Inventory Management with Advance Demand Information and Flexible Delivery

Transcription

1 Submtted to Management Scence manuscrpt MS R1 Inventory Management wth Advance Demand Informaton and Flexble Delvery Tong Wang Decson Scences Area, INSEAD, Fontanebleau 77305, France, Berl L. Toktay College of Management, Georga Insttute of Technology, Atlanta, Georga , USA, Ths paper consders nventory models wth advance demand nformaton and flexble delvery. Customers place ther orders n advance, and delvery s flexble n the sense that early shpment s allowed. Specfcally, an order placed at tme t by a customer wth demand leadtme T should be fulflled by perod t + T ; falure to fulfll t wthn the tme wndow [t, t + T ] s penalzed. We consder two stuatons: (1) customer demand leadtmes are homogeneous and demand arrvng n perod t s a scalar d t to be satsfed wthn T perods. We show that state-dependent (s, S) polces are optmal, where the state represents advance demands outsde the supply leadtme horzon. We fnd that ncreasng the demand leadtme s more benefcal than decreasng the supply leadtme. (2) Customers are heterogeneous n ther demand leadtmes. In ths case, demands are vectors and may exhbt crossover, necesstatng an allocaton decson n addton to the orderng decson. We develop a lower-bound approxmaton based on an allocaton assumpton, and propose protecton level heurstcs that yeld upper bounds on the optmal cost. Numercal analyss quantfes the optmalty gaps of the heurstcs (2% on average for the best heurstc) and the beneft of delvery flexblty (14% on average usng the best heurstc), and provdes nsghts nto when the heurstcs perform the best and when flexblty s most benefcal. Key words : Stochastc Inventory Model; Advance Demand Informaton; Flexble Delvery 1. Introducton Order An Introducton to Probablty Theory and Its Applcatons by Feller from Amazon, and you wll be promsed that the book wll shp wthn 14 days. Order a popular tem such as the Apple Pod Nano and Emnem s CD, and you wll be told that your purchase usually shps wthn 24 hours. In addton to the standard delvery opton (e.g. 14 days for Feller s book), there are optons lke Guaranteed Accelerated 1-day Delvery and Guaranteed Accelerated 2-day Delvery at dfferent shppng costs. Shppng fees are guaranteed to be refunded f tems fal to arrve on or 1

2 2 Artcle submtted to Management Scence; manuscrpt no. MS R1 before the quoted due dates. Flexble delvery arrangements where the customer accepts delvery of the product at any tme before ts quoted due date are prevalent n many nstances where companes sell drectly to consumers. In contrast, the nventory management lterature on advance demand nformaton (ADI) focuses almost exclusvely on exact delvery: When customers place an order wth a gven due date, t s assumed that they wll not accept early delvery. In other words, early shpment s forbdden. At the same tme, delayed shpment s penalzed. Harharan and Zpkn (1995), one of the frst papers to ncorporate advance demand nformaton nto nventory management, provdes the followng justfcaton: Ths assumpton s realstc n many though not all stuatons. The costs to customers of early delveres are now wdely apprecated, partly due to the JIT movement. The authors argue that early payment assocated wth early delvery s a deterrent. The addtonal nventory cost borne by the customer, and the uncertanty n delvery tmng may also make flexble delvery unappealng to the customer. Much of the lterature followng has taken the exact delvery assumpton for granted. However, n many stuatons where companes nteract wth end users drectly (e.g. onlne retalng, servces), t s customary for frms to tell ther customers that the product/servce wll be delvered by a partcular due date. It s easy to see why ths s acceptable: If the product s for use or consumpton, customers would typcally prefer recevng ther goods earler rather than later. In ths case, early delvery offers frms a powerful mechansm to reduce ther nventory costs by transformng the frm s nventory cost nto the customers utlty. Some frms recognze the varety n customer preferences and offer a range of optons. For example, Dell s Intellgent Fulfllment program ncludes both delvery wthn fve days and delvery on an exact date n ts delvery optons (Özer and We 2004). In ths paper, we analyze nventory management wth advance demand nformaton and the possblty of early shpment, whch we call flexble delvery. The model we use s closely related to the dscrete-tme, uncapactated, advance demand nformaton model of Gallego and Özer (2001), except that we allow for delvery flexblty. We frst consder a model where customers are homogeneous n that they all have an dentcal demand leadtme T : Demand d observed n perod needs

3 Artcle submtted to Management Scence; manuscrpt no. MS R1 3 to be satsfed on or before perod + T. The supply leadtme s L. Flexble delvery ntroduces a nonlnearty nto the system evoluton equatons. Nevertheless, we show that the structure of the optmal soluton parallels that of Gallego and Özer (2001): If T L + 1, the system reduces to the tradtonal model by replacng the nventory poston wth the modfed nventory poston, and a modfed (s, S) polcy s optmal; f T > L + 1, a state-dependent (s( ˆV ), S( ˆV )) polcy s optmal, where the state ˆV represents nformaton about advance demands beyond the supply leadtme. We next turn to the more general model where customers are heterogeneous n ther demand leadtmes: There are T + 1 categores of customers, wth demand leadtmes rangng from 0 to T. Demand n perod s now a vector (d, d +1,..., d +T ), where d j stands for orders receved n perod and to be satsfed by the end of perod j. Unlke the homogeneous demand case, t s no longer optmal to satsfy orders as early as possble snce future orders may be due earler (called demand cross-over ) wth T 2. Fulfllng observed advance orders early reduces holdng cost, but at the same tme, ncreases the probablty of shortage as unobserved urgent orders may arrve n the future. Besdes choosng when and how much to order (orderng decson), now nventory managers have to decde when and by how much to fulfll advance orders (allocaton decson). As the analyss becomes ntractable n ths case, we develop an approxmaton that relaxes the nonnegatvty constrants on delvery quanttes. Ths s equvalent to allowng the frm to take prevously msallocated unts back and to reuse them to satsfy urgent demands. Imposng such a relaxaton helps bypass the allocaton decsons, and ensures that myopc allocaton s optmal for the relaxed problem. Ths approxmaton yelds a lower bound on the optmal objectve functon value. We then propose three protecton level heurstcs (PL(0), PL(σ), and PL(Σ)) that use dfferent levels ( zero, optmal, and maxmal ) of stock to protect aganst shortages due to ms-allocaton. These heurstcs yeld upper bounds on the optmal cost. We benchmark ther performance by determnng the optmalty gap between the upper bounds they yeld and the lower bound obtaned from the relaxaton. Numercal experments yeld structural results concernng the state-dependent (s( ˆV ), S( ˆV )) polces, some of whch we prove for a specal case. These experments quantfy the performance of the

