We analyze the problem faced by companies that rely on TL (Truckload) and LTL (Less

Transcription

1 Effecve Zero-Invenory-Orderng Polces for he Sngle-Warehouse Mulrealer Problem wh Pecewse Lnear Cos Srucures LapMu Ann Chan Ana Murel Zuo-Jun Max Shen Davd Smch-Lev Chung-Paw Teo School of Managemen, Unversy of Torono, Torono, Onaro, Canada Deparmen of Mechancal and Indusral Engneerng, Unversy of Massachuses, Amhers, Massachuses Deparmen of Indusral and Sysems Engneerng, Unversy of Florda, Ganesvlle, Florda The Engneerng Sysems Dvson and he Deparmen of Cvl and Envronmenal Engneerng, Massachuses Insue of Technology, Cambrdge, Massachuses Deparmen of Decson Scences, Naonal Unversy of Sngapore, Sngapore We analyze he problem faced by companes ha rely on TL (Truckload) and LTL (Less han Truckload) carrers for he dsrbuon of producs across her supply chan. Our goal s o desgn smple nvenory polces and ransporaon sraeges o sasfy mevaryng demands over a fne horzon, whle mnmzng sysemwde cos by akng advanage of quany dscouns n he ransporaon cos srucures. For hs purpose, we sudy he cos effecveness of resrcng he nvenory polces o he class of zero-nvenory-orderng (ZIO) polces n a sngle-warehouse mulrealer scenaro n whch he warehouse serves as a cross-dock facly. In parcular, we demonsrae ha here exss a ZIO nvenory polcy whose oal nvenory and ransporaon cos s no more han 4/ (5.6/4.6 f ransporaon coss are saonary) mes he opmal cos. However, fndng he bes ZIO polcy s an NPhard problem as well. Thus, we propose wo algorhms o fnd an effecve ZIO polcy: An exac algorhm whose runnng me s polynomal for any fxed number of realers, and a lnear-programmng-based heursc whose effecveness s demonsraed n a seres of compuaonal expermens. Fnally, we exend he wors-case resuls developed n hs paper o sysems n whch he warehouse does hold nvenory. (Approxmaon Algorhms; Zero-Invenory-Orderng Polces; Invenory; Transporaon; Truckload (TL); Less han ruckload (LTL)). Inroducon In recen years, many companes have realzed ha mporan cos savngs can be acheved by negrang nvenory conrol and ransporaon polces hroughou her supply chans. Thus, he problem faced by hese companes s o fnd an opmal replenshmen plan,.e., an nvenory and ransporaon sraegy, so as o mnmze oal nvenory and ransporaon coss over a fne plannng horzon. The dffculy n desgnng a coordnaed sraegy, however, s compounded by he fac ha ypcally hese companes rely on exernal hrd-pary logscs provders for he Managemen Scence 2002 INFORMS Vol. 48, No., November 2002 pp /02/48/446$ elecronc ISSN

2 Zero-Invenory-Orderng Polces ransporaon of goods from supplers hrough warehouses o realers. These problems are dfferen from radonal nework flow problems, as he ransporaon cos srucure, also referred o as orderng cos, offered by he carrers s usually pecewse lnear bu no necessarly convex. Ths cos srucure, represenng quany dscouns, volume-based prce ncenves, and oher forms of economes of scale, has a major mpac on he replenshmen sraegy. I usually reflecs eher ncremenal or all-un dscoun effecs, leadng o he followng ypes of cos funcons. Fgure Cos Modfed All-Un Dscoun Cos Srucure α α α 2 Incremenal Dscoun Cos Srucures. Ths class can be fully characerzed by pecewse lnear concave cos funcons. Of course, a specal case of hs cos funcon s he fxed-charge funcon, where a fxed cos, ndependen of he shpmen sze, s ncurred whenever here s a shpmen. All-Un Dscoun Cos Srucures. Such a cos funcon mples ha f he facly orders Q uns, he ransporaon cos funcon s 0 f Q =0, c f 0<Q<M, G Q = Q f M Q<M 2, Q f M Q<M + =2 n, n Q f M n Q, where > 2 > 0, and M = c. Thus, c s a mnmum charge for shppng a small volume,.e., c s he oal cos when he number of uns shpped s no more han M. Ineresngly, n pracce, when he shpper s plannng o shp Q uns, M Q<M +, he cos s calculaed as F Q = mn G Q G M + = mn Q + M + Tha s, f he order quany s greaer han a ceran