QUANTITY DISCOUNTS, CAPACITY DECISIONS, AND CHANNEL STRUCTURE CHOICES IN SUPPLY CHAINS JONATHAN EUGENE JACKSON JR.

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1 QUANTITY DISCOUNTS, CAPACITY DECISIONS, AND CHANNEL STRUCTURE CHOICES IN SUPPLY CHAINS By JONATHAN EUGENE JACKSON JR. A dssertaton submtted n partal fulfllment of the requrements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Carson College of Busness MAY 2015 Copyrght by JONATHAN EUGENE JACKSON JR., 2015 All Rghts Reserved

2 Copyrght by JONATHAN EUGENE JACKSON JR., 2015 All Rghts Reserved

3 To the Faculty of Washngton State Unversty: The members of the Commttee apponted to examne the dssertaton of JONATHAN EUGENE JACKSON JR. fnd t satsfactory and recommend that t be accepted. Charles L. Munson, Ph.D., Char Sung K. Ahn, Ph.D. Stergos B. Fotopoulos, Ph.D.

4 ACKNOWLEDGEMENTS Completng ths dssertaton would not have been possble wthout the help, support, and gudance of my famly, frends, and colleagues. For that, I am extremely thankful. I am especally grateful for all the support and gudance that my commttee has provded over the past fve years. Wthout ther push and nspraton I could never have become the young scholar I am today. They gave me the freedom to choose my own research topcs, but smultaneously drected and refned my research plans to ensure that I was successful. Indvdually, I would lke to thank Drs. Sung Ahn and Stergos Fotopoulos for ther support and teachngs throughout my tme n the Ph.D. program. Your trust and belef n me has opened numerous doors durng the last fve years, and I am ncredbly grateful for those opportuntes. Addtonally, I would lke to thank my commttee char and mentor, Dr. Chuck Munson. My success as a young scholar would not be possble wthout hs tutelage and backng. Hs crtcal analyss of my research has asssted me n understandng the key characterstcs of qualty research. Furthermore, hs contnuous effort to mprove and nnovate n the classroom has been an nspraton for my own teachng, and led to my desre for excellence n not only research, but teachng as well. Outsde of my commttee members, I would lke to thank the rest of the faculty members n the Department of Fnance and Management Scence at Washngton State Unversty. In partcular, the department char Dr. Gene La who contnually provded opportuntes for me to teach, and for always havng an open door f I had any questons or concerns. Addtonally, I am grateful for Dr. Vctor Sh from Wlfrd Laurer Unversty who reached out as a colleague to work on research projects throughout the last fve years. Beyond hs thoughtful comments and wllngness to be a soundng board for my research projects, I am also ncredbly thankful for hs

5 wllngness to provde fnancal support to allow me to attend conferences. Those experences were nvaluable as a Ph.D. student. Even wth the unwaverng support and gudance from the faculty members, ths dssertaton would not have been possble wthout my famly and frends. Mom and Dad, your support throughout my lfe has made me the ndvdual I am today, and words cannot do justce for how grateful I am. I could not have made t wthout your well-tmed words of advce and encouragement as well as your wllngness to lsten to me vent after a bad day. I would also lke to thank my fellow Ph.D. students n the Department of Fnance and Management Scence for your support and ablty to make the day-to-day grnd bearable. I consder you all frends and I am proud to call you future colleagues. Fnally, I would lke to thank my wonderful wfe-to-be Andrea. You have always been there to keep me grounded, and I could not have survved ths journey wthout you. I am ncredbly proud of us both completng advanced degrees, and I cannot wat to see what the future holds for us. v

6 QUANTITY DISCOUNTS, CAPACITY DECISIONS, AND CHANNEL STRUCTURE CHOICES IN SUPPLY CHAINS Abstract by Jonathan Eugene Jackson Jr., Ph.D. Washngton State Unversty May 2015 Char: Charles L. Munson Ths dssertaton takes a two-pronged approach nto explorng best practces to optmze supply chans through procurement and dstrbuton. I develop three models to help managers make better busness decsons regardng ther procurement polces when facng quantty dscounts and ther usage of dfferent dstrbuton channels when there are varyng types and levels of retal competton. Chapters Two and Three develop soluton technques to the shared resource allocaton problem n the presence of quantty dscounts and wth the opportunty to expand or reduce the capacty level. Ths type of mult-tem nventory system has two common orderng structures. The frst consders each tem ndependently, and thus the frm must keep ts capacty level such that t can handle the worst-case stuaton where all tems are ordered smultaneously. The second orderng structure lnks the tmng of orders for all tems to avod a stuaton where all tems are ordered smultaneously. I develop models to determne effcent procurement polces when facng these two orderng structures n Chapters Two and Three, respectvely. v

7 In Chapter Four I develop a model to help manufacturers determne ther equlbrum dstrbuton polcy. My model prescrbes whether they should sell ther product drectly through a manufacturer-owned retaler or onlne, or ndrectly through an ntermedary such as an ndependent retaler. Introducng asymmetres n product substtutablty and brand equty, as well as dfferent forms of competton, causes the equlbrum channel structure to vary wdely. Dependng on the levels of asymmetry and competton, there are multple equlbrum channel structures, ncludng a case where t s benefcal to utlze dual channels. Outsde of the modelng chapters, n Chapter Fve I report results from two manageral surveys related to common practces of quantty dscounts from both buyers and sellers perspectves. The benefts of these surveys are two-fold: frst, they provde an up-to-date perspectve nto ndustry s usage of quantty dscounts, and second, they dentfy research gaps that can help unfy the nterests of researchers and practtoners. A major area for future exploraton s the development of quantty dscount tranng scenaros. The survey results ndcate a lack of fundamental understandng of the basc gudelnes assocated wth quantty dscounts. v

8 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS... ABSTRACT...v LIST OF TABLES...x LIST OF FIGURES... x CHAPTERS 1. INTRODUCTION SHARED RESOURCE CAPACITY EXPANSION DECISIONS FOR MULTIPLE PRODUCTS WITH QUANTITY DISCOUNTS...6 Introducton...7 Lterature Revew...8 Model...12 Capacty Reducton Opton...20 Incremental Dscounts...21 Numercal Analyss...24 Conclusons COMMON REPLENISHMENT CYCLE ORDER POLICIES FOR MULTIPLE PRODUCTS WITH CAPACITY EXPANSION OPPORTUNITIES AND QUANTITY DISCOUNTS...38 Introducton...39 Lterature Revew...41 Model...42 v