4 4 Artcle submtted to Management Scence; manuscrpt no. MS R1 heurstc over an experment wth 540 nstances, we fnd the average optmalty gap obtaned from the best heurstc PL(σ) to be 2.08%; and dentfy the cost beneft of advance demand nformaton and delvery flexblty on average a 14.06% cost reducton was acheved n our experments by ntroducng flexble delvery to an ADI system. An nterestng fndng s that ncreasng the demand leadtme by one perod has a hgher beneft than shortenng the supply leadtme by one perod. Ths s n contrast to prevous research (Harharan and Zpkn 1995) showng that the two are equvalent for systems wth ADI but no delvery flexblty. We show that delvery flexblty and ADI are complements: The beneft of delvery flexblty s hgher when there are hgher degrees of advance demand avalablty. The remander of ths paper s organzed as follows. Secton 2 postons our work n the context of the advance demand nformaton lterature. In Sectons 3 and 4, we develop and analyze models wth homogeneous and heterogeneous customers, respectvely. Each secton ncludes numercal analyss followed by structural and manageral nsghts obtaned from them. Concludng remarks are presented n Secton 5. All the proofs can be found n the e-companon to ths paper, unless otherwse noted. 2. Lterature Revew Our model drectly contrbutes to the stream of research that analyzes uncapactated nventory systems (where the supply leadtme s exogenous) wth advance demand nformaton and exact delvery. In addton to Harharan and Zpkn (1995) s contnuous-revew model dscussed above, Gallego and Özer (2001) study a perodc-revew model wth heterogeneous advance demand nformaton. They show that t s optmal to adopt a modfed (s, S) polcy, where replenshments are made to rase the modfed nventory poston (=nventory poston mnus advance demands, hereafter MIP) to S whenever MIP reaches or drops below s. Gallego and Özer (2003) and Özer (2003) extend ths analyss to mult-echelon models, and dstrbuton systems, respectvely. Other related models that all demonstrate the benefts of ADI are Bourland et al. (1996) n a two-stage supply system, Güllü (1997) n a two-echelon, sngle-depot, multple-retaler problem, Decrox and

5 Artcle submtted to Management Scence; manuscrpt no. MS R1 5 Mookerjee (1997) n a costly nformaton acquston settng, and van Donselaar et al. (2001) n a project-based (pure make-to-order) envronment. Lu et al. (2003) study an assemble-to-order system wth stochastc leadtme and advance demand nformaton. Our analyss establshes the structure of the optmal polcy under flexble delvery wth a determnstc supply leadtme, and quantfes both the magntudes of and the nteracton between the values of ADI and delvery flexblty. An artcle that allows for a flexble tme-wndow fulfllment scheme s Wang et al. (2005) that studes nventory management wth a servce level constrant. Assumng the nventory polcy s of the (s, S) type, the authors develop algorthms for searchng for the optmal (s, S) levels and demonstrate the trade-off between nventory cost and demand leadtme. Our model proves the optmalty of state-dependent (s, S) polces wth respect to the modfed nventory poston wth homogeneous customers. A related stream of lterature consders prcng and strategc nteractons wth ADI. Chen (2001) studes retaler market segmentaton strateges wth advance demand nformaton. By offerng dfferent prces and delvery schedules, the company s able to segment customers accordng to dfferent demand leadtmes. Thonemann (2002) and Zhu and Thonemann (2004) analyze the benefts of obtanng dfferent levels of future demand nformaton from multple customers. Ho and Zheng (2004) provde nterestng examples of flexble delvery n practce, and dscuss the role of delvery tme commtment and customer expectatons n market competton. Tang et al. (2004) model an advance bookng program for pershable seasonal products and present the optmal dscountng polcy to nduce customers to pre-commt. McCardle et al. (2004) extend the model to a duopoly envronment and dentfy condtons such that both retalers mplementng advance bookng program s the unque equlbrum. By quantfyng the value of flexble delvery, our model can provde the bass for market segmentaton and contract negotaton wth flexble delvery. The value of advance demand nformaton has also been analyzed n capactated productonnventory systems (modeled as queues, where the supply leadtme s endogenous). Buzacott and

6 6 Artcle submtted to Management Scence; manuscrpt no. MS R1 Shanthkumar (1994) analyze a sngle-stage make-to-stock queue wth advance demand nformaton, and nvestgate the relatonshp between safety stock and safety leadtme. Karaesmen et al. (2002) present a dscrete-tme verson of Buzacott and Shanthkumar (1994). They show that generalzed base-stock polces are optmal and conjecture the optmalty of order base-stock polces for leadtmes below a threshold; these polces are further characterzed and evaluated n Karaesmen et al. (2003) and Karaesmen et al. (2004). Wjngaard and Karaesmen (2005) prove the conjecture for an M/D/1 queue. Güllü (1996), Toktay and Wen (2001), and Hu et al. (2003) use the Martngale Model of Forecast Evoluton (developed n Heath and Jackson 1994, Graves et al. 1998), and Özer and We (2004) use addtve forecast updates to model advance demand nformaton n capactated dscrete-tme producton-nventory systems. They characterze or provde approxmatons for the optmal order base-stock level, and nvestgate the value of ADI. In ths stream of lterature, Karaesmen et al. (2004) and Jema (2003) are partcularly relevant as they allow for early delvery, and delvery wthn a gven tme wndow, respectvely, assumng homogeneous demand leadtmes. Based on the homogenety of the customers, Karaesmen et al. (2004) consder a base-stock polcy where arrvng orders trgger mmedate producton releases and all outstandng orders are satsfed as soon as possble n a frst-come-frst-served manner. They show that the model wth advance demand nformaton and delvery flexblty s then equvalent to one wth no advance demand nformaton and a modfed backorder cost. Jema (2003) generalzes the analyss to delvery wthn a tme wndow, of whch the analyss n Karaesmen et al. (2004) s a specal case. He shows that n decentralzed producton-nventory systems operatng under basestock polces, a tme wndow contract can reduce the neffcences and even coordnate the system. In ths paper, we also explot the frst-come-frst-served characterstc of homogeneous leadtmes. Ths characterstc s key n showng the optmalty of the modfed state-dependent base-stock polcy. Wth heterogeneous leadtmes, demand cross-over can occur, n whch case we develop and evaluate approxmate order and fulflment polces. Fnally, the concept of our protecton level heurstcs s closely related to the nventory ratonng lterature (e.g. Venott 1965, Topks 1968, Ha 1997, de Vércourt et al. 2002). These models are

