The Retail Planning Problem Under Demand Uncertainty

Transcription

1 Vol., No. 5, September October 013, pp ISSN EISSN DOI /j x 013 Producton and Operatons Management Socety The Retal Plannng Problem Under Demand Uncertanty George Georgads, Kumar Rajaram UCLA Anderson School of Management, 110 Westwood Plaza, Los Angeles, Calforna , USA W e consder the retal plannng problem n whch the retaler chooses supplers and determnes the producton, dstrbuton, and nventory plannng for products wth uncertan demand to mnmze total expected costs. Ths problem s often faced by large retal chans that carry prvate-label products. We formulate ths problem as a convex-mxed nteger program and show that t s strongly NP-hard. We determne a lower bound by applyng a Lagrangan relaxaton and show that ths bound outperforms the standard convex programmng relaxaton whle beng computatonally effcent. We also establsh a worst-case error bound for the Lagrangan relaxaton. We then develop heurstcs to generate feasble solutons. Our computatonal results ndcate that our convex programmng heurstc yelds feasble solutons that are close to optmal wth an average suboptmalty gap at 3.4%. We also develop manageral nsghts for practtoners who choose supplers and make producton, dstrbuton, and nventory decsons n the supply chan. Key words: retalng; faclty locaton; nventory management; stochastc demand; nonlnear nteger programmng Hstory: Receved: May 011; Accepted: August 01 by Vshal Gaur, after revsons. 1. Introducton Retal store chans typcally carry prvate-label merchandse. For example, the department store chan Macy s carres several prvate-label brands such as Alfan, Club Room, Hotel Collecton, and others. Smlarly, Target, J. C. Penney, and others carry ther own prvate-label brands. Other retal store chans such as GAP, H&M, and Zara carry prvate-label products exclusvely. Prvate labels allow frms to dfferentate ther products from those of ther compettors and enhance customer loyalty, and they typcally provde hgher proft margns. However, these benefts are accompaned by addtonal challenges. The retaler must plan the entre supply chan by selectng supplers and by makng decsons on producton, dstrbuton, and nventory at the retal (and possbly other) locatons for each of these prvate-label products to mnmze total costs. Ths problem can be complcated when there s a large number of products wth uncertan demand that can be sourced from varous supplers, and they are dstrbuted across varous demand zones. An example of such a supply chan s llustrated n Fgure 1. Prvate-label products can be produced n-house, or producton can be outsourced to thrd-party supplers. Wthout loss of generalty, we refer to these optons as supplers. Suppler choce entals fxed costs such as buldng and staffng a plant when 100 producng n-house or negotatng, contractng, and control costs when outsourcng t. Each producton faclty can manufacture multple products nterchangeably, and there are economes of scale n manufacturng and dstrbuton. Demand at each zone (.e., store or cty) s stochastc, and nventory s carred at every demand zone. Here, demand zones can be nterpreted ether as retal stores or as dstrbuton centers (DCs). 1 The retaler ncurs overstock and understock costs for leftover nventory and unmet demand, respectvely. In ths context, there are three types of decsons. Frst, the retalers need to decde whch supplers to choose. Second, they need to conduct producton and logstcs plannng. Thrd, nventory management decsons on how much of each product to stock at each demand zone need to be made. We develop the retal plannng problem (RPP) under uncertanty to address these decsons. In ths problem, we model the selecton of supplers, producton, dstrbuton, and nventory decsons faced by the retaler as a nonlnear mxed-nteger program that mnmzes total expected costs. We show that ths problem s convex and strongly NP-hard. An nterestng attrbute of ths problem s that t combnes two well-known subproblems: a generalzed mult-commodty faclty locaton problem and a newsvendor problem. We explot ths structure to develop computatonally effcent heurstcs to generate feasble solutons. In addton, we apply a Lagrangan relaxaton

2 Georgads and Rajaram: The Retal Plannng Problem Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 101 Fgure 1 The Retal Supply Chan for Prvate-label Products to obtan a lower bound, whch we use to assess the qualty of the feasble solutons provded by the heurstcs. We show that the feasble solutons of a convex programmng heurstc are close to optmal, on average wthn 3.4% of optmal, whle n the majorty of cases they are closer to optmal, as evdenced by the.8% medan suboptmalty gap. Furthermore, the performance gap of ths heurstc mproves wth larger problem szes, and the computatonal tme of ths heurstc scales up approxmately lnearly n the problem sze. We also conduct robustness checks and fnd that the performance of ths heurstc, as well as ts advantage relatve to the benchmark practtoner s heurstc, s not senstve to changes n the problem parameters. All these are desrable attrbutes for potental mplementaton n large-szed, real applcatons. Our analyss enables us to draw several manageral nsghts. Frst, the optmal nventory level when solvng the jont suppler choce, producton, dstrbuton, and nventory problem s smaller than when the nventory subproblem s solved separately. Ths s because when solvng the jont problem, the soluton accounts for the fact that a larger downstream nventory level rases producton quanttes, whch ncreases upstream producton and dstrbuton costs as well as the costs assocated wth establshng producton capacty. In contrast, these costs are not consdered when the nventory subproblem s solved separately, and hence result n a larger nventory level. Thus, to mnmze total supply chan costs, one needs to adopt an ntegrated approach to solve the jont problem by consderng the effect of downstream nventory decsons on upstream producton and dstrbuton costs. Our model provdes a framework to analyze these decsons. Second, the two major costs that nfluence total (expected) supply chan costs are producton costs and the understock costs assocated wth the varance n demand. Therefore, retalers should focus on reducng these costs frst before consderng the effects of suppler capacty and contractng costs. Thrd, t s mportant to consder establshment, producton, dstrbuton, and nventory costs together when choosng supplers, because a suppler who s desrable n any one of these aspects may n fact not be the best