9 Refnement Polcy...53 Numercal Analyss...61 Conclusons MANUFACTURERS CHANNEL STRUCTURES WHEN SELLING ASYMMETRIC COMPETING PRODUCTS...72 Introducton...73 Lterature Revew...76 Model...79 Results and Dscussons...85 Extensons...96 Conclusons THE RHYMES AND REASONS OF QUANTITY DISCOUNTS: A PRACTICAL PERSPECTIVE Introducton Lterature Revew Data Research Questons Results Opportuntes for Future Research Conclusons BIBLIOGRAPHY APPENDICES A. ALGORITHMS ADAPTED FROM PIRKUL AND ARAS (1985) v

10 B. PARAMETERS FOR THE BENTON EXAMPLE AND THE FORTUNE 500 EXAMPLE FROM RUBIN AND BENTON (1993) C. DETERMINATION OF THE NASH EQUILIBRIUM UNDER COURNOT COMPETITION WITH SYMMETRIC PARAMETERS D. PROOF OF PROPOSITION x

11 LIST OF TABLES Page Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table x

12 Table Table Table Table Table Table Table Table Table B Table B Table B Table B Table B x

13 LIST OF FIGURES Page Fgure Fgure Fgure Fgure Fgure Fgure Fgure Fgure Fgure Fgure Fgure x

14 Dedcaton Ths dssertaton s dedcated to my mom and dad, who provded unwaverng support throughout ths journey. x

15 CHAPTER ONE INTRODUCTION There are many defntons for supply chan management, but one of my favortes s: 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 (Smch-Lev et al. 2008). There are countless approaches to attempt to optmze a supply chan: some approaches have been studed extensvely, others are stll n ther nfant stages of analyss and understandng, and some have yet to be explored. Snce the 1980s, supply chan management has been a hot topc n operatons management research, and gven the multfaceted nature of the topc, I envson t wll contnue to be a vbrant research feld. Ths dssertaton approaches the general theme of supply chan management usng a twopronged approach. Frst, I analyze the procurement polces of frms n the presence of quantty dscounts. Quantty dscounts reman omnpresent n both busness-to-busness (B2B) and busness-to-consumer (B2C) transactons. Munson and Rosenblatt (1998) report that a majorty of B2B transactons nvolve some form of quantty dscounts. Addtonally, as consumers, we cannot go nto town wthout notcng quantty dscounts whether we upsze our fast food lunch, make sure to get our punch on our punch card at our favorte local coffee shop, or overload on tems at Costco to get a cheaper unt cost. Despte ths overwhelmng popularty of quantty dscounts, there s stll evdence that frms are not optmzng ther procurement and sales polces to mnmze costs ether locally or wthn ther supply chan. My dssertaton focuses on understandng the needs of practtoners wth respect to quantty dscounts and seeks to provde applcable and benefcal quantty dscount models for real-world stuatons. 1

16 My second approach to better understand how to optmze supply chans s through the determnaton of approprate dstrbuton polces. I look downstream from the manufacturer s perspectve, where every competng manufacturer must decde whch channel(s) to use for the dstrbuton of ts products. The manufacturer can choose to vertcally ntegrate and thus sell the product drectly. Alternatvely, the manufacturer can sell the product through an ntermedary, such as an ndependent retaler, who then resells the tem to the end consumer. Lastly, the manufacturer can take advantage of both dstrbuton channels and utlze a dual channel setup. Through the ntroducton of multple forms of competton and asymmetry between the competng supply chans on brand equty and product substtutablty, I help manufacturers determne the deal dstrbuton plan. Chapter Two entals my frst dssertaton essay n whch I develop a mathematcal model to solve the tradtonal shared resource allocaton problem n the presence of quantty dscounts wth the ablty to ncorporate the capacty level as a decson varable. My model s the frst of ts knd to be able to approxmate any functonal form of capacty cost (assocated wth a capacty expanson or potentally a reducton) through the use of a pecewse-lnear approxmaton. Ths allows me to smultaneously solve for the approprate order quantty for each tem and the shared resource capacty level n the presence of all-unts and ncremental quantty dscounts. The model s able to effcently and accurately solve large-scale problems wth up to 15,000 tems n under an hour. Ths has potental to be an nvaluable tool for manufacturers or retalers who routnely receve quantty dscounts on ther orders but also face capacty restrctons. These can be specal restrctons (e.g., warehouse square footage) or budgetary restrctons (e.g., avalable cash). 2

17 Chapter Three presents an extenson of Chapter Two. In Chapter Two, the mathematcal model soluton apples a commonly used orderng structure for whch each tem s treated ndependently; thus, the capacty level must be able to handle a worst-case scenaro where all tems are ordered smultaneously. An alternatve orderng structure lnks all the tems replenshment ponts through a common replenshment cycle. For example, the frm could reorder nventory weekly. To prevent the worst-case scenaro from Chapter Two, the replenshment pont of each tem s phased wthn the cycle; therefore, some tems are ordered Monday, some Tuesday, etc. In Chapter Three I develop a mathematcal model to solve the same tradtonal shared resource allocaton problem, but utlzng a new orderng structure. Agan, I allow for capacty expanson (or reducton) and consder all-unts quantty dscounts. In the numercal analyss, I compare the two orderng structures. In general, the ndependent cycle approach outperforms the common replenshment cycle approach, but the common replenshment cycle approach s valuable as t mrrors the concepts of a perodc revew system seen n nventory management, whch s covered n most ntroductory operatons management texts (e.g., Hezer and Render 2014). The prmary advantage of such a system s the convenence factor of placng orders and recevng goods on a fxed and repeatable schedule. The common replenshment cycle approach allows for ths type of setup, and the convenence factor lkely outweghs the added cost for many managers who do not want to montor ther nventory replenshment needs constantly. Chapter Four ncorporates my thrd dssertaton essay, whch helps manufacturers determne n whch channel(s) to dstrbute ther products. More specfcally, the model prescrbes f they should sell ther product drectly through a manufacturer-owned retal store or through an onlne outlet, or f they should sell ther product through an ntermedary, such as an 3

