A FAMILY OF MARKET-BASED SHIPMENT METHODOLOGIES FOR DELIVERY SUPPLY CHAIN

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1 Proceedngs of the 2005 Wnter Smulaton Conference M. E. Kuhl, N. M. Steger, F. B. Armstrong, and J. A. Jones, eds. A FAMILY OF MARKET-BASED SHIPMENT METHODOLOGIES FOR DELIVERY SUPPLY CHAIN Seog-Chan Oh Soundar R. T. Kumara Industral & Manufacturng Engneerng Department The Pennsylvana State Unversty Unversty Park, PA 16802, U.S.A. Shang-Tae Yee Jeffrey D.Tew Manufacturng Systems Research Laboratory General Motors Research and Development Center Warren, MI 48088, U.S.A. ABSTRACT Load buldng s an mportant step to make the delvery supply chan effcent. We present a famly of load makeup algorthms usng market control-based strategy, named LoadMarket, n order to buld effcent loads where each load conssts of a certan number of fnshed products havng destnatons. LoadMarket adopts mnmum spannng tree graph algorthm for generatng ntal owment for Load Traders who cooperate to mnmze ether total travel dstance or the varance wth respect to the travel dstances of loads through a spot market or double-sded aucton market mechansm. For the smulated load shpment market, the effcency of the LoadMarket algorthms s analyzed usng smulaton experments. 1 INTRODUCTION Load buldng s a process of assgnng a set of products to a number of transportaton carrers lke truck or ralcar. Companes have made an effort to acheve more effcent shpment plannng and executon of fnshed products on ther delvery supply chan. The effcency n shpment plannng and executon can mprove customer fulfllment wth fast delvery and reduce total transportaton cost. We present LoadMarket, a famly of load assgnment methods usng market-based control mechansm. Let us consder an example scenaro where the optmzaton of load assgnment s crucal ssue. Suppose an automotve company produces cars and transports them to the customers natonwde (Yee, 2002). As shown n Fgure 1(a), whenever an ordered car s produced n the plant, t moves to the yard where cars wat to be assgned to a truck, the transportaton carrer. Mantanng cars n the yard results n costs. Each truck has a lmtaton for the maxmum number of vehcles to carry and after contanng one load of cars, t starts to vst each cty to delver cars. The transportaton cost of each truck s proportonal to the total travel dstance. In summary, the daly cost model for ths scenaro has the followng: Cost model = (average # of cars n the yard mantenance fee per a day) + (daly # of trucks departng from the yard average travel dstance of trucks) The above cost model mples that the man costs savng problems are: (1) fast shppng out cars watng n the yard and (2) fndng a shortest average travel dstance. In ths paper, we consder the second problem that s, fndng a shortest average travel dstance. Plant Yard Transportaton Load1 Load2 Load3 Load4 Load5 Fgure 1(a): A scenaro of the load assgnment If there s a large truck wth the ablty to carry all daly produced cars and cover all destnaton ctes, ths problem s reduced to a travelng salesman problem (TSP) (.e., fndng a path through a weghted graph whch starts and s at the same vertex) (Flood, 1956). However, n the real world, a truck can carry at most ten cars at one shpment and can travel at most over thrty neghborng ctes. Thus, the problem needs to employ multple trucks n order to delver many daly produced cars, whch leads to the capactated vehcle routng problem (CVRP) (Clark and Wrght, 1964). For solvng CVRP, as shown n Fgure 1(b), we can use the k-tree coverng algorthm whch s NP-complete proven by reducton from the Bn-Packng Problem (Even et al., 2003). 1759