overall choce. Our analyss provdes a mechansm to ntegrate these aspects and pck the best set of supplers. As one of the decsons consdered n the RPP under demand uncertanty s the establshment of producton capacty by the explct choce of supplers, ths problem can be placed n the broad category of faclty locaton problems under uncertan demand. Akens (1985), Drezner (1995), Melo et al. (009), Owen and Daskn (1998), and Snyder (006) provde extensve revews. The problem wth stochastc demand was frst studed by Balachandran and Jan (1976) and Le Blanc (1977), who developed a branch and bound procedure and a Lagrangan heurstc, respectvely. Ths study generalzes ther models by consderng multple products, as well as ncorporatng economes of scale n producton and dstrbuton. Ths study can also be placed n the general area of ntegrated supply chan models. Shen (007) provded a comprehensve revew of ths area. In partcular, ths study s related to Daskn et al. (00) and Shen et al. (003), who studed a locaton-nventory problem n a suppler DC retaler network. Here,

3 Georgads and Rajaram: The Retal Plannng Problem 10 Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety the planner s problem s to determne whch DCs to establsh, the nventory replenshment polcy at each DC, and logstcs between DCs and retalers. Daskn et al. (00) and Shen et al. (003) solved ths problem by usng a Lagrangan relaxaton and a column-generaton approach, respectvely. Shen (005) studed a mult-commodty extenson of Daskn et al. (00) wth economes of scale but wthout explctly modelng nventory decsons and wthout capacty constrants. Relatve to these papers, we ncorporate economes of scale n both producton and dstrbuton as well as capacty constrants at each suppler. Moreover, we explctly model the nventory problem. Here, by usng the newsvendor nstead of a replenshment model to make nventory decsons, we capture features of the retal fashon ndustry, where lead tmes are long relatve to product lfe cycles so that nventory cannot be replenshed mdseason, and unmet demand s lost, resultng n underage costs. A related problem was also studed by Oszen et al. (008), who studed a capactated extenson of Shen et al. (003). However, unlke these papers, we focus on the jont suppler choce, logstcs, and nventory plannng problem, as opposed to the rsk poolng effects from strategcally locatng DCs. Ths s because manufacturng s often outsourced to thrd-party supplers and contracts are volume based, and producton, and nventory decsons are best made smultaneously (Fsher and Rajaram 000). 3 Fnally, n contrast to all these papers, we consder an mportant problem faced by retal chans carryng prvate-label products, propose an effectve methodology to generate feasble solutons for ths problem, test t on realstc data to assess ts performance, and develop nsghts that practtoners can use for choosng supplers and makng producton, dstrbuton, and nventory decsons. The artcle s organzed as follows: In secton, we present the basc model formulaton; n secton 3, we dscuss the correspondng Lagrangan relaxaton; whle n secton 4, we propose heurstcs. In secton 5, we present results from our numercal study. In secton 6, we summarze and provde future research drectons.. Model Formulaton We formulate the RPP under uncertanty as a nonlnear mxed-nteger program. To provde a precse statement of ths problem, we defne: Indces: I,J,K: The set of possble supplers, demand zones, and products, respectvely.,j,k: The subscrpts for supplers, demand zones, and products, respectvely. Parameters: f : Fxed annualzed cost assocated wth choosng suppler. d k : Setup cost assocated wth producng product k at suppler. e j : Setup cost assocated wth shppng from suppler to demand zone j. c jk : Margnal cost to produce and shp product k from suppler to demand zone j. L ; : Mnmum acceptable throughput and capacty of suppler, respectvely. a jk : Unts of capacty consumed by a unt of product k at suppler that s shpped to demand zone j. h jk =p jk : Per unt overstock/understock cost assocated wth satsfyng demand for product k at demand zone j. U jk ðnþ=/ jk ðnþ: The cumulatve/probablty densty functon of the demand dstrbuton for product k at demand zone j. Decson varables: z :0 1 varable that equals 1 f suppler s chosen to supply products and 0 otherwse. w k :0 1 varable that equals 1 f product k s produced n suppler and 0 otherwse. v j :0 1 varable that equals 1 f suppler shps to demand zone j and 0 otherwse. x jk : Quantty of product k shpped from suppler to demand zone j. y jk : Inventory level of product k carred at demand zone j. To capture economes of scale so that per-unt producton and shppng costs decrease n quantty, we approxmate these costs by a setup cost d k that s ncurred to ntate producton for each product k at every suppler, a setup cost e j that s ncurred to shp from each suppler to every demand zone j, and a constant margnal cost (c jk ) that s ncurred to produce and dstrbute each addtonal unt. Whle a more complex cost structure could be desrable n some applcatons, we employ ths structure as t captures economes of scale and t permts structural analyss of the problem. To model the nventory problem faced by the retaler, we employ the newsvendor model. In contrast to Daskn et al. (00) and Shen et al. (003), who use a (Q,r) replenshment model, ths study s motvated by the fashon retal ndustry, where merchandse s often seasonal and lead tmes are long relatve to the season length. Consequently, the retaler cannot replensh nventory md-season, so unmet demand s lost, whle leftover demand needs to be salvaged va mark-downs at the end of the season. Therefore, the standard sngle-perod newsvendor model would seem most

