Dynamic Cost Allocation for Economic Lot Sizing Games

Transcription

1 Dynamc Cost Allocaton for Economc Lot Szng Games Alejandro Torello H. Mlton Stewart School of Industral and Systems Engneerng Georga Insttute of Technology Atlanta, GA elson A. Uhan Mathematcs Department Unted States aval Academy Annapols, MD ovember 6, 2013 Abstract We consder a cooperatve game defned by an economc lot szng problem wth concave orderng costs over a fnte tme horzon, n whch each player faces demand for a sngle product n each perod and coaltons can pool orders. We show how to compute a dynamc cost allocaton n the strong sequental core of ths game,.e. an allocaton over tme that exactly dstrbutes costs and s stable aganst coaltonal defectons at every perod of the tme horzon. 1 Introducton and Motvaton Producton and nventory management are areas n whch cooperaton among ndependent agents has an ntutve appeal. For example, ndependent retalers sharng a warehouse may want to combne orders from a common suppler to enjoy economes of scale derved from larger order quanttes. Vewed from a system-wde perspectve, combnng orders lowers the total cost, but the order consoldaton cannot be acheved unless the ndependent retalers can agree on a far way to splt ths cost. Over the last decade or more, cooperatve game theory research n producton, dstrbuton and nventory models has endeavored to determne far cost allocatons under a varety of settngs. The economc lot szng problem s one of the canoncal producton and nventory models studed n operatons research and management scence, almost snce the feld s ncepton Wagner and Whtn In a cooperatve settng, ths model would capture the stuaton descrbed above, wth each retaler facng ts own demand over the plannng horzon that must be satsfed wth product orders or nventory, but wth coaltons of retalers beng able to order product together. The cooperatve game derved from ths model was proposed n van den Heuvel et al and subsequently studed n Chen and Zhang 2006, Gopaladeskan et al However, all of these papers approach the problem from a statc perspectve; that s, the authors assume an optmal soluton s mplemented over the entre horzon, and seek an up-front statc cost allocaton n the core of ths cooperatve game. 1

2 Unfortunately, a statc cost allocaton even one n the core suffers from a number of sgnfcant drawbacks. Frst, t assumes each retaler would be able to cover ts porton of the entre plannng horzon s cost up front, a sgnfcant fnancal burden that s unrealstc n many settngs. Second, once the allocated costs are collected from the partcpatng retalers, there may be ncentves for ndvdual retalers or coaltons to defect later on n the plannng horzon f they fnd themselves n an advantageous poston. Fnally, a statc allocaton gnores the typcal rollng horzon approach to these models, n whch one or a few perods solutons are mplemented, and then a new model s formulated wtpdated parameters. Wthn a rollng horzon approach, t s unclear how a statc allocaton could be mplemented. Our goal n ths paper s to develop dynamc cost allocatons for the economc lot szng problem. Unlke ther statc counterparts, dynamc allocatons can be constraned to allocate costs as they are ncurred. They can also be desgned so that no coalton ever has an ncentve to defect throughout the entre plannng horzon. Furthermore, because the costs are allocated as they are ncurred, dynamc allocatons can also be ncorporated nto a rollng horzon framework. Our allocaton depends conceptually on extendng the core concept to dynamc settngs e.g. see Kranch et al. 2005, and contnues our work n Torello and Uhan 2013 for mult-perod models wth lnear costs. Cooperatve game theory has been successfully appled n many related models, and we cannot hope to adequately revew them here. Instead, we refer the nterested reader to Cachon and etessne 2006, agarajan and Sošć Two n-depth references on lot szng and more general producton plannng and nventory management are Pochet and Wolsey 2006, Zpkn Ths paper has two remanng sectons. In Secton 2, we ntroduce the model and revew some relevant results. Then, n Secton 3, we defne the strong sequental core, the dynamc verson of the core, and show how to compute such a dynamc allocaton. 2 The Economc Lot Szng Game We consder the followng determnstc economc lot szng settng wth multple retalers. A set of retalers : {1,..., n} faces determnstc demand for a sngle product over a fnte dscrete tme horzon. In partcular, each retaler needs to satsfy demand d t n perods t r,..., T ; we start n perod r nstead of perod 1 because we wll eventually vary the startng perod. In every perod t r,..., T, each retaler can order x unts at a total cost of c t x; the functon c t s concave and nondecreasng wth c t 0 0. Each retaler can also hold one unt of product n nventory n perod t r,..., T at a cost of h t. Fnally, each retaler has an ntal nventory of ŝ r 1. For notatonal convenence, we defne d [j,k] : k tj d t for, r j k T, ŝ r 1 : ŝ r 1. Here and throughout, we take a summaton n whch the ntal ndex s greater than the endng ndex to be vacuous and equal to zero; for example, d [j,j 1] 0. The economc lot szng problem for a subset of retalers seeks to satsfy the demands d r,..., dt wth ntal nventory ŝr 1 n a way that mnmzes the total orderng and nventory cost. The economc lot szng game s a transferable utlty cooperatve game, f r, ŝ r 1 n whch each retaler corresponds to a player, and the cost f r, ŝ r 1 to a subset of players s the optmal value of ts economc lot szng problem. Followng standard termnology n the cooperatve game theory lterature, we refer to a subset of players as a coalton, and the set of all players as the grand coalton. 2

