No. of Pages 15, Model 5G ARTICLE IN PRESS. Contents lists available at ScienceDirect. Computers & Industrial Engineering

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1 Computers & Industral Engneerng xxx (8) xxx xxx Contents lsts avalable at ScenceDrect Computers & Industral Engneerng journal homepage: Heurstc approaches for the nventory-routng problem wth backloggng 3 Tamer F. Abdelmagud a, *, Maged M. Dessouky b, Fernando Ordóñez b 4 a Caro Unversty, Faculty of Engneerng, Mechancal Desgn & Producton Department, Gza 63, Egypt 5 b Danel J. Epsten Department of Industral and Systems Engneerng, 375 McClntock Avenue, Unversty of Southern Calforna, Los Angeles, CA , USA 6 artcle nfo 8 9 Artcle hstory: Receved 6 February 8 Receved n revsed form 7 September 8 3 Accepted 7 September 8 4 Avalable onlne xxxx 5 Keywords: 6 Heurstcs 7 Inventory management 8 Vehcle routng 9 Inventory routng 3 3. Introducton abstract 33 Recent decades have seen ferce competton n local and global 34 markets, forcng manufacturng enterprses to streamlne ther lo- 35 gstc systems, as they consttute over 3% of the cost of goods sold 36 for many products (Thomas & Grffn, 996). The major compo- 37 nents of logstc costs are transportaton costs, representng 38 approxmately one thrd, and nventory costs, representng one 39 ffth (Buffa & Munn, 989). The transportaton and nventory cost 4 reducton problems have been thoroughly studed separately; 4 whle, the ntegrated problem has recently attracted more nterest 4 n the research communty as new deas of centralzed supply 43 chan management systems, such as vendor managed nventory 44 (VMI), have ganed acceptance n many supply chan 45 envronments. 46 The ntegraton of transportaton and nventory decsons s 47 represented n the lterature by a general class of problems re- 48 ferred to as dynamc routng and nventory (DRAI) problems. 49 As defned by Bata, Ukovch, Pesent, and Favaretto (998), ths 5 class of problems s characterzed by the smultaneous vehcle 5 routng and nventory decsons that are present n a dynamc 5 framework such that earler decsons nfluence later decsons. 53 They classfy the approaches used for DRAI problems nto two 54 categores. The frst category operates n the frequency doman 55 where the decson varables are replenshment frequences, or * Correspondng author. Tel./fax: E-mal address: tabdelmagud@alumn.usc.edu (T.F. Abdelmagud). We study an nventory-routng problem n whch multperod nventory holdng, backloggng, and vehcle routng decsons are to be taken for a set of customers who receve unts of a sngle tem from a depot wth nfnte supply. We consder a case n whch the demand at each customer s determnstc and relatvely small compared to the vehcle capacty, and the customers are located closely such that a consoldated shppng strategy s approprate. We develop constructve and mprovement heurstcs to obtan an approxmate soluton for ths NP-hard problem and demonstrate ther effectveness through computatonal experments. Ó 8 Elsever Ltd. All rghts reserved. headways between shpments. Examples n the lterature nclude the work of Blumenfeld, Burns, Dltz, and Daganzo (985), Hall (985), Daganzo (987), and Ernst and Pyke (993) (for more references see Daganzo, 999). Anly and Federgruen (99) ntroduced the dea of fxed-partton polces (FPPs) for solvng the frequency-doman DRAI problems. FPPs are polces that solve the problem by parttonng the set of customers nto a number of regons such that each regon s served separately and ndependently from all other regons. In addton to that, whenever a customer n a partton s vsted, all other customers n that partton are vsted by the same vehcle. The soluton s consdered optmal n the set that ncludes all the FPPs f, wth respect to vehcle capactes, t defnes regons that mnmze the average of the sum of nventory holdng costs and transportaton costs. Examples n the lterature nclude Anly and Federgruen (993), Bramel and Smch Lev (995). However, Hall (99) ponts out that the FPPs approach can not model the case n whch delveres are coordnated. As a consequence, the results t provdes are ether vald only n the case of ndependent delveres, or can be just consdered as provdng upper bounds for real costs. The second category, referred to as the tme doman approach, determnes the schedule of shpments. Wth dscrete tme models, quanttes and routes are decded at fxed tme ntervals. Wthn ths category the most famous problem s the nventory routng problem (IRP), whch arses n the applcaton of the dstrbuton of ndustral gases. The man concern for ths knd of applcaton s to mantan an adequate level of nventory for all customers and to avod any stockout. In the IRP, t s assumed that each customer has a fxed demand rate and the focus s on /$ - see front matter Ó 8 Elsever Ltd. All rghts reserved. do:.6/j.ce.8.9.3

2 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx 86 mnmzng the total transportaton cost; whle nventory costs 87 are mostly not of concern. Examples of ths applcaton n the 88 lterature nclude Bell, Dalberto, Fsher, and Greenfeld (983), 89 Golden, Assad, and Dahl (984), Dror, Ball, and Golden (985), 9 Dror and Ball (987), Campbell, Clarke, and Savelsbergh () 9 and Adelman (3). 9 In ths paper, we consder a DRAI problem that addresses the 93 ntegrated nventory and vehcle routng decsons n the tme 94 doman at the operatonal plannng level. Ths problem, referred 95 to as the nventory-routng problem wth backloggng (IRPB), 96 consders multple plannng perods, both nventory and trans- 97 portaton costs, and a stuaton n whch backorders are permt- 98 ted. The knd of applcaton that permts backorders s, of 99 course, dfferent from the dstrbuton of ndustral gases, where stockout s not allowed. The proposed model s sutable to ndustral applcatons n whch a manufacturer dstrbutes ts product to geographcally dsbursed factores/retalers whch 3 are located n ctes close to ts warehouse. At the operatonal 4 plannng level, backorder decsons are generally justfed n 5 two cases. The frst s when there s a transportaton cost sav- 6 ng that s hgher than the ncurred shortage cost by a cus- 7 tomer. The second case s when there s nsuffcent vehcle 8 capacty to delver to a customer gven that rentng addtonal 9 vehcles s not an opton due to technologcal or economc constrants. In the lterature, the ntegraton of vehcle routng and nven- tory decsons wth the consderaton of nventory costs n the 3 tme doman approaches of the DRAI problems has taken dfferent 4 forms. In a few cases a sngle perod plannng problem has been 5 addressed as found n Federgruen and Zpkn (984) and Chen, 6 Balakrshnan, and Wong (989). In the mult-perod problem, 7 the decsons are conducted for a specfc number of plannng 8 perods, or the problem s reduced to a sngle perod problem 9 by consderng the effect of the long term decsons on the short term ones. Examples nclude Dror and Ball (987), Trudeau and Dror (99), Vswanathan and Mathur (997), and Herer and Levy (997). 3 Other researchers take nto consderaton varous forms such 4 as dstrbutng pershable products (Federgruen, Prastacos, & 5 Zpkn, 986), and the consderaton of the tme value of money 6 for long-term plannng (Dror & Trudeau, 996). Some work fo- 7 cused on dfferent structures of the dstrbuton network such 8 as Bard, Huang, Jallet, and Dror (998) n the case of satellte 9 facltes, Chan and Smch-Lev (998) n the case where ware- 3 houses act as transshpment ponts n a 3-level dstrbuton net- 3 work, and Hwang (999and ) n the case of a mult-depot 3 problem. 