4 Georgads and Rajaram: The Retal Plannng Problem Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 103 approprate here. Under ths model, let S jk ðyþ denote the expected overstock and understock cost assocated wth carryng y unts of nventory for product k at demand zone j. Ths can be wrtten as Z y S jk ðyþ ¼h jk ðy nþ/ jk ðnþdn 0 Z 1 þ p jk ðn yþ/ jk ðnþdn ¼) S jk ðyþ ¼ðh jk þ p jk Þ y Z y 0 U jk ðnþdn þ p jk ½EðnÞ yš: ð1þ The problem of suppler selecton, producton, dstrbuton, and nventory plannng faced by the retaler can be expressed by the followng nonlnear mxednteger program, whch we call the RPP: ( ðrppþ X Z P ¼ mn f z þ X X d k w k þ P P e j v j I I I þ P P P c jk x jk þ P P ) S jk y jk I subject to X x jk ¼ y jk 8j J; k K I L z X X a jk x jk z 8 I j k X a jk x jk w k 8 I; k K X a jk x jk v j 8 I; j J x jk 0; y jk 0 8 I; j J; k K ðþ ð3þ ð4þ ð5þ ð6þ w k f0; 1g; v j f0; 1g; z f0; 1g8 I; j J; k K: ð7þ The objectve functon of the RPP conssts of four terms. The frst term represents the annualzed fxed cost assocated wth securng capacty at suppler. The second term represents the setup cost assocated wth producton, whle the thrd term represents the setup cost assocated wth dstrbuton. The fourth term represents the correspondng (constant) margnal producton and dstrbuton costs. The ffth term represents the total expected cost assocated wth carryng nventory at the demand zones. Constrant () ensures that total nventory level for each product at every demand zone equals the total quantty produced and shpped to that zone. Note that t s also a couplng constrant. Were t not for Equaton (), the RPP would decompose by suppler nto a set of mxed-nteger lnear problems, and by demand zone j and product k nto a set of newsvendor problems. Ths observaton suggests that ths may be a good canddate constrant to use n any eventual decomposton of the problem. The left-hand sde nequalty of Equaton (3) mposes a lower bound on the mnmum allowable throughput of a suppler, f the suppler s selected. A lower bound on a suppler s throughput may be desrable to acheve suffcent economes of scale. The rght-hand sde nequalty of Equaton (3) mposes the capacty constrant (.e., ) for each suppler that s selected, and t enforces that no producton wll take place wth supplers that are not selected. Constrant (4) enforces the condton that x jk [ 0 f and only f product k s produced at suppler (.e., ff w k ¼ 1 for some j J), whle Equaton (5) enforces the condton that x jk [ 0 f and only f some quantty s shpped from suppler to demand zone j (.e., ff v j ¼ 1forsomek K). Fnally, Equaton (6) are nonnegatvty constrants, whle Equaton (7) are bnary constrants. Observe that the RPP s a convex mxed-nteger program, as t conssts of a lnear generalzed faclty locaton subproblem and a convex nventory plannng subproblem. By notng that the capactated plant locaton problem (CPLP) s strongly NP-hard (Cornuejols et al. 1991), t can be shown that the RPP s also strongly NP-hard. 4 Therefore, t s unlkely that realszed problems can be solved to optmalty. We verfy ths n our computatonal results. Consequently, t s desrable to develop heurstcs to address ths problem. The qualty of these heurstcs can be assessed by comparng them to a lower bound, whch we establsh n the next secton. 3. Decomposton and Lower Bounds To obtan a tght lower bound, we apply a Lagrangan relaxaton to the RPP (see Fsher 1981, Geoffron 1974). An mportant ssue when desgnng a Lagrangan relaxaton s decdng whch constrants to relax. In makng ths choce, t s mportant to strke a sutable compromse between solvng the relaxed problem effcently and yeldng a relatvely tght bound. Observe that by relaxng (), the problem can be decomposed nto a mxed-nteger lnear program (MILP) contanng the x jk, w k, v j, and z varables, and nto a convex program contanng the y jk varables. Moreover, ths relaxaton enables us to further decompose the MILP by suppler (.e., by ), and the convex program by demand zone and product (.e., by j and k) nto multple subproblems. A key attrbute of ths decomposton s that all subproblems can be solved analytcally. On the other hand, a potental

5 Georgads and Rajaram: The Retal Plannng Problem 104 Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety concern s that ths decomposton generates a relatvely large number of dual multplers: J 9 K of them, whch we denote by k jk. Relaxng () for a gven J 9 K matrx k of multplers, the Lagrangan functon takes the followng form: LðkÞ ¼mn X " 0 1 f z þ d k w k þ X ðc jk k jk Þx jk A I where 8 >< y jk ðkþ ¼ >: U 1 jk ð1þ fk jk h jk f h jk k jk p jk U 1 jk p jk k jk p jk þh jk U 1 jk ð0þ fk jk p jk : ð10þ þ X # e j v j þ X X ½k jk y jk þ S jk ðy jk ÞŠ ð8þ subject to Equatons (3) (7). Note that (L k )decomposesby nto I ndependent producton and dstrbuton subproblems, and by j and k nto J 9 K ndependent nventory subproblems. More specfcally, Equaton (8) can be rewrtten as follows: where L mlp LðkÞ ¼ X I L mlp ðkþþ X X L cvx jk ðkþ ðkþ ¼ < mn f z þ X 4d k w k þ X ðc jk k jk Þx jk 5 þ X = e j v : j ; and L cvx jk ðkþ ¼mnfk jky jk þ S jk ðy jk Þg: Note that the Lagrangan multplers n the producton and dstrbuton subproblems (.e., L mlp ðkþ) can be nterpreted as the cost saved (or cost ncurred f k jk \ 0) from producng and dstrbutng an addtonal unt of product k to demand zone j. On the other hand, the Lagrangan multplers n the nventory subproblems (.e., L cvx jk ðkþ) can be nterpreted as the change n holdng cost assocated wth carryng an addtonal unt of nventory of product k at demand zone j. For any gven set of multplers k, Proposton 1 determnes the optmal soluton for ðl k Þ, thus provdng a lower bound for the RPP. PROPOSITION 1. For gven set of multplers k R JK, a lower bound for the RPP s gven by LðkÞ ¼ X ( mn (mn e j I ( )) ) þ mn d k þðc jk k jk Þ þ f ; 0 þ X a jk " X Z # yjk ðk jk Þ p jk E jk ðnþ ðp jk þ h jk Þ n/ jk ðnþdn ; 0 ð9þ PROOF. To begn, fx k R JK. Let us frst consder each producton and dstrbuton subproblem. To solve each subproblem, we apply the nteger lnearzaton prncple by Geoffron (1974). Frst, observe that f z ¼ 0, then L mlp ¼ 0. Hence, the optmal soluton must satsfy L mlp ðkþ 0. As a result, we fx z ¼ 1 and solve 3 L mlp ðk; z ¼ 1Þ,mn X 4d k w k þ X ðc jk k jk Þx jk 5 þ X e j v j þ f subject to Equatons (3) (7). Because the problem s lnear, usng Equaton (3) t can easly be shown that x jk ðkþ f0; a jk g. Usng that L mlp ðkþ 0, Equatons (4), (5), and (7), t follows that ( ( ( L mlp ðkþ ¼mn mn þðc jk k jk Þ a jk e j þ mn )) d k ) þ f ; 0 : Next, consder each nventory subproblem. It s easy to show that ths problem s convex n y jk and by solvng the frst-order condton wth respect to y jk, we obtan Equaton (10), where U 1 jk ðþ denotes the nverse of U jk ðþ. Fnally, by usng Equaton (1) and y jk ðk jk Þ, t s easy to show that for each j J and k K, L cvx jk ðk jkþ can be wrtten as L cvx jk ðk jkþ ¼p jk E jk ðnþ ðp jk þ h jk Þ Z yjk ðk jk Þ 0 n/ jk ðnþdn: By notng that a lower bound can be obtaned by LðkÞ ¼ P I Lmlp ðkþ þ P P Lcvx jk ðkþ, the proof s complete. h Note that the Lagrangan soluton wll chose a suppler (.e., set z ðkþ ¼1) f and only f the cost savngs assocated wth producng and dstrbutng an addtonal unt of product k to demand zone j exceed the fxed cost assocated wth choosng ths suppler fornat least some n j and k (.e., f and ooonly f mn e j þ mn d k þðc jk k jk Þ a jk f.