18 ndependent retaler. Each manufacturer may also choose a mx of both, or a dual channel strategy where they sell a proporton of ther products through a drect channel and the rest through an ndrect channel. My essay s unque n that t s the frst of ts knd to ntroduce asymmetry among the characterstcs of each supply chan and ts products. I determne the equlbrum channel structure for each manufacturer n a duopoly where the manufacturers compete va dfferent forms of competton and wth varyng levels of asymmetry n brand equty and product substtutablty. All three factors play an mportant role n determnng the equlbrum channel structure for each manufacturer. Intuton leads us to beleve that, f possble, sellng drectly and elmnatng the mddle man s preferable, but t turns out that under ntense competton at the retal level, manufacturers beneft from removng themselves from that competton by sellng and proftng through ther sales to an ndependent retaler. Ths study demonstrates the need to ncorporate these asymmetres and dfferent forms of competton nto the channel structure decson facng many manufacturers. Chapter Fve summarzes the fndngs from two manageral surveys regardng the common practces assocated wth quantty dscounts from both buyers and sellers perspectves. The am of the surveys s to update the common practces and uses of quantty dscounts n ndustry to help motvate and dentfy future research opportuntes. Recently, academc research on quantty dscounts and the needs of practtoners who face quantty dscounts seem to be gong n dfferent drectons. Ths essay and these surveys ntend to help dentfy ways to brdge the gap between academa and ndustry. Addtonally, I hope to gan a better grasp of practtoners knowledge of some basc quantty dscount gudelnes as well as the extent of ther use of proft or cost analyses n ther decson makng n the presence of quantty dscounts. The results are nformatve and provde numerous potental avenues for future research. One avenue 4

19 s the development of quantty dscount tranng scenaros. A major fndng n my study s that purchasng and sales managers do not fully grasp the bascs of quantty dscounts. For example, when facng all-unts quantty dscounts t s often benefcal to order at a prce breakpont; meanwhle, when facng ncremental quantty dscounts t s never benefcal to order at a prce breakpont. A surprsng number of survey respondents do not follow these gudelnes, perhaps ndcatng a lack of nsttutonal knowledge about quantty dscounts. Ideally, these future scenaros can help us better understand the ratonale behnd decsons made by practcng managers n the presence of quantty dscounts. 5

20 CHAPTER TWO SHARED RESOURCE CAPACITY EXPANSION DECISIONS FOR MULTIPLE PRODUCTS WITH QUANTITY DISCOUNTS Abstract I analyze the tradtonal shared resource capacty allocaton problem by ncorporatng the exstence of quantty dscounts for multple products and convertng the capacty level nto a decson varable. By utlzng a pecewse-lnear approxmaton for capacty cost, the algorthms can generate solutons regardless of the functonal form of capacty cost (.e., concave or convex). The model can accommodate both all-unts and ncremental quantty dscounts, or even a mxture of both. I utlze numercal examples and senstvty analyss to understand the key factors that nfluence the capacty expanson decson and the performance of the algorthms. The algorthms can ncorporate smultaneous lot-szng decsons for thousands of products n reasonable soluton tme. 6

21 2.1 Introducton Wth ever-shrnkng margns n today s marketplace, proper nventory management has become more vtal than ever before. Purchasng managers typcally procure many products smultaneously, often subject to an nventory-based resource constrant (e.g., warehouse space). Basc nventory models, such as the economc order quantty (EOQ) model, assume that procurement decsons relate to a sngle product wth no constrants on the order sze. These basc models are well-known and easy to solve analytcally. However, the addton of a common resource constrant, multple products, and quantty dscounts sgnfcantly complcates the determnaton of optmal order quanttes. Inventory-based common resource constrants typcally come n one of two forms: warehouse constrants or fnancal constrants. A warehouse can be constraned n terms of volume (or square footage), weght, or number of unts. The most common warehouse constrant s square footage, as the drve toward lean n many ndustres today severely lmts avalable warehouse space. Fnancal constrants mght arse, for example, from a maxmum nventory value that nsurance covers or from a credt lne avalable to purchase nventory. A natural, but often forgotten, queston s, Would t be valuable for our frm to ncrease the capacty of our common resource? Too often managers may smply assume that capacty s fxed, but n realty, more warehouse space may be avalable for purchase, or unused warehouse space may be convertble for lease. Any expanson decson comes wth a cost, so what s the beneft? The vast majorty of busnesses have opportuntes to receve quantty dscounts for at least some of ther purchased products (Munson and Rosenblatt 1998). By ncreasng the resource capacty, frms open up more opportuntes to take advantage of quantty dscounts. The model n ths chapter 7

22 accommodates both of the common quantty dscount forms: all-unts and ncremental (Hadley and Whtn 1963). Specfcally, I address the followng research queston: When a frm faces allunts or ncremental quantty dscount schedules for multple products n a common resourceconstraned nventory system, how much capacty should exst, and how many unts of each tem should the frm order gven that capacty lmtaton? The remander of ths chapter s organzed as follows. In Secton 2.2, I revew relevant lterature on quantty dscounts and the capactated common resource problem. In Secton 2.3, I ntroduce the baselne model and algorthm when facng all-unts quantty dscounts. Ths s followed n Secton 2.4 by an extenson of the baselne model to ncorporate capacty reducton decsons (leadng to potental capacty cost savngs) along wth capacty expanson decsons. The model and soluton algorthm are then modfed to handle ncremental quantty dscounts n Secton 2.5. In Secton 2.6, I descrbe numercal studes that test the performance of the algorthms along wth senstvty analyss that dentfes the key parameters nfluencng the capacty expanson decson. Fnally, n Secton 2.7, I conclude the study and dentfy drectons for future research. 2.2 Lterature Revew Several hundred academc artcles on quantty dscounts have appeared. Benton and Park (1996) as well as Munson and Rosenblatt (1998) summarze the work publshed through the turn of the century and provde an overvew of the landscape of the quantty dscount lterature. A steady stream of quantty dscount papers has contnued snce then (e.g., Rubn and Benton 2003, Munson and Hu 2010, Manerba and Manss 2012, Hamman et al. 2014) and are summarzed n an updated and expansve revew of the quantty dscount lterature by Munson and Jackson 8