33 Soluton heurstcs that have been proposed n the lterature for 34 the dfferent varatons of the nventory routng problem are ether 35 based on subgradent optmzaton of a Lagrangan relaxaton (see 36 Bell et al., 983 and Chen et al., 989) or constructve and 37 mprovement heurstcs. The constructve heurstcs are broadly 38 classfed nto heurstcs that allocate customers to servce days 39 and then solve a VRP to generate vehcle routes for each day (Dror 4 & Ball, 987); and heurstcs that allocate customers to days and 4 vehcles and then solve a travelng salesman problem for every 4 assgnment (Dror et al., 985). Improvement heurstcs found n 43 the lterature (Dror & Levy, 986 and Federgruen & Zpkn, n the sngle perod case) are generally consdered as extensons 45 to the arc-exchange and node-exchange heurstcs as found n 46 the vehcle routng lterature. 47 In the lterature of the tme doman approaches of the DRAI 48 problems, some models n the case of mult-perod plannng 49 may nclude shortage or stockout costs; however, backorder 5 decsons are generally not explctly consdered. Instead, the 5 shortage or stock-out cost s treated as the penalty cost that s ncurred due to makng drect delveres to customers whose demand s not fulflled n the regular delvery route n a gven perod. Examples of such models n the lterature nclude Herer and Levy (997) and Jallet, Bard, Huang, and Dror (). In ths paper, we consder a stuaton n whch backorder decsons are ether unavodable or more economcal, and they have to be coordnated wth other nventory holdng and vehcle routng decsons over a specfc plannng horzon. We ntroduce constructve and mprovement heurstc approaches for solvng the problem wth backorders, and benchmark t aganst lower and upper bounds found by a commercal software package, CPLEX. The rest of ths paper s organzed as follows. In Secton, we formulate the problem as a mxed nteger lnear program. The motvatng deas and search plan for the developed heurstcs are presented n Secton 3. Sectons 4 and 5 provde descrptons of the constructve and mprovement heurstcs, respectvely. In Secton 6, the expermental results are presented followed by the concluson and drectons for future research n Secton 7.. Problem defnton and mxed nteger programmng model ger varable x v jt, whch equals f vehcle t travels from to j n trp s represented by y v. At customer, the nventory and In the IRPB, we study a dstrbuton system consstng of a 7 depot, denoted, and geographcally dspersed customers, n- 73 dexed,...,n. Each customer faces a dfferent demand d t for 74 a sngle tem per tme perod t (day/week). As tradtonally con- 75 sdered, a sngle tem does not restrct the problem to the case 76 of a sngle product dstrbuton, as the word tem can refer to 77 a unt weght or volume of the dstrbuted products and each 78 customer can be vewed as a consumpton center for packages 79 of unt weght or volume (Daganzo, 999). Accordngly, the pro- 8 posed model can be appled to the case of multple products g- 8 ven that the values of the nventory holdng and shortage costs 8 per unt volume/weght have small varance among the dffer- 83 ent products. We consder the case n whch the demand of 84 each customer s relatvely small compared to the vehcle 85 capacty, and the customers are located closely such that a con- 86 soldated shppng strategy s approprate. Delveres to custom- 87 ers,...,n are to be made by a capactated heterogeneous fleet 88 of V vehcles, each wth capacty q v startng from the depot at 89 the begnnng of each perod. Vehcles must return to the depot 9 at the end of the perod, and no further delvery assgnments 9 should be made n the same perod. In ths model, we consder 9 the case n whch rentng addtonal vehcles durng the short 93 plannng horzon s not an opton, and t s assumed that the 94 fleet of vehcles remans unchanged throughout the plannng 95 horzon. 96 Each customer mantans ts own nventory up to capacty 97 C and ncurs nventory carryng cost of h per perod per unt 98 and a backorder penalty (shortage cost) of p per perod per 99 unt on the end of perod nventory poston. We assume that the depot has suffcent supply of tems that can cover all cus- tomers demands throughout the plannng horzon. The plan- nng horzon consders T perods. Transportaton costs nclude 3 f vt a fxed usage cost for vehcle t, whch depends on the perod 4 t, and c j a drect transportaton cost between and j, whch sat- 5 sfes the trangular nequalty. The objectve s to mnmze the 6 overall transportaton, nventory carryng and backloggng costs 7 ncurred over a specfc plannng horzon. We consder an nte- 8 9 perod t, and f t does not. The amount transported on that jt backorder at the end of tme t s I t and B t, respectvely. The followng s a mxed nteger programmng formulaton for the 3 problem

3 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx [IRPB] Inventory routng problem wth backloggng 6 3 mn XT X V X V 6 4 f vt x v jt þxn C j x v jt þxn ðh I t þp B t Þ7 5 ðþ 8 t¼ subject to j¼ j k¼ k j¼ v¼ ¼ j¼ j ¼ ¼ x v jt 6 ¼ ;...;N;t ¼ ;...;T and v ¼ ;...;V ðþ x v kt XN l¼ l x v lt ¼ ¼ ;...;N; t ¼ ;...;T and v ¼ ;...;V Y v jt q vx v jt 6 ¼ ;...;N;j¼;...;N; j;t ¼ ;...;T and v ¼ ;...;V l¼ l y v lt XN k¼ k I y B t I t þb t þ XV ðþ ð3þ y v kt ¼ ;...;N;t ¼ ;...;T and v ¼ ;...;V ð4þ v¼ l¼ l y v lt XN k¼ k y v kt C A ¼ dþt ¼ ;...;N and t ¼ ;...;T ð5þ I t 6 C ¼ ;...;N and t ¼ ;...;T ð6þ I t ¼ ;...;N and t ¼ ;...;T ð7þ B t ¼ ;...;N and t ¼ ;...;T ð8þ Y v jt ¼ ;...;N; j ¼ ;...;N; j; t ¼ ;...;T and v ¼ ;...;V ð9þ Y v jte ; ; ¼ ;...;N; j ¼ ;...;N; j; t ¼ ;...;T and v ¼ ;...;V ðþ 9 The objectve functon () ncludes transportaton costs and nventory carryng and shortage costs on the end of perod nventory poston. Constrants Eq. () make sure that a vehcle wll vst a loca- ton no more than once n a tme perod, and constrants Eq. () en- 3 sure route contnuty. Constrants Eq. (3) serve for two purposes. 4 The frst one s to ensure that the amount transported between 5 two locatons wll always be zero whenever there s no vehcle mov- 6 ng between these locatons, and the second s to ensure that the 7 amount transported s less than or equal to the vehcle s capacty. 8 Constrants Eq. (4) along wth the other elements of the model en- 9 sure that effcent solutons wll not contan subtours. We llustrate 3 n the Appendx A how ths condton s acheved. Constrants Eq. (5) 3 are the nventory balance equatons for the customers. Constrants 3 Eq. (6) lmt the nventory level of the customers to the correspond- 33 ng storage capacty. It s assumed that the amount consumed by 34 each customer n a gven perod s not kept n the customer s storage 35 locaton; accordngly, t s not accounted for n constrants Eq. (6). 36 Constrants Eqs. (7) () are the doman constrants Motvatng deas and heurstc desgn 38 The IRPB s NP-hard snce t ncludes the capactated vehcle 39 routng problem (VRP) as a subproblem. In ths secton, we present 4 the key deas n the proposed constructve and mprovement heu- 4 rstcs for ths NP-hard problem. 