value, he shppers pay as f hey were shppng M + uns. Ths s called n he ndusry shppng Q bu declarng M +. Ths commonly used pracce mples ha he rue ransporaon cos funcon, F, has he srucure descrbed by he heavy sold lne n Fgure. As he dashed lnes ndcae, he assocaed sold lnes orgnae a pon 0 0. We refer o such cos funcons as modfed all-un dscoun cos funcons. Noce ha hese cos funcons sasfy he followng properes: M0 M M M2 M2 M Quany () hey are nondecreasng funcons of he amoun shpped, and () he cos per un s nonncreasng n he amoun shpped. Indeed, hese wo properes are necessary and suffcen o derve our resuls. These cos funcons are common n ndusry pracce. Mos LTL carrers use an ndusry sandard ransporaon-rang engne called CZAR (Souhern Moor Carrer s Complee Zp Audng and Rang engne). Ths engne allows he shpper o fnd he ransporaon cos of every shpmen whch s a funcon of he source, desnaon, produc class, and dscoun. The carrer and he shpper conracually agree on he produc class (ypcally class 00) and on he level of dscoun, whch mples ha he shpper wll pay only a gven fracon, say 90%, of he cos generaed by he rang engne. Now, gven hs npu, he ransporaon cos as a funcon of he amoun shpped enjoys he all-un dscoun srucure descrbed above. In hs paper, we sudy a class of mulperod dsrbuon problems wh ransporaon cos srucures ha model boh he ncremenal and all-un dscoun cos funcons. Specfcally, we consder a classcal nvenory dsrbuon model n whch a sngle warehouse receves nvenory from a sngle suppler and replenshes he nvenory of n realers. Each realer provdes he warehouse wh forecas demand for he nex T me perods. We assume ha shorages and backloggng are no allowed eher a he warehouse or a he realers. Managemen Scence/Vol. 48, No., November

3 Zero-Invenory-Orderng Polces Furhermore, we assume ha he warehouse uses a common logsc sraegy, referred o as cross-dockng, n whch he warehouse acs merely as a coordnaor of he supply process and as a ransshpmen pon for ncomng orders from he suppler, bu does no hold any socks. In hese suaons, (large) shpmens from he suppler o he warehouse are ofen delvered by TL carrers whose coss can be approxmaed by pecewse lnear concave funcons (for example, a fxed-charge cos funcon). Economes of scale n producon can also be modeled n hs way. Henceforh, we wll assume ha he orderng cos funcon from he suppler o he warehouse s of he ncremenal dscoun ype. By conras, snce shpmen szes from he warehouse o a realer are relavely small, hese shpmens are ypcally delvered by LTL carrers whose coss follow he modfed all-un dscoun cos srucure. The objecve s o fnd an opmal shpmen plan ha explos he quany dscoun effec and, a he same me, conrols he nvenory holdng cos a he realers end. Observe ha he sngle-warehouse mulrealer problem descrbed here can also be used o model he jon replenshmen problem; see Joneja (990). In hs problem, a sngle facly replenshes a se of ems over a fne horzon. Whenever he facly places an order for a subse of he ems, wo ypes of coss are ncurred: a jon se-upcos and an em-dependen se-upcos. The objecve n he jon replenshmen problem s o decde when and how many uns o order for each em so as o mnmze nvenoryholdng and orderng coss over he plannng horzon. Because he jon replenshmen problem s NP-hard (see Arkn e al. 989), he sngle-warehouse mulrealer problem s also NP-hard even f all ransporaon cos funcons are fxed-charge cos funcons. Evdenly, he fxed-charge cos funcon s a specal case of he all-un dscoun cos funcon consdered n hs paper. Ths mples ha he problem analyzed n hs paper s NP-hard n general. An neresng queson s wheher s NP-hard for a sngle or fxed number of realers. Ths queson was answered by Chan e al. (2002), who show ha a specal case of our problem, n whch a sngle realer s replenshed by a sngle warehouse wh zero ransporaon cos for shpmens o he warehouse and modfed allun dscoun ransporaon coss for shpmens o he