3 It s well-known that any nstance of the economc lot szng problem can be transformed nto an equvalent nstance wth zero nventory by usng the ntal nventory to greedly satsfy demand n the begnnng tme perods Zabel It s also well-known that when the ntal nventory s zero, there exsts an optmal soluton that satsfes the zero-nventory property n whch orders only occur n perods when the nventory level s zero Wagner and Whtn 1958, Wagner Therefore, when ntal nventores are of the form ŝ r 1 d [r,τ 1] for for some τ r, we can model the economc lot szng problem for coalton wth the followng lnear program Chen and Zhang 2006: τ 1 f r, ŝ r 1 h t d [t1,τ 1] tr mn x s.t. T T [ jτ kj t jr kt c j d [j,k] k tj ] h t d [t1,k] x jk 1a T x jk 1 for t τ,..., T, 1b x jk 0 for j, k : τ j k T, 1c where x jk s a decson varable that ndcates an order n perod j to meet demands n perods j,..., k, for all j, k such that τ j k T. Although the decson varables are contnuous, ths nterpretaton s well-defned: Chen and Zhang 2006 showed that there always exsts an optmal soluton to 1 such that x jk {0, 1} for all j, k. The frst term of the objectve 1a s the cost of greedly usng the ntal nventory ŝ r 1 to satsfy demand n perods r,..., τ 1, and the coeffcent of x jk n the second term of the objectve s the cost of satsfyng demand n perods j,..., k wth an order n perod j. The constrants 1b ensure that demand s satsfed n each perod τ,..., T. We slghtly abuse notaton and use f r, ŝ r 1 to refer to the model 1 as well as ts optmal value. The dual of f r, ŝ r 1 s τ 1 f r, ŝ r 1 h t d [t1,τ 1] tr max α s.t. T d t α t tτ k tj d t α t c j d [j,k] k tj h t d [t1,k] for j, k : τ j k T. 2a 2b Gopaladeskan et al proposed a polynomal-tme combnatoral algorthm that solves f r, ŝ r 1 and ts dual actually, an affne transformaton of ts dual, and used the prmal and dual optmal solutons n conjuncton wth some structural propertes establshed by Chen and Zhang 2006 to compute an allocaton n the core of the economc lot szng game, f r, ŝ r 1. We summarze these results n the lemma below. Lemma 1 Chen and Zhang 2006, Gopaladeskan et al Let x and α respectvely be optmal solutons to f r, s r 1 and ts dual, obtaned usng the algorthm by Gopaladeskan et al Then, x and α satsfy the followng propertes: a α t h t1 α t1 for t τ,..., T. 3

4 b For all d d r,..., d T such that d t dt for t r,..., T, k tj d t α t c j d [j,k] k tj h t d [t1,k] for j, k : τ j k T. c There exst replenshment ntervals {a 1, b 1,..., a m, b m } such that a 1 τ, b m T, a l b l and b l 1 a l1 for all l 1,..., m, and x a l b l 1 for l 1,..., m, d t α [al,bl] t c al d 3 Dynamc Cost Allocaton b l h t d [t1,bl] for l 1,..., m. ow suppose that the tme horzon starts n perod r 1, wth each player havng zero ntal nventory, and the players agree to mplement an optmal soluton x to f 1, 0 wth correspondng dual optmal soluton α and replenshment ntervals {a 1, b 1,..., a m, b m } wth a 1 1 and b m T, obtaned by the algorthm of Gopaladeskan et al In prevous approaches to economc lot szng games Chen and Zhang 2006, Gopaladeskan et al. 2012, van den Heuvel et al. 2007, the authors assume the entre cost of the optmal soluton s allocated up front at the start of the horzon. Instead, we make the more realstc assumpton that orderng costs must be allocated n the same perod an order takes place, and holdng costs for each perod must be allocated n that same perod. Fnally, we assume the players agree that at any pont n tme, each player owns the nventory that was ordered to meet ts own demand; n other words, for each l 1,..., m, the nventory of player s ŝ a l 1 0, 3a ŝ r 1 d [r,b l] for r a l 1,..., b l. 3b Defnton 2 Kranch et al. 2005, Torello and Uhan In a dynamc allocaton χ χ t,t1,...,t, player s allocated a cost of χ t n perod t. The strong sequental core of the economc lot szng game, f r r1,...,t s the set of dynamc allocatons χ that satsfy the followng condtons for some optmal soluton x to f 1, 0: a Stage-wse effcency: For each l 1,..., m, χ a l [a c al d l,b l ] hal d [a l1,b l ], 4a χ t h t d [t1,b l] for t a l 1,..., b l. 4b In other words, the costs ncurred n each perod must be fully dstrbuted among the grand coalton. b Tme-consstent stablty: For each perod r a l,..., b l for some l 1,..., m, tr T χ t f r, ŝ r 1 for, 4c 4

