Competition under Capacitated Dynamic Lot Sizing with Capacity Acquisition

Transcription

1 Competton under Capactated Dynamc Lot Szng wth Capacty Acquston Hongyan L 1 and Joern Messner 2 1 CORAL, Department of Busness Studes, Aarhus School of Busness, Aarhus Unversty, Denmark. hojl@asb.dk 2 Department of Management Scence, Lancaster Unversty Management School, Unted Kngdom. joe@mess.com August 20, 2010 Abstract Lot-szng and capacty plannng are mportant supply chan decsons, and competton and cooperaton affect the performance of these decsons. In ths paper, we look nto the dynamc lot szng and resource competton problem of an ndustry consstng of multple frms. A capacty competton model combnng the complexty of tme-varyng demand wth cost functons and economes of scale arsng from dynamc lot-szng costs s developed. Each frm can replensh nventory at the begnnng of each perod n a fnte plannng horzon. Fxed as well as varable producton costs ncur for each producton setup, along wth nventory carryng costs. The ndvdual producton lots of each frm are lmted by a constant capacty restrcton, whch s purchased up front for the plannng horzon. The capacty can be purchased from a spot market, and the capacty acquston cost fluctuates wth the total capacty demand of all the competng frms. We solve the competton model and establsh the exstence of a capacty equlbrum over the frms and the assocated optmal dynamc lot-szng plan for each frm under mld condtons. Keywords: Game theory, capacty optmzaton, competton, lot szng, approxmaton, equlbrum Please reference ths paper as: Hongyan L, Joern Messner. Competton under Capactated Dynamc Lot Szng wth Capacty Acquston. Workng Paper (avalable at Lancaster Unversty Management School, BbT E X and plan text references are avalable for download here:

2 L and Messner: Lot Szng Competton wth Capacty Acquston 1 1 Introducton One of the fundamental problems n operatons management s determnng the nvestment n capacty. A frm s capacty determnes ts maxmal potental producton. To acqure capacty s usually cost and tme consumng, and once the nvestment s made, the cost s often partally or completely rreversble, as nstalled capacty s dffcult to adjust n the short term. Moreover, the decson on how much capacty to acqure also strongly nfluences the acton space for future operatons plannng. To nvest n too much capacty wastes resources that could be used for other mportant operaton actvtes, such as new product development and marketng; to nvest n too lttle capacty means long watng tmes, mssed sales opportuntes and lost revenue. Therefore, t s necessary to fnd an effectve and comprehensve method to determne the proper capacty confguraton for operatons. Increasng the capacty does not necessarly mprove the operatonal performance, even f the product proft margns are large, because capacty acquston cost s usually negatve correlated to the producton cost and often affected by the compettve resource envronment. In addton, the compettors other decsons, such as the tmng of producton and quantty, also affect capacty acquston cost and nvestment performance. Game theoretc modellng has been an effectve method of descrbng and solvng competton problems. In ths paper, we solve a game-theoretc model of capacty competton problem over a fnte-perod plannng horzon for a multple-frm ndustry that uses a common resource to produce ts products. For each frm, ts bestresponse problem s a sngle-tem capacty acquston and lot-szng problem. The best-response problem consders a sngle-producton faclty that produces a sngle product tem to satsfy a determnstc demand stream. The best-response problem for ndvdual frms smultaneously determnes an optmal capacty and a lot sze plan over the plannng horzon. The capacty acquston, producton and nventory holdng costs are consdered. We formulate the problem as a cost mnmzng Mxed Integer Non-Lnear Programmng (MINLP) model. Ths general problem class s mpossble to solve usng a polynomal tme algorthm. Thus, we dscretze the possble capacty choces and solve t for each of those. The major dfference between the best response problem and the classcal capactated lot-szng problems s that the capacty level s an nternal decson n our model. Gven the capacty competton model, we dscuss the capacty equlbrum and assocated optmal dynamc lot-szng plans by analyzng the resulted best-response problem. We ntroduce an approxmaton for a frm s best response functon, showng through a numercal study that ts use results n only a mnor dfference to the actual

3 L and Messner: Lot Szng Competton wth Capacty Acquston 2 cost fgures but stll has desrable propertes. We then proceed to analyze the compettve problem and show the exstence of an equlbrum under modest assumptons. To the best of our knowledge, ths s the frst study to address lot-szng problems consderng resource competton. Moreover, snce the complexty of the capacty competton problem, the approxmated solutons are acceptable n practce. The remander of ths paper s organzed as follows. We revew the relevant studes n Secton 2. Secton 3 ntroduces the relevant notaton and the basc compettve model. Secton 4 frst descrbes the best-response problem that an ndvdual frm faces when makng ts purchasng and lot-szng decsons. In Secton 5, we show our suggested soluton n a structure of the game whch results n an equlbrum followng a standard procedure. Fnally, a computatonal study and numercal examples are dscussed n Secton 6. 2 Lterature Revew The am of capacty acquston decsons s to select the proper capacty that not only satsfes demand completely, but also mnmzes the total capacty acquston and lotszng cost. The research on capacty nvestment problems ncludes two man streams, the tradtonal mathematcal programmng models and the economc models. Tradtonal mathematcal programmng methods have been appled to capactyacquston problems ever snce research efforts frst took notce of them. The flexble capacty nvestment and management problems arose and were addressed at a relatvely early stage. Fne and Freund (1990) present a two-stage stochastc programmng model and an analyss of the cost-flexblty trade-offs nvolved n the nvestment n product-flexble manufacturng capacty for a frm. They address the senstvty of the frm s optmal capacty nvestment decson to the costs of capacty, demand dstrbuton and rsk level. Also, van Meghem (1998) studes the optmal nvestment problem of flexble manufacturng capacty as a functon of product prces, nvestment costs and demand uncertanty for a two-product producton envronment. He suggests fndng the optmal capacty by solvng a mult-dmensonal news-vendor problem assumng contnuous demand and capacty. Netessne et al. (2002) propose a one-perod flexble-servce capacty optmzaton and allocaton model takng the capacty acquston, usage, and shortage costs nto account. Whle each paper consders the multple products and multple resources problems wth demand uncertantes, ther focus s lmted to sngle-perod models. Apart from the studes whch focus on flexble capacty nvestment, many efforts to solve generalzed capacty-nvestment problems have also been made. Harrson and van