6 Georgads and Rajaram: The Retal Plannng Problem Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 105 However, ths soluton may not be feasble. Thus, the purpose of ths soluton s more to establsh the value of the objectve functon of (L k ), whch s a lower bound on the value of the optmal soluton of the RPP. Ths lower bound can then be used to evaluate the qualty of any feasble soluton generated by heurstcs for ths problem. In the unlkely event that the correspondng soluton s feasble for the orgnal problem, t then solves the RPP optmally. In Lemma 1, we show that the Lagrangan problem (L k ) does not possess the ntegralty property (see Geoffron 1974). Therefore, the Lagrangan bound s lkely to be strctly better than that of a convex programmng relaxaton (.e., the relaxaton that s obtaned by replacng the bnary constrants n Equaton (7) by the contnuous nterval [0,1] for the RPP). We confrm ths n our computatonal results n secton 5. LEMMA 1. The Lagrangan problem (L k ) does not possess the ntegralty property. PROOF. It suffces to show that a convex programmng relaxaton of the RPP where Equaton (7) s replaced by 0 w k 1; 0 v j 1, and 0 z 1 8 I; j J; k K does not yeld a soluton such that the w, v, and z varables are ntegral. We prove ths by constructng a counterexample as follows: Let J = K =1, e 1 ¼ d 1 ¼ L ¼ 0 8 I; a 11 ¼ 1 8 I, and U 11 ðnþ ¼ n. To smplfy exposton, n the remander of ths proof we drop the subscrpts j and k. Observe that by cost mnmzaton, we wll have that z ¼ x. As a result, t suffces to show that there exsts an nstance of the convex programmng relaxaton of the RPP wth optmal soluton x 6 f0; g for some I (and hence z 6 f0; 1g. To proceed, by notng that Slater s condton s satsfed for the prmal problem, we dualze () and wrte the Lagrangan LðmÞ ¼mn 0 x ( X I þðh þ pþ Z y 0 c þ f þ m x ndn þ p ðm þ pþy ): It s straghtforward to check that for any gven dual multpler m, the Lagrangan program assumes the followng optmal soluton: 8 f c þ f >< þ m\0 x ðmþ ¼ ½0; Š f c þ f þ m ¼ 0; and yðmþ ¼ >: mþp hþp 0 otherwse : Observe that a soluton of the form x f0; g wll be optmal (and hence z f0; 1g) f and only f there exsts a dual multpler m such that X n o m þ p ¼ I 1 c þ f þm 0 h þ p : By notng that the RHS s a smooth functon strctly ncreasng n m whle the LHS s a step functon decreasng n m, t follows that there may exst at most one m such that the above equalty s satsfed. We now construct an example n whch there exsts no m such that the above equalty s satsfed. Lettng h = p = 1, I =, ¼, and c þ f ¼ 3, observe that f 1 3 \m mþ1 ðlhsþ ¼0\ ¼ðRHSÞ 1\m 1 3 then ðlhsþ ¼1 [ mþ1 ¼ðRHSÞ m\ 1 ðlhsþ ¼ 3 [ mþ1 ¼ðRHSÞ: We have thus constructed an nstance for whch the convex programmng relaxaton does not yeld an optmal soluton that s ntegral and hence proven that the Integralty Property does not hold. h We next consder the problem of choosng the matrx of Lagrangan multplers k to tghten the bound L(k) as much as possble. Specfcally, we are nterested n the tghtest possble lower bound, whch can be obtaned by solvng LB LR ¼ max kr JKLðkÞ: One way to maxmze L(k) s by usng a tradtonal subgradent algorthm (see Fsher 1985 for detals). However ths technque may be computatonally ntensve n our problem, as we have J 9 K Lagrangan multplers. To overcome ths dffculty, we explot the structure of the dual problem to demonstrate how the optmal set of Lagrangan multplers k can n some cases be fully or partally determned analytcally. In preparaton, we establsh Lemma. LEMMA. The optmal set of Lagrangan multplers k J K satsfy mn mn I c jk þ a jk d k þ e j þ f ; p jk k jk U p jk 8j J and k K: PROOF. Frst, t s easy to check from the frst lne of Equaton (9) that L mlp ðkþ decreases n k, and L mlp ðkþ ¼0 f d k þ e j þðc jk k jk Þ a jk þ f 0 8; j; k. By rearrangng terms, one can show that L mlp ðkþ ¼0 f k jk c jk þ a jk ðd k þ e j þ f Þ 8; j; k. It s also easy to verfy from the second lne of (8)

7 Georgads and Rajaram: The Retal Plannng Problem 106 Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety that L cvx jk ðk jkþ ncreases n k jk, and L cvx jk ðk jkþ ¼p jk E jk ðnþ f k jk p jk 8j; k. n n o To show that mn mn I c jk þ a jk ðd k þ e j þ f Þ ; p jk gk jk p jk, frst suppose that the LHS nequalty s not satsfed for some j,k. Then, L mlp ðk Þ¼ L mlp nð^kþ and n L cvx jkn ðk jk ÞLcvx jk ð^k jk Þ, where o o ^k o ¼ max k ; mn mn I c jk þ a jk ðd k þ e j þ f Þ ; p jk. As a result, Lðk ÞLð^kÞ and hence k cannot be optmal. Now suppose that k jk [ p jk for some j, k. Then L cvx jk ðk jk Þ¼Lcvx jk ðp jkþ and L mlp ðk ÞL mlp ðk jk Þ, where k jk denotes the set of Lagrangan multplers k, n whch the j k th element has been replaced by p jk. As a result Lðk ÞLðk jk Þ, and hence k cannot be optmal. Ths completes the proof. h Lemma states that the optmal set of Lagrangan multplers k les n a well-defned compact set. Observe from the left-hand sde expresson n Lemma that k jk [ 0 8j; k. From Equaton (10), observe that the optmal nventory level y jk ðk jkþ s strctly smaller than the optmal nventory level that would be determned from solvng the nventory subproblem separately from the suppler choce and producton plannng subproblem. Ths s a drect consequence of performng producton, dstrbuton, and nventory plannng n an ntegrated manner. The second mplcaton of Lemma s that the optmal soluton of the P Lagrangan relaxaton wll always satsfy I x jkðk Þy jk ðk Þ, and f the set defned n Lemma s a sngleton for some j and k, then t s possble to partally characterze the optmal set of Lagrangan multplers ex ante. When these sets are sngletons for all j and k, then t s possble to completely characterze k ex ante. Ths s establshed by Proposton. n o PROPOSITION. If mn I c jk þ a jk ðd k þ e j þ f Þ p jk, then the optmal Lagrangan multpler k jk ¼ p jk. If ths nequalty holds j J and k K, then k ¼ p and Z P ¼ LB LR (.e., the Lagrangan relaxaton solves the RPP). n PROOF. For any j and k, f mn I c jk þ a jk ðd k þ e j þ f Þg p jk, then by Lemma k jk ¼ p jk. If ths condton holds for all j and k, then t follows that k jk ¼ p jk, and by substtutng k jk ¼ p jk nto Equaton (8), t s easy to check that ðl k Þ s feasble for RPP. Ths completes the proof. h n o Observe that mn I c jk þ a jk ðd k þ e j þ f Þ can be nterpreted as the lowest margnal cost assocated wth establshng capacty at some suppler, producng product k, and dstrbutng t to demand zone j. As a result, when ths margnal cost exceeds the margnal underage cost, t s optmal not to produce any quantty of product k for demand zone j and ncur the expected underage cost; that s, set k jk ¼ p jk, whch yelds y jk ðp jk Þ¼0 by applyng (10). By usng Lemma and Proposton we now establsh a worst-case error bound for the Lagrangan relaxaton studed n ths secton. COROLLARY 1. The worst-case error bound for ths Lagrangan relaxaton satsfes 8 LR 1 þ max P j;k ðp jk þ h jk Þ R 1 yðk < jk Þ 0 n/ jk ðnþdn P j;k p ; : jke jk ðnþ P n n n oo o9 mn mn e j þ mn d k þðc jk p jk