7 Artcle submtted to Management Scence; manuscrpt no. MS R1 7 also concerned wth the optmal orderng polcy and how to allocate on-hand nventory to dfferent demand classes. The optmal allocaton polcy normally conssts of a protecton level for each segment. In these models, customers dffer n ther senstvty to stock-outs, represented by dfferent shortage costs or fll rate requrements, whle n our model, customers dffer n ther wllngness to wat, represented by the demand leadtmes. 3. Analyss wth a Homogeneous Customer Base We consder a sngle-tem, fnte-horzon, perodc-revew nventory system. The nventory manager makes an orderng decson at the begnnng of each perod to mnmze dscounted expected nventory holdng and backorder costs over a fnte plannng horzon of N perods. The sequence of events n any perod s as follows: nventory revew, placement of new order, recept of replenshng delvery, demand arrval, and fulfllment of demand. All quanttes (eg. demand, nventory level/poston, replenshment order) are assumed to be ntegers. In ths secton, we analyze the case wth a homogeneous customer base where the demand leadtmes of all customers are dentcal and denoted by T. Demand arrvng n perod s denoted by the scalar d. Ths demand s due by perod +T, snce fulfllng t n any perod wthn the tmewndow [, + T ] s consdered a successful fulfllment. Partal fulfllment of an order s allowed. All unsatsfed overdue demands are fully backlogged, and a backorder penalty s appled per perod. Demands n dfferent perods are ndependent. The nventory manager determnes the order quantty z n perod. The supply leadtme L s assumed to be a known nonnegatve constant, whch means that a replenshng order placed at the begnnng of perod wll arrve at the begnnng of perod + L. Outstandng supply arrvng n perod j s denoted by w j. At the begnnng of perod, the system state s gven by (x, W, V ). Here, the scalar x s the nventory level (on-hand nventory mnus backorders); the vector W = (w, w +1,..., w +L 1 ) s the supply ppelne, and vector V = (v, v +1,..., v +T 1 ) s the advance demand profle, where v j, j s the unsatsfed advance demand at the begnnng of perod that s due by perod j.

8 8 Artcle submtted to Management Scence; manuscrpt no. MS R1 We next derve the state evoluton equatons. The evoluton of the vector W s smple snce there s no possble control on the supply stream: the whole ppelne just moves one poston forward, and the new order s nserted n last poston,.e., W +1 = (w +1, w +2,..., w +L 1, z ). (1) However, delvery flexblty sgnfcantly modfes the dynamcs of x and V, whch have a lnear structure n nventory models wthout delvery flexblty. In basc nventory models wthout advance demand nformaton, x +1, the nventory level at the begnnng of perod + 1, s equal to x + w d, where w and d are the replenshment quantty to be receved n perod and the demand arrvng n perod, respectvely. Smlarly, n nventory models wth advance demand nformaton but no flexblty (see Fgure 1 for a schematc representaton), the evoluton equaton s x +1 = x + w v, whch preserves lnearty. Fgure 1 Inventory Model wth Advance Demand Informaton and a Homogeneous Customer Base Advance Demand Profle V Replenshment z Supply Ppelne W w +L-1... w +1 w Inventory x +1 v v... v +T-1 d d +1 Demand... In contrast, lnearty s lost when flexble delvery s possble, because the nventory manager now has the freedom to satsfy future demands earler than ther due dates. In fact, f stock remans after fulfllng the current perod s demand (x + w v > 0), t s optmal to shp as many of the exstng orders as possble to mnmze the nventory holdng cost. Snce the total exstng orders are gven by +T 1 v j + d, we obtan an end-of-perod nventory of (x + w +T 1 v j d ) +. If x + w v < 0, a stock-out occurs and the unflled demand (x + w v ) s backlogged. Here, x +. = max{x, 0} and x. = max{ x, 0}. Combnng the two, we have +T 1 x +1 = (x + w v j d ) + (x + w v). (2)

9 Artcle submtted to Management Scence; manuscrpt no. MS R1 9 Clearly, as orders arrve nto the system one by one and form a seral ppelne, t s optmal to fll future orders prortzed by earlest due date (or equvalently n a frst-come-frst-served manner) to mnmze the expected dscounted backorder costs. Ths observaton can be used to wrte the evoluton equatons for demand profle V : { } k v k +1 = mn (x + w v j ), v k, k = + 1,..., + T 1; (3) v +T +1 = mn { (x + w +T 1 v j d ), d }. (4) To nterpret (3) and (4), note that three outcomes are possble. If the on-hand nventory (x +w ) s suffcent to cover all advance demands up to and ncludng perod k ( k vj ), then v k +1 = 0. If t s suffcent to cover all demands up to and ncludng perod k 1 and only part of the perod k advance demand, then v k +1 = k vj x w. Fnally, f the nventory on hand can only cover at most the advance demand up to but not ncludng perod k, the advance demand for perod k s unchanged and v k +1 = v k. Fgure 2 plots x +1 (sold lne) and v k +1, k = +1,..., +T (dashed lnes). Note that although the evoluton of x s not lnear, x v evolves lnearly: x +1 +T +1 vj +1 = x +T 1 v j + w d. The observaton s crtcal when we re-defne system states and collapse the dmenson n the followng subsectons. Applyng the standard DP formulaton, we can now wrte the optmal cost-to-go functon n perod as C (x, W, V ) = mn z 0 {c(z ) + E d [L(x +1 ) + αc +1 (x +1, W +1, V +1 )]}. (5) Here, perods are ndexed n ncreasng order. α [0, 1] s the dscount factor. c(z) s the orderng cost when orderng z unts. Snce varable cost does not change the nature of the problem (Venott 1966), we assume that there s only a fxed order cost and no varable cost,.e., c(z) = K 1 {z>0}, where 1 {A} s equal to 1 f A s true, and zero otherwse. L(x) s the sngle-perod holdng and backorder cost (ncurred at the end of each perod): L(x) = h x + + p x.