4 L and Messner: Lot Szng Competton wth Capacty Acquston 3 Meghem (1999) develop a sngle-perod plannng model to ncorporate both capacty nvestment and producton decsons for a multple-product manufacturng frm. Ther study yelds a mult-dmensonal descrptve model generated from the news-vendor model, and gves qualtatve nsghts nto real-world capacty-plannng and captalbudgetng practces. Nevertheless, the decsons on optmal capacty nvestment are hghly generalzed, and the producton plan decsons are not explctly presented. van Meghem and Rud (2002) extend the work of Harrson and van Meghem (1999) to nclude an operatons envronment wth multple products, producton processes, storage facltes and nventory management. Moreover, they nvestgate how the structural propertes of a sngle perod extend to a mult-perod settng. They also mprove prevous studes by consderng some nventory-management ssues. Many studes have made extensve use of game theoretc models n the development of product prcng and compettve strategc nvestment models, among others. For nstance, van Meghem (1999) uses a game-theoretc approach to model the coordnaton process of smultaneous nvestment, producton, and subcontractng decsons. The model s objectve s to maxmze the overall supply chan system proft and to analyze the sze and tmng of capacty nvestment. Whle capacty acquston problems have been studed extensvely, each paper mentoned above focused on snglefrm operatons. The competton for resources, however, s a common phenomenon n real-world operatons n a mult-frm ndustry nvolvng a partcular product but s generally gnored n the lterature because t often ncreases the ntractablty of the models, regardless of whether the model s stochastc or game theoretc. Increasng global competton and cost pressure force busnesses to dscover undetected cost-savng potentals on nvestment n resources. Arnold et al. (2009) presents a determnstc optmal control approach optmzng the procurement and nventory polcy of a company that s processng a raw materal when the purchasng prce, holdng cost, and the demand rate fluctuate over tme. However, they do not consder the effect of resource competton. The three papers lsted below address capacty decson problems emphaszng realworld capacty competton. Roller and Sckles (2000) propose a two-stage prcng and capacty-decson model consderng prce and capacty competton smultaneously. In the frst stage, the capacty s determned and a prce-settng game s performed n the second stage. Chen and Wan (2005) also study a servce capacty competton problem for two make-to-order frms that are modeled as sngle-server queueng systems. They characterze the Nash equlbrum of the competton. The frms make ther capacty choce based on the equlbra. Cheng et al. (2003) study the prce and capacty competton of two applcaton-servce provders. The authors suggest that the provders

5 L and Messner: Lot Szng Competton wth Capacty Acquston 4 wth hgher capacty would charge a hgher prce and enjoy a larger market share. Although capacty competton problems have not completely escaped notce, the aforementoned studes focus strctly on servce ndustres modeled as queueng systems. The specal operatons nature of the servce ndustry restrcts the methods from beng generalzed to other ndustres, such as manufacturng or other more complcated servce systems. Whle great progress has been made n the development of capacty-nvestment models and approaches, most studes have focused on macro analyss rather than practcal applcatons. Many complcated decson factors, such as tme-varyng costs and nventory management, have been left unconsdered. In ths sense, lot-szng methods can compensate perfectly for ths defcency n the game theoretc models, wth the combnaton approach resolvng the real-world capacty-nvestment and producton problems more realstcally. Lot-szng problems have been studed extensvely for the past half century. Wagner and Whtn (1958) gve a forward algorthm for a general dynamc verson of the uncapactated economc lot-szng model. Snce then, varous varants, ncludng sngle-tem and mult-tem, uncapactated and capactated lot-szng problems, reman an mportant topc n Operatons Research felds. More recent results nclude Federgruen and Tzur (1991), who consder a dynamc lot-szng model wth general cost structure. The authors gve a smple forward algorthm whch solves the general dynamc lot-sze model no(t logt) tme and wtho(t) space requrement. Ths s an mportant mprovement over the well-known shortest path algorthm soluton no(t 2 ) space, advocated prevously. Wagelmans et al. (1992) extend the range of allowable cost data to allow for coeffcents that are unrestrcted n sgn. They developed an algorthm to solve the resultng problem no(t logt) tme. However, the uncapactated lot-szng problem s an deal case and hardly applcable to real-world operatons. Capacty constrants always heavly nfluence productonplan decson makng. Furthermore, the general capactated lot-szng problem snphard, see Btran and Yanasse (1982). For the specal case of a constant lmt over our decson perod, a number of effcent algorthms are capable of calculatng an optmal producton plan. For example, Floran and Klen (1971) present an algorthm wth the computatonal complexty (O(T 4 )) for the capactated lot-szng problem wth constant capacty lmts, explorng the mportant propertes of an optmal producton plan, the optmal plan consstng of a sequence of optmal sub-plans. Baker et al. (1978) dscover some mportant propertes of an optmal soluton to the problem when the producton and nventory-holdng costs are constant.