Þ a jk ;0 = P j;k p jke jk ðnþ ; n n where LR ¼ LB LR Z P and k 1 jk ¼ mn mn I c jk þ a jk ðd k þ e j þ f Þg; p jk g8j and k. Moreover, there exsts a problem nstance of the RPP such that the bound s tght (.e., LR ¼ 1). PROOF. Frst note that the Lagrangan dual s a concave maxmzaton problem, and recall from Lemma that k jk mn mn a I c jk þ jk ðd k þ e j þ f Þg; p jk g¼k 1 jk. Moreover, t s easy to check that a trval feasble soluton can be obtaned by settng z ¼ w k ¼ x jk ¼ y jk ¼ 0 8; j; k, n whch case the objectve functon s equal to P j;k p jke jk ðnþ. As a result, the followng nequaltes hold: maxflðk 1 Þ; LðpÞg LB LR Z P X j;k p jk E jk ðnþ: Hence, LP ¼ LB LR Z P maxflðk1 P Þ;LðpÞg, and the result p j;k jke jk ðnþ follows by substtutng Lðk 1 Þ and L(p) from Equaton (9). To show that there exsts an nstance such that ths bound s tght, for every I, pck f such that mn j;k n d k þ e j þðc jk p jk Þ a jk þ f 0. Then, t s easy to check that LR 1. Because LR 1 by defnton, we conclude that LR ¼ 1 n ths nstance. Ths completes the proof. h 4. Heurstcs and Upper Bounds In ths secton, we develop heurstcs, whch can be used to obtan feasble solutons for the RPP. These heurstcs can be used n conjuncton wth the lower bound developed n secton 3 to provde upper bounds for a branch and bound algorthm or to generate a feasble soluton for the RPP. We ntally propose two ntutve heurstcs. The frst s a practtoner s o

8 Georgads and Rajaram: The Retal Plannng Problem Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 107 heurstc developed based on observed practce at a large retal chan. The second s a sequental heurstc, whch solves the nventory management subproblem frst, and then t solves the remanng standard faclty locaton problem by applyng the well-known Drop procedure (Klncewcz and Luss 1986). These two heurstcs can be used to benchmark the performance of the analytcally more rgorous heurstcs we develop. The frst s a convex programmngbased heurstc, whch generates a feasble soluton by solvng a sequence of convex programs. We also propose a smpler LP-based heurstc, whch s computatonally more effcent. Ths heurstc uses the nventory levels from the Lagrangan problem (.e., yðk Þ), and t generates a feasble soluton by solvng a sequence of lnear programs. We next present these heurstcs, and we evaluate ther performance n secton Practtoner s Heurstc Ths heurstc frst chooses the nventory level for every product at each demand zone to equal the respectve expected demand; that s, y jk ¼ l jk 8j J; k K. Second, supplers are sorted accordng to the rato R ¼ f, whch captures the fxed cost perunt of capacty assocated wth choosng suppler. Thrd, the algorthm establshes suffcent capacty to satsfy the total nventory by choosng supplers that have the lowest R. For example, f R 1 R... R I, then the algorthm wll set z ¼ 1 8 f1;...; ng and z ¼ 0 otherwse, where n ¼ mn n I : P n P P ¼1 y jkg. Fnally, producton and transportaton decsons are made by solvng a relaxed verson of the RPP, where the fxed cost varables w k and v j are relaxed to le n [0,1]. Here, a feasble soluton s obtaned by roundng to 1 the fractonal w k and v j varables and by re-solvng the lnear program wth respect to x jk 0. Note that ths heurstc does not take nto account the underage and overage costs due to the varaton n demand as nventory levels are set to smply equal the mean demand. We denote the objectve functon of ths heurstc by UB Pr. Ths procedure s formalzed n Algorthm 1. A more sophstcated verson of ths heurstc can be obtaned by choosng the nventory levels accordng to the newsvendor model, and then usng Algorthm 1. Practtoner s Heurstc 1: Let R ¼ f, and sort canddate facltes such that R 1 R... R I. : Fx y jk ¼ l jk n8j and k. 3: Let n ¼ mn n I : P n ¼1 P P o y jk. 4: Fx z ¼ 1 8 ¼ 1;...; n and z ¼ 0 otherwse. 5: Solve the RPP wth relaxed varables v j ; w k ½0; 1Š. 6: Fx to 1 any v j [ 0 and w k [ 0, re-solve LP, and compute objectve functon UB Pr. the same approach as descrbed n Algorthm 1 to choose supplers and conduct logstcs plannng. We call ths the newsvendor-based practtoner s heurstc, and we denote ts objectve functon by UB Pr NV. 4.. Sequental Heurstc Ths heurstc obtans a feasble soluton for the RPP n two stages: n the frst stage, t fxes the nventory level for each product at every demand zone by solvng J 9 K newsvendor problems. Ths reduces the problem to a standard capactated faclty locaton problem wth pece-wse lnear costs. Then, n the second stage, t uses a Drop heurstc a well-known constructon heurstc for faclty locaton problems to determne whch supplers to choose. The general dea of the Drop heurstc s to start wth a soluton n whch all canddate supplers are chosen (.e., z ¼ 1 8), teratvely deselect one suppler at a tme, and solve the remanng subproblem n whch the fxed cost varables w k and v j are relaxed to le n [0,1]. Then, any fractonal w k and v j varables are rounded to 1, and the problem s resolved wth respect to the x jk varables. In each loop, the heurstc permanently deselects the suppler who provdes the greatest reducton n total expected costs, and t termnates f no further cost reducton s possble. Snce exactly one z s dropped n each loop, and at least one suppler must be selected n any feasble soluton, the algorthm needs at most I(I 1) teratons n total, and two convex programs are solved n each teraton. We denote the objectve functon of ths heurstc by UB Seq. Ths procedure s formalzed n Algorthm. For completeness, we also consder a varant of the sequental heurstc that fxes the nventory level for each product at every demand zone to equal the respectve expected demand. We call ths the smplfed sequental heurstc, and we denote ts objectve functon by UB Seq Smple. Algorthm. Sequental Heurstc 1: Fx y jk ¼ yðnewsvendorþ 8j and k : Fx z ¼ 1 8 and UB Seq ¼þ1. 3: for n = 1 toido 4: for m = 1 toido 5: f z m ¼ 1do 6: Fx z m ¼ z 8 6¼ m and zm m ¼ 0. 7: Solve the RPP wth z m and relaxed varables v j ; w k ½0; 1Š. 8: Fx to 1 any v j [ 0 and w k [ 0, and re-solve RPP to fnd x jk varables. 9: Compute objectve functon UBSeq m. 10: end f 11: end for 1: f mn m UBSeq m \ UB Seq do 13: UB Seq ¼ mn m UBSeq m and z m ¼ 0, where m ¼ arg mn m UBSeq m. 14: termnate 15: end f 16: end for