4 A key decson n solvng the IRPB s the amount delvered to 43 customer n perod t, as ths quantty, let us defne t by 44 w t ¼ PV v¼ P N l¼ l y v PN lt k¼ k y v kt C A, effectvely separates the routng 45 and nventory problems. In fact, gven delvery values w t for all customers and perods, the nventory and backorder values are determned by constrants Eqs. (5) (8). At the same tme, the best routng soluton for these w t s obtaned by solvng T separate capactated vehcle routng problems. Each VRP computes the optmal transportaton costs to delver W t ¼ðw t : ¼ ;...; NÞ n perod t by solvng the followng feasble problem whenever the delvery amounts satsfy ¼ w t 6 Xv v¼ q v : TC t ðw t Þ¼mn XN Subject to : j¼ j k¼ k x v jt j¼ X V v¼ f vt x v jt þ XN X V ¼ j¼ v¼ j c j x v jt 6 ¼ ;...N and v ¼ ;...V ð Þ x v XN kt x v lt l¼ l ¼ ¼ ;...N and v ¼ ;...V ð Þ Y v jt q vx v jt 6 ; j ¼ ;...N; j and v ¼ ;...V ð3 Þ l¼ l X v v¼ y v XN lt k¼ k l¼ l y v kt ¼ ;...N; and v ¼ ;...V ð4 Þ 56 y v XN lt y v kt k¼ k C A ¼ w t ¼ ;...N ðþ Y v jt and xv jt ¼ or ; j ¼ ;...N; j and v ¼ ;...V ðþ 58 Therefore, the key n solvng IRPB s to be able to dentfy the optmal delvery amounts w t snce what s left s a vehcle routng problem for whch there exst several effcent algorthms. Our proposed heurstcs buld on ths observaton by focusng on how to determne the w t varables effcently. The procedure used to determne the w t values must take nto consderaton the tradeoff exstng between nventory and transportaton costs. In Secton 4, we propose a constructve heurstc that sets the delvery amounts by balancng ths tradeoff. The dea of the heurstc s to estmate a transportaton cost value for each customer n each perod from an approxmate routng soluton. Actual delvery amounts, w t, are then decded by comparng these transportaton cost estmates wth the correspondng nventory costs. Ths process s done sequentally from the frst perod onward and n each perod the comparson of transportaton and nventory costs s done n two phases. The frst phase looks nto backorder decsons that are ether mposed by nsuffcent vehcle capacty or preferred due to savngs n transportaton costs that are hgher than backorderng costs. The second phase nvestgates nventory decsons that would cover demand requrements n future perods n the case that excess vehcle capacty s avalable at the current perod. The heurstc looks nto nventory decsons that provde savngs n future transportaton costs that are hgher than nventory carryng costs. The mprovement heurstc ntroduced n Secton 5 nvestgates possble mprovements to the solutons generated by the constructve heurstc by lookng nto modfcatons to the delvery quanttes that would reduce transportaton and/or nventory costs and result n overall cost savngs. In partcular the mprovement heurstc relaxes the requrement made n the constructve heurstc to reduce the search space, that s all demand satsfed n a gven perod must be satsfed exactly not partally

4 4 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx 9 A key step n ths heurstc s to be able to effectvely estmate 9 the transportaton cost of each customer. Below we present a re- 9 sult that provdes nsght nto the structure of the total transporta- 93 ton cost n perod t as a functon of the delvery amount W t. 94 Proposton. TC t (W t ) s a mult-dmensonal monotonc ncreasng 95 step functon. 96 Proof. Gven that the defnton of TC t (W t ) s based on an MIP 97 model for the capactated vehcle routng problem (VRP) n whch 98 trangular nequalty holds. Startng from an optmal soluton of a 99 specfc VRP at an ntal W t ¼ðw t : ¼ ; :::; NÞ, and by addng 3 DW þ t ¼ðow t : ow t P ; ¼ ; :::; NÞ to W t (.e. ncreasng the 3 demand values for a subset of the customers) such that 3 P N ¼ ðw t þ ow tþ 6 P V v¼ q v, one of two possble consequences wll 33 occur: () new arc or arcs wll be added to the current soluton 34 to satsfy the vehcle capacty constrants Eq. (3), whch wll 35 ncrease TC t W t by the correspondng cj and/or ft amounts as 36 needed, or ) the current VRP soluton remans optmal. Thus 37 TC t ðw t þ DW þ t ÞTC tðw t Þ. Snce the changes of TC t (W t ) occur at 38 dscrete ponts accordng to the vehcle capactes, TC t (W t ) takes 39 the form of a multdmensonal step functon. h 3 As a result of proposton, the soluton scheme can focus 3 only on those values of the contnuous varables, w t, at whch 3 changes to the transportaton cost occur. We can look at ths re- 33 sult from another perspectve. Gven planned delvery amounts 34 to customers n a perod, by reducng the delvery quantty of 35 a specfc customer, the transportaton costs wll be reduced at 36 dscrete ponts and the maxmum possble reducton wll occur 37 when the delvery to that customer s dropped to zero. Although 38 proposton s proven for optmal solutons to the VRP, ths re- 39 sult can stll be used for solutons generated by effcent heurs- 3 tcs as an approxmaton, such as the savngs algorthm (Clarke & 3 Wrght, 964) Constructve heurstc 33 As mentoned earler, the constructve heurstc s based on the 34 dea of estmatng a transportaton cost value for each customer n 35 each perod, whch s necessary to facltate the comparson be- 36 tween transportaton and nventory carryng and shortage costs. 37 We therefore refer to the constructve heurstc as the Estmated 38 Transportaton Costs Heurstc (ETCH). In Subsecton 4., we de- 39 scrbe how the transportaton cost estmates are evaluated and 33 contnuously updated throughout the course of the heurstc. Usng 33 these estmates, we show n Subsecton 4. how the nventory 33 problem n IRPB can be decomposed nto two subproblems that 333 are solved by the heurstc n two phases. The soluton technques 334 for these subproblems are llustrated n Subsecton Estmatng transportaton costs 336 Let w PL t be the planned delvery amount for customer n perod 337 t. For perod s n whch P N j¼ wpl js 6 P V v¼ q v; let W s ¼ðw js : w js ¼ 338 w PL js ; j ¼ ; :::; NÞ. For customer whose wpl s > ; letwðþ s ¼ðw js : w s 339 ¼ ; w js ¼ w PL js ; j ¼ ; :::; N; j Þ. Then, the transportaton cost reduc- 34 ton that would result from reducng customer s delvery n per- 34 od s to zero can be calculated as TC s ðw s Þ TC s ðw ðþ s Þ. Snce the 34 transportaton cost functon nvolves the soluton of a VRP, whch 343 s known to be NP-hard, t may not be possble to calculate ts exact 344 value, especally for large problem szes. Instead, an effcent heu- 345 rstc can be used to approxmate t. In our mplementaton, the 346 savngs algorthm s used for ths purpose. Let ATC s (W s ) be an approxmaton for TC s (W s ) when the savngs algorthm s used to solve the assocated VRP. The transportaton cost estmate for customer n perod s s calculated as ETC ðw s Þ¼ATR s ðw s Þ ATR s ðw ðþ s Þ. However, resolvng a VRP every 35 tme the transportaton cost estmate for each customer s calcu- 35 lated s n fact computatonally neffcent. Instead, a faster approx- 35 maton scheme can be constructed by evaluatng the 353 transportaton cost savng that wll result when a customer s re- 354 moved from ts delvery tour assgned to t n a gven VRP soluton. 355 Ths means that for gven delvery amounts, W s, the assocated VRP 356 wll be solved only once and the resultng vehcle tours wll be 357 used for generatng transportaton cost estmates. 358 ATC t (W t ) and ETC (W t ) are functons of the planned delvery 359 amounts w PL t whch are determned based on the customers net 36 demand requrements n perod t. However, the values of w PL 36 t must be defned such that the vehcle capacty constrant, PN w PL 36 jt 6 PV q v, j¼ v¼ s satsfed. Gven the nventory poston at the begnnng of perod 363 t, I,t - B,t, and the demand requrements d s for all perods 364 s P t, ETCH evaluates the net demand requrement for each cus- 365 tomer, and based on that t estmates w PL s. If the vehcle capacty 366 constrant s not satsfed n a gven perod, the w PL s values are ad- 367 justed such that customers wth the lowest unt shortage costs, p, 368 wll have part of ther demand requrements postponed to future 369 perods. The followng lst descrbes the steps of ths approach. 