2 Cty2 Cty3 Cty4 Cty5 Cty9 Cty1 Cty10 Cty11 Cty6 Cty7 Cty8 Cty12 K=1 Cty13 Cty14 Cty15 Cty2 Cty3 Cty4 Cty5 Cty9 Cty1 Cty10 Cty11 Cty6 Cty7 Cty8 Cty12 K=3 Cty13 Cty14 Fgure 1(b): k-tree coverng The presented scenaro often occurs n many other ndustres that am to Order to Delvery (OTD) servce. However, dentfyng mnmzed cost value when the number of ctes to vst and sze of tems to delver ncrease, s a challengng task, snce n real applcaton, such numbers can be many and non-trval. Therefore, there s an mmnent need for the method that systematcally and mechancally helps to plan a load assgnment such that average travel dstance for delverng the loads s mnmzed or load balancng between the loads s satsfed. Market-based automated negotaton, or more commonly market-based control, s a paradgm for controllng complex system that would otherwse be very dffcult to handle, by takng advantage of some desrable features of a market (especally a free market) ncludng decentralzaton, nteractng agents, and some noton of resources that need to be allocated (Clearwater, 1996). Ths approach has been appled to a wde range of felds such as supply chan management (Hnkkanen et al., 1997; Sauter et al., 1999), vehcle routng (Sandholm, 1993), manufacturng schedulng (Tlley, 1996; Baker, 1996), and process control (Jose and Ungar, 1998). The load assgnment problem doman can be classfed nto the case of dstrbuted resource allocaton havng a team objectve. In ths paper, we consder two market control mechansms (Lee, 2002) that are desgned as follows: Double Aucton (DA): Wthn the control wndow, multple sellers and multple buyers place or ask bds for the exchange of a desgnated commodty n a vrtual market whch matches buyers and sellers mmedately on detecton of compatble bds; SpotMarket: A market s establshed for a seller and a buyer wthn the control wndow, meanng that the seller sells any commodty that s most beneft to the buyer. Based on ths settng, we can defne our load problem n detal. In general, when consderng the load assgnment problem, the followng defntons hold: Defnton 1 Let Load denote the graph < V, E > whch contans V as a set of products nted to be delvered to ther destnatons and E as a set of dstances between destnatons, v ( V ) and v j ( V ). Defnton 2 Let TD(Load ) denote the length of the shortest path of Load =1,,k. Defnton 3 LoadMakeup s a set of Load =1,,k. Let W products nted to be delvered from the yard to customers. Then, U k V ( Load ) = W. When a = 1 LoadMakeup s obtaned, = TD( Load 1 ) s the total dstance requred to delver all the products of W. Defnton 4 Load Trader s an agent that processes a Load and partcpates n DA or SpotMarket n order to k mnmze: (1) or (2) = TD( Load 1 ) by cooperaton wth other Load Traders. In ths paper, we am at solvng the followng problems. Gven a set of products, W, effectvely fnd a LoadMakeup such that for Load LoadMakeup (1) k = 1 TD( Load ) s mnmzed; or (2) varance w.r.t TD Load ) s mnmzed. ( In partcular, the second problem of fndng a LoadMakeup wth mnmzed varance over s what we refer to as the Load Balancng problem, and can be essentally formulated by balance spannng tree (BST). A spannng tree T of a graph G s called a BST f t mnmzes the dfference between the most costly arc and the least costly arc selected (that s, from among all spannng trees of G, the dfference between the maxmum arc cost n T and the mnmum arc cost n T s as small as possble). Obvously, one very neffcent (f not ntractable) method of fndng a BST s to enumerate all spannng trees of G and then select the balanced spannng tree from among these. Snce we am at obtanng a set of Loads rather than dscoverng just a sngle balanced shortest path, such exstng BST soluton s, however, not feasble. In the followng secton, we nvestgate alternatve, non-exhaustve, but approxmate soluton based on both Prm s MST algorthm (Prm, 1957) and aforementoned DA or SpotMarket. 2 OVERVIEW OF LOADMARKET In handlng our problems wth market-based control, two ssues are crtcal: (1) the ntal Loads to be assgned to Load Traders for ther owment; and (2) the market mechansm to control Load Traders. To address the ssues, we propose technques generatng ntal load assgnment usng the tree coverng technque adoptng Prm s MST and two market control-based algorthms. k 1760