9 Georgads and Rajaram: The Retal Plannng Problem 108 Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 4.3 Convex Programmng-Based Heurstc One dsadvantage of the practtoner s and the sequental heurstcs s that nventory decsons are made ndependent of suppler selecton and logstcs decsons. Moreover, the Drop approach used n the sequental heurstc can be computatonally ntensve. Therefore, we construct a convex programmng-based heurstc as an alternatve way to obtan a feasble soluton for the RPP. The heurstc begns by solvng a relaxed RPP where the fxed cost varables z, w k, and v j have been relaxed to le n [0,1]. Frst, t temporarly fxes the largest fractonal z to 1, solves the remanng (relaxed) problem, and rounds to 1 any fractonal w k and v j varables. Second, t temporarly fxes the smallest fractonal z to 0, and agan t solves the remanng (relaxed) problem and rounds to 1 any fractonal w k and v j varables. The algorthm then permanently fxes the z that yelded the lowest total expected costs, and t contnues to terate untl all z varables have been fxed to 0 or 1. The assumpton behnd ths approach s that the fractonal value of z s a good ndcator of the worthness of choosng suppler. Snce at least one z s fxed n each loop, the algorthm needs at most teratons n total, and two convex programs are solved n each teraton. We denote the objectve functon of ths heurstc by UB Cvx. Ths procedure s formalzed n Algorthm 3. To gauge the value of jont logstcs and nventory plannng, we also consder a smplfed verson of the convex programmng heurstc n whch nventory levels are selected n advance usng the soluton correspondng to the lower bound from the Lagrangan relaxaton (.e., y jk ðk Þ8j and k). Then, the problem of fndng a feasble soluton reduces to solvng a sequence of lnear programs, whch are easer to solve than convex programs. We denote the objectve functon assocated wth ths LP-based heurstc by UB Lp. 5. Computatonal Results Algorthm 3. Convex Programmng-Based Heurstc 1: Intate z mn ¼ 0 and z max ¼ 1 8 : whle z max [ z mn for some do 3: Solve the RPP wth relaxed varables v j ; w k ½0; 1Š and z mn z z max 4: f z f0; 1g do 5: Set z mn ¼ z max ¼ z 6: end f 7: Let max ¼ arg maxfz : z ð0; 1Þg and mn ¼ arg mnfz : z ð0; 1Þg. 8: Solve the RPP wth relaxed varables v þ z mn z þ z max, and z þ max 9: Fx to 1 any v þ j [ 0 and w þ k UB þ CVX. 10: Solve the RPP wth relaxed varables vj ; w z mn z z max, and z mn ¼ 0. 11: Fx to 1 any vj UBCVX. [ 0 and w k j ; w þ k ½0; 1Š, ¼ 1. [ 0, and compute objectve functon k ½0; 1Š, [ 0, and compute objectve functon 1: f Z þ [ Z do 13: z mn max ¼ 1 14: else 15: z max mn ¼ 0 16: end f 17: end whle 18: Fx to 1 any v j [ 0 and w k [ 0, re-solve the convex program, and compute UB Cvx. In ths secton, we present a computatonal study to evaluate the performance of the heurstcs. In addton, we nvestgate the key factors that drve ther performance and also examne ther robustness. In addton, we nvestgate the key factors that drve ther performance and we examne ther robustness. To test our methods across a broad range of data, we randomly generated the parameter values usng a realstc set of data made avalable to us by a large retaler. We generated 500 random problem nstances, each comprsng between 5 and 0 canddate supplers, 10 and 40 demand zones, and 1 and 5 products (.e., I U{5,,0}, J U{10,,40} and K U{1,,5}). The parameters we used n our computatonal study are summarzed n Table 1. To solve the optmzaton problems assocated wth the boundng technques we propose, we used the CVX solver for Matlab (CVX 011) runnng on a computer wth an Intel Core 7-670QM.GHz processor and 6 GB of RAM memory. To evaluate the performance of the Lagrangan lower bound, we benchmark t aganst a standard convex programmng relaxaton, n whch the ntegralty constrants are relaxed so that Equaton (7) s replaced by 0 w k 1; 0 v j 1; 0 z 1 8 I; j J; k K: In every one of the problem nstances tested, the Lagrangan relaxaton generated a better lower bound than the convex programmng relaxaton, on average by.34%. Ths s consstent wth Lemma 1, whch asserts that the Lagrangan problem L k does not possess the Integralty Property. To test the performance of the heurstcs developed n secton 4, we evaluate the suboptmalty gaps relatve to the Lagrangan lower bound. Let D x ¼ 100% UB x LB LR LB LR, where x {Cvx,Lp,Seq,Seq Smple,Pr,Pr NV} denote the convex programmng based, the LP-based, the sequental, the smplfed sequental, the practtoner s, and the newsvendorbased practtoner s heurstcs, respectvely. The average and medan values, as well as the range of these metrcs, are llustrated n Fgure.

10 Georgads and Rajaram: The Retal Plannng Problem Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 109 Table 1. Parameters Fxed cost of choosng a suppler Setup cost assocated wth producton Setup cost assocated wth dstrbuton Margnal producton and dstrbuton cost Suppler capacty Summary of Parameters Used n Our Computatonal Study Dstrbuton of values Parameters Dstrbuton of values f Nð50; 10Þ Overstock h Nð5; 1Þ f Nð JK 3I f ; 3JK I f Þ cost hjk U 3 h; 3 h d Nð00; 40Þ Understock p Nð50; 10Þ d k U½0; dš cost p jk U 3 p; 3 p e Nð00; 40Þ Mean l Nð0; 4Þ e j U½0; eš demand l jk U 3 l; 3 l c Nð10; Þ Demand r Nð5; 1Þ c jk U 3 c; 3 c varance r jk U 3 r; 3 r U Nð100; 0Þ Weghts a jk ¼ 1 N 40JK I U; 90JK I U mn. L ¼ 0 throughput In addton, to get an dea of the computatonal complexty of these heurstcs, Table reports the mean, medan, and maxmum computatonal tmes for the problem nstances tested. Frst, observe from Fgure that the convex programmng-based heurstc unambguously outperforms the other heurstcs. In partcular, t provdes feasble solutons that are on average wthn 3.44% of optmal, and range from 0.41% to 18.76%. Whle the gap of the LP-based heurstc s hgher than the convex programmng-based heurstc on average, n the majorty of cases t generates a feasble soluton that s qute close to optmal as evdenced by the medan gap of 4.3%. The practtoner s heurstcs generate feasble solutons that are on average 19.95% and 36.3% from optmal for the standard and the newsvendor-based versons, respectvely. On the other hand, the suboptmalty gap for the sequental heurstcs s on average 36.7% and 3.7% for the standard and the smplfed versons, respectvely. Interestngly, wth both the practtoner s and the sequental heurstcs, the verson n whch nventory levels are set equal to the mean demand outperforms the verson n whch nventory levels are chosen accordng to the newsvendor soluton. Ths s because the understock costs are generally larger than the overstock costs, and, hence, the newsvendor model leads to a larger stockng quantty than the average demand. Ths n turn ncreases producton and dstrbuton costs as well as the fxed costs assocated wth establshng capacty n excess of the beneft of reducng underage costs. In addton, the nventory levels correspondng to the soluton of the convex programmng-based heurstc are always lower than those determned by the newsvendor soluton, and they are often lower than those chosen by the LP-based heurstc. Ths s because the convex programmng heurstc solves the jont problem n contrast to the LP-based heurstc as Fgure Suboptmalty Gap Table. Computatonal Tmes (sec) UB Pr UB Pr NV UB Seq UB Seq Smple UB Cvx UB Lp Mean Medan Max