10 10 Artcle submtted to Management Scence; manuscrpt no. MS R1 Fgure 2 State Evoluton v x +1 v +1 +T v +1 +T v v... v +T-1 d x + w Note. Ths graph demonstrates the state evoluton equatons as a functon of x + w. The dstance between tck marks s v, v +1,..., v +T 1, d. When x + w s less than v, on-hand nventory s nsuffcent to cover the current demand and x + w v unts are backlogged, wth advance demands v +1,..., v +T 1 unchanged. At the other extreme, when x + w s large enough to cover all observed demands, x + w P +T 1 v j d unts reman on-hand and v +1 +1,..., v+t +1 are zero. For ntermedate levels of on-hand nventory, x + w can cover v and part of the other advance demands, so no nventory remans, and advance demands are fulflled n a FCFS manner. Note that the tmng of payment may make a bg dfference on unt holdng cost h. h represents both physcal and fnancal holdng costs. Physcal holdng cost s ncurred untl the unt s delvered, whle the fnancal part s ncurred untl the unt s pad for. Here we assume that customers pay at the tme of delvery, so that the same unt holdng cost (both physcal and fnancal) wll be ncurred before and after customers place ther orders. All cost parameters (K, h, p) are ndependent of tme. When the supply leadtme s postve, the nventory manager s orderng decson z has no effect on the system n perods, + 1,..., + L 1. We adopt the standard technque of shftng the system by L perods and studyng the nventory level at the end of perod + L (.e., x +L+1 ). Because of the dfference between the cases T L + 1 and T > L + 1, we analyze ther dynamcs and optmal polces separately n the next two subsectons Case 1: T L + 1 When T L+1, advance demands flled by ther due date are only satsfed from prevously placed orders. Gven the system state (x, W, V ), we can recursvely derve x j and V j for all j (, T ]. Followng the last remark above, we are nterested n x +L+1. After some algebrac manpulaton, we obtan

11 Artcle submtted to Management Scence; manuscrpt no. MS R1 11 ( ) +L + ( x +L+1 = y d j y +L T d j ), (6) where y. = u + z and +L 1. +T 1 u = x + w j v j. (7) Here u s smply the nventory poston less all the advance demands, and s called Modfed Inventory Poston (MIP) n perod (before order z s placed). y = u + z s then the MIP after orderng. Note that ths defnton dffers from Gallego and Özer (2001), who defne MIP as nventory poston less the observed advance demands wthn the protecton perod [t, t + L],.e., MIP. = x + +L 1 w j +L vj. Proposton 1. For T L + 1, a modfed (s, S) polcy s optmal, where S = max{y : G (y) G (x), x}; s = max{y < S : G (y) > K + G (S )}, wth ( ) G (y ) =. +L +L T α L E d,...,d +L L (y d j ) + (y d j ) + αe d f +1 (y d ), (8) f (u ). = mn y u {K 1 {y >u } + G (y )}. (9) Here, the system state n perod collapses to the scalar u wth lnear evoluton u +1 = u + z d (recall the observaton that x v evolves lnearly). Therefore, the replenshment decson s made based on the modfed nventory poston. An order of S u unts should be placed whenever u s Case 2: T > L + 1 In ths case, t can be smlarly derved that ( ) +L + ( x +L+1 = y d j y + +T 1 +L+1 v j ). (10)

12 12 Artcle submtted to Management Scence; manuscrpt no. MS R1 Now x +L+1 can no longer be expressed by the modfed nventory poston and the demands only. An addtonal (T L 1)-dmensonal vector (eg. (v +L+1 recorded. Defne ˆV = (ˆv +L+1,..., v +T 1 ) n perod ) needs to be +T 1,..., ˆv ) (11) and let ˆv j 1 = v j 1 for j = L + 2,..., T. ˆV1 s then the vector of advance demands whose due dates exceed the supply leadtme n perod 1. Proposton 2. For T > L + 1, a state-dependent (s( ˆV ), S( ˆV )) polcy s optmal, where S ( ˆV ) = max{y : G (y, ˆV ) G (x, ˆV ), x}; s ( ˆV ) = max{y < S ( ˆV ) : G (y, ˆV ) > K + G (S ( ˆV ), ˆV )}, wth ( G (y, ˆV ) =. +L α L E d,...,d +L L (y d j ) + (y + +T 1 +L+1 ˆv j ) ) + αe d f +1 (y d, ˆV +1 ), (12) f (u, ˆV ). = mn y u {K 1 {y >u } + G (y, ˆV )}. (13) The redefned system state varables evolve lnearly: u +1 = u + z d and ˆV +1 = (ˆv +L+2,..., ˆv +T 1, d ). Now the re-order pont s and order-up-to level S are functons of the state vector ˆV. For any gven ˆV, there exst two crtcal numbers s ( ˆV ) and S ( ˆV ) such that the modfed nventory poston should be rased up to S ( ˆV ) once t falls to s ( ˆV ) or below Structural Results and Manageral Insghts In ths secton, we mplement the dynamc program for a number of experments and pont out some structural propertes of the optmal polcy. We then llustrate the cost beneft of advance demand nformaton and delvery flexblty. Our man manageral nsght s that the cost beneft of extendng the demand leadtme s much larger than that of shrnkng the supply leadtme by the same amount. Ths s n contrast to exstng lterature on advance demand nformaton n uncapactated systems that shows that these two are equvalent wthout fulfllment flexblty.