6 L and Messner: Lot Szng Competton wth Capacty Acquston 5 Some studes have tred to relax the strct cost-structure restrctons n the algorthms revewed above. Krca (1990) presents a dynamc programmng-based algorthm wth the computatonal complexty ofo(t 4 ), and Shaw and Wagelmans (1998) develop a dynamc programmng algorthm for the capactated lot-sze problem wth general holdng costs and pecewse lnear producton costs. The algorthm of the latter reduces the computaton tme too(t 2 d), whereds the average demand when producton cost s lnear. Akbalk and Penz (2009) study a specal case of the capactated lot szng problem (CLSP) where the producton cost s assumed to be pece-wse lnear wth dscontnuous steps. They propose an exact pseudo-polynomal dynamc programmng algorthm whch makes tnp-hard n the ordnary sense. All the studes mentoned above address capacty competton, capacty nvestment and lot-szng problems ndvdually. The mplcatons of combnng these problems are, however, rarely dscussed. An excepton, Atamturk and Hochbaum (2001), studes capacty acquston, subcontractng, and lot-szng ntegrally. Although ther approach makes the producton plan and capacty acquston decsons smultaneously, the authors smply dscuss some specal cases of producton and holdng-cost structure. Moreover, the study stll focuses on solvng a seres of capactated lot-szng problems dscretely, causng the computatonal complexty to ncrease exponentally wth the number of plannng perods and demands. Addtonally, Ahmed and Garca (2004) study a dynamc capacty-acquston and assgnment problem n a smplfed operatons settng to determne the resource capacty and allocaton of the resources to tasks. The study actually proposes a capacty-expanson and plannng approach wthout consderng nventory carryover and the determnaton of producton plans. In summary, whle the progress has been made nvestgatng the questons of capacty acquston decsons and lot-szng separately, few results are avalable that address strateges that jontly optmze capacty acquston and lot-szng decsons under a competton envronment. 3 The Competton Model and Notaton We consder an ndustry wth N frms, and each produces a sngle tem. Ther productons requre a common resource, measured here by capacty. The capacty level purchased by a frm s assumed to be the capacty restrcton n a dynamc lot-szng settng. Examples for ths nclude the number of trucks to lease or a scarce raw materal. The frms have to purchase the capacty at the begnnng of the plannng horzon and can then use the capacty n each followng perod. The capacty must satsfy the

7 L and Messner: Lot Szng Competton wth Capacty Acquston 6 demand constrants, and the excess capacty wll be dsposed of wthout extra dsposal costs. The producton plan wll be consdered n a plannng horzon oft perods. If the frms face a natural sellng season to ntroduce a new model or varant, a natural choce of T arses, e.g. T = 52 weeks n the automoble manufacturng ndustry operatng wth a weekly producton and sales schedule. Otherwse,T s chosen to be large enough to ensure that the frms decsons pertanng to the ntal perods of the plannng horzon are not affected by ths truncaton of the plannng process. We use the followng ndces: = 1,...,N, the ndex for each frm n the ndustry; t = 1,...,T, the ndex for each perod. Each frm has a demand stream durng the plannng horzon, known only to the frm tself and followng some predctable seasonalty pattern (we present and dscuss sx common seasonalty patterns n Secton 6). Thus, let d t = the demand faced by frmn perodt,=1,...,n,t= 1,...,T; β t = the seasonalty factor n perodt,t= 1,...,T; d = the average demand of frm,=1,...,n; andd t =d β t. The frms produce ther goods va a process that, n prncple, allows for nventory replenshment at the begnnng of each perod. As n standard dynamc lot-szng problems, we assume that fxed as well as varable producton costs are ncurred as well as nventory carryng costs, whch are proportonal to each end-of-the-perod nventory. We assume cost parameters may fluctuate n arbtrary ways, and they are defned as f t = the fxed setup cost for a producton batch delvered to frmn perodt, =1,...,N;t= 1,...,T; a t = the varable producton cost for a unt product n frm n perod t, = 1,...,N;t= 1,...,T; h t = the nventory carryng cost for each unt of tem at the end of perod t, =1,...,N;t= 1,...,T. At the begnnng of each plannng horzon, each frmselects the level of capacty to acqure, as well as a complete producton schedule for the entre plannng horzon to satsfy the gven demand stream{d t }. We denote ths capacty as

8 L and Messner: Lot Szng Competton wth Capacty Acquston 7 C = the capacty acqured by frm. We assume that the capacty n queston s traded on a spot market. The market prce for a unt of capacty s relevant wth the demand of the frm and ts compettors for capacty. We denote the market prce as p and assume t s convex n capacty. For example, a smple lnear form capacty acquston cost can be modelled as below: N p=λ+θ =1 C (1) whereλandθare non-negatve constants known to all players. Note that ths cost of capactypaffects the profts earned by all frms n the ndustry, as the capacty acquston s a cost factor n each frm s proft functon. At the same tme, the producton schedule selected by frmaffect only ts own proft measure. It s thus possble to conceptualze the compettve model as a sngle-stage game between N frms, n whch each frm makes a sngle compettve choce,.e. the capacty level to acqure n each season. The game s characterzed by the cost functons below, where refers to the compettors of the frmn the ndustry. π (C C ) = the cost ncurred by frmunder choce of capacty C, assumng frm adopts an optmal dynamc lot-szng schedule to respond to ts own demand stream, and gven that the compettors choose to purchase the capactesc ={C 1,...,C 1,C +1,...,C N }. For ease of exposton, we rewrte ths functon as follows: π (C C )=pc +K (C ) =C Λ+θ N =1 =A (C C )+K (C ) C +K (C ), =1,...,N (2) where A(C C ) denotes the acquston cost of capacty, and K (C ), the mnmum total operatng costs for frmto serve the demand under the capacty levelc. In the competton model, we assume that the frms know about the total number of frmsn, and be able to observe the rval frms decsons on capacty acquston levels. In addton, the prces of the product produced by dfferent frms are assume to be same and constant over the entre plannng horzon. Therefore, the proft maxmzng objectve s equvalent to the cost mnmzng objectve. In the rest of the paper we use the cost mnmzng objectve functon.