23 (2014). Ths chapter draws from the current quantty dscount lterature along wth the capactated common resource problem lterature to determne both the correct common resource level and order quanttes for each tem. Prevously publshed models that address the common resource capacty problem have three key characterstcs: prcng structure, orderng structure, and capacty flexblty. Table 2.1 provdes a summary of pror lterature and the characterstcs of each artcle. There are two common prcng structures: fxed prcng (.e., no quantty dscounts) and varable prcng wth quantty dscounts. Hadley and Whtn (1963) and Johnson and Montgomery (1974) were among the frst to analyze the fxed prcng (undscounted) problem. They use an orderng structure whereby each product has an ndependent cycle length,.e., tme between orders. Ths type of orderng structure forces the frm to prepare for the worst-case scenaro when all products are ordered at once and arrve smultaneously. Lagrangan relaxaton s the most popular soluton technque to solve the undscounted common resource problem wth ndependent cycle tmes. Rosenblatt (1981) and Rosenblatt and Rothblum (1990) modfy the orderng structure to have every product on a fxed cycle length. The replenshment ponts are then phased wthn the fxed cycle length. Ths technque lowers the maxmum nventory level by elmnatng the possblty of the worst-case scenaro that ndependent cycle lengths mght produce. There are several papers dedcated to solvng the capactated common resource problem wth quantty dscounts, but all of them assume that capacty s fxed. Prkul and Aras (1985) wrote the semnal paper analyzng all-unts quantty dscounts, whch solves the problem wth each product havng an ndependent cycle length. The authors ntroduce a Lagrangan relaxaton on the capacty constrant to develop a lower bound on the orgnal objectve functon. Once the Lagrangan problem s solved, a bsecton method fnds a near-optmal soluton to the orgnal 9

24 problem. Rubn and Benton (1993) extend the research area by ntroducng multple constrants (e.g., budget and space). Güder and Zydak (2000) combne the prevous work done on the common resource-constraned problem wth quantty dscounts and the approach taken by Rosenblatt (1981) by examnng the problem wth a fxed cycle length for all products. All of the pror models ncorporate the typcal EOQ assumptons, one of whch s constant and known demand for each product. Mnner and Slver (2007) ntroduce stochastc demand nto the resource-constraned problem, and Zhang (2010) expands ther research by addng all-unts quantty dscounts. Sh and Zhang (2010) formulate the Zhang (2010) problem as a mxed nteger nonlnear program and solve t usng Lagrangan relaxaton. Table 2.1: Summary of pror lterature. Reference Prcng Structure a Orderng Structure b Capacty Flexblty c Güder et al. (1994) I I F Güder and Zydak (1997) A NS F Güder and Zydak (2000) A F F Hadley and Whtn (1963) F I F Haksever and Moussouraks (2005) F F,I F Haksever and Moussouraks (2008) I F F Hall (1988) F F V Johnson and Montgomery (1974) F I F Mnner and Slver (2007) F I F Moussouraks and Haksever (2008) A F,I F Prkul and Aras (1985) A I F Rosenblatt (1981) F F,I F Rosenblatt and Rothblum (1990) F F V Rubn and Benton (1993) A I F Rubn and Benton (2003) I I F Zhang (2010) F NV F a A = all-unts, F = fxed, I = ncremental b F = fxed, I = ndependent, NS = non-statonary, NV = newsvendor c F = fxed, V = varable 10

25 Fewer artcles have addressed the ncremental quantty dscount case. Güder et al. (1994) modfy the approach from Prkul and Aras (1985) for ncremental dscounts. Ther heurstc assumes that each product has an ndependent cycle length, and t fnds a near-optmal soluton through the use of Lagrangan relaxaton. Rubn and Benton (2003) extend pror research by allowng multple constrants (.e., fnancal and space lmtatons). Ther methods are adopted from ther earler paper (Rubn and Benton 1993) that analyzes ths problem when facng allunts dscounts. Ther ncremental dscount artcle also nvokes a Lagrangan relaxaton, coupled wth partal enumeraton, to solve for near-optmal order quanttes. Hall (1988) and Rosenblatt and Rothblum (1990) were among the frst to analyze the common resource capacty as a decson varable. Recently, Gham et al. (2013) and Dye et al. (2007) have developed models that allow the opportunty to rent extra warehouse space when the nventory level exceeds the capacty of the orgnal warehouse. To the best of my knowledge, no lterature to date has ntroduced quantty dscounts nto the common resource capacty problem when the common resource capacty s a decson varable. Ths chapter flls that research gap. I develop algorthms to determne effcent solutons for a constraned, mult-product nventory problem that ncorporates quantty dscounts (both all-unts and ncremental) and the resource capacty as decson varable. The man contrbuton of ths chapter s the consderaton of quantty dscounts smultaneously wth a dynamc nventory-based resource constrant. A second key contrbuton of ths chapter s the ablty of the soluton algorthm to ncorporate any functonal form of capacty cost. Prevous studes (e.g., Rosenblatt and Rothblum 1990, Hall 1988) have lmtatons on the functonal form of capacty cost (e.g., convex or lnear, respectvely). Ths unrestrcted capacty cost model allows, among other thngs, the capacty cost to ncorporate annualzed economes of scale benefts seen n larger expansons. 11