37 Procedure PLNDLV(t) 37. Let OC = ordered set of all customers n whch customers are 37 sorted n a non-ncreasng order of ther p values; 373. For every customer e OC, let nv = I,t- B,t ; For perod s = t to T do Let Q max ¼ PV q v ; 376 v¼ 5. For every customer e OC usng the order n set OC do ðw PL s ¼ mnðq max ; maxðd s nv ; ÞÞ; Q max ¼ Q max w PL s ; Inv ¼ nv þ w PL s d s; End-Loop; End-Loop;The resultant w PL s values can be safely used n evalu- 383 atng both functons ATC s (W s ) and ETC (W s ). Durng the course of 384 the algorthm, f a change n the delvery amounts occurs, a VRP 385 for the perod n whch the change occurred s nstantated and 386 solved to update the values of the transportaton cost estmates Problem decomposton and soluton scheme In the ETCH, the comparson between the transportaton cost estmates and nventory carryng and shortage costs s separated nto two subproblems that are solved sequentally. Ths comparson s conducted for every perod t startng from the frst perod onward. The frst subproblem s concerned wth decdng whether to have backorders on perod t and the second subproblem s concerned wth decdng whether to use remanng vehcle capacty n perod t, f any, to cover future customer demand. Backorders can be proftable for two reasons; t s ether cheaper to pay the backorder cost than the transportaton cost, or there s nsuffcent capacty n the vehcles to satsfy demand. Let d,t =- max(d,t I,t + B,t, ) be the outstandng demand at customer at the begnnng of perod t, and CD be the set of customers that have d,t >. The followng subproblem decdes whether to delver to customer n perod t or not (z = or, respectvely) and the quantty r to delver such that the sum of backorder cost and estmated transportaton cost s mnmzed and vehcle capacty constrants are satsfed. [SUB] Backorder decsons subproblem

5 49 4 Mn ATC t ðx t Þþ X CD p ðd ;t r Þ Subject to : X q v ðþ r 6 X CD v¼ r ¼ d ;t z 8 CD ð3þ X t ¼ðx t : x t ¼ r ; CDÞ 8 CD ð4þ z ¼ or 8 CD ð5þ 4 In SUB, the objectve functon s composed of two parts, an 43 approxmaton of the transportaton costs n perod t and backor- 44 der penalty costs. Both parts are functons of the decson varables 45 r. Constrant () ensures that we do not exceed the total vehcle 46 capacty, and constrants Eq. (3) enforce that we delver the exact 47 amount of the outstandng demand only to customers ncluded n 48 the delvery n perod t. Constrant Eq. (4) defnes the vector of 49 delvery amounts used n approxmatng the transportaton cost 4 functon. 4 The man outcome from solvng SUB s the backorder decsons 4 evaluated as B t = d,t for every customer CDthat has z =, and 43 accordngly w t =, n the soluton of SUB. The delvery amounts, 44 w t, for customers n set CD that have z = n the soluton of SUB 45 are not decded yet as future demand requrements may be added. 46 These decsons are nvestgated through subproblem SUB. For 47 every other customer jcd, w jt =,B jt = and I jt = I jt d jt. 48 Let FD be the set of customers that have z = n the soluton of 49 SUB. Consder the nteger varable u s to decde whether to delver 43 customer s demand for perod s n the current perod t, where 43 s > t. Let Q r denote the total remanng vehcle capacty,.e. Q r ¼ PV q v P r ; andlett max be the latest perod where customer 43 v¼ CD 433 s demand can be consdered wthout volatng ts storage capacty constrant,.e. T max P ¼ mn arg max Ls¼tþ d s 6 C ; T We also 434 L 435 defne T max ¼ max ðt max Þ. 436 Let w PL s be the planned delvery amount for customer n a fu- 437 ture perod s > t. The values of w PL s are ntally calculated usng 438 the PLNDLV(t + ) procedure as descrbed n Subsecton 4. wth 439 a small modfcaton to make sure that for every customer j FD, 44 ntal values of w PL js ¼ d jt. If t s not possble to acheve ths cond- 44 ton n a future perod s for customer j FD, T max j s set to s. The 44 w PL s values for customers that do not belong to set FD are fxed; 443 however, the values of w PL s for customers n set FD change wth 444 the change of the u s decson varables. The followng problem de- 445 cdes whether to nclude future demand for any customer n the 446 current delvery by mnmzng the total transportaton and nven- 447 tory costs and satsfyng capacty lmts. Ths part s formulated as 448 follows: 449 [SUB] Inventory decsons subproblem Mn XT max ATCðX s Þ X s¼tþ FD Subject to : X XT max d s u s 6 Q r FD s¼tþ U s u s W PL s ¼ d sð u s Þ X s ðx s : x s ¼ w PL U s ¼ or XT max s¼tþ ½ðs tþh d ;s Šu s ð6þ s ¼ t þ ;...; T max 8 FD ð7þ s ¼ t þ ;...; T max 8 FD ð8þ s ; ¼ ;...; NÞ s ¼ t þ ;...; Tmax ð9þ s ¼ t þ ;...; T max 8 FD ðþ 453 Constrant Eq. (6) represents the avalable vehcle capacty lmt. 454 For smplfcaton, the customers storage lmts are represented T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx 5 by the tme ndex ðt max Þ, whch s computed n advance as descrbed earler. The precedence constrants Eq. (7) are added to represent the fact that future demand n a certan perod s to be consdered only f the customer s precedng perod demand s fulflled. Constrants Eq. (8) defne the relatonshp between the future planned delvery amounts for customers n set FD and the decson varables u s. When the delvery amount n perod t changes, there may be changes n the transportaton cost n that perod. The formulaton of SUB neglects such changes.by solvng SUB, the delvery amounts for customers n set FD can be calculated as w t ¼ r þ P T max s¼tþ d s u s. Accordngly, the nventory and backorder decson varables n perod t can be easly calculated. Fnally, delvery routes n perod t are decded by solvng a VRP usng the resultng delvery amounts. The flow chart n Fg. summarzes the major steps of the proposed heurstc. The followng subsecton provdes the algorthmc solutons for both subproblems and ther related analyses Solvng subproblems The two subproblems are resource allocaton problems n whch the scarce resource s the assocated avalable vehcle capacty and the man decson varables, z and u t, are bnary varables. Accordngly, both of them can be solved optmally usng dynamc programmng (DP) as descrbed n Taha (99). However, wth the ncrease of the problem sze, manly due to the number of customers and the plannng horzon, the DP mplementatons suffer from the curse of dmensonalty. In ths secton, we present effcent heurstcs that can be used nstead. Frst, we present the followng result that characterzes optmal solutons to subproblem SUB. Proposton. There s an optmal soluton to SUB that makes delveres to customer only f the quantty delvered satsfes r > ETC ðx t Þ=p Also, every optmal soluton to SUB only makes delveres f r > ETC ðx t Þ=p. Proof. Assume that n the optmal soluton to SUB, some customer s delvered r that satsfes r > ETC ðx t Þ=p or equvalently p ðd ;t r ÞþATR t ðx t Þp d ;t þ ATR t ðx ðþ t Þ. If we consder the modfed soluton obtaned by settng z = r =, then the prevous nequalty shows that the modfed soluton, whch s feasble, s at least as good as the optmal soluton. In the case when r > ETC ðx t Þ=p then the modfed soluton s strctly better. Thus, the orgnal soluton cannot be optmal. h Proposton gves a necessary condton for the optmalty of the delvery decson made for a specfc customer; however, satsfyng ths condton for all customers that have planned delveres does not guarantee optmalty for the soluton of SUB. Yet, snce backorder decsons are generally not preferable, we wll consder solutons that have ths characterstc suffcently good. We desgn the followng algorthm that utlzes ths rule. Let DL k ={dl: dl # CD and dl = CD k}, where. denotes the sze of a set. We defne f SUB (dl) as the objectve functon value of subproblem SUB when z = for every customer dl and z j = for every customer j CD dl, where dl DL k for some k. If the vehcle capacty constrant of SUB assocated wth settng z = for all customers n a set dl s not satsfed, we defne f SUB (dl)=.the followng lst descrbes the steps of a breadth-frst-based heurstc approach that searches for effcent solutons to SUB. Procedure SUBALG. Let k = and dl mn = CD;. If f SUB ðdl mn Þ and r ETC ðx t Þ=p 8 dl mn then go to 9; 3. For every dl DL k evaluatef SUB ðdlþ;