3 2.1 LOADMARKET MST Input: W : all products, E : dstance matrx, and # : one load sze Output: L : a set of loads l φ, L φ and δ W ; whle δ φ do Pr m l Matchng ( δ, E,#); δ δ \ l prnt l, ; prnt L ; Algorthm: LOADMARKET MST whle Converged true do DA( ); prnt L( ) ; Algorthm: LOADMARKET DA Buyer (Load Trader) (a1) Do Sync (a2) I m a Buyer Vrtual Market (match maker) (a1) Do Sync (a2) I m a Seller (a3) Ask Seller (Load Trader) LOADMARKET MST uses a naïve tree coverng algorthm for obtanng ntal assgnment. δ denotes the set of products watng ther assgnment to some Load and Pr m l denotes a Load whch product sze s #. Matchng adopts Prm s MST algorthm to cut out one feasble Load l wth a sze of # from δ. It s fxed pont algorthm because when W = #, t has a computatonal complexty O (# 2 ) whch s the same as that of the partally-ordered set problem (.e., lattce). Input: V, E, and # Output: U ( V ) T φ and U = {1 } ; whle U # do δ {( u, v) u U, v V \ U}; mn ( u, v) ( u, v)( δ ) wth MIN( d( u, v)( E)); T T U{( u, v)}; U U U{v}; prnt U ; Pr m Algorthm: Matchng 2.2 LOADMARKET DA The man dea of ths algorthm les on the post process to run DA (.e., double-sded market between Load Traders) to enhance a ntal assgnment generated by LOADMARKET MST such that each Load can make ther shorter. Input: W : all products, E : dstance matrx, and # : one load sze Output: L : a set of loads L φ ; LOADMARKET MST ( W, E, # ) (a4) Bd (a6) Confrm (a5) Reject (a5) Confrm Fgure 2(a): DA Protocol X Iterate untl convergent As shown n Fgure 2(a), when a new teraton of DA begns, a Vrtual Market ss out Do Sync sgnals to all Load Traders. Next, snce Load Traders have an opton to choose one role from ether a buyer or a seller, they pck up one role and respond to the Vrtual market. Subsequently, the Vrtual market matches buyers and sellers. For matched buyer and seller, they start a negotaton to decde on exchangng ther products. If exchangng actvty s benefcal to both of them, they mmedately conduct the exchangng actvty. All traders have a common polcy, that s, mnmze. Wth ths polcy, the DA protocol exchangng process s terated untl the market converges or a predefned teraton s satsfed. The teratng DA protocol process s expected to reach emergent behavor that s, a more effcent load assgnment (.e., more reduced travel dstance). 2.3 LOADMARKET SpotMarket Lke LOADMARKET DA, t also apples a post process to the ntal assgnment created by the MST based tree coverng algorthm. One thng dfferent from LOADMARKET DA s that t runs afore-defned SpotMarket mechansm whch allows only two Load Traders to trade each other. In ths context, one Load Trader wth the most costly becomes a buyer whle the other Load Trader wth the least costly becomes a seller. The seller s role s focused on supportng the buyer by allowng the buyer to exchange any product whch the buyer wants as long as her loss s not greater 1761

4 than some threshold. In other words, the seller s polcy s to mnmze maxtd( Load ) such that load balancng on the correspondng LoadMakeup s obtaned even though she can take some loss. The defnton of LoadMakeup s found n the prevous secton. Input: W : all products, E : dstance matrx, and # : one load sze Output: L : a set of loads L φ ; MST LOADMARKET ( W, E, # ) whle Converged true do Aucton( ); prnt L( ) ; Algorthm: LOADMARKET SpotMarket Compared wth LOADMARKET DA whch nvokes the Load Traders to seek ther mmedate ncreased beneft (.e., greedy behavor), LOADMARKET SpotMarket drves the Load Traders to cooperate each other. Fgure 2(b) shows the protocol between the buyer and seller. Ths SpotMartket protocol exchangng process s terated untl the market converges or a pre-defned teraton s satsfed. car has ts delvery destnaton ndexed by cty. We assume that the daly avalable number of trucks s fve so that a LoadMackup can be generated daly meanng that ten cars are wrapped n one Load together and contaned n one truck. Each truck carres ten cars and delvers them to destnaton ctes. In order to analyze the effect of the number of ctes to vst, we smulated the aforementoned scenaro for 3 cases where the number of ctes s 50, 100, and 200 respectvely. For example, n the case of 50 ctes, each cty s assgned wth unque ndex between 1 and 50, and the dstance between any two ctes, and j are defned by -j. For example, cty 2 and cty 28 has the dstance of 26. At the, overall, we expermented a combnaton of 3 dfferent numbers of ctes 3 dfferent algorthms = 9 combnatons. Each combnaton has fve replcatons. Furthermore, we assume that the protocol exchange teraton per a tradng s 100. Fnally, we note that each Load s assumed as an un-rooted tree for the convenence of analyss. That s, n, the dstance from the yard to the frst and last vstng cty s gnored. In addton, a shortest travel path wthn a Load starts from a cty wth the smallest ndex and goes along ctes wth ncreasng number of ndex and s at a cty wth the largest ndex because all ctes are assumed to be connected. 