13 Artcle submtted to Management Scence; manuscrpt no. MS R1 13 A few words on the mplementaton are n order. In Case 1 (T L + 1), the state s one dmensonal, so the DP can be solved easly. Snce the model s the same as Scarf s where the nventory poston s replaced wth the modfed nventory poston, exstng algorthms to search for the optmal (s, S) parameters are also readly applcable (for references, see Venott and Wagner 1965, Zheng and Federgruen 1991). In the second case, the state has dmenson 1+(T L 1). As solvng such a hgh-dmensonal DP s computatonally prohbtve, we lmt our numercal analyss to the two-dmensonal case, whch s the smplest non-trval case. Specfcally, we consder combnatons of (L, T ) pars, where L = 0, 1, 2, 3, 4 and T = 0, 1, 2. Among the 15 combnatons, L = 0, T = 2 s the only one that s two-dmensonal. In ths case, the vector ˆV n the state (u, ˆV ) reduces to a scalar, denoted by ˆv below. Recall that u s the modfed nventory poston before orderng, and y s the modfed nventory poston after orderng. The optmal polcy has some nterestng structural propertes. Proposton 3. When T L = 2, the system state reduces to (u, ˆv). The followng propertes hold for = 1,..., N: 1. The order-up-to level S (ˆv ) s ndependent of ˆv ; 2. The re-order pont s (ˆv ) s decreasng n ˆv. The propertes can be observed n the (s 1 (ˆv), S 1 (ˆv)) polcy n perod 1 for the case L = 0, T = 2 (Fgure 3), where demand has a Posson dstrbuton wth mean λ = 6, plannng horzon N = 30, dscount factor α = 1, orderng cost K = 100, holdng cost h = 1, and shortage cost p = 9. In the fgure, we can see that a replenshment s made once the MIP s below some threshold s 1 (ˆv), and the MIP s rased up to level S 1 (ˆv). Fgure 3 confrms that the order-up-to level S(ˆv) s ndependent of ˆv. The ntuton s the followng: By defnton, the modfed nventory poston s equal to the nventory poston mnus all the known advance demands. Thus, no matter what the advance demand ˆv s, by rasng the modfed nventory poston x ˆv up to S, we are able to frst satsfy all the backorders and then clear all the known advance demands ˆv, and fnally have S unts remanng on-hand. No

14 Wang and Toktay: Advance Demand Informaton and Flexble Delvery 14 Artcle submtted to Management Scence; manuscrpt no. MS R1 Fgure 3 Optmal State-dependent (s 1(ˆv), S 1(ˆv)) Fgure 4 Optmal Costs wth and wthout Delvery Polcy Flexblty as Functons of L and T Total Cost L T 2 matter what ˆv s, the remanng on-hand nventory level S, whch captures the cost trade-off n the followng perods, wll not change. Ths property does not hold n Gallego and Özer (2001), where flexble delvery s not allowed. They do observe a smlar pattern when ˆv s small, but when ˆv s large, snce early fulfllment of ˆv s not possble, t does not pay to order and hold unts to cover a large ˆv: Any order one places above and beyond what can mmedately be shpped to satsfy exstng demand ncurs nventory holdng cost. Fgure 3 also shows that s(ˆv) s decreasng n ˆv. In other words, as the advance demand level ncreases, the reorder pont decreases, whch may appear counterntutve at frst. To understand why, note that for a gven MIP, the expected nventory holdng cost s ndependent of ˆv, whle the expected penalty cost decreases n ˆv, accordng to Equaton (12). Thus a larger ˆv allows the nventory manager to postpone her order by choosng a lower reorder pont s(ˆv). Gallego and Özer (2001) also observe that s(ˆv) s decreasng n ˆv; ths structure s not drven by delvery flexblty. In Fgure 4, we plot C1 (L, T ) (n sold lnes), the optmal cost as a functon of L and T (where the ntal state s (x 1, W 1, V 1 ) = (0, 0, 0)), together wth the optmal cost of Gallego and Özer (2001) s ADI model (n dashed lnes). It suggests that ADI and flexblty are complements: One gans more from delvery flexblty when T becomes larger. Another nterestng fndng s that n our model, the cost reducton from extendng the demand leadtme T to T + 1 s much larger than

15 Artcle submtted to Management Scence; manuscrpt no. MS R1 15 that from shrnkng the supply leadtme L to L 1. In contrast, Harharan and Zpkn (1995) show that n ther contnuous-revew model, what really matters s the effectve leadtme L T,.e., C (L, T + 1) = C (L 1, T ). In other words, the cost savngs should be equal from ncreasng the demand leadtme or decreasng the supply leadtme by an equal amount, f there s no delvery flexblty. The symmetry s also observed n Gallego and Özer (2001) s perodc-revew ADI model (see the surface plotted n dashed lnes n Fgure 4). Intutvely, the only dfference here s that we have delvery flexblty n our model, and the gan from such flexblty s always nonnegatve. Therefore ntroducng delvery flexblty breaks the symmetry and favors the drecton of extendng the demand leadtme. Ths s a useful manageral nsght, whch says that all else beng equal, effort should frst be concentrated on ncreasng the demand leadtme. Our analyss can be partcularly useful n the strategc nteractons wth upstream supplers and downstream customers by provdng quanttatve estmates of the benefts of shortenng the supply leadtme and extendng the demand leadtme, whch are crtcal when negotatng supply contracts wth supplers and when prcng delvery optons for customers. 4. Analyss for a Non-Homogeneous Customer Base We generalze our prevous analyss by allowng customers to be heterogeneous n terms of demand leadtme. Specfcally, there are T + 1 segments, wth demand leadtmes rangng from 0 to T. In any perod, a demand vector D = (d,..., d +T ) s observed, where d j s the demand arrvng n perod and to be fulflled by perod j. We assume D s are ndependent of each other. Note that the homogeneous customer case analyzed n the prevous secton s equvalent to D = (0,..., 0, d +T ). The supply leadtme L s agan a gven nonnegatve constant, and the dynamc programmng formulaton s the same as (5), wth (x, W, V ) as the system state. Now v j s the cumulatve unsatsfed demand at the begnnng of perod that needs to be fulflled by perod j. The evoluton of the supply ppelne remans the same as (1), snce the supply part s unchanged. However, the evoluton equatons of x and V, namely (2), (3), and (4), do not apply anymore. Remember that n wrtng these equatons, we nvoked the optmalty of satsfyng orders on a FCFS bass