11 Georgads and Rajaram: The Retal Plannng Problem 110 Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety well as other heurstcs n whch the nventory levels are chosen separately from the jont problem. When one solves for the jont problem, the soluton accounts for the fact that a larger downstream nventory level rases producton quanttes, whch ncreases upstream producton and dstrbuton costs as well as the costs assocated wth establshng producton capacty. In contrast, these costs are not consdered when the nventory subproblem s solved separately, and hence result n a larger nventory level. The takeaway from ths s that when plannng the entre supply chan, t s mportant to consder the effect of downstream nventory decsons to the upstream producton and dstrbuton costs. Retalers often underestmate the mpact of upstream costs n ther urge to have a hgher market share assocated wth hgher fll rates. When such costs are adequately represented, a lower fll rate may actually be preferable to lower total costs. Fnally, note that the cost reducton resultng from the convex programmng-based heurstc relatve to the other heurstcs s mportant, because retalers operate n a hghly compettve envronment wth very low margns and even a small cost reducton can lead to a large proft ncrease. Next, consder the computaton tme for each heurstc. Observe that both practtoner s heurstcs are computatonally very fast, whle both sequental heurstcs are qute slow. Also note that the standard sequental heurstc s computatonally less ntensve than ts smplfed counterpart. Because the standard sequental heurstc chooses the stockng quanttes accordng to the newsvendor model, whch n general are hgher than the expected demand, the Drop procedure needs fewer teratons n the standard sequental heurstc. Fnally, observe that the convex programmng-based heurstc s about as computatonally ntensve as the smplfed sequental heurstc, but leads to much lower average gaps. Thus, t clearly domnates both versons of the sequental heurstc. However, as expected, t s computatonally more ntensve than the LP-based heurstc. As the convex programmng heurstc domnates the other heurstcs n terms of the gap from the lower bound, we focus on ths heurstc to examne () how the computatonal tme scales up wth the sze of the problem, () how the suboptmalty gap and ts performance advantage relatve to the practtoner s heurstc depend on the parameters of the problem, and () whch parameters have the greatest mpact on the total expected costs. To conduct ths analyss, we regress the computatonal tmes, the suboptmalty gap of the convex programmng heurstc (.e., D Cvx ), the gap between the convex programmng and the practtoner s heurstc (.e., 100% UB Cvx UB Pr UB Pr ), and the total expected cost assocated wth the convex programmng heurstc (.e., UB Cvx ) of the 500 problem nstances tested earler on the sze (.e., I,J,K), and the parameters of the problem (.e., l; r; h; p;c; d;e; f; U). Table 3 summarzes the results. Frst note that the computatonal tme of the convex programmng heurstc s strongly dependent on the problem sze (.e., I, J, and K), whle t s nsenstve to the other parameters of the problem. More nterestngly, the relatvely large R rato mples that the computatonal tme of the convex programmng heurstc s explaned by a lnear model well, whch n turn suggests that the computatonal tme scales up approxmately lnearly n the problem sze. From the second column, observe that the suboptmalty gap decreases n the sze of the problem (I, j, and K), and ths effect s sgnfcant at the 1% level. Ths fndng s encouragng: t predcts that the convex programmng heurstc wll perform even better n larger problem nstances that could be expected n some applcatons. The suboptmalty gap ncreases n the capacty of the canddate supplers ( U), whle t decreases n the mean demand (l) and the fxed costs assocated wth choosng a suppler ( f). The suboptmalty gap also ncreases n the demand varance (r), the underage and overage costs (p and h), and the producton costs (c, d, and e), but ths effect s not sgnfcant at the 10% level. Fnally, note that a 95% confdence nterval for each regresson coeffcent can be obtaned from (regresson coeff.) ± (SE). 5 Therefore, as seen n Table 3, because the values of all regresson coeffcents and ther respectve standard errors are close to zero, (regresson coeff.) ± (SE) s close to zero for all parameters. Ths shows that the performance of the convex programmng heurstc s robust to changes n the parameters of the RPP. The thrd column examnes how the performance advantage of the convex programmng heurstc relatve to the practtoner s heurstc depends on the parameters of the problem. Observe that the performance advantage of the convex programmng heurstc becomes larger n the sze of the problem, whle t s nsenstve to the cost parameters, as evdenced by the small regresson coeffcents and the small (respectve) standard errors. Ths, together wth the fndng that the value of the ntercept s negatve at the 1% sgnfcance level, renforces the benefts from usng the convex programmng heurstc, as one could expect even larger problems wth dfferent cost parameters n certan applcatons. The fourth column consders the relatonshp between the total expected cost of the feasble solutons generated by the convex programmng heurstc and the parameters of the problem. Predctably, the expected cost ncreases n the sze of the problem (I, J, and K), n the mean demand (l), n the producton costs (c, d, and e), n the fxed costs assocated wth

12 Georgads and Rajaram: The Retal Plannng Problem Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety 111 Table 3. Suboptmalty Gap vs. Problem Parameters (Convex Programmng Heurstc) Computatonal tme Suboptmalty gap Cvx. vs. Pract. H. Expected cost I 3.93 (1.301) 0:13 (0.034) 0.3 (0.089) ( ) J 7.66 (0.651) 0:09 (0.016) 0.06 (0.04) (517.59) K (0.803) 0:16 (0.06) 0.15 (0.067) 19, (8.44) l 0.35 (0.893) 0:1 (0.017) 0.5 (0.044) 11,04.93 (543.84) r 0.04 (3.73) (0.071) 0.1 (0.185) (78.) h 7.79 (5.706) 0.1 (0.109) 0.1 (0.87) (351.77) p 0.03 (0.548) 0.01 (0.0104) 0.5 (0.07) (334.45) c 6.13 (.69) (0.051) (0.134) ( ) d 0.0 (1.108) 0.06 (0.004) (0.054) (657.6) e 1.59 (1.099) (0.01) (0.055) (67.77) f 1:40 (0.551) 0:00003 ( ) 0:00008 ( ) :9 (0.1) U 1.38 (0.66) (0.0001) 0:00 (0.0003) 37:8 (3.35) Intercept 689: (88.8) 8:5 (1.469) 36:6 (3.851) 830; 954:6 (47,333.18) R The values before the parentheses denote the regresson coeffcents correspondng to the parameter n the left column. The values n parentheses denote standard errors. *Sgnfcance at 10% level. **Sgnfcance at 5% level. ***Sgnfcance at 1% level. choosng a suppler ( f), as well as n the underage and overage costs (p and h). On the other hand, the expected cost decreases n the capacty of the canddate supplers ( U), whle the effect of the demand varance (r) s nsgnfcant. Therefore, our fndngs suggest that besdes the problem sze (.e., I, J, K, and l), the two most mportant factors that affect the expected cost of a feasble soluton are () the margnal producton cost and () the nventory underage and overage costs. The latter observaton emphaszes the value of an mproved demand forecast. On the other hand, the capacty of a suppler and the fxed contractng costs appear to have a secondary effect. Ths s consstent wth the ntatves undertaken at several retalers to reduce the mpact of producton, nventory underage, and overage costs (Fsher and Raman 010). As the gaps of the convex programmng-based heurstc are the smallest, we analyze the solutons to develop some nsghts about how t chooses supplers. Ths could be useful for practtoners who make such decsons. We fnd that supplers are chosen n ncreasng order of the rato r, where 3 r ¼ 1 f þ 1 X 4 jjjjkj j;k d k þ v j þ c jk a jk 5: The term n brackets represents the sum of fxed establshment costs and the average producton and dstrbuton costs across products and demand zones when a suppler s fully utlzed. Therefore, the rato r can be nterpreted as the average total cost per unt of capacty at suppler. Ths suggests that t s mportant to consder establshment, producton and dstrbuton costs together when choosng supplers, and t s benefcal to choose supplers wth the lowest total average cost per unt of capacty. 