9 L and Messner: Lot Szng Competton wth Capacty Acquston 8 Knowledge of the cost functonk (C ) s mportant to be able to analyze the competton model and characterze ts equlbrum behavor. However, dffculty arses from the fact that the functon K (C ) cannot be represented n a closed analytcal form. We wll deal wth ths problem n the Subsecton 5.1. Under the assumpton that the frm has knowledge of ts functonk (C ) and gven that the frm knows ts compettors capacty chocesc, the best response problem (functon) of the frm can then be expressed as: C (C )=arg mn C π (C C ) (3) Before provdng a complete characterzaton of the ndustry s equlbrum behavor, we frst analyze an ndvdual frm s best-response problem n the followng Secton. 4 Best Response Problem Gven the capacty decsons of other frms, a frmhas to determne ts own capacty acquston level and a correspondng lot szng plan so that the total cost s mnmzed. Here, the defned lot-szng and capacty acquston problem s the best response problem of frm. In ths secton, we analyze frm s best response problem (3), whch s crucal to descrbe the ndustry equlbrum. 4.1 Formulaton In order to model the best response problem, we further defne the followng decson varables: x t = the producton quantty of product, =1,,N produced n perodt, y t = t= 1,...,T; 1 x t > 0 0 otherwse ; I t = the nventory amount of product, = 1,,N at the end of perod t, t= 1,...,T. Ths gves rse to the followng formulaton of the best response problem P b of a frm: T P b :π (C C )=mn (a t x t +h t I t +f t y t )+A (C C ), =1,,N (4) t=1

10 L and Messner: Lot Szng Competton wth Capacty Acquston 9 subject to I t =x t d t +I t 1, t= 1,...,T x t C max y t, =1,,N, t= 1,...,T x t C, =1,,N, t= 1,...,T I 0 =I T = 0, =1,,N (5a) (5b) (5c) (5d) x t 0, I t 0, y t {0, 1}, C 0, t= 1,...,T, =1,,N. (5e) where the objectve functon (4) mnmzes the producton and nventory-holdng costs as well as the capacty acquston costs. Constrants on the problem nclude: Equaton (5a) ensures that nventory s balanced; Producton s restrcted by (5b) and (5c), wherec max s the mnmum capacty whch allows the optmal uncapactated lot sze plan; Constrans (5d) set ntal and fnal nventores to zero; and the bounds of the varables are restrcted by (5e). Solvng the model entals smultaneously determnng the optmal capacty, setup perods, and producton amount n each order perod. Capacty s assumed to be a contnuous varable, meanng that capacty can be acqured at any non-negatve level. If t s assumed that the frms observe ther compettors capacty decsons, the best response problem can be solved accordng to the analyss n followng subsecton. 4.2 Calculaton of the optmal capacty and lot szes The smultaneous calculaton of the optmal capacty and producton plan, as explaned above, s a MIP model P b wth quadratc objectve functon but wth all constrants as lnear. Ths problem class s generally N P-hard accordng to Garey and Johnson (1979) and Poljak and Wolkowcz (1995). Whle the general capactated lot-szng problem s N P hard (see Btran and Yanasse (1982)), polynomal tme algorthms are avalable n the specal case of a gven constant capacty. For example, Floran and Klen (1971) suggest an O(T 4 ) algorthm, and alternatve approaches are also suggested by van Hoesel and Wagelmans (1996) and Chen et al. (1994) that run no(t 3 ) tme. Therefore, the problem P b can be solved by dscretzng the nterval of potental values for the capactes and solvng for each of those values. Consequently, the best response problem s notnp-hard n the strong sense, and n prncpal, t can therefore be solved n pseudo-polynomal tme. Whle a commercal package such as CPLEX can n theory handle the best response problem as descrbed above wth a quadratc objectve and lnear constrants, computatonal tmes on a HP 2.0 GHz wth 1 GB memory were typcally n excess of a few