26 2.3 Model The baselne model smultaneously solves for the approprate common resource capacty level and the order quantty for each tem gven the capacty lmtaton when facng all-unts quantty dscounts (for ncremental dscounts, see Secton 2.5). Recall that the common resource could be an nventory-based resource (e.g., warehouse space) or a fnancal-based resource (e.g., a credt lne for nventory). In an unconstraned system, quantty dscounts ncentvze purchasng managers to purchase larger quanttes, but that potentally requres a larger capacty. Ths model attempts to dentfy the deal balance between the purchasng cost savngs from quantty dscounts and the capacty cost ncrease. I nvoke a pecewse-lnear approxmaton of capacty cost to determne effcent order quanttes and the approprate capacty level. Consstent wth pror lterature, ths model uses many tradtonal assumptons to assst wth analytcal tractablty. I nvoke the typcal EOQ assumptons of determnstc and ndependent demand for each tem, constant demand over tme, no backorders, and nstantaneous replenshment (.e., nfnte producton rate and zero lead tme). My model assumes that each tem has an ndependent cycle length (or tme between orders). Under ths orderng structure, the capacty level must be able to accommodate the worst-case scenaro where all tems are ordered smultaneously, resultng n the hghest possble nventory levels. I assume that the resource capacty s a decson varable that can be reduced (reducng capacty cost) or expanded (ncreasng capacty cost). Wthout loss of generalty, from ths pont forward I assume that the constraned common resource s the space avalable n a warehouse (other shared resource constrants would be modeled smlarly). Fnally, I assume that the capacty cost functon s monotoncally non-decreasng wth respect to capacty level. 12

27 2.3.1 Notaton Indces: tem to be purchased, = 1, 2,, I m segment of the pecewse-lnear approxmaton of capacty cost, m = 0, 1, 2,, M Parameters: D annual demand for tem S setup (or orderng) cost for tem h annual holdng cost as a percentage of purchase prce for tem k amount of the common resource used per unt of tem K0 ntal resource capacty level ym capacty breakpont n the pecewse-lnear approxmaton of capacty cost between segments m 1and m Vm capacty cost assocated wth a capacty level of ym bm margnal capacty cost for each addtonal unt of capacty n segment m Decson Varables: K resource capacty Q order quantty for tem Cost Equatons: H(Q) annual holdng cost for tem P(Q) annual purchasng cost for tem G(K) annual capacty cost assocated wth havng capacty level K Modelng Prelmnares The goal s to determne the optmal combnaton of resource capacty level and order quanttes to mnmze annual setup, nventory holdng, purchasng, and capacty costs when facng quantty dscounts. For an all-unts quantty dscount schedule wth J + 1 prces, I defne qj as the j th prce breakpont n the prce schedule for tem, j = 0, 1,, J (q0 = 0, q(j+1) = ), and A p j as the perunt prce for all unts purchased when qj Q < q(j+1). The general objectve functon s: I D Z= S + HQ ( ) + PQ ( ) + GK ( ), (2.1) = 1 Q 13

28 A where qj Q < q(j+1), H( Q ) ( Q )( hp j ) A j = /2, and PQ ( ) = p D. Note that the objectve functon (2.1) s not contnuous due to the exstence of all-unts quantty dscounts. If the problem dd not nclude quantty dscounts and f G(K) were convex, then the objectve functon would be contnuous and convex hence solvable usng frst order condtons. Rosenblatt and Rothblum (1990) and Hall (1988) assume that G(K) s convex. However, the shape of G(K) s dependent upon the nterpretaton of the resource capacty cost. Both artcles defne G(K) as the yearly cost of mantanng a capacty level K. Hall (1988) assumes a specal case where G(K) s a lnear functon, whereas Rosenblatt and Rothblum (1990) assume a more general convex functon. Alternatvely, G(K) could well represent the annualzed cost assocated wth buldng, purchasng, or leasng extra warehouse space. Due to the prevalence of economes of scale, the cost of a capacty ncrease mght easly be a concave functon of the sze of expanson (Nahmas 2009), as opposed to a convex functon. Another potental feature of the capacty cost functon would be the exstence of a potentally large fxed cost assocated wth any capacty expanson. The potentally dverse nature of the capacty cost functon leads to multple analytcal complcatons. Let me begn wth the general formulaton of the problem: I D mn S + HQ ( ) + PQ ( ) + GK ( ) = 1 Q s.t. I = 1 kq Q 0 K 0 K (2.2) Snce K s a decson varable and G(K) s assumed to be monotoncally non-decreasng n K, the capacty constrant wll be bndng at the optmal soluton: 14

29 I kq = K. (2.3) = 1 Ths result allows me to absorb (2.3) nto the objectve functon (Rosenblatt and Rothblum 1990). Updatng the optmzaton problem: I I D mn S + H ( Q ) + P ( Q ) + G ( kq ) = 1 Q = 1 s.t. Q 0 (2.4) From here, I frst analyze a smplfed verson of the problem wth lnear capacty cost and then extend to an approxmaton that can handle any functonal form of G(K) Smplfed Case: Lnear Capacty Cost wth No Intal Capacty Defne G(K) as the annual cost of mantanng a capacty level K. Followng Hall (1988), let me assume that G(K) s a lnear functon of K. Ths smplfed setup assumes there s no ntal capacty K0. Let b represent the margnal cost of mantanng one addtonal unt of capacty per year. Updatng the optmzaton problem: mn Z = D S + H( Q) + P( Q) + b kq s.t. Q 0 I I = 1 Q = 1 (2.5) Due to the dscontnuous objectve functon for each tem, each prce level must be consdered ndvdually to determne the optmal order quantty for that prce. Consder the frst and second dervatves of the objectve functon ( Z ) wth respect to Q where qj Q < q(j+1): δ Z DS h = + + (2.6) δq Q A p, 2 j bk 2 15

30 δ δ = > 0. (2.7) 2 Z 2DS 2 3 Q Q Note that the objectve functon s separable wth respect to each tem s order quantty. Ths allows me to optmze each tem ndvdually. As Z s convex (from (2.7)) the frst order condtons are suffcent to fnd the mnmum pont xj for prce level j and tem : x j = 2DS. A h p + 2bk j (2.8) However, xj may not be feasble,.e., xj may not le wthn prce nterval j. If xj < qj, then qj s the best order quantty for that nterval. If xj q(j+1), then the global optmal order quantty wll not le wthn nterval j. Adaptng from Munson and Hu (2010), the optmal order quantty for tem s: where j * =Argmn{j=0,1,, J} Zj, and Q * j* qj* = max(x, ), (2.9) D max(, ) xj qj A A Zj = S + h pj + pj D + bk max( xj, qj ). max( xj, qj ) 2 (2.10) The value of Zj* represents the total cost of Q *. Upon the completon of ths process for each tem, the optmal capacty (K * ) s: K I * * kq = 1 =. (2.11) Whle managers could mplement the soluton to the smplfed case drectly nto spreadsheets, practcal applcatons of lnear capacty costs are lmted. I now move to a more general soluton procedure that s applcable regardless of the functonal form of capacty cost. 16