6. Conclusons We analyze a mult-product RPP under demand uncertanty n whch the retaler jontly chooses supplers, plans producton and dstrbuton, and selects nventory levels to mnmze total expected costs. Ths problem typcally arses n retal store chans carryng prvate-label products, who need to plan the entre supply chan by makng decsons wth respect to () suppler selecton for ther prvate-label products, () dstrbuton of products from supplers to demand zones (.e., stores or DCs), and () the nventory levels for every product at each demand zone. Ths problem s formulated as a mxed-nteger convex program. As the RPP s strongly NP-hard, we use a Lagrangan relaxaton to obtan a lower bound, and we develop heurstcs to generate feasble solutons. Frst, we develop an analytc soluton for the Lagrangan problem (Proposton 1), and we establsh condtons under whch the Lagrangan dual can be solved analytcally (see Proposton ). We frst develop a practtoner s and a sequental heurstc. We then propose two heurstcs, whch reduce the problem of generatng a feasble soluton to solvng a sequence of convex or lnear programs. To test the performance and the robustness of our methods, we conduct an extensve computatonal study. The convex programmng-based heurstc and ts LP-based counterpart yeld feasble solutons that are on average wthn 3.4% and 10.% from optmal, respectvely. Senstvty analyss suggests

13 Georgads and Rajaram: The Retal Plannng Problem 11 Producton and Operatons Management (5), pp , 013 Producton and Operatons Management Socety that the computatonal tme of the convex programmng heurstc scales up approxmately lnearly n the problem sze, whle t s stable to changes n problem parameters. Fnally, these heurstcs outperform both the sequental and the practtoner s heurstcs, and the performance advantage of the convex programmng-based heurstc relatve to the practtoner s heurstc s robust to the parameters of the problem. All these are desrable features for any eventual mplementaton n large-szed real applcatons. Several manageral nsghts can be drawn from ths work. Frst, solvng the more complcated jont suppler choce, producton, dstrbuton, and nventory problem leads to a leaner supply chan wth lower nventory levels than solvng the nventory subproblem separately from the suppler choce and logstcs subproblem. Ths hghlghts the mportance of consderng the effect of nventory decsons on upstream producton and dstrbuton costs. Our methodology provdes an effectve approach to solve ths jont problem. Second, the major costs that nfluence supply chan costs across the retaler are producton costs, as well as the understock and overstock costs assocated wth carryng nventory at the demand zones. Therefore, retalers should focus on reducng these costs frst before consderng the effects of suppler capacty and contractng costs. Thrd, t s mportant to consder establshment, producton, dstrbuton, and nventory costs together when choosng supplers, because a suppler who s desrable n any one of these aspects may n fact not be the best overall choce. Our analyss provdes a mechansm to ntegrate these aspects and pck the best set of supplers. Ths study opens up several opportuntes for future research. Frst, ths problem could be extended to explctly model nonlnear producton and shppng costs, whch s of partcular nterest for applcatons that exhbt sgnfcant economes of scale. In that case, the problem formulaton s a mxed-nteger nonlnear program that s nether convex nor concave (see Caro et al. 01 for detals about addressng a related problem n the process ndustry wth uncertan yelds). Second, our model could be extended to ncorporate multple echelons n the supply chan (.e., wholesalers and DCs) and allow multple echelons to carry nventory. Thrd, t may be desrable to ncorporate sde constrants pertanng to facltes, producton, and dstrbuton (.e., v, x, w, and z varables) as n Geoffron and McBrde (1978). Undoubtedly, all these extensons would requre sgnfcant, nontrval modfcatons to our model. Fnally, further work could be done to mprove the heurstcs to further reduce the suboptmalty gap. In concluson, we beleve the methods descrbed n ths artcle provde an effectve methodology to address the RPP under demand uncertanty. Acknowledgments We thank two anonymous referees, the assocate edtor, and the department edtor Vshal Gaur for suggestons that mproved ths artcle. We also thank Professors Reza Ahmad, Arthur Geoffron, and Chrstopher Tang for useful comments. Notes 1 Wth the latter nterpretaton we mplctly assume () that the locatons of DCs and the assgnment of stores to DCs are predetermned and () that stores mantan only a mnmal amount of nventory so that nventory costs at ndvdual stores are neglgble. Ths latter assumpton s consstent wth the exstng lterature (e.g., Shen et al. 003). Whle t s plausble that management must also determne the locaton of DCs and allocate stores to DCs, we leave ths mportant problem for future research. Specfc lead tmes faced by manufacturers are reported to be 7 months for Oxford shrts ordered by J. C. Penney, and 5 months for Benetton apparel (Iyer and Bergen 1997). 3 For example, leadng retalers H&M and GAP outsource 100% of ther manufacturng, whle Zara outsources approxmately 40% of ts manufacturng to the thrd-party supplers (Tokatl 008). Anecdotal evdence suggests that Macy s outsources all of ts manufacturng. 4 Ths result can be shown by reducng an nstance of the RPP to the CPLP. Specfcally, n ths reducton, let () demand assume a degenerate probablty dstrbuton, () the overage and underage costs to be arbtrarly large (.e., h jk and p jk!18j; k), () d k ¼ 0 and e j ¼ 0 8; j; k, (v) L ¼ 0 8 I, and (v) s take values from the set {1,, p} for any fxed p 3 I. 5 ± corresponds to the.5 and 97.5 percentle of a t-dstrbuton wth degrees of freedom. References Akens, C. H Faclty locaton models for dstrbuton plannng. Eur. J. Oper. Res. : Balachandran, V., S. Jan Optmal faclty locaton under random demand wth general cost structure. Nav. Res. Logstcs Q. 3: Caro, F., K. Rajaram, J. 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