11 L and Messner: Lot Szng Competton wth Capacty Acquston 10 hours for our examples as descrbed n Secton 6 and the optmzer even often ran out of memory and delvered no result. Therefore, n ths paper, we dscretze the potental capacty space and evaluate the total cost functonπ (C C ) at each pont to fnd the optmal soluton of the problem P b under the assumpton that the varable of capacty s contnuous. For each potental capacty value, we fnd the best capactated lot-szng replenshment plan by a standard MIP solver such as CPLEX. We then pck the capacty resultng n the least sum of capacty acquston cost and cost resultng from the capactated lot-szng producton plan. Frst, we need to determne the possble capacty range of each frm. The range s defned by an nteger lower boundc mn that allows a feasble soluton of the best response problem and an nteger upper boundc max. They can be calculated as descrbed below: C mn = max t=1,...,t { D (t) t }, =1,,N (6) whered (t)= t j=1 d j. The upper boundc max can be determned by solvng the uncapactated lot-szng problem, and t can be calculated by the classc Wagner-Whtn algorthm or by other algorthms proposed by Federgruen and Tzur (1991) and Wagelmans et al. (1992). C max equals the maxmum lot sze over the plannng horzon. If capacty ncreases up toc >C max, the lot-szng cost s no longer decreasng, and the capacty acquston cost s ncreasng. Thereby,π(C)>π(C max ), whenc >C max, and the frm wll not ever be better off by acqurng a capacty levelc >C max. Therefore, t s not necessary to consder the capacty values whch are greater thanc max. In the remander of the paper, we focus our analyss on the range[c mn,c max ]. Next, gven the capacty decsons of other competng frms C, we consder each nteger capacty levelc =C mn,c mn +,C mn + 2,,C max, where s assumed to be an nteger such as 1, 2,, and apply a standard solver to solve the capactated lot-szng problem so that the optmal capacty level and lot-szng plan are obtaned over all the capacty levels. Gven the unt ncrement of capacty level, t s reasonable to assume the total cost of a frm s pece-wse lnear functon n capacty. Upon the obtaned pecewse-lnear functons for each dscretzed capacty level, the pareto-optmal capacty s obtaned when the total capacty acquston and lot-szng cost reaches the mnmal. Furthermore, the cost functonk (C ) has followng property n Proposton 1. Proposton 1 Gven the other competng frms capacty decsonsc, the lot szng and capacty acquston cost functonk (C ) of frms non-ncreasng and quas-convex n ts own capacty levelc.

12 L and Messner: Lot Szng Competton wth Capacty Acquston 11 Proof: Wth respect to the best response problem P b, all constrants are lnear, and thus, constrant set s concave. Suppose there exst two capacty levelsc 1 <C 2, and C 1,C2 [Cmn,C max ], becausek (C ) s non-ncreasng nc, we have K (αc 1 +(1 α)c2 ) max{k (C 1 ),K (C 2 )} (7) whereα [0, 1], and thus,k (C ) s quas-convex nc, snce a frm s lot szng cost wll not be affected by other frms capacty decsons, we can also seek (C ) s quasconvex n ts own capacty levelc. We have conducted an extensve numercal study to llustrate the property. For an llustraton of Proposton 1, see Fgure 1 selected from a numercal example dscussed n Secton 6. We also present an example of total cost curve n Fgure 1 over the capacty range[c mn,c max ], whch s selected from a numercal example dscussed n Secton 6. 7 x Cost 3 2 A(C ) 1 K(C) Total cost Capacty Fgure 1: An llustraton of convexty on cost functon n capacty ncrease A suffcent condton for the exstence of an equlbrum s from Theorem 2.1 n Vves (1999) orgnally attrbuted to Debreu (1952): the strategy sets are convex and compact, and the payoff to frms contnuous n the actons of all frms and quasconcave n ts own control varable. Snce we exclusvely deal wth cost n ths paper, the proft functon s just the negated cost functon and hence concave n the control varable, namely the capacty choce. However, the quas-convexty and contnuty of each frm s total cost functon n our capacty competton game cannot be guaranteed, and therefore, we are not able to show any explct equlbrum results of the frms.

13 L and Messner: Lot Szng Competton wth Capacty Acquston 12 5 Equlbrum analyss In order to address the competton problem, we frst ntroduce an approxmaton of lot-szng cost functon K (C ). We show that the approxmaton ndeed results n values very close to the actual cost functon and then proceed to nvestgate the equlbrum behavor. 5.1 Approxmaton of lot-szng cost Accordng to Proposton 1, the lot szng cost functon K (C ) s non-ncreasng and quas-convex, but ths does not suffce to establsh equlbrum behavor. Therefore, we seek to approxmate the lot szng cost functon by a convex functon. Ubhaya (1979) provdes an algorthm to fnd the optmal approxmaton convex functon of a quas-convex functon. Because the approxmaton process s not our focus, n analogy to Federgruen and Messner (2009), we apply an approxmaton functon model of the lot-szng costk (C ) as below: K (C ) K (C )=Td 2 [ η + ζ (C ) γ ], =1,...,N, C [C mn,c max ]. (8) whereη,ζ > 0 andγ > 0 are approprate constants. Snce the functonk(c ) has no closed form, we consder the dscrete functon values n the vald doman[c mn,c max ], and apply the dea of least square curve fttng method to determne the parameters of the approxmaton functon K (C ) whch mnmzes the sum of squared dfferences between the left and the rght sdes of equaton (8). In order to estmate the approxmaton functon more accurately, we calculate the constants of the approxmaton functon based on dfferent demand seasonalty patterns and fxed setup cost levels. Assumng that frm faces a plannng horzon of T perods, and the demand behaves accordng tod t =β t d, sx seasonalty patterns {β t :t= 1,...,T} are typcal n realty as follows. The frst pattern reflects a stuaton where demand functons are tme-nvarant. The second pattern shows one wth lnear growth, whle the thrd shows lnear declne. The fourth and ffth patterns represent a plannng horzon wth a sngle season of peak demands ether at the begnnng or at the end of the plannng horzon. The last pattern (VI) s cyclcal wth a cycle length of sx perods, such that demands n the two mddle perods of each cycle are 7 tmes ther value n the frst and last perod, whleβ t = 1 n the remanng two perods of the cycle. (1) Tme-nvarant demand functons:β t = 1; t= 1,...,T

