sze I =10, J =30, K =2500 Fgure A.10. Soluton convergence results, we can clam the sgnfcance of the approach and computatonal effectveness of the proposed method. A.4.3. Results over wde range of devatons To analyze the effect of nventory condton aganst demand devatons, we carred out a parametrc study regardng orderng 165

175 Inventory cost unt [-] ponts usng a small model lke I =2, J =5, K =100 over 30 days. Actually, we solved every par of problem wth fve dfferent orderng ponts (β={0.1, 0.2, 0.3, 0.4, 0.5}) and four dfferent ranges of demand varaton (α={0.2, 0.3, 0.4, 0.5}). Ths comes to that 600 optmzaton problems were totally solved. Now, we show the results n Fgure A.11 and A.12 by applyng the same value settng as before. Fgure A.11 shows feature of total cost regardng the range of demand devaton and the level of orderng pont. Due to the nondetermnstc parameter settng, complcatedly wndng profle s observed. But ts trend s plausble snce the regon where the mnmum cost locates wll move on the hgher orderng pont accordng Fgure A.11. Total cost wth demand devaton range and orderng pont Demand devaton range α [-] Orderng pont β [-] Fgure A.12. Inventory cost wth demand devaton range & orderng pont 166

176 to the ncrease n the devaton ranges as a whole. Ths fact suggests that t s mportant to control the orderng pont or nventory level accordng to the demand devaton n the cost management. When we cut off only the nventory cost from the total cost, ts changes are rather smple as shown n Fgure A.12. Snce hgher stock level needs more holdng cost, the cost ncreases proportonally wth the orderng pont regardless of the devaton ranges of demand. Fnally, from these parametrc studes, we clam the adequateness of the appled model behnd the mathematcal formulaton. Plausblty of the results can support the sgnfcance of the approach f the realstc parameters were used n real world optmzaton. A.4.4. Prospect as a workng tool To realze the plannng system llustrated n Fgure A.1 as a goal, t s essental to provde a user frendly nterface to manage the system. At the plannng secton n producton sde, ths goal s closely related to data handlng and vsualzaton of the crcumstance at hard. Regardng ths matter, we can effectvely utlze some software developed by Google Maps API. By now, we have deployed the followng step-wse procedure by makng Java scrpts and approprate free software. Step 1: Collect the address of members n an Excel spread sheet or text sheet. Step 2: Add the nformaton of longtude and lattude of every one nto the sheet. We are avalable a free software named AGtoKML [157]) for ths purpose. Step 3: Calculate the dstance between every par of members usng a procedure of Google Map API known as Go-codng. 167

177 Step 4: Solve the optmzaton problem by the developed method. Step 5: Dsplay the routes obtaned from Step 4 n Google map. In the result of the llustratve problem wth I =1, J =3 and K =17, every depot has a sngle route. In Fgure A.13, for smplcty, only the routng paths from depot 1 are shown n terms of the popup marks (A(=L) B K L(=A)). We can see ths knd of vsual nformaton s upmost for some tasks at operatonal level. However, there stll reman many possbltes to add more amenable servce nformaton usng GIS applcatons and Google Map API. A.5. Concluson We have proposed a herarchcal approach to optmze a daly logstcs problem ncludng nventory control at depots and vehcle routng for customer delvery. For ths purpose, we have extended Fgure A.13. A part of dsplay of result (Route from Depot 1 (Popup mark L)) 168