5 where the ntal nventores ŝ r 1 are as defned n 3. In other words, at any pont n the tme horzon, the cost allocated to any coalton from that pont forward does not exceed ts cost f t abandons the grand coalton and contnues on ts own. We defne the dynamc allocaton ˆχ as follows: ˆχ a l d t ˆχ t h t d [t1,b l] t 1 ua l h al d [a l1,b l ] for t a l 1,..., b l for l 1,..., m. 5 Theorem 3. The dynamc allocaton ˆχ defned n 5 s n the strong sequental core of the economc lot szng game, f r r1,...,t. Proof. Frst, we show that ˆχ s stage-wse effcent. For any l 1,..., m, ˆχ a l [ bl d t d t d t d t d t c al d t [al,b l ] t 1 t 1 ua l ua l t 1 ] h al d [a l1,b l ] h al d [a l1,b l ] d t h al d [al1,bl] ua l ut1 h t d u h al d [a l1,b l ] h t d [t1,bl] h al d [a l1,b l ] hal d [a l1,b l ], where follows from Lemma 1c. In addton, we have for all l 1,..., m: ˆχ t h t d [t1,b l] h t d [t1,b l] for t a l 1,..., b l. ext, we show that ˆχ s tme-consstently stable. Fx r {a p 1,...,, a p1 }, for some p {0, 1,..., m} gnorng a 0 1,..., b 0 and a m1. For any, we have that tr T ˆχ t tr tr tr h t d [t1,bp] h t d [t1,bp] h t d [t1,bp] m lp1 m lp1 m lp1 bl d t bl d t bl d t t 1 ua l t 1 ua l d t ut1 h t d [t1,bl] h t d u h t d [t1,bl] ut1 h t d u 5

6 tr h t d [t1,bp] f r, ŝ r 1 m lp1 d t α t where holds because Lemma 1b mples that α s a dual feasble soluton for f r, ŝ r 1. Acknowledgements The authors work was partally supported by the U.S. atonal Scence Foundaton va grant CMMI eferences G.P. Cachon and S. etessne. Game theory n supply chan analyss. In Tutorals n Operatons esearch: Models, Methods, and Applcatons for Innovatve Decson Makng, pages IFOMS, X. Chen and J. Zhang. Dualty approaches to economc lot szng games. IOMS: Operatons Management Workng Paper OM , Stern School of Busness, ew York Unversty, M. Gopaladeskan,.A. Uhan, and J. Zou. A prmal-dual algorthm for computng a cost allocaton n the core of economc lot-szng games. Operatons esearch Letters, 406: , L. Kranch, A. Perea, and H. Peters. Core concepts for dynamc TU games. Internatonal Game Theory evew, 7:43 61, M. agarajan and G. Sošć. Game-theoretc analyss of cooperaton among supply chan agents: evew and extensons. European Journal of Operatonal esearch, 187: , Y. Pochet and L.A. Wolsey. Producton Plannng by Mxed Integer Programmng. Sprnger, A. Torello and.a. Uhan. Dynamc lnear producton games under uncertanty. optmzaton-onlne.org/db_html/2013/10/4064.html, W. van den Heuvel, P. Borm, and H. Hamers. Economc lot-szng games. European Journal of Operatonal esearch, 176: , H.M. Wagner. A postscrpt to Dynamc Problems n the Theory of the Frm. aval esearch Logstcs Quarterly, 7:7 12, H.M. Wagner and T.M. Whtn. Dynamc verson of the economc lot sze model. Management Scence, 51:89 96, E. Zabel. Some generalzatons of an nventory plannng horzon theorem. Management Scence, 10 3: , P.H. Zpkn. Foundatons of Inventory Management. McGraw-Hll,