14 L and Messner: Lot Szng Competton wth Capacty Acquston 13 (2) Lnear Growth:β t =β 0 +(β T β 0 ) (t 1) T 1 ; t= 1,...,T. whereβ 0 s the base seasonalty factor. We take example ofβ 0 = 0.25, and then accordng to T t=1 β t=t,t = 54,β T can be calculated as Therefore, we have lnear growth seasonalty pattern β t = (t 1) t = 1,...,T. Smlarly, T 1 ; other parameter sets could be derved and appled, but the results of our model wll not be affected. (3) Lnear Declne:β t =β T (β T β 0 ) (t 1) T 1 ; t= 1,...,T. Analogous wth seasonalty pattern (II), letβ 0 = 0.25, and we obtanβ t = (t 1) T 1 ; t= 1,...,T. (4) Holday Season at the Begnnng of the Plannng Horzon: β t = (1+P)β 0 Pβ 0 (L/2 1) β 0 + Pβ 0 L/2 1w (t 1),t= 1,...,L/2 (t 1 L/2),t=L/2+1,...,12 β 0,t=L+1,...,T (9) wherelrepresents the length of the peak season, andp descrbes the degree of peak seasonalty factor over base, and for example, f P = 10, the hghest demand s 10 tmes of the base demand. Based on the condton T t=1 β T = T, we have β 0 = T T+L P. LetL=12,P = 10; we obtan the exact holday seasonalty pattern 2 formula as below. β t = (t 1),t= 1,...,6 (t 7),t= 7,..., ,t= 13,...,T (10) (5) Holday Season at the End of the Plannng Horzon: β 0,t= 1,...,T L β t = β 0 + Pβ 0 (L/2 1)(t T+L 1),t=T L+1,...,T L/2 (11) (1+P)β 0 Pβ 0 (L/2 1) (t T+L/2 1),t=T L/2+1,...,T

15 L and Messner: Lot Szng Competton wth Capacty Acquston 14 Smlarly wth seasonalty pattern (IV), we apply the seasonalty pattern (V) as follows: β t = ,t= 1,..., (t 43),t= 43,...,48 (12) 570 (t 49),t= 49,...,T (6) Cyclcal Pattern: β t = Pβ 0 (P 1)β 0 L/2 1 β 0 + (P 1)β 0 L/2 1 (t 1),t= 1,...,L/2 (t L/2 1),t=L/2+1,...,L β tmod6,t=l+1,...,t (13) wheretmod6 denotes t modulo 6. In ths case,lrepresents the cycle length, and P denotes the multpler of peak demand over base demand. LetL=6,P= 7 n ths paper, and then t s calculatedβ 0 = we obtan (t 1),t= 1,...,3 β t = (t 4),t= 4,...,6 β tmod6,t= 7,...,T (14) Assumng that the setup, producton and nventory costs, and capacty acquston costs are dentcal, the values ofk (C ) are dsplayed as a functon of feasble capacty levels for the sx demand patterns (DP) n Fgure 2a and 2b. Cost DP1 DP2 DP Cost DP4 DP5 DP Capacty Capacty Fgure 2:K (C ) as functon of capacty n the sx demand patterns

16 L and Messner: Lot Szng Competton wth Capacty Acquston 15 Not surprsngly, the total cost curves dffer n demand seasonalty pattern. Therefore, t s necessary to estmate the constants of the approxmaton functon wth respect to each seasonalty pattern. Furthermore, we dstngush three levels of the fxed setup cost (expressed by Tme Between Orders (TBO), see further defnton n Secton 6) and nvestgate the fnal form of an approxmaton functon. We frst pck producton costa t = 15; nventory holdng costh t = 5; and TBO=[Hgh, Medum, Low] for three frms = 1, 2, 3, t = 1,,T; and average demand d = [12, 10, 8] for three frms=1, 2, 3. Gven the dentcal cost parameters and seasonalty patterns, the frms have the same lot-szng costsk (C ). Smlarly, for the tmevaryng producton and nventory holdng cost cases, the approxmaton curve can also be estmated. Table 1 below exhbts the parameters generatng the best possble ft:

17 Demand d = 12 d = 10 d = 8 TBO Pattern γ η ζ Gap γ η ζ Gap γ η ζ Gap % % % % % % Low % % % % % % % % % % % % Average 0.76% 0.62% 0.94% % % % % % % Medum % % % % % % % % % % % % 0.88% 0.85% 0.98% % % % % % % Hgh % % % % % % % % % % % % 1.19% 1.18% 1.24% Table 1: Approxmatng curves for demand patterns (I) to (VI) L and Messner: Lot Szng Competton wth Capacty Acquston 16

18 L and Messner: Lot Szng Competton wth Capacty Acquston 17 The columns of Gap n Table 1 dsplay the average relatve dfference between the exact and the approxmate curve. The narrow gaps ndcate that approxmatons of the type (8) are very close for any combnaton of fxed setup cost levels and seasonalty patterns. In addton, the performance of the approxmaton functon s relevant wth the capacty ncremental value. The smaller of, the closer the approxmaton functon s to the actual lot szng and capacty acquston cost curve. 5.2 Exstence of Equlbrum, Unqueness and Convergence Treatng the capacty C as contnuous varable, and substtutng the cost functon K (C ) by the close approxmaton functon (8), the total lot-szng and capacty acquston cost functonπ (C C ) can be expressed as π (C C ) π (C C )= K (C )+A (C C ) [ ] n =Td 2 η + ζ +(Λ+θ C )C, =1,,N. (15) (C ) γ Accordng to the functon (15), we show that (1) Nash equlbrum exsts for the competton game; (2) The Nash equlbrum s unque, and, (3) The equlbrum converges by an teratve Tatônnement scheme and can be computed effcently. Theorem 1 A Nash equlbrum exsts for the competton game. Proof: Snce =1 2 π (C C ) C 2 =Td 2 γ (γ + 1)ζ (C ) γ θ 0, =1,,N. (16) Therefore, the approxmaton payoff functon s contnuous n the actons of all frms and convex n ts own control varable. In addton, the strategy sets are convex and compact, so the concluson holds. Theorem 2 The (approxmate) cost functons π (C C ) satsfes the domnance condton 2 π (C C ) C 2 2 π (C C ) C C j, j,,j= 1,,n (17)