3.2 Results Buyer (most costly load Trader) (b2) Bd (b4) Confrm (b3) Reject (b1) Ask (b3) Confrm X Seller (least costly Load Trader) Fgure 2(b): SpotMarket Protocol 3 EXPERIMENTAL VALIDATION 3.1 Set-up Iterate untl convergent To valdate the effcacy of our proposals, we conducted experments usng smulaton. Frst, we bult a smulator whose scenaro s dentcal to the example scenaro n the Introducton secton. In other words, the smulator manly conssts of three parts: (1) a plant, (2) a yard and (3) a transportaton system (.e., a set of trucks). Every day, the plant produces 50 cars that move nto yard. Each produced Fgure 4(a) and Table 1(a) summarze the average travel dstance tme of LoadMakeup w.r.t three algorthms over three dfferent numbers of ctes, where all algorthms performance get worse reasonably as the number of ctes ncreases. When t comes to the sample sze of the smulaton, we generate 18,250 data (.e., 50 cars per a day 365 days) wth 5 replcatons usng the smulator mentoned n the prevous secton. Assgnng a cty ndex to a generated data (.e., a car) follows the unform dstrbuton whch s specfed by the number of ctes. The dstrbuton LOADMARKET DA slghtly outperforms LOADMARKET MST whle LOADMARKET SpotMarket performs worse than others. Fgure 4(b) and Table 1(b) summarze the standard devaton (STD) wth respect to the travel dstance between loads wthn the same LoadMakeup, where all algorthms performance get worse as the number of ctes ncreases. Table 1(a): Average Travel Dstances Number of ctes Algorthm LOADMARKET DA LOADMARKET MST LOADMARKET SpotMarket

5 Table 1(b): STD w.r.t Travel Dstances Number of ctes Algorthm LOADMARKET DA LOADMARKET MST LOADMARKET SpotMarket Interestngly, LOADMARKET SpotMarket outperforms other algorthms wth a bg dfference. When the number of cty s 200, the dfference of STD between LOADMARKET SpotMarket and LOADMARKET DA reaches around 13. Ths s somewhat ntutve snce LOADMARKET DA stmulates Load Traders to act a greedy behavor seekng ther mmedate ncreased beneft whle LOADMARKET SpotMarket drves the least costly Load Trader to yeld the most costly Load Trader ts beneft. Travel dstance (average) # of ctes Travel dstance (average) Load1 Load2 Load3 Load4 Load5 Traders Fgure 4(a): Comparson of Average Travel Dstance w.r.t Load Traders (# of Ctes = 50) Travel dstance (average) Load1 Load2 Load3 Load4 Load5 Traders Fgure 4(b): Comparson of Average Travel Dstance w.r.t Load Traders (# of Ctes = 100) Fgure 3(a): Comparson of Travel Dstance for Dfferent # of Ctes standard devaton (average) # of ctes Fgure 3(a): Comparson of STD w.r.t Travel Dstance for Dfferent # of Ctes Travel dstance (average) Load1 Load2 Load3 Load4 Load5 Traders Fgure 4(c): Comparson of Average Travel Dstance w.r.t Load Traders (# of Ctes = 200) Fgure 4(a)-(c) llustrates the average total dstance w.r.t each Load Trader on dfferent # of ctes. Consstently, LOADMARKET DA outperforms other algorthms but as far as the Load Balancng problem s concerned, t performs worse. For example, n the case of 200 ctes, the unbalance of travel dstance between Load 5 and Load 1 exceeds

6 4 CONCLUSION In ths paper, we have consdered the problem of fndng an effcent load assgnment, and suggested a soluton usng (1) Prm s mnmum spannng tree algorthm, and (2) market controls such as DA and SpotMarket. Despte ts worse performance of average travel dstance, LOADMARKET SpotMarket algorthm showed the best performance as far as the balanced load assgnment s concerned. Several drectons exst for further research. Rather than the one-shot assgnment we dealt wth, dynamc load assgnment can be consdered next. For example, whenever a product s produced from the plant, our proposals can be appled and the load assgnment s updated contnuously. Also, we can address a problem to fnd an optmal teraton number of protocol exchange per a tradng. ACKNOWLEDGMENTS Electronc Commerce at Agents 99, Seattle, WA, USA. Sandholm, T An mplementaton of Contract Net Protocol based on margnal cost calculatons. In Proceedngs of the Eleventh Natonal Conference on Artfcal Intellgence, Washngton DC, USA. Tlley, K. J Machnng task allocaton n dscrete manufacturng systems. In Market-Based Control - A Paradgm for Dstrbuted Resource Allocaton. World Scentfc, Sngapore. Yee, S.