6 6 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx Start Set day ndex t= Instantate and solve subproblem SUB to decde the delvery amounts for customers n day t and decde whether backorder decsons wll be made Fnd dl s from set DL k that has the mnmum f SUB (dl) selected 56 from the members of DL k that satsfy the followng condtons 57 (te-breakng s arbtrary): 58 a. f SUB ðdlþ and r ETC ðx t Þ=p 8 dl; If dl s Ø then let dl mn = dl s go to 9; 5 6. Fnd dl* from set DL k that has the mnmum f SUB (dl); te-break- 5 ng s arbtrary If f SUB ðdlþ < f SUB ðdl mn Þ then let dl mn ¼ dl; If k < CD then let k = k +,goto3; Generate a soluton for SUB n whch delveres are only 56 made to customers n set dl mn ; SUBALG evaluates the f SUB (dl) value for every set dl DL k at 59 values of k =,..., CD. If at some level of k, the condton that 53 r > ETC ðx t Þ=p s satsfed for all dl, we fnd an approxmate 53 soluton and the algorthm termnates. However, f steps, 4 and 53 5 are removed, the algorthm guarantees that an optmal soluton 533 for SUB has been dentfed. 534 Subproblem SUB can be llustrated graphcally. Consder the 535 sample case for SUB llustrated n Fg.. The decson varables 536 u s are represented by drected arcs, where the cost savng assoc- 537 ated wth each arc S s t ¼ ETC ðx s Þ ðs tþh d s. A sold vertcal 538 lne s drawn to represent the tme lmt T max for customer. Start- 539 ng from node, arcs are to be selected usng the order gven by 54 ther drectons, such that the total cost savng s maxmzed and 54 the vehcle capacty constrant s satsfed. We note here that f 54 one or more arcs n a gven perod are selected, the savng values 543 S s t of the unselected arcs n the same perod wll be changed 544 due to changes n the transportaton cost estmates and therefore 545 have to be recalculated. If t < T, nstantate and solve subproblem SUB to decde whether to use the remanng vehcle capacty to ncrease the delveres decded n day t such that future demand requrements are covered. Yes Is there remanng vehcle capacty n day t? Usng customers decded delvery amounts for day t, w t, calculate the nventory and backorder varables, I t and B t, and solve a VRP to generate feasble vehcle tours. t = t+ Yes t T No Stop Fg.. An outlne of ETCH. Inspred by ths graphcal representaton, subproblem SUB can be dealt wth as precedence constraned knapsack problem (PCKP) n whch the coeffcents of the objectve functon, S s t, are dependent on the decson varables. The PCKP s known to be NP-hard (Garey & Johnson, 979); however, Johnson and Nem (983) provde a dynamc programmng algorthm for the PCKP that can solve the problem n a pseudo-polynomal tme, gven that the underlyng precedence graph s a tree, whch s fortunately a property of SUB as can be seen n Fg.. We present here a smpler algorthm based on a greedy search that selects the next possble arc (see Fg. ) that has the maxmum postve savng. Ths algorthm does not guarantee optmalty to the soluton of SUB; however, t can produce relatvely good solutons n polynomal tme. The followng steps descrbe the algorthm. Procedure SUBALG. Let D max = Q r and TD = FD;. For every customer n set TD, Let Dt =; 3. Fnd customer j n set TD that has the largest postve value of ðetc j X tþdtj Þ Dt j h j d j;tþdtj Þ; If none found then termnate; 4. If D max d j;tþdtj then Let D max ¼ D max d j;tþdtj ; Add d j;tþdtj to customer j s delvery amount and updatetransportaton cost estmates n perod t+dt j ; Let Dt j = Dt j +; If D j > T max j then remove customer j from set TD; End-If Else remove customer j from set TD; 5. If TD = Ø then termnate; Else go to step 3. No

7 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx 7 perod Customers n set FD t+ t+ t+3 t+4 S Improvement heurstc There are two apparent lmtatons of ETCH. The frst s due to 58 the myopc nature of the decsons conducted. Ths myopc nature 58 stems from the strategy used at each perod for solvng the two 583 subproblems, as t ams to optmze nventory allocaton decsons 584 at the studed perod wthout consderng the mpact of such dec- 585 sons on the optmalty of the overall soluton. The second lmta- 586 ton s concerned wth not allowng for partal fulfllment of 587 demand, that s exact demand requrements n the current and fu- 588 ture perods must be consdered n the delvery schedule. Ths may 589 prevent ETCH from achevng further savngs, especally n trans- 59 portaton and backorderng costs. To overcome these lmtatons, 59 we ntroduce n ths secton an mprovement heurstc. In ths heu- 59 rstc, transtons from a gven soluton to ts neghborhood are 593 conducted usng the dea of exchangng customers delvery 594 amounts between perods. These delvery exchanges are conducted 595 through gudng rules that tend to reduce the total cost Neghborhood search structure 597 Frst, we defne n ths secton a neghborhood search structure 598 to be used n the developed mprovement heurstc. The act of 599 reducng a customer s delvery n a gven perod t and addng the 6 reduced amount to another perod s referred to as delvery ex- 6 change. If a delvery exchange s made to perod s < t, t s referred 6 to as backward delvery exchange. In ths case, there may be an n- 63 crease n the nventory carryng cost for the customer or a reduc- 64 ton n backorderng cost when a backorder exsts n a precedng 65 perod to whch the transferred amount s added. A forward delv- 66 ery exchange occurs when a delvery exchange s made to perod 67 s > t. In ths case, ether a reducton n the nventory carryng cost 68 wll be ganed or a shortage cost wll be ncurred dependng on the 69 amount exchanged. A forward delvery exchange may be needed to 6 create more capacty n perod t, whch could be more proftable for 6 other customers backward delvery exchanges. 6 Let ^w ;t!s denote the amount of delvery exchange made from 63 perod t to perod s for customer. Let DICð ^w ;t!s Þ represent the 64 overall change n nventory carryng and shortage costs (postve 65 f ncreased) assocated wth the delvery exchange ^w ;t!s. From 66 Proposton, we know that the reducton (ncrease) of the delv- 67 ery amount made to a specfc customer s assocated wth ether 68 reducton (ncrease) n the transportaton costs or the transporta- 69 ton costs reman unchanged. Let TCR t ð ^w ;t!s Þ and TCI s ð ^w ;t!s Þ de- S S S 3 S S S 3 S 3 S 4 S 44 Fg.. Graphcal llustraton of subproblem SUB for a sample case. note, respectvely, the amounts of transportaton cost reducton n perod t and the transportaton cost ncrease n perod s that wll result from a delvery exchange ^w ;t! s. By usng backward and forward delvery exchanges, a transton from an ncumbent soluton to ts neghborhood can be acheved. A sngle delvery exchange may not be proftable, yet a combnaton of delvery exchanges, appled n a specfc sequence, can lead to a reducton n the total cost. Generally, a better soluton can be obtaned by searchng for an ordered set of exchanges, DX, that maxmze the resultant cost savng P ^w ;t!s TCR tð ^w edx ;t!s Þ TCI s ð ^w ;t!s Þ DIC s ð ^w ;t!s Þ whle mantanng the vehcle and customer capacty constrants. The followng subsecton dscusses some of the gudng rules that can use for ths purpose. 