19 L and Messner: Lot Szng Competton wth Capacty Acquston 18 Therefore, the Nash equlbrum s unque. Proof: π (C C ) C = K (C ) C +Λ+θ N C +θc, =1,,N (18) =1 Snce 2 K (C ) C 2 =Td 2 γ (γ +1)ζ (C ) γ +2, and 2 K (C ) C C j = 0, we have 2 π (C C ) C 2 2 π (C C ) C C j =Td 2γ (γ + 1)ζ (C ) γ θ 0 =θ 0, j,,j= 1,,N (19) Thus, 2 π (C C ) C 2 2 π (C C ) C C, and a unque equlbrum s guaranteed. j Furthermore, the unque equlbrum can be effcently computed as the lmt pont of the followng smple Tatônnement scheme: Tatônnement scheme: Startng wth an arbtrary capacty vector C (0), n thek th teraton of the scheme, each frm determnes t capactyc (k) and lot szes whch solves the best response problem (4). Assume all competng frms capactes are set accordng to ther value n the capacty vector C (k 1). Convergency of ths Tatônnement scheme s guaranteed when the game s supermodular. Theorem 2 shows that the supermodularty property holds because 2 π (C C ) C C j =θ 0, =1,,N. (20) 6 Numercal example A complete numercal experment nvestgatng our approach to make capacty acquston and producton decsons wth mult-frm capacty competton s performed n ths secton. Usng dfferent combnatons of demand pattern and TBO, we generate a number (18) of hypothetcal test problems. The algorthm on the competton game s also coded usng MatLab and calls for the approxmaton functon of the best response problem of each frm. The problem nstances are solved on a Pentum 4 PC wth 1G RAM.

20 L and Messner: Lot Szng Competton wth Capacty Acquston 19 Three frm games are set based on the followng data: demand d = (8, 10, 12), producton costa =(17, 15, 13) and nventory holdng costh =(6, 5, 4). For the sake of smplcty and replcablty, we use the constant producton and nventory holdng costs n the test problems, but t wll not be much more dffcult to solve the problems wth tme varyng costs. Gven that the frms produce an dentcal product, the demand patterns n each game nstance are dentcal. The hypothetcal test problems vary wth the combnaton of the dfferent demand patterns and fxed setup cost levels. We determne the fxed setup cost ndrectly by frst choosng the EOQ-cycle tme Tme-between-Orders (TBO) 2f = hd and determne the fxed setup costf from ths dentty. Let TBO value be 2 for low level fxed setup cost, 5 for medum level fxed setup cost and 8 for hgh level fxed setup cost. Fnally, let the fxed capacty acquston costλ=250 and varable capacty acquston cost θ = 3. Based on the approxmaton results, we calculate each frm s best response capacty, cost and lot-szng plan teratvely accordng to the Tatônnement scheme, and reach the equlbrum untl not a sngle frm wll devate further from ts decson. The computatonal results are presented n Table 2. Test Demand TBO Capacty Cost Problem Pattern Level Equlbrum Equlbrum(x10 4 ) Setups (1) (2) (3) (4) (5) (6) 1 DP1 [17;21;25] [1.61;1.79;1.85] [27;27;27] 2 DP2 [29;36;43] [1.93;2.15;2.33] [25;24;24] 3 DP3 [L, L, L] [29;36;43] [2.64;2.18;2.32] [24;24;24] 4 DP4 [43;53;64] [3.08;3.61;3.97] [23;23;23] 5 DP5 [43;53;64] [2.78;3.14;3.35] [24;24;24] 6 DP6 [27;35;40] [1.77;2.04;2.11] [19;19;19] 7 DP1 [41;51;61] [3.58;3.94;4.12] [11;11;11] 8 DP2 [56;70;83] [3.73;4.16;4.31] [11;11;11] 9 DP3 [M, M, M] [56;70;83] [3.76;4.18;4.36] [10;10;10] 10 DP4 [119;148;178] [5.73;6.54;6.99] [10;9;10] 11 DP5 [77;96;115] [4.55;5.05;5.26] [10;10;10] 12 DP6 [51;73;76] [3.76;4.20;4.31] [9;9;9] 13 DP1 [65;81;97] [5.94;6.53;6.79] [7;7;7] 14 DP2 [82;102;122] [6.18;6.88;7.09] [7;7;7] 15 DP3 [H, H,H] [95;106;127] [6.27;6.99;7.25] [6;6;6] 16 DP4 [120;150;179] [7.34;8.22;8.63] [6;6;6] 17 DP5 [180;224;269] [8.31;9.10;9.39] [6;6;6] Contnued on next page