31 2.3.4 General Case: General Capacty Cost wth Intal Capacty In the smplfed case, I assumed that there s no ntal capacty (e.g., buldng a new warehouse) and that G(K) was the annualzed lnear cost of mantanng a capacty level K. Now let me consder an ntal capacty level K0 and the potental to expand the capacty through an nvestment that s non-decreasng n the expanson sze. I convert the nvestment to an annualzed payment to contnue to express the objectve functon as the annual total cost. For now let me lmt ourselves to capacty expanson opportuntes (I extend ths to ncorporate capacty reducton n Secton 2.4). To elmnate restrctons on the functonal form of the capacty cost, I ntroduce a pecewse-lnear and monotoncally non-decreasng approxmaton of G(K). The pecewse-lnear functon can be broken nto M + 1 segments. Obvously, more segments lead to a more accurate, but more computatonally cumbersome, soluton. When analyzng the pecewse-lnear functon segment by segment, each segment of the objectve functon (2.1) becomes convex regardless of the orgnal functonal form of capacty cost (.e., convex or concave), allowng me to fnd approxmate analytcal solutons. The best soluton s the feasble segment soluton wth the lowest total cost. Consder a segment m that goes from capacty level ym to ym+1 and has a margnal cost of bm for each addtonal unt of capacty added. Let Vm be the capacty cost of mantanng capacty level ym (Chopra and Mndl 2010). Defne y0 = 0, y1 = K0, ym+1 =, and b0 = 0 to provde one segment n case the fnal capacty s less than or equal to the ntal capacty level. Ths results n V0 = V1 = 0. I then calculate Vm for 2 m M as follows: Then: V = b ( y y ) b ( y y ). (2.12) m m 1 m m 1 17

32 0 f K < K0 GK ( ) = Vm + bm( K ym) f y m K < ym+ 1, for m= 1,2,..., M. (2.13) Note that f there s no ntal capacty level, one can smply elmnate segment m = 0. Addtonally, capacty expansons could nclude a fxed upfront cost. Such a cost mght ncorporate a myrad of expendtures, e.g. a contractor s fxed fee or the cost of shuttng the lne down durng producton. Managers may also ncorporate a theoretcal fxed cost as a mechansm to prevent the algorthm from recommendng nsgnfcant changes n capacty. To nclude a fxed cost n (10), one can smply add a fxed cost F to G(K) when K K0. Updatng the objectve functon Z for each segment m, where m= 0, 1,, M: I I m D Z = S + H( Q) + P( Q) + Vm + bm kq ym. = 1 Q = 1 (2.15) Smlar to the smplfed case, the approxmate analytcal mnmum pont ( x ) for tem n prce level j n segment m s found usng frst order condtons: m j x = 2DS. A hp + 2b k m j j m (2.16) The order quantty for tem s: m m j* j* Q = max( x, q ), (2.16) m j where j * =Argmn{j=0,1,, J} Z, and m m D max( xj, qj ) A A m Zj = S + hp j + pj D + bmk max( xj, qj ). m max( xj, qj ) 2 (2.17) Note that Z m becomes: m I m j* m m m. (2.18) = 1 Z = Z + V b y 18

33 Next, I must check that the resultng capacty falls nto the m th segment of the pecewselnear approxmaton for the capacty cost. The adjustment to assure that the capacty level falls wthn the correct segment s dffcult because the capacty level s dependent on I decson varables, namely, the order quanttes for each tem. Lemma 2.1 allows me to gnore certan segments based on a smple check of the segment soluton s fnal capacty level. Lemma 2.1 If the soluton provded by (13) for a partcular segment m results n a capacty level K < y, then the overall optmal capacty level K * wll not unquely le n segment m. m m Proof. Assume the condton from Lemma 2.1 s true. Identfy the segment w such that w m yw Km y w 1, + and let Q = Q, resultng n a feasble capacty level for segment w of K w Km. = Snce the capacty cost s non-decreasng n capacty level, there exsts a feasble soluton n segment w wth dentcal nventory-related costs and wth capacty costs that are less than or equal to the capacty costs for segment m. Algorthm 2.1 allows me to adjust the orgnal soluton for segment m to fall wthn the approprate capacty nterval. Whle ths soluton procedure does not guarantee an optmal soluton to the common resource capacty decson problem when facng all-unts quantty dscounts, t does provde an effcent soluton that s solvable regardless of the functonal form of the cost of capacty expanson. In Step 3, I utlze Lemma 2.1 to only consder capacty adjustments where the fnal capacty level requres reducton to acheve feasblty. 19

34 Algorthm 2.1: All-Unts Quantty Dscounts m m m Step 1 For each segment m (m = 0, 1,, M), calculate order quanttes ( Q1, Q2,..., Q I ) from (2.16), m m Z from (2.18), and K = kq. m Step 2 Defne segment m * m = mn Z. m Step 3 If Km* y m* 1, + then go to Step 5. Otherwse, go to Step 4. Step 4 Let m = m *. Usng a fxed capacty y m + 1 for segment m, solve for Q m usng Algorthm A.1 n Appendx A. Update Step 5 Set ( * * * m ) ( * m * m Q * 1, Q2,..., QI Q1, Q2,..., QI ) m Z m usng (2.18), and re-calculate m. = and set * K = K. m* K kq = Go to Step Capacty Reducton Opton I now relax the assumpton that capacty can only expand from the ntal capacty level, allowng frms to lease out excess capacty as another form of revenue. The soluton technque s smlar to the procedure descrbed n Secton 2.3.4, wth mnor modfcatons to (2.12) and (2.13) to ncorporate capacty reductons. Let r1, r2,, rr be the segments of the pecewse-lnear capacty cost approxmaton that result n capacty reducton, and let e1, e2,, ee be the segments assocated wth capacty expanson. Addtonally, let yk0 = ye1 = yr1 = K0 (thus VK0 = Vr1 = Ve1 = 0), yr(r+1) = 0, ye(e+1) =, and remove the condtons from Secton 2.3 that V0 = V1 = 0 and b0 = 0. Updatng (2.12) and (2.13): br1( yr2 yr1) + br2( yr3 yr2) bm 1( ym ym 1) for m = r2, r3,..., rr Vm = 0 for m= r1, e1 be1( ye2 ye1) + be2( ye3 ye2) bm 1( ym ym 1) for m = e2, e3,..., ee, (2.19) and 20