5.. Gudelnes for delvery exchanges For a gven soluton, the frst step n constructng useful delvery exchanges s to look for reductons to the delvery amounts at a selected perod so that savngs n transportaton costs n that perod can be acheved, and addtons of delvery amounts to customers that have backorders at the end of that perod such that ther assocated shortage costs s reduced. In Proposton, t s shown that the reducton n transportaton costs as a result of reducng the delvery amount to a customer occurs at dscrete values of the amount reduced. The reason for such dscrete changes s due to changes n the vehcle tours whch are drectly related to the usage of vehcle capactes. Therefore, reductons to delvery amounts that wll result n reducng transportaton cost can be determned by studyng the relatonshp between the total delvery amount and the vehcle capactes. In the case when there s a backorder decson for a customer n a gven perod, reducton to shortage costs can be acheved by ncreasng the delvery made to the customer. The amount of ncrease s bounded by the total amount of backorder. In ths case backward delvery exchanges from future perods are needed. After decdng the sutable amounts of delvery reducton and addton, the next step s to select the mechansm by whch the reduced or added amount can be exchanged to or from another perod (the ordered set of delvery exchanges) such that reducton n the total cost can be acheved. Abdelmagud and Dessouky (6) lst a set of dfferent delvery exchange rules that can be used to effectvely gude a neghborhood search algorthm. These rules are adopted here for the developed mprovement heurstc. S

8 8 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx The mprovement heurstc 663 The developed mprovement heurstc can be consdered as a 664 complementary phase to ETCH n whch partal fulfllments of 665 demand and ther assocated cost reductons are nvestgated. Con- 666 trary to ETCH, the mprovement heurstc conducts ts teratons 667 startng from perod T backward to perod. The reasons for such 668 backward movement s to provde a remedy for the myopc dec- 669 sons of ETCH. In ts search for the best ordered set of delvery 67 exchanges, the mprovement heurstc employs the prevously 67 descrbed delvery exchange rules repeatedly at a gven perod n 67 a systematc fashon. We refer to the mprovement heurstc as 673 the Backward Delvery Exchanges Heurstc (BDXH). The followng 674 lst descrbes ts man steps. 675 Procedure BDXH 676. Let s* be the ntal soluton obtaned by ETCH; 677. Let t = T; Let set S = {s*} and set S =Ø; For every soluton n set S do Let R represent the set of customers that ether have 68 scheduled delveres or backorders n perod t For each customer n set R fnd all possble reductons/ 683 addtons (Dw t ) to the delvery amount of customer n 684 whch ether a transportaton cost savng or a reducton 685 n the shortage cost can be acheved For every possble Dw t found n step 4., generate sut- 687 able delvery exchanges for that amount n perod t usng 688 the delvery exchange rules descrbed earler Generate all the resultng neghborhood solutons for the 69 delvery exchanges found n step 4.3, and add them to set 69 S such that solutons are stored n an ncreasng order of 69 ther costs and solutons are not repeated. If the allowed 693 maxmum sze of set S s exceeded, solutons wth the 694 worst costs are elmnated If the cost of the best soluton n set S s less than the cost of s*, 697 then let s* be that soluton Let S = S, repeat step 4 untl there s no further delvery 699 exchanges n perod t that can be made 7 7.Let t = t,ft > then go to step 3, otherwse STOP. The best 7 soluton found s s* The most tme consumng part of the BDXH s the generaton of 74 the neghborhood solutons n step 4.4, as a vehcle routng problem 75 has to be solved for every perod n the plannng horzon n whch a 76 change n the delvery schedule occurs. The loop conducted n step 77 4 has a polynomal tme complexty that s a functon of the max- 78 mum sze allowed for set S. In the conducted experments ths 79 maxmum sze s selected to be twce the number of customers Expermentaton and results 7 Two versons of ETCH have been mplemented. In the frst one, 7 optmal solutons for the two subproblems are generated usng a 73 complete breadth-frst search for SUB and a dynamc programmng 74 algorthm for SUB. We refer to ths mplementaton as ETCH-O. The 75 second verson uses the provded breadth-frst heurstc for SUB 76 and the greedy-search algorthm for SUB, and s referred to as 77 ETCH-H. The mprovement heurstc s then appled usng the ntal 78 solutons generated by each verson of ETCH. Accordngly, we refer 79 to the results obtaned by the mprovement heurstc as BDXH-O 7 and BDXH-H dependng on the ntal soluton used. These heurstcs 7 are programmed and compled usng Borland C++ Bulder verson 3 and benchmarked aganst the lower and upper bounds obtaned by AMPL-CPLEX 8. runnng under an Intel Pentum 4 processor runnng wth a clock speed of.4 GHz wth GB RAM. 6.. Expermental desgn Frst, we consder two dfferent scenaros to examne the effectveness of the developed heurstcs under dfferent crcumstances. These scenaros smulate the ntegrated nventory-dstrbuton decsons faced by manufacturng companes that deal wth small number of customers, each located n a dfferent major cty. An example for smlar cases n the lterature can be found n Fumero and Vercells (999). The frst scenaro s desgned to test the qualty of the nventory holdng decsons of ETCH; whle, n the second one, some parameters are tuned to provde condtons n whch backorder decsons are economcal, so that the backorder decsons of the ETCH are assessed. The man factors that are controlled to produce such cases are the rato of the avalable vehcle capacty to the average daly demand by customers, the average unt shortage cost and the transportaton cost per unt dstance. In both scenaros, customers are allocated n a square of dstance unts and ther coordnates are generated usng a unform dstrbuton wthn these lmts. The depot s located n the mddle of the square. Customers unt holdng costs are generated usng a normal dstrbuton wth a mean of. and a standard devaton of., and each customer has a storage capacty of tems. A constant value of for the vehcle usage cost (f vt ) s used. In the frst scenaro, the transportaton cost per unt dstance s set to, the customers unt shortage costs are generated usng a normal dstrbuton wth a mean of 5 and a standard devaton of.5, and the customers demands are generated usng a unform dstrbuton from 5 to 5 tems per day. In the second scenaro, we set the parameter values so t s optmal to carry backorders. In ths scenaro, the transportaton cost per unt dstance s set to, the customers unt shortage costs are generated usng a