realer, s NP-hard. Thus, he sngle-warehouse mulrealer problem descrbed above s NP-hard even for a fxed number of realers. Our focus n hs paper s on a class of polces referred o as zero-nvenory-orderng (ZIO) polces, n whch orders are placed by he realers only when her nvenory levels dropo zero. I s easy o see ha n he case of concave ransporaon cos funcons here exss an opmal ZIO polcy for he snglewarehouse mulrealer problem. Unforunaely, he resul of Arkn e al. (989) mples ha fndng he bes ZIO s n self NP-hard. Of course, our model s more general and, as shown by Chan e al. (2002) for he sngle-warehouse sngle-realer case, he opmal polcy may no be a ZIO polcy. Ths paper bulds upon resuls developed n Chan e al. (999), whch consders more general producon-dsrbuon problems wh pecewse lnear concave coss, and n Chan e al. (2002), whch sudes a smpler dynamc lo-szng problem wh a sngle realer and modfed all-un dscouns. In we exend he resuls n Chan e al. (2002) o he onewarehouse mulrealer case and show ha () here s a ZIO polcy whose assocaed cos s no more han 4/ mes he cos of he opmum replenshmen plan, and () f he orderng cos funcon does no vary over me, hen he cos of he opmal ZIO polcy s no more han 5 6/4 6 mes he opmal cos. Ths leads o he developmen of effcen algorhms. Secon 4 descrbes an exac algorhm ha fnds he bes ZIO polcy and whose runnng me s polynomal for a fxed number of realers. The second algorhm, descrbed n 5, s based on formulang he problem of fndng he bes ZIO polcy as an neger program. Ths neger program has a smlar srucure o he one developed n Chan e al. (999) for he concave case. In fac, we fnd ha he properes of he lnear programmng relaxaon derved n ha paper can be appled o our new neger programmng formulaon as well. The lnear programmng relaxaon of hs model s consequenly solved and s soluon and srucural properes are used o generae a ZIO polcy. An emprcal sudy shows (see 6) ha he heursc algorhm s compuaonally effcen and generaes soluons very close o he opmal ZIO polcy. 448 Managemen Scence/Vol. 48, No., November 2002

4 Zero-Invenory-Orderng Polces Fnally, n 7 we pon ou exensons of he worscase resuls o more general cos srucures and o he case of a radonal nvenory dsrbuon sysem n whch he warehouse may hold nvenory. 2. Noaon and Man Resuls Le n be he number of realers served by he warehouse and T be he lengh of he plannng horzon under consderaon. For each = 2 T,wele K 0 Q0 be he pecewse lnear concave ransporaon cos funcon assocaed wh shppng a quany Q 0 from he suppler o he warehouse a me. Smlarly, for each = 2 n and = 2 T, we denoe by K Q he modfed all-un dscoun ransporaon cos funcon assocaed wh shppng a quany Q from he warehouse o realer a me. Fnally, for each = 2 n and = 2 T, le I denoe he nvenory a realer a he end of perod, h he cos of holdng an em a realer a he end of perod, and d he demand of realer a me. Our objecve s o fnd he sze and mng of shpmens so as o mnmze oal ransporaon and nvenory coss whle sasfyng all demands whou shorages. In wha follows, we wll refer o hs problem as he sngle-warehouse mulrealer problem (SWMR): Problem SWMR s.. Q 0 = n Mn Q = T = [ K 0 Q0 + n K Q + h I = = 2 T Q + I = d + I = n = 2 T I 0 = 0 = 2 n Q 0 I = 0 n = 2 T 0 = n = 2 T () where whou loss of generaly we assume ha nal nvenory s 0; ha s, I0 = 0, for all = 2 n. Le Z be he cos of he opmal soluon o he snglewarehouse mulrealer problem and for any heursc H, le Z H be he cos of he soluon generaed by heursc H. We frs show ha unless P = NP, s no possble o developan algorhm ha runs n polynomal me and generaes, for any nsance of he problem, a soluon whch s whn a facor of O log n from opmaly. Theorem 2.. Suppose here exss a >0 and a polynomal me heursc, H, for he sngle-warehouse mulrealer problem such ha for all nsances hen P = NP Z H Z log n Proof. The proof s based on showng ha he se-coverng problem can be reduced o he snglewarehouse