35 Vm + bm( K ym) f ym+ 1 K < ym for m = r1, r2,..., rr GK ( ) = 0 f K= K0 Vm + bm( K ym) f ym K < ym+ 1 for m = e1, e2,..., ee, (2.20) whch are llustrated n Fgure 2.1. I mantan non-negatvty for all the slopes (bm) of the pecewse-lnear approxmaton. To acheve the reducton n cost assocated wth a capacty reducton, the order of the segment endponts s reversed for segments r1, r2,, rr. Once the modfcatons are mplemented, utlze Algorthm 2.1 to solve for the approprate capacty level and order quanttes. Fgure 2.1: Illustraton of the pecewse-lnear approxmaton for capacty cost when there s an opton for capacty reducton. 2.5 Incremental Dscounts In ths secton, I explore the necessary changes to mplement and solve the capactated common resource problem when facng ncremental as opposed to all-unts quantty dscounts. There are two dstnct dfferences between all-unts and ncremental quantty dscounts. Frst, wth all-unts quantty dscounts, the optmal order quantty often falls on a prce break quantty; therefore, I 21

36 m m j j calculate the order quanttes n Secton as Q = max( x *, q *). Alternatvely, wth ncremental dscounts the optmal order quantty never falls on a prce break quantty. Second, ncremental dscounts only apply each specfed dscount to any unts that fall wthn that dscount s respectve quantty nterval, whereas all-unts dscounts apply to all unts n the order. Followng standard conventon, let Rj represent the total purchasng cost assocated wth qj unts when facng ncremental dscounts: I I I I j = ( j 1) j ( j 1) R p q p ( q q ) p ( q q ) p ( q q ), (2.21) where I p j s the per-unt prce for unts (Q qj) purchased when facng ncremental quantty dscounts and qj Q < q(j+1). Thus, the total purchasng cost assocated wth orderng Q unts when facng ncremental dscounts and qj Q < q(j+1) s: I j j j PQ ( ) = p ( Q q ) + R. (2.22) Gven the two dfferences between all-unts and ncremental quantty dscounts, a modfcaton to the soluton algorthm s requred. Smlar to the all-unts case, I calculate the mnmum pont m x j for tem n prce level j n segment m usng frst order condtons: x m j = I 2 D ( S + Rj pq j j ) I. hp + 2b k j m (2.23) Adaptng from Munson and Hu (2010), the optmal order quantty for tem among these canddate values m x j s: Q m m j*, = x (2.24) m j where j * =Argmn{j=0,1,, J} Z, and 22

37 m D I h I m I m Zj = ( S Rj pjqj ) ( Rj pj ( Q qj )) pjd bmkq. m Q 2 (2.25) Agan, Z m follows (2.18). Algorthm 2.2 ncorporates the necessary modfcatons to Algorthm 2.1 to handle ncremental quantty dscounts. Note that, due to the smlartes between Algorthms 2.1 and 2.2, t s straghtforward to combne them for a scenaro where a frm s supplers offer a mx of allunts and ncremental quantty dscounts. Ths s possble because the order quantty calculatons n Algorthms A.1 and A.2 (see Appendx A) are separable by tem. Thus, as long as those algorthms are solved smultaneously usng the same λ, tems wth both forms of quantty dscounts can be combned. Algorthm 2.2: Incremental Quantty Dscounts m m m I Step 1 For each segment m (m = 0, 1,, M), calculate order quanttes ( Q1, Q2,..., Q ) from (2.24), m m Z from (2.18) and K = kq. m Step 2 Defne segment m * m = mn Z. m Step 3 If K * y * 1, + then go to Step 5. Otherwse, go to Step 4. m m Step 4 Let m = m *. Usng a fxed capacty y m + 1 for segment m, solve for Q m usng Algorthm A.2 n Appendx A. Update Step 5 Set ( * * * m ) ( * m * m Q * 1, Q2,..., QI Q1, Q2,..., QI ) m Z m usng (2.18), and re-calculate m. = and set * K = K. m* K kq = Go to Step 2. 23

38 2.6 Numercal Analyss Algorthm Performance Analyss The two key attrbutes that defne the effectveness and applcablty of an algorthm are ts accuracy and computatonal run-tme. For these algorthms, the accuracy s dffcult to measure for multple reasons. Frst and foremost, when there are more than a handful of tems, the true optmal soluton becomes too computatonally cumbersome to solve. Second, to the best of my knowledge, there are no other algorthms for comparson that allow for capacty expansons n the capactated common resource problem when facng quantty dscounts. The best alternatve to test the accuracy of ths algorthm s to fx capacty and compare these solutons to the fxed capacty problems from pror lterature. Rubn and Benton (1993) and Moussouraks and Haksever (2008) solve the fxed capacty problem when facng all-unts quantty dscounts, and, smlarly, Rubn and Benton (2003) and Haksever and Moussouraks (2008) solve the problem when facng ncremental quantty dscounts. For the comparsons, I defne an arbtrarly hgh capacty expanson cost to prevent the algorthms from consderng capacty expanson. Rubn and Benton (1993) orgnally provded two examples wth unque demands and cost structures. The frst example, the Benton Example, has 10 tems and 3 prce levels, and t s ncorporated nto Studes 1 and 2 wth dfferent fxed capacty levels. Study 1 uses a capacty of 2350 square feet, whereas Study 2 uses a capacty of 1350 square feet. The second example, the Fortune 500 Example, has 15 tems and 4 prce levels, and t s utlzed for Study 3 wth a fxed capacty level of 35,000 square feet. All demand and cost nformaton for the two examples s presented n Appendx B. In all sx studes (three all-unts and three ncremental dscounts), f an tem has fewer than the maxmum number of prce levels, I ntroduce dummy prce levels where the prce does not change. Table 2.2 llustrates the compettveness of ths algorthm compared to the pror studes of Rubn and 24