21 L and Messner: Lot Szng Competton wth Capacty Acquston 20 Table 2 contnued from prevous page Test Demand TBO Capacty Cost Problem Pattern Level Equlbrum Equlbrum(x10 4 ) Setups (1) (2) (3) (4) (5) (6) 18 DP6 [89;112;133] [6.35;7.13;7.27] [6;8;7] 19 DP1 [17;51;97] [1.85;4.01;6.01] [27;11;7] 20 DP2 [29;70;122] [2.36;4.23;6.28] [25;11;7] 21 DP3 [L,M,H] [29;70;127] [3.38;4.28;6.27] [24;10;6] 22 DP4 [43;148;179] [4.78;5.77;7.68] [23;9;6] 23 DP5 [43;96;269] [3.98;5.85;6.98] [24;10;6] 24 DP6 [27;73;133] [2.18;4.40;6.31] [19;9;7] 25 DP1 [17;81;61] [1.84;5.78;4.16] [27;7;11] 26 DP2 [29;102;83] [2.33;6.09;4.34] [25;7;11] 27 DP3 [L,H,M] [29;106;83] [3.33;6.04;4.42] [24;6;10] 28 DP4 [43;150;178] [4.79;7.43;6.10] [23;6;10] 29 DP5 [43;224;115] [3.86;6.75;5.96] [24;6;10] 30 DP6 [27;112;76] [2.13;6.09;4.41] [19;8;9] 31 DP1 [41;21;97] [3.61;2.08;5.95] [11;27;7] 32 DP2 [56;36;122] [3.75;2.64;6.21] [11;24;7] 33 DP3 [M,L,H] [56;36;127] [3.81;2.78;6.20] [10;24;6] 34 DP4 [119;53;179] [4.96;5.55;7.45] [10;23;6] 35 DP5 [77;53;269] [5.17;4.51;6.88] [10;24;6] 36 DP6 [51;35;133] [3.85;2.51;6.17] [9;19;7] 37 DP1 [41;81;25] [3.54;5.66;2.16] [11;7;27] 38 DP2 [56;102;43] [3.69;5.97;2.83] [11;7;24] 39 DP3 [M,H,L] [56;106;43] [3.74;5.92;2.94] [10;6;24] 40 DP4 [119;150;64] [4.82;7.04;6.05] [10;6;23] 41 DP5 [77;224;64] [4.98;6.59;4.68] [10;6;24] 42 DP6 [51;112;40] [3.77;5.98;2.57] [9;8;19] 43 DP1 [65;21;61] [5.19;2.04;4.08] [7;27;11] 44 DP2 [82;36;83] [5.42;2.58;4.25] [7;24;11] 45 DP3 [H,L,M] [95;36;83] [5.43;2.75;4.39] [6;24;10] 46 DP4 [120;53;178] [6.55;5.55;5.86] [6;23;10] 47 DP5 [180;53;115] [6.07;4.24;5.72] [6;24;10] 48 DP6 [89;35;76] [5.37;2.43;4.31] [6;19;9] 49 DP1 [65;51;25] [5.14;3.87;2.13] [7;11;27] 50 DP2 [82;70;43] [5.37;4.07;2.80] [7;11;24] 51 DP3 [H,M,L] [95;70;43] [5.38;4.17;2.95] [6;10;24] 52 DP4 [120;148;64] [6.39;5.38;6.03] [6;9;23] 53 DP5 [180;96;64] [6.01;5.41;4.53] [6;10;24] Contnued on next page

22 L and Messner: Lot Szng Competton wth Capacty Acquston 21 Table 2 contnued from prevous page Test Demand TBO Capacty Cost Problem Pattern Level Equlbrum Equlbrum(x10 4 ) Setups (1) (2) (3) (4) (5) (6) 54 DP6 [89;73;40] [5.39;4.21;2.57] [6;9;19] Table 2: The equlbrum soluton of the competton games In Table 2, Column (3) shows the possble fxed setup cost levels for the three frms. We consder the nstances when the frms have dentcal TBO levels and dfferent TBO levels respectvely. The results show that the frms choose smlar producton strategy when the fxed setup costs are at the same level. However, when fxed setup costs dffer from each other, the frms choose rather dfferent setup polcy. In addton, Column 4 represents the frms decsons on capacty, and column 5 shows the total costs of the frms. Column 6 shows the setup numbers of the frms. In general, all nstances converge wthn a small number of teratons. 7 Concluson Ths paper consders a multple frm lot-szng problem wth resource competton. We model and solve the competton game and dscuss the equlbrum behavors of the frms. As a best response problem of a frm, a typcal capacty acquston and lot-szng problem s solved by lne search. The algorthm solves the capacty acquston, producton, and nventory decsons smultaneously for multple frms teratvely. In order to tackle the complexty of dynamc lot szng problem and potental dscontnuty of ts cost functon, a close approxmaton s appled to substtute the dynamc lot szng cost. Under the mld condtons, we show the exstence and unqueness of equlbrum, and furthermore, the equlbrum converges wthn fnte teratons of computaton. In addton, the extenson of multple products share a common resource can be easly adapted nto our method by solvng the approxmaton problem of multple product lot szng problem. In the present study, we only consder a rather smple structure of the resource competton and dynamc lot szng problem. Frst, the analyss s lmted to the determnstc supply and demand. Ths leaves the future research opportuntes on the capacty acquston and competton problem under random supply and demand uncertanty. In addton, we only consder a constant capacty settng over the plannng

23 L and Messner: Lot Szng Competton wth Capacty Acquston 22 horzon. It would also be nterestng to analyze the tme varyng capacty stuaton. If the capacty can be purchased or dsposed of n each perod, t could lead to a soluton for a dynamc competton game and the problem would be much more complcated.

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