mulrealer problem. I s well known (see Fege 998 or Arora and Sudan 997) ha here s no polynomal me algorhm for he se-coverng problem wh wors-case bound beer han log n, for >0, unless P = NP. Consder an nsance of he se-coverng problem: mn m = x Ax, where A = a s a n m 0- marx. I can be reduced o he sngle-warehouse mulrealer problem wh n realers and m + perods as follows. Le K x = { M x f a = 0 0 f a = K m+ x = M x K 0 x = x { 0 d = for all and = 2 m for all for = 2 m m+ f = 2 m f = m + h = 0 for all for all where M s some large number no less han m, and x = when x>0, and 0 oherwse. The hgh se-upcos a me m + forces realers o order n earler perods. In addon, an order for realer s placed a me only f a = because here s a large fxed cos assocaed wh shpmens Managemen Scence/Vol. 48, No., November

5 Zero-Invenory-Orderng Polces n perods n whch a = 0. Thus, fndng he bes nvenory orderng polcy n hs suaon s equvalen o fndng he mnmum number of orderng perods, whch s deermned by cluserng realers ha wll be served ogeher a a ceran me. Thus, n he remander of hs paper we focus on he analyss of a class of polces referred o as zeronvenory-orderng (ZIO) polces. In hs class, orders are placed only a mes when on-hand nvenory has been fully depleed; ha s, Q I = 0. Le ZZIO be he cos assocaed wh he opmal ZIO polcy. Of course, he opmal polcy for he sngle-warehouse mulrealer problem may no be a ZIO polcy. However, n we show he followng wo heorems. Theorem 2.2. For every nsance of he snglewarehouse mulrealer problem, Z ZIO 4 Z and hs bound s gh. In pracce, he orderng cos funcon does no vary from perod o perod;.e., for all, K 0 = K0 and K = K, = 2 n. In hs case, we show ha he wors-case rao of he cos of he bes ZIO polcy o he opmal cos s no more han 5 6/ Tha s, Theorem 2.. For every nsance of he snglewarehouse mulrealer problem n whch he ransporaon cos funcons are saonary, Z ZIO Z The opmal ZIO polcy can be found n polynomal me for any fxed number of realers, usng he algorhm presened n 4. No surprsngly, however, he compuaonal complexy of hs mehod grows exponenally as he number of realers ncreases. To overcome hs problem, n 5 we propose a lnearprogrammng-based heursc ha runs n polynomal me. Ths algorhm s shown o be very effcen n our compuaonal sudy.. The Effecveness of Zero-Invenory-Orderng Polces In hs secon we analyze he effecveness of ZIO polces and prove Theorems 2.2 and 2.. We sar by showng some srucural properes of feasble soluons o he sngle-warehouse mulrealer problem. Le S be a feasble replenshmen plan for he sysem. Le m be he number of shpmens from he suppler o he warehouse and T 0 = = 2 m be he me epochs n whch hese shpmens occur. Smlarly, le m be he number of orders placed by realer, = 2 n, and T = = 2 m T 0 be he me epochs n whch goods are delvered from he warehouse o realer. Le Q denoe he sze of he shpmen o he realer a me perod and Q 0 = n = Q be he quany requred a he warehouse. Obvously, Q = 0 for all T, = 0 2 n. Observe ha all hese parameers are assocaed wh he nvenory polcy S under consderaon. We om he reference o S n our noaon because s clear wha polcy hey are referrng o a any pon. However, we add he superscrp * o he parameers correspondng o an opmal polcy, S. Consder he nvenory level a realer. Whou loss of generaly, we can assume ha n any replenshmen plan, he shpmen ha arrved a me l+ wll only be used afer he depleon of he shpmen ha arrved a me l. Tha s, whou loss of generaly we assume ha orders are used o sasfy demand n a frs-n-frs-ou bass. Thus, le s l, = 2 n and l = 2 m, s l l, be he earles (or frs) me a shpmen ha arrves n me l s used o sasfy cusomer demand. Observe ha n he opmal sraegy, a shpmen ha arrves a me l may only be needed a me s l (s l > l ). The early shpmen may be due o he need o explo he characerscs of he ransporaon cos funcon o he warehouse or he realer. Tha s, he early delvery may ake advanage of changes n he cos funcon as a funcon of me, and/or he effec of he ransporaon dscoun on he order quany. Consder realer and le H k denoe he cos of holdng an em a realer from me k o he begnnng of perod k+. Tha s, H k h + h + + +h + In addon, le K 0 Q 0 K 0 Q 0 Q + K Q A k Q k 450 Managemen Scence/Vol. 48, No., November 2002