-T Smulaton applcatons n the automotve ndustry: establshment of product offerng and producton levelng prncples va supply chan smulaton under order-to-delvery envronment. In Proceedngs of the 2002 Wnter Smulaton Conference, ed. E. Yücesan, C.-H. Chen, J. L. Snowdon, and J. M. Charnes, Pscataway, New Jersey: Insttute of Electrcal and Electroncs Engneers. Ths research has been done wth support from General Motors Research and Development Center. REFERENCES Clark, G., and Wrght, J. W Schedulng of vehcles from a central depot to a number of delvery ponts. Operatons Research 12: Clearwater, S. H., Costanza, R., Dxon, M., and Schroeder, B Savng energy usng market-based control. In Market-Based Control - A Paradgm for Dstrbuted Resource Allocaton. World Scentfc, Sngapore. Even, G., Garg N., Konemann, J., Rav, R., and Snha, A Coverng graphs usng trees and stars. In Proceedngs of RANDOM-APPROX. Flood, M. M The Travelng Salesman Problem. Operatons Research 4: Hnkkanen, A., Kalakota, R., Saengcharoenrat, P., Stallaert, J., and Whnston, A. B Dstrbuted decson support systems for real-tme supply chan management usng agent technologes. Readngs n Electronc Commerce. AW Computer and Engneerng Publshng Group. Jose, R. A., and Ungar, L. H Aucton-drven coordnaton for plantwde optmzaton. In Proceedngs of Foundatons of Computer-Aded Process Operaton (FOCAPO), Snowbrd, UT, USA. Lee, Y.-H Market-based dynamc resource control of dstrbuted multple projects. Ph.D. thess, Department of Industral Engneerng. The Pennsylvana State Unversty, PA, USA. Prm, R.C Shortest connecton networks and some generalzatons. Bell System Techncal Journal 36: Sauter, J. A., Parunak, H., Van, D., and Goc, J ANTS n the supply chan. In Workshop on Agent for AUTHOR BIOGRAPHIES SEOG-CHAN OH s a Ph.D. canddate n Industral and Manufacturng Engneerng, snce Before Ph.D. student, he worked as an IT consultant for seven years at Daewoo Informaton Systems. Hs research nterests nclude Artfcal Intellgence, Mult Agent System and Web Intellgence. He has a Certfed Professonal Engneer of Informaton Management from the Korean government. Hs e-mal address s sxo160@psu.edu. SHANG-TAE YEE s Staff Research Engneer of Manufacturng Systems Research Laboratory at the General Motor's Research and Development Center n Warren, MI. He has been leadng a research program at General Motors that develops and mplements real-tme enterprse decson systems framework usng wreless, advanced software, and decson technology. He receved hs Ph.D. n Industral Engneerng from the Pennsylvana State Unversty n He s a member of INFORMS and IIE. Hs emal address s mshangtae.yee@gm.com. SOUNDAR R. T. KUMARA s the Dstngushed Professor of Industral Engneerng, wth jont appontments n the department of Computer Scence and Engneerng, and School of Informaton Systems and Technology. Hs research nterests are n: Intellgent systems research wth emphass on sensor based equpment montorng and dagnoss, Software Agents, and Complexty n Supply Chans. He has over 150 publcatons. He has won several awards and s an elected member of The Internatonal Insttuton of Producton Research (CIRP) skumara@psu.edu. 1764

7 JEFFREY D. TEW s Group Manager of the Manufacturng Enterprse Modelng group n the Manufacturng Systems Research Laboratory at General Motor's Research and Development Center n Warren, MI. He receved hs Ph.D. n Industral Engneerng from Purdue Unversty n He s a member of Alpha P Mu, The Assocaton for Computng Machnery, The Amercan Statstcal Assocaton, The Insttute of Industral Engneers, The Insttute for Mathematcal Statstcs, INFORMS, The Socety of Computer Smulaton, and Sgma X. Hs emal address s jeffrey.tew@gm.com. 1765