16 16 Artcle submtted to Management Scence; manuscrpt no. MS R1 for homogeneous customers, whch s equvalent to servng them n earlest due date order. Ths property no longer holds wth heterogeneous customers as demand cross-over can take place: A demand arrvng later can have an earler due date than some exstng orders. As a result, n addton to the orderng decson, the nventory manager now faces an allocaton decson: If there s surplus nventory on hand, should she use t to satsfy observed orders that are due later n the future and reduce nventory cost, or carry the nventory over for future orders that may have earler due dates? The answer s not straghtforward. It may depend on the nventory level, cost parameters, demand dstrbuton, etc. The key ssue here s how to balance the trade-off between the holdng cost that can be saved n the current perod and the potental shortage costs that could be ncurred n the followng perods. Fgure 5 provdes a vsual llustraton of demand cross-over. Fgure 5 Inventory Model wth Advance Demand Informaton and a Heterogeneous Customer Base Advance Demand Profle V Replenshment Supply Ppelne W z w +L-1... w +1 w Inventory x v v v +T-1 d d d +T D d d +1 +T+1 D +1 Demand... The jont optmzaton of orderng and allocaton decsons wth demand cross-over s a dffcult problem, whose optmal polcy could be qute complcated. In the followng, we develop heurstcs that are easy to mplement and perform well. Note that when T = 1, demand D s a two-dmensonal vector (d, d +1 ). In ths case, there s no cross-over and the problem can be solved as before. In the remander of ths secton, we focus on the case T > L + 1, snce the other case s essentally a specal case where the state space reduces to one dmenson Descrpton of Heurstcs We frst develop an approxmaton (AP) by ntroducng the allocaton assumpton, whch s wdely appled n the mult-echelon dstrbuton system lterature: We assume that unts that have been

17 Artcle submtted to Management Scence; manuscrpt no. MS R1 17 delvered to fulfll advance demands can be taken back and resent to other customers wth urgent demand wthout ncurrng any costs or penaltes. Mathematcally, ths s equvalent to allowng the delvery of negatve unts aganst advance demands (where the quantty of negatve unts s bounded by the quantty of postve unts shpped earler). Then the nventory manager has no reason to care about the future, snce even f urgent demands arrve, she can always take the msallocated unts back. Therefore, t s optmal for her to to use a myopc allocaton polcy that uses all the on-hand nventory to satsfy the observed demands accordng to the earlest-due-date rule. As we demonstrate n 4.2, the allocaton assumpton allows us to solve for the orderng polcy wth prevously developed technques. Snce ths approxmaton s a relaxaton of the orgnal problem, the cost obtaned consttutes a lower bound on ts optmal cost. However, the approxmaton s not mplementable because allocated unts can hardly be taken back n practce. For ths reason, we propose three mplementable heurstcs that refne the allocaton polcy by ntroducng protecton levels to balance the holdng cost n the current perod and potental shortage costs n the future. In partcular, we assgn protecton stock σ between each adjacent par of upcomng demands d +1, d +2,..., d +T. Any on-hand nventory above ths protecton stock level can be used to satsfy advance demands n a frst-come-frst serve manner. After demand s fulflled, any remanng nventory s carred over to the next perod and can be used to fulfll future urgent demands. Ths approach reduces shortage costs n the future, but ncreases the holdng cost n the current perod. We develop three protecton level heurstcs, PL(Σ), PL(0) and PL(σ), as explaned n detal n 4.3. PL(Σ) uses a protecton level that s large enough to cover the whole support of urgent demand. Because ths avods demand cross-over, we can solve a dynamc program to obtan the optmal orderng polcy. PL(0) uses no protecton stock and PL(σ) uses an ntermedate protecton stock level that balances shortage and holdng costs. Because of demand cross-over, we cannot solve for the optmal orderng polcy for these two levels of protecton stock. Instead, we use the orderng polcy obtaned n AP for these heurstcs and evaluate ther performance usng smulaton. Table 1 summares all four models.

18 18 Artcle submtted to Management Scence; manuscrpt no. MS R1 Table 1 Approxmaton and Protecton Level Heurstcs Model Descrpton Allocaton Polcy Orderng Polcy Cost AP PL(0) PL(σ) PL(Σ) Approxmaton based on allocaton assumpton Protecton level heurstc wth zero protecton Protecton level heurstc wth balanced protecton Protecton level heurstc wth maxmal protecton FCFS wth reallocaton allowed (s( ˆV ), S( ˆV )) by solvng DP from DP reserve no protecton stocks (equvalent to FCFS wthout reallocaton) reserve optmal unts of protecton stocks and then fulfll the demands wth the surplus reserve maxmum unts protecton stocks and then fulfll the demands wth the surplus adopt the polcy of AP adopt the polcy of AP (s( ˆV ), S( ˆV )) by solvng DP va smulaton va smulaton from DP The protecton-level heurstcs provde upper bounds on the cost of the orgnal problem. By comparng then wth the lower-bound obtaned from AP, we are able to benchmark the optmalty gaps. Numercal analyss s presented n 4.4, and s the bass for structural and manageral nsghts Approxmaton Based on the Allocaton Assumpton As explaned earler, we assume that unts that have been delvered to fulfll advance demands can be taken back and re-sent to other customers wthout ncurrng any costs or penaltes. Mathematcally, ths s equvalent to allowng the delvery of negatve unts aganst advance demands. Our assumpton parallels the allocaton assumpton made n the analyss of mult-echelon dstrbuton models. It s well known that the decomposton method by Clark and Scarf (1960) can be appled to seral and assembly systems, but not to dstrbuton systems, due to the addtonal decson on how to allocate nventory to multple downstream retalers optmally. Eppen and Schrage (1981) derve a closed-form optmal polcy for a dstrbuton system by makng what they call the allocaton assumpton. The dea s essentally relaxng the nonnegatvty constrants on allocaton varables,.e., negatve delvery s allowed. Then, the allocaton problem s straghtforward myopc allocaton to mnmze the expected cost n the current perod wthout consderng the future. Federgruen and Zpkn (1984) also make the same assumpton to solve allocaton problems n a smlar context. Özer (2003) studes a dstrbuton system wth ADI, and once agan relaxes the nonnegatvty constrant. To the best of our knowledge, the allocaton assumpton s stll the key to solve such problems, and t s beleved that n general t wll not hurt system performance sgnfcantly (Doğru et al. 2005). Gven the allocaton assumpton, the nventory level at the begnnng of perod + L + 1 wll be