6 Zero-Invenory-Orderng Polces Observaon.. A k represens he orderng cos per un assocaed wh shppng he quany Q o realer a perod k, and s also an upper bound on he un cos assocaed wh orderng addonal uns n ha perod for realer. Ths s due o he concavy of K 0 k and he fac ha cos per un resulng from he LTL charges s nonncreasng wh volume. Lemma.2. Gven any feasble polcy S, here exss a feasble polcy wh lower or equal cos n whch a posve fracon of any order s used o sasfy demand for perods pror o he arrval of he subsequen order. Tha s, k+ > s k k for all = 2 n and k = 2 m. Proof. Suppose ha he curren polcy S does no sasfy hs propery;.e., here exss an ndex k such ha k+ s k. Because boh he orders a perods k and k+ cover demand occurrng on or afer perod k+, he wo orders can be combned and sen n he same perod, eher k or k+. The oal coss assocaed wh orderng he combned quany and holdng he uns n nvenory unl perod k+ s no more han Q + Q mn A + k + H k A k+ Q A k + H k + Q +A k+ whch s he cos assocaed wh orderng hose quanes and holdng he uns n nvenory unl perod k+ n he curren polcy S. Snce all oher coss reman he same when combnng he orders, he above argumen shows ha we can always oban a polcy wh lower or equal cos sasfyng he propery. Gven a polcy S sasfyng he condon n Lemma.2, he order placed by realer n any perod j can be wren as Q j = j Q + j j Q j where j Q 0 < j j denoes he poron of he jh shpmen ha s used o sasfy demands from some me s j < j+ unl he arrval of he j + h shpmen. Ths s used n provng Theorems 2.2 and 2., whch are he subjecs of he subsequen secons... Tme-Varyng Orderng Cos Funcons In wha follows we show ha gven an opmal polcy of he one-warehouse mulrealer problem we can consruc a ZIO polcy whose cos s no more han 4/ mes he cos of he orgnal soluon. For hs purpose, we frs show ha we can focus on a subse of feasble polces. Lemma.. Gven any feasble polcy S, here exss a feasble polcy wh lower or equal cos sasfyng he followng properes.. A fracon of any order s used o sasfy demand for perods pror o he arrval of he subsequen order. Tha s, k+ >s k k for all = 2 n and k = 2 m. 2. If he order a k covers demands occurrng afer he nex order has arrved a perod k+, hen he cos per un assocaed wh orderng and holdng hose uns n nvenory n he earler perod s hgher. Tha s, A k +H k > A k+. Proof. The frs propery s ha n Lemma.2. If he second propery does no hold, hen followng he same reasonng as n Lemma.2, he wo orders can be consoldaed and delvered n perod k whou ncreasng oal coss. For = 2 n and j = 2 m, le K 0 n C j = j x= Qx K 0 n j j x=+ Qx + K Q j j j In wha follows, we allocae an orderng cos of C j o each un of he jh order of realer. Usng hs noaon and addng a superscrp o denoe values a opmaly, he oal cos assocaed wh an opmal polcy S, sasfyng he condons n Lemma., can be wren as: Z = n = [ C Q + C j m j=2 Q j + C j + H j j Q j j Q j + H (2) where H = oal holdng cos of polcy S mnus he cos of carryng, for each = 2 n and j = m, he poron of demand delvered n perod j bu used only afer j+ from perod j o j+. Tha s, H = (oal holdng cos of polcy S n m = j= H j. Managemen Scence/Vol. 48, No., November