39 Benton (1993) and Moussouraks and Haksever (2008). The algorthm penalty compares the total cost found here wth the lowest cost found by the two pror algorthms. For example, n Study 2 my algorthm provdes a total cost that s 0.005% hgher than the best soluton from pror lterature. For each all-unts quantty dscount study, my algorthm performs comparably to the pror fxed capacty algorthms. Table 2.2: All-unts quantty dscount algorthm performance compared to the bestperformng algorthm between Rubn and Benton (1993) and Moussouraks and Haksever (2008). Study Total Cost Penalty 1 0% % % Rubn and Benton (2003) and Haksever and Moussouraks (2008) analyze smlar examples to those n Rubn and Benton (1993), except that they use ncremental dscounts and a slghtly dfferent quantty dscount schedule for the Benton Example. Studes 4 and 5 ncorporate the Benton Example wth the ncremental quantty dscount schedule and fxed capacty levels of 2350 square feet and 1350 square feet, respectvely. The ncremental quantty dscount schedule s presented n Appendx B. Study 6 utlzes the Fortune 500 Example wth a fxed capacty level of 35,000 square feet. There s no adjustment n the quantty dscount schedule for the Fortune 500 Example. Table 2.3 dsplays the compettveness of my algorthm compared to Rubn and Benton (2003) and Haksever and Moussouraks (2008). Once agan, my algorthm performs comparably wth the pror algorthms. I performed a computatonal run-tme analyss to show that my algorthms have reasonable computatonal run-tme requrements even for large problems and to understand the 25

40 key drvers that ncrease the algorthms run-tme. Followng Nahmas (2009) I utlzed a capacty cost functon wth the followng functonal form: a 0 GK ( ) = rk ( K), (2.26) where r s a constant of proportonalty and a s a measure of the rato between the ncremental and average cost of an addtonal unt of capacty. Ths functonal form of capacty cost allows for the functon to be strctly concave (a < 1), lnear (a = 1), or strctly convex (a > 1). Table 2.4 shows the structural nput values consdered n ths study that I expected to potentally nfluence run tme. For each combnaton, ten trals were run wth randomly generated data for the other system parameters (Table 2.5). There are a total of 1920 trals splt evenly between the all-unts and ncremental quantty dscounts. The soluton procedure was mplemented n Mcrosoft s Excel usng VBA and all trals were run on a PC wth an Intel Core 2 Duo 3.00GHz. The computatonal tmes are reported n CPU tme of ths machne. Table 2.3: Incremental quantty dscount algorthm performance compared to the bestperformng algorthm between Rubn and Benton (2003) and Haksever and Moussouraks (2008). Study Total Cost Penalty % % % Tradtonally, the prmary drver of varablty n computatonal run tme for algorthms determnng order quanttes n a mult-tem nventory system s the number of dfferent tems. Table 2.6 summarzes the run tme as a functon of the number of unque tems for both all-unts and ncremental quantty dscounts. The algorthms are able to effcently solve every tral n under two mnutes wth up to 5000 tems, fve prce levels, and ten segments n the pecewse- 26

41 lnear approxmaton of capacty cost. The summary statstcs nclude all 80 trals for each form of quantty dscount and number of tems combnaton. Table 2.4: Parameter values used n the run-tme analyss. Parameter Values Number of tems (I) 50, 100, 500, 1500, 3000, 5000 Number of prce levels (J) 3, 5 Number of pecewse-lnear segments (M+1) 5, 10 Capacty Cost Parameter (a) a 0.4, 2 Intal Capacty (K0) 0.8 EOQ,1.25 EOQ I I a The proportonalty parameter (r) s adjusted approprately for changes n a, the number of tems, and the correspondng requred capacty. Table 2.5: Dstrbutons used to generate data for other system parameters used n the run-tme analyss. D ~ U(400, 2500) S ~ U(10p0, 20p0) h ~ U(0.20, 0.40) k ~ U(0.5, 1.5) p0 ~ U(1, 15) q0 = 0 p1 = p0 α1, where α1 ~ U(0.01, 0.30) q1 = (EOQ0)γ1, where γ1 ~ U(0.2, 2.0) pj = p(j-1) βj, where βj ~ U(0.00, 0.20) qj = q(j-1) (1+δj), where δj ~ U(0.5, 1.0) Table 2.6: Summary statstcs on computatonal run tme (n seconds) for each combnaton of number of tems and form of quantty dscounts. # of Items Average Standard Devaton Max All-Unts Increm. All-Unts Increm. All-Unts Increm After fxng the number of tems, further analyss found that the number of segments M + 1 n the pecewse-lnear approxmaton of capacty cost s another prmary drver of varablty 27

42 n the computatonal run tme. Table 2.7 llustrates the mpact, where on average I found a 39% ncrease n run tme assocated wth ncreasng the number of segments from 5 to 10. The mpact of the ncrease n the number of segments seems to be enhanced by an ncrease n the number of tems. An ncrease n the number of segments n the pecewse-lnear functon ncreases the accuracy of the capacty cost approxmaton, but t sgnfcantly ncreases run-tme. Table 2.7: Impact on the number of pecewse-lnear segments on run tme (n seconds). # of Items All-Unts Dscounts Incremental Dscounts 5 Segments 10 Segments 5 Segments 10 Segments The number of prce levels n the quantty dscount schedule, the functonal form of capacty cost (concave vs. convex), and the ntal capacty level dd not show consstent results n terms of ther mpact on the varablty of computatonal run-tme. Table 2.6 llustrates an nterestng nsght n that the form of quantty dscount mpacts the computatonal run tme of the algorthms. It s apparent that all-unts quantty dscounts result n longer computatonal run tmes n comparson wth ncremental dscounts. Ths result s counterntutve n that tradtonal all-unts quantty dscount problems are typcally easer to solve. I beleve that ths counterntutve fndng s a result of the reduced lkelhood of an expanson wth ncremental dscounts (see Secton 2.6.2). Therefore the algorthm consders fewer segments n the cases where the capacty expanson costs outwegh the savngs from the ncremental quantty dscounts. 28