19 Artcle submtted to Management Scence; manuscrpt no. MS R1 19 ( ) +L l+t + ( ) +L +L +L x +L+1 = y d j l y + ˆv j d j l. (14) l= where u and ˆV are defned n (7) and (11), and ˆv j 1 = v j 1 for j = L + 2,..., T. j=l The DP can be formulated smlarly as n 3.2, where the only dfference s that the state (u, ˆV ) evolves as follows: +T u +1 = y d j, (15) ˆv j +1 = ˆv j + d j, j = + L + 2,..., + T, (16) ˆv +T +1 = d +T. (17) l= j=l The sngle-perod loss functon EL(x +L+1 (y, ˆV )) s convex n y for any gven ˆV, so the optmalty of the state-dependent (s( ˆV ), S( ˆV )) polcy s preserved. The techncal detals can be found n the e-companon EC Protecton Level Heurstcs As dscussed above, we propose Protecton Level heurstcs where protecton stocks are kept aganst urgent demand. In partcular, we assgn protecton stocks between each adjacent par of upcomng demands d +1, d +2,..., d +T. To understand how ths works, consder the smplest case where T = 2, where a sngle protecton stock s suffcent. The allocaton polcy works as follows: Gven nventory s avalable, frst satsfy the demands due n the current perod (v + d ) and the next perod (v +1 + d +1 ) (nether of these demands wll be crossed over by future demands); then, f anythng remans, reserve some unts, σ, as safety stock n perod to protect from beng unable to satsfy d n the next perod; fnally use the surplus, f any, to fll the remanng non-urgent advance demands d +2. When protecton levels are used, the system states evolve n a much more complcated manner. In the followng, we demonstrate the case wth T = 2 and L = 0. For other cases where T > 2 and/or L > 0, the result stll follows, but the notaton becomes very cumbersome. There are x unts on-hand at the begnnng of perod, and z unts are ordered and arrve mmedately (snce L = 0), brngng the total avalable nventory to x + z. The observed advance

20 20 Artcle submtted to Management Scence; manuscrpt no. MS R1 demand profle s (v, v +1 ), and the demand vector arrvng n perod s D = (d, d +1, d +2 ). As the level of avalable nventory vares, there could be fve dfferent stuatons: (1) x + z v d 0. v + d s the amount due n perod, and cannot be fully satsfed from nventory. The unsatsfed quantty wll be backlogged, x +1 = x + z v d, and ncur backorderng penalty. Other components n the demand ppelne are unchanged, so v = v +1 + d +1 and v = d +2. (2) 0 x + z v d < v +1 + d +1. Now the nventory s enough to cover the current perod s demand, whle the surplus can all be used to satsfy part of the demand due n the next perod. So x +1 = 0, and v = (x + z v d v +1 d +1 ), the remanng demand of the next perod. v = d +2 agan. (3) 0 x + z v d v +1 d +1 < σ. The nventory level s hgh enough so that all the demand due n perod and +1 can be covered, but the surplus s less than σ, the protecton level. The surplus s carred to next perod and d +2 s not satsfed. So x +1 = x +z v d v +1 d +1, v = 0, and v = d +2. (4) 0 x + z v d v +1 d +1 σ < d +2. The nventory level s even hgher, and the surplus, after satsfyng demand n and + 1, s more than σ. Then only σ unts are carred to perod + 1, whle the remanng quantty, whch s less than d +2, s delvered to fulfll d +2 partally. So x +1 = σ, v = 0, and v = (x + z v d v +1 d +1 σ d +2 ). (5) x + z v v dj σ 0. The nventory level s so hgh that all the demand due n perods, + 1 and + 2 can be satsfed, and there are stll more than σ unts remanng for protecton from beng penalzed n perod + 1. So x +1 = x + z v v dj and v = v = 0. To summarze, we have the followng state evoluton equatons, where (x, v, v +1 ) s the system state; see Fgure 6 for a graphcal demonstraton. x + z v d f (1); 0 f (2); x +1 = x + z v d v +1 d +1 f (3); σ f (4); x + z v v dj f (5), (18)

21 Artcle submtted to Management Scence; manuscrpt no. MS R1 21 Fgure 6 State Evoluton of the Protecton Level Heurstc v+1 +2 x+1 v v +1 + d +1 v +1 + d +1 +σ v +1 + d +1 +d +2 v +1 + d +1 +σ+d +2 x + z - v -d (1) (2) (3) (4) (5) Note. Ths graph demonstrates the state varables n perod + 1 as functons of x + z v d. The horzontal axs can be segmented nto fve ntervals (1) to (5), correspondng to the fve cases dscussed above. x +1 s the pecewse lnear functon plotted wth a sold lne, and v and v+2 +1 are the two dashed lnes. (v +1 +1, v +2 +1) = (v +1 + d +1, d +2 ) f (1); ( (x + z v d v +1 d +1 ), d +2 ) f (2); (0, d +2 ) f (3); (0, (x + z v d v +1 d +1 σ d +2 )) f (4); (0, 0) f (5). Ideally, we would jontly optmze the orderng decson varable z and the allocaton decson varable σ, or at least determne the optmal orderng polcy for gven protecton levels. Unfortunately, we fnd that the prevous technque to reduce the dmensonalty and reformulate the DP wth state (u, ˆV ) no longer apples, unless demand cross-over s fully avoded (ths requres that σ to cover the whole support of d +1 +1). Ths suggests the followng heurstc that we call PL(Σ). Heurstc PL(Σ). In ths heurstc, we take the protecton level hgh enough to cover the whole support of the urgent demand (d +1 +1), or f the support s nfnte, large enough to make the probablty that d > σ arbtrarly small. As demand cross-over s avoded n ths manner, t can be shown that the optmal orderng polcy s stll a modfed state-dependent (s(ˆv), S(ˆv)) polcy, f the demand probablty densty/mass functons are strongly unmodal 1 (or equvalently, log-concave). The formulaton and proof of ths result can be found n the e-companon. (19) 1 Most commonly used dstrbutons (eg. unform, normal, Posson, bnomal) are strongly unmodal, see Dharmadhkar and Joag-dev (1988) for more detals.