7 Zero-Invenory-Orderng Polces We now ransform he opmal polcy S no a ZIO polcy S whose cos s whn 4/ of he opmal cos (2). For ha purpose, we apply he followng procedure. Transformaon Procedure. Sep 0. Le S = S. Sep. Fnd he smalles ndex of a realer, say realer, ha does no sasfy he ZIO polcy and he smalles ndex, k, such ha k < ; ha s, k s he earles perod n whch realer places an order before nvenory has been fully depleed. Sep 2. Eher, Combne : Move k Q from he order a perod k o ha a perod k,or Combne 2: Move k Q from he order a perod k o ha a perod k and k Q from he order a perod k o ha a perod k+, whchever resuls n a lower cos. Sep. If necessary, combne orders whou ncreasng oal cos unl he curren polcy sasfes he condons n Lemma.. Sep 4. Repea Seps o unl all realers sasfy he ZIO propery. Noe ha hs requres a mos nt eraons because he soluon generaed afer each eraon ensures ha one more orderng perod a a realer sasfes he ZIO propery. See Fgures 2,, and 4 for llusraon. Observe ha before he execuon of Sep2 for a ceran ndex k and realer, he oal order quan- Fgure Polcy Obaned f Combne Is Performed ( αk ) Q k d d2 d d4 5 d6 k Q k + d d2 d d4 d 5 d6 d Q k + k Noe. The op depcs he combne operaon: Add he srped par of he order a perod o ha a perod k k. The boom fgure represens he new polcy obaned, wh demand sasfed by each order n a dfferen shade of gray. y a any perod k n he curren polcy S sasfes Q 0 n x= Q x. Ths s rue because all he orders of realers + hrough n, and hose of a perods greaer han k, have no been modfed and he order of realer a perod k may have been ncreased n he prevous eraon. Noe ha f combne 2 s performed for a prevous ndex l, he order a perod l s reduced bu l+ wll no be consdered n Sep of he subsequen sages as he nvenory (ordered a l ) would have been fully depleed exacly a perod l+. Furhermore, even hough he order a perod l+ may have been ncreased, he poron ha wll be used afer perod Fgure 2 Inal Polcy Fgure 4 Polcy Obaned f Combne 2 Is Performed Q Q Q k k k + d d2 d d4 d5 d6 (α k )Q k ( αk )Q k k + d d2 d d4 d5 d6 k + sk s k sk+ Noe. Each shaded block represens he demand (here denoed as d d 2 ) faced by realer a subsequen me perods, from perod k onwards. The sze of he block depcs he magnude of he correspondng demand. The curren polcy S s supermposed n he pcure by ponng ou he perods, k k k+, a whch orders are placed and he parcular demands ha each of he ordered quanes (Q Q Q ) wll cover. k k k+ d d2 d d4 d5 d6 k Noe. The op depcs he combne operaon: Add he vercally srped par of he order a perod k o ha a perod k and he horzonally srped par of he order a perod k o ha a perod k+. The boom fgure represens he new polcy obaned, wh demand sasfed by each order n a dfferen shade of gray. 452 Managemen Scence/Vol. 48, No., November 2002

8 Zero-Invenory-Orderng Polces l+2 (.e., l+ Q remans unchanged under he l+) modfed polcy. Noe ha a each eraon of he ransformaon procedure no new orderng perods are added. Thus, we can consder whou loss of generaly ha he se of orderng perods k remans he same over he eraons;.e., k = k for k = 2 m and = 2 n, even f he order quany n some of hose perods becomes 0 afer Seps 2 and are execued. The followng wo lemmas demonsrae ha a each eraon of Sep2 for a parcular ndex k and realer, he ncrease n cos accrued s no more han onehrd of he correspondng kh erm assocaed wh he realer under consderaon n he expresson of he opmal cos (Equaon (2)), [ C k + H k k Q + C k k Q Ths proves ha he cos of he ZIO polcy generaed by he ransformaon procedure s no more han 4 Z, snce he ndex k s srcly ncreasng n he number of eraons performed and he sum of hose erms for all k and all s no larger han he opmal value, Z. Thus, he cos of he opmal ZIO polcy sasfes he 4 Z bound, as saed n Theorem 2.2. and Lemma.4. Le A x = K0 Q 0 K 0 Q 0 Q + K Q x x x x x x x Q x x+ H x = h = x The ncrease n cos a he combne sep for any realer and ndex k s no more han. A k k Q f combne s execued, 2. A k + H k A k k Q f combne 2 s execued. Proof.. The cos ncrease s no more han A k k Q f combne s execued. Ths s rue because A k s an upper bound on he cos per un assocaed wh exra uns sen a perod k ; see Observaon.. 