normal dstrbuton wth a mean of 3 and a standard devaton of.5, and the customers demands are generated usng a unform dstrbuton from 5 to 5 tems per day. For each scenaro, sxty problems have been generated by varyng the number of customers (N), the number of plannng perods (T) and the number of homogenous vehcles (V). We generate three levels of N (5), (), and (5), two levels of T (5) and (7), and two levels of V () and (). For each problem settng defned by a combnaton of N, T, and V, we randomly generate fve problems. The total vehcle capacty n the frst scenaro s selected to be fxed at 5,, and 5 for each level of N, respectvely. In the second scenaro, the selected total vehcle capactes are 5, 3, and 45. The namng conventon used for the test problems starts wth a number that refers to the scenaro. After a hyphen, two dgts are assgned for the number of customers, followed by a dgt representng the length of the plannng horzon. The next dgt represents the number of vehcles. Fnally, the replcate number s gven at the last dgt after a hyphen. Thus, the problem -55- represents the frst replcate of the frst scenaro wth 5 customers, a plannng horzon of 5 perods and vehcle. 6.. Results and dscusson The detaled expermental results are lsted n Tables 3 and 4 n the Appendx A. These tables lst the cost components of solutons obtaned by each verson of the constructve and mprovement heurstcs along wth the CPLEX lower and upper bounds. An * next to the lower bound n the tables ndcates that CPLEX was able to fnd the optmal soluton wthn the one hour tme lmt

9 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx 9 Table Average computatonal tmes (n mnutes) for the developed heurstcs N T V # Bnary varables Frst scenaro Second scenaro ETCH-O BDXH-O ETCH-H BDXH-H ETCH-O BDXH-O ETCH-H BDXH-H Table Average results for the thrd scenaro problems N T V # Bnary varables CPLEX UB LB dff% ETCH-H BDXH-H LB dff% Tme (mn) LB dff% Tme (mn) The percentage dfferences between the total cost obtaned by 785 each heurstc and the lower bound are used as performance nd- 786 cators. The percentage dfference, also referred to as optmalty 787 gap, s calculated by takng the rato of the dfference between 788 the heurstc s total cost and the lower bound to the lower bound. 789 A comparson aganst the lower bound provdes a measure of 79 devaton from optmalty. The CPLEX upper bound n a maxmum 79 of one-hour runnng tme s used as an alternate heurstc and ts 79 percentage dfference aganst the lower bound s smlarly 793 calculated. 794 For each problem settng, the average of the percentage dffer- 795 ences of the 5 replcates s calculated and plotted aganst the num- 796 ber of bnary varables of that settng as shown n Fg. 3 and Table summarzes the average computatonal tme of the devel- 798 oped heurstcs for each problem set n both scenaros. 799 As shown n Fgs. 3 and 4, the combned constructve and 8 mprovement heurstcs outperform the CPLEX upper bound for n- 8 stances wth ten customers and more n the frst scenaro and 5 8 customers n the second scenaro. Whle the growth of the CPLEX 83 optmalty gap seems to be exponental wth the ncrease of the 84 number of bnary varables, the optmalty gap for the developed 85 heurstcs s below 3% on average and remans almost level wth 86 the ncrease of the number of bnary varables. 87 In the frst scenaro, the ETCH-O verson of the constructve 88 heurstc s on average % closer to the lower bound than ETCH- 89 H. However, after applyng the mprovement heurstc on both ver- 8 sons ths dfference reduces to only.4%. In the second scenaro, 8 ths dfference s % and slghtly ncreases wth the applcaton of 8 the mprovement heurstc to.5%. 83 Reductons to the total cost as a result of applyng the mprove- 84 ment heurstc are evdent. In the frst scenaro the mprovement 85 heurstc provdes reductons n the optmalty gap of 6.% and % on average over solutons generated by ETCH-O and ETCH- 87 H, respectvely. Whle n the second scenaro, these fgures are 88 4% and 3.8%, respectvely. 89 In the case of small problem nstances of 5 and bnary 8 varables, for whch CPLEX was able to fnd optmal solutons 8 wthn the one-hour tme lmt, we can see that the mprovement heurstc can reach a relatvely good optmalty gap of less than % n the case of 5 bnary varables and less than 5% for the case of bnary varables for the frst scenaro problems. These fgures are hgher n the case of the second scenaro problems. However for larger problem nstances, t s hard to judge the qualty of the lower bounds obtaned by CPLEX, and so the optmalty gaps obtaned can not gve a clear cut measure of how far the results obtaned by the developed heurstcs are from the optmal solutons. The computatonal tme for ETCH-H s found to be less than one second n all the cases tested. For ETCH-O, due to the dynamc programmng part of the algorthm, the computatonal tme ncreases wth the ncrease of the problem sze; however on average, t has not reached the 9 seconds lmt n all problem sets. The ncrease of computatonal tme of ETCH-O s mostly attrbuted to the ncrease n both N and T; whle, the number of vehcles, V, does not seem to have a sgnfcant effect on computatonal tme. The computatonal tme of the mprovement heurstc seems to ncrease at a hgher rate wth the ncrease of the problem sze n the frst scenaro as compared to the second one. We attrbute ths to the ncrease of the rato of the total vehcle capacty to the average daly customers demand, whch ncreases the number of possble delvery exchanges and neghborhood solutons generated at each teraton of BDXH. From the prevous results we conclude the followng. The ETCH- O verson of the constructve heurstc s capable of generatng slghtly better solutons compared to ETCH-H wth up to % dfference on average n the optmalty gap. However, wth the ncrease of the problem sze, especally the number of customers and the number of plannng perods, the computatonal tme of ETCH-O wll be sgnfcantly hgher than the computatonal tme of ETCH- H. The consderaton of partal fulfllment of demand and the mechansm of delvery exchanges mplemented by the mprovement heurstc seem to offer mprovements to the optmalty gap that can reach more than 3.8% on average. However, the computatonal tme of BDXH wll be consderably hgher wth the ncrease of the rato between vehcle capacty and the average customers demand per perod Expermental results for larger problem nstances To nvestgate the performance of the developed heurstcs wth larger problem szes, we construct an addtonal expermental set based on a thrd scenaro. In ths scenaro, medum vehcle capacty to average daly demand rato s used such that a stuaton n the mddle of the frst two extreme scenaros s addressed. Ths scenaro consders smlar parameters as n the second one wth some modfcatons to reduce the frequency n whch backorder decsons are needed. The man dfference between the parameters

10 Table 3 Detaled costs for the frst scenaro problems Problem CPLEX bounds ETCH-O BDXH startng wth ETCH-O ETCH-H BDXH startng wth ETCH-H UB LB Hold Short Transp Total Hold Short Transp Total Hold. Short. Transp Total Hold Short Transp Total * * * * * * * * * * * * * T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx CAIE 435