22 22 Artcle submtted to Management Scence; manuscrpt no. MS R1 Heurstc PL(0). Wth zero protecton stock, we recover the myopc allocaton polcy (wthout re-allocaton). For the orderng polcy, we use the one obtaned n AP. Federgruen and Zpkn (1984) show that the polcy obtaned ((s, S) orderng polcy and myopc allocaton polcy) n ther approxmaton model for dstrbuton systems s near-optmal. In contrast, myopc allocaton, whch mnmzes nventory holdng cost but omts possble future shortage cost, could potentally be far from optmal n our model. The cost of ths heurstc s obtaned va smulaton. Heurstc PL(σ). Clearly, PL(Σ) and PL(0) are the two extremes: the former heurstc avods future shortage costs wthout consderng the holdng costs mposed by the protecton stocks, whle the latter mnmzes the holdng costs but omts the shortage costs. These polces may perform well under some extreme settngs (such as very low holdng or penalty cost), whle for others, a properly chosen protecton level that balances the two costs would be preferable. Ths motvates the PL(σ) allocaton polcy, where protecton level σ s chosen to mnmze a specfc newsvendor-lke objectve functon H(σ ) = h σ + p E[(d σ ) + ]. (20) Ths s based on the observaton that the protecton level affects cost only when the nventory level s n regon (4) of Fgure 6. In that case, ncreasng σ by one unt ncurs one unt of holdng cost for sure, but ncurs penalty cost only f n the next perod, the protecton stock plus the arrvng replenshment z +1 s not enough to cover the urgent demand d Here z +1 s dffcult to estmate or predct, so we conservatvely take t as zero to obtan (20). Essentally, the PL(0) heurstc mnmzes the frst term of H( ) by settng zero protecton levels, PL(Σ) mnmzes the second term by settng protecton levels large enough to cover d +1 +1, and PL(σ) strkes a balance between the two parts by settng σ to mnmze H( ). Equaton (20) s presented for the T = 2 case. When T > 2, more than one protecton level are needed. The protecton levels can be defned smlarly. For example, when T = 3, we need protecton stock, σ +1, coverng future demand d and protecton stock, σ +2, coverng d

23 Artcle submtted to Management Scence; manuscrpt no. MS R1 23 σ +1 can stll be defned as the mnmzer of (20), whle σ +2 h σ +2 + p E[(d d σ +1 σ +2 ) + ]. can be defned as the mnmzer of Note that the functon H( ) s just one of many that could capture the trade-off between the holdng cost due to keepng protecton stocks and the penalty cost due to not beng able to satsfy d H( ) can be further refned, for example, by ncorporatng the effect of the fxed orderng cost K (hence ndrectly capturng the effect of z +1 ), nventory level x, or even the whole system state. Nevertheless, (20) s smple, t provdes a statonary protecton level that s easy to mplement, and t captures the most crtcal trade-off between the holdng cost and the penalty cost. As mentoned before, snce the allocaton polcy PL(σ) does not rule out cross-over, t s dffcult, f not mpossble, to solve for the optmal orderng polcy and estmate ts cost analytcally. Instead, we use the orderng polcy obtaned n the approxmaton model, based on the belef that the optmal orderng polces are relatvely nsenstve to dfferent allocaton polces adopted, and calculate the cost of PL(σ) va smulaton. The robustness of the orderng polcy to the allocaton rules and the performance of the heurstcs are tested n the next subsecton Structural Results and Manageral Insghts To quantfy the mpact of delvery flexblty, and to evaluate the performance of the proposed heurstcs, we start by mmckng an experment n Gallego and Özer (2001), found n Table 3 of ther paper. All the cost parameters reman the same as the prevous numercal analyss,.e., K = 100, h = 1, p = 9, and we agan focus on the case where L = 0 and T = 2. The plannng horzon s 12 perods. The dscount factor s α = 1. Dfferng from the homogeneous model, now the demand s a 3-dmensonal vector of Posson random varables wth mean (λ 0, λ 1, λ 2 ). We follow the settng n Gallego and Özer (2001), where the total demand rate s constant, wth λ 0 + λ 1 + λ 2 = 6, and the demand scenaro (λ 0, λ 1, λ 2 ) vares from (5, 1, 0) to (4, 1, 1),..., to (0, 1, 5) (labeled Expr 1 to 6). In ths manner, dfferent degrees of advance demand nformaton avalablty can be modeled, and ther benefts can be measured. One extreme case s (6, 0, 0) (labeled Expr 0), whch can be regarded as the tradtonal case where no advance demand nformaton s avalable at all. The other

24 24 Artcle submtted to Management Scence; manuscrpt no. MS R1 s (0, 0, 6) (labeled Expr 7), whch s exactly the homogeneous case we consdered n the prevous secton. Propertes of the Orderng Polces. We frst calculate the optmal orderng polces for the approxmaton (AP for short) and the PL(Σ) heurstc, whch can be analyzed by solvng the correspondng dynamc programs. Snce PL(Σ) requres a large enough protecton level, whle Posson demand s unbounded, we set Σ to be such that P {d > Σ} < Table 2 reports the protecton levels used n PL(Σ), as well as PL(0) and PL(σ) for future reference. Table 2 Protecton Levels Expr. Demand Protecton Level No. λ 0 λ 1 λ 2 PL(0) PL(σ) PL(Σ) L = 0, T = 2, K = 100, h = 1, p = 9, N = 12 Table 3 shows the perod-1 orderng polcy for AP n all sx demand scenaros (Expr 1 to 6). Optmal orderng polces for Expr 0 and 7 are also reported for ease of comparson. It s strkng that when we compute the optmal polces of heurstc PL(Σ), we fnd that they are exactly the same as those n Table 3. Ths provdes some support for our conjecture that the optmal orderng polces are nsenstve to the allocaton polcy. Once agan, we observe that S(ˆv) s ndependent of ˆv and s(ˆv) s decreasng n ˆv. We also observe that both the order-up-to level and the re-order pont decrease as advance demand nformaton becomes avalable earler (as we progress from Expr 0 to 7). Performance of the Heurstcs. For the same parameter set as above, we now compare sx models for a range of advance demand nformaton scenaros (Expr 1-6): (1) tradtonal nventory model wthout ADI, (2) nventory model wth ADI, (3) the approxmaton of ADI wth flexble delvery ( ADI-F AP ), (4) PL(0) heurstc, (5) PL(σ) heurstc, and (6) PL(Σ) heurstc. Among