11 * CAIE 435 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx Optmal soluton found. used n the thrd scenaro as compared to the second one s that the travel cost per unt dstance s set to and customers daly demand s generated usng a unform dstrbuton between and 5. We consder three dfferent levels of the number of customers, N:, 5, and 3, and a total vehcle capacty of 3, 35, and 4 at each level of N, respectvely. We only consder one level for both T and V at 7 and, respectvely. Fve random replcates are generated at each level of N. We use the prevously defned namng conventon for the thrd scenaro problems. The detaled cost results for the thrd scenaro problems are shown n Table 5 n the Appendx A. CPLEX lower and upper bounds are obtaned after a runnng tme of three hours. Due to the nablty of the optmzaton routnes to fnd solutons for the two subproblems for these large problem nstances, we only ran the heurstc versons, ETCH-H and BDXH-H. The average cost and tme results for the ETCH-H verson and the mprovement heurstc BDXH-H are shown n Table. We can see that the rate of ncrease of the heurstcs optmalty gaps s almost constant wth the ncrease of the number of customers. When we compare ths wth the exponental rate of ncrease for the CPLEX upper bound percentage dfference, we can see the potental beneft of the developed constructve and mprovement heurstcs for larger problem szes. In terms of computatonal tme, the ETCH-H verson of the constructve heurstc remans below one second for larger problems wth up to 3 customers; whle, the mprovement heurstc has an ncreasng computatonal tme. 7. Concluson and future work Ths artcle addressed the nventory routng problem wth backloggng n whch multperod vehcle routng and nventory holdng and backloggng decsons for a set of customers are to be made. We consdered an envronment n whch the demand at each customer s relatvely small compared to the vehcle capacty, and the customers are closely located such that a consoldated shppng strategy s approprate. We presented a constructve heurstc based on the dea of allocatng sngle transportaton cost estmates for each customer. Two subproblems, comparng nventory holdng and backloggng decsons wth these transportaton cost estmates, are formulated and ther soluton methods are ncorporated n the developed heurstc. The man dea behnd the constructve heurstc as seen n the formulaton of the two subproblems s to consder only delvery plans n whch fulfllment of part of the current or the future demand requrements n a currently studed perod s not allowed. An mprovement heurstc s developed to overcome some of the lmtatons of the constructve heurstc. Ths mprovement heurstc s based on the dea of exchangng delvery amounts between perods to allow for partal fulfllments of demands and explot assocated reductons n costs. A mxed nteger programmng formulaton s provded and used to obtan lower and upper bounds usng AMPL-CPLEX to assess the performance of the developed heurstcs. For small szed problems wth up to 5 customers, the expermental results show that the developed constructve heurstc can acheve solutons that are on average not farther than 3% from the optmal n a few mnutes. Ths fgure can be reduced to 5% by applyng the mprovement heurstc whch shows the sgnfcance of allowng partal fulfllment of demand. Wth the ncrease of problem sze, the optmalty gap of the developed heurstcs ncreases wth almost a constant rate and results can be obtaned n a few mnutes. Ths shows the potental beneft of the developed heurstcs for larger problem szes. The studed problem and the developed heurstc approaches can gve nsghts for solvng other problems n the manufacturng

12 Table 4 Detaled costs for the second scenaro problems Problem CPLEX bounds ETCH-O BDXH startng wth ETCH-O ETCH-H BDXH startng wth ETCH-H UB LB Hold Short Transp Total Hold Short Transp Total Hold. Short. Transp Total Hold Short Transp Total * * * * * * * * * * T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx CAIE 435

13 * CAIE 435 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx Optmal soluton found. ndustry that have wder scope. The ntegraton of manufacturng and logstc decsons at the operatonal plannng level, as found n Chandra and Fsher (994), Fumero and Vercells (999) and Le, Lu, Ruszczynsk, and Park (6), s a good example of one such problem. Appendx A. Appendx A.. An llustraton of the subtour elmnaton mechansm n the developed MILP model To llustrate the role of constrants Eq. (4) n elmnatng subtours n the proposed MILP model, let us start wth an MILP model for the IRPB that does not contan constrants Eq. (4) and call t IRPB -(4). Consder a case n whch there s only one vehcle and three customers. Then, constrant Eq. (5) can be rewrtten as follows: X 3 l¼ l y X3 lt k¼ k y kt ¼ d t I t þ B t þ I t B t ¼ ;...; 3 and t ¼ ;...; T ðþ 948 The rght-hand-sde of the above equaton represents the amount that wll be delvered to customer n perod t. Let us denote ths quantty by t. Notce that t s unrestrcted n sgn snce constrants Eq. (4) are excluded. Let us consder one tme perod and let us drop the ndexes for both the tme perod and the vehcle for brevty. Then, the above equaton s reduced to: X 3 l¼ y l X3 k¼ k y k ¼ ¼ ;...; 3 ðþ The above equaton s qute famlar n network flow models as t s equvalent to sayng that the dfference between the amount of nflow and the amount of outflow to and out of node equals the quantty delvered to that node. Now, let us consder a smple numercal example n whch =, = 5 and 3 =3. Fg 5(a) llustrates one feasble vehcle tour for ths case that satsfes all the vehcle routng constrants of IRPB -(4), yet t contans the subtour Notce that constrants Eq. (4) are non-negatvty constrants for the delvery quanttes whch when added to the MILP model we would not obtan a negatve value for. The two feasble vehcle tours llustrated n Fg. 5(b) and (c) represent two dfferent feasble solutons when the value of equals zero and greater than zero, respectvely. Based on the requred delvery quanttes,, and 3, the values for the contnuous varables y j wll be determned as requred by constrants Eq. (5), whch n turn wll force the bnary decson varables x j to take the value of one as necesstated by constrants Eq. (3). Accordngly, new arcs wll be added to the vehcle tour, whch n turn must satsfy constrants Eq. () and (). It can be easly shown that the subtour --3- n both cases shown n Fg. 5(b) and (c) can not occur, for otherwse constrants Eq. () and () wll be volated. Ths logc can be easly extended for the case of more than one vehcle. Furthermore, for the cases n whch nodes (o, o,..., o N ) have zero delvery quanttes, subtours that come n the form o o x... o would not be effcent snce an addtonal unnecessary transportaton cost assocated wth the ther arcs wll be added. From the above analyss, It s evdent that constrants Eq. (4) whch mandates that the quantty delvered to any node by a gven vehcle should be greater than or equal to zero s necessary for elmnatng subtours n the developed MILP model

14 4 T.F. Abdelmagud et al. / Computers & Industral Engneerng xxx (8) xxx xxx 9 UB LB dff % ETCH-O LB dff % BDXH-O LB dff % ETCH-H LB dff % BDXH-H LB dff % Table 5 Detaled costs for the thrd scenaro problems No. of bnary varables Fg. 3. Average percentage dfferences aganst lower bounds for the frst scenaro problems. UB LB dff % ETCH-O LB dff % BDXH-O LB dff % ETCH-H LB dff % BDXH-H LB dff % No. of bnary varables Fg. 4. Average percentage dfferences aganst lower bounds for the second scenaro problems. Problem CPLEX bounds ETCH-H BDXH startng wth ETCH-H UB LB Hold Short Transp Total Hold Short Transp Total