2. The cos ncrease s no more han A k +H k A k k Q f combne 2 s execued. We prove hs n hree seps. () By reducng he quany ordered a perod k by Q orderng coss n ha perod are reduced by, exacly A k Q by defnon of A, k. () Movng k Q from perod k o k ncreases orderng coss and nvenory coss a perod k by a mos A k + H k k Q. () Movng k Q from perod k o k+ resuls n an ncrease n cos a perod k+ of a mos A k+ k Q A k +H k k Q where he, nequaly s rue because he soluon a any eraon sasfes he second propery n Lemma., and a reducon n nvenory coss of H k k Q. Lemma.5. A each eraon of he combnng sep for a ceran ndex k and realer, he ncrease n cos s no more han [ C k + H k k Q + C k k Q Proof. The proof of hs resul uses he followng key resul n Chan e al. (2002, see proof of Lemma 4.): mn A B + A + B for all A B 0 () The ncrease n cos n he combne sepassocaed wh he kh order of realer s no more han { } mn A k k Q A k + H k A k k Q [ A k + H k k Q + A k k Q where he nequaly s a drec consequence of (). Now, because K 0 s concave and a each eraon x Q 0 n x y= Qy for x k, we have ha x ( ) ( ) n n K 0 Q 0 K 0 Q 0 Q K 0 K 0 x x x x x x x Q y x y= Therefore, for all x k, A x K0 Q 0 K 0 Q 0 Q + K Q x x x x x x x Q x Q y x y=+ K0 n x j= Qj K x 0 n x j=+ Qj + K x x Q x C Q x x Managemen Scence/Vol. 48, No., November

9 Zero-Invenory-Orderng Polces Now observe ha Cx C x for all x k because x = x, Q x = Q x for x>k, and Q k Q k. Thus, he ncrease n cos s bounded by [ C k + H k k Q + C k k Q = [ C k + H k k Q + C k k Q The las equaly s due o he fac ha he curren soluon S, before he eraon s performed, sasfes s x = s x, x = x, x = x, Q x k Q k = k Q k. = Q x for x k, and I remans o show ha he bound s gh. Chan e al. (2002) show ha here exs nsances of he onewarehouse mulrealer problem wh a sngle realer for whch he rao Z ZIO /Z s arbrarly close o 4/. Thus, he 4/ bound canno be mproved n he case of mulple realers. Observe ha he only properes of he modfed all-un dscoun funcon used n he proof of Theorem 2.2 are ha s nondecreasng n he quany shpped and ha he cos per un s nonncreasng n ha quany. Hence, he heorem holds rue for any LTL ransporaon funcon sasfyng hose properes. In a smlar way, holdng coss can be generalzed o be any funcon of he quany held ha sasfes hose wo properes..2. Saonary Orderng Cos Funcons In hs case, we need o consder soluons sasfyng condons slghly dfferen han hose n he prevous secon. For hs purpose, le and K 0 Q = lm 0 + K 0 Q K 0 Q K Q B k = K0 Q 0 + k Q k for each realer, = 2 n and orderng ndex k = 2 m. Noe ha we have dropped he me subndex n he ransporaon cos funcons because hey are consan over me. Tha s, K 0 = K0 and K = K for all and. Observaon.6. K 0 Q 0 k s an upper bound on he per-un cos ha addonal uns o be delvered o he warehouse a me k would ncur and, a he same me, a lower bound on he cos per un ncurred by he curren Q 0 uns beng sen n ha perod. Ths s due o he concavy of he warehouse orderng cos funcon, K 0. Observaon.7. Smlarly, B k s an upper bound on he per-un ransporaon cos ha addonal uns delvered o he realer a me k would ncur, and, a he same me, a lower bound on he cos per un ncurred by he curren Q 0 uns beng sen n ha perod. Ths s explaned by he prevous observaon and he fac ha he cos per un resulng from he LTL charges does no ncrease as he shpmen becomes larger. We are now ready o nroduce he counerpar of Lemma. for he saonary case. Lemma.8. Gven any feasble polcy S, here exss a feasble polcy wh lower or equal cos sasfyng he followng properes.. A posve fracon of any order s used o sasfy demand for perods prevous o he arrval of he subsequen order. Tha s, k+ >s k k for all = 2 n and k = 2 m. 2. If he order a k covers demands occurrng afer he nex order a perod k+ has arrved, hen B k + H k >B k+. Proof. The proof s smlar o ha of Lemma.. Please refer o he echncal appendx for deals ( mansc.pubs.nforms.org/ecompanon.hml ). As n he prevous secon, we ransform an opmal polcy whch sasfes he properes n Lemma.8 no a ZIO polcy and show ha he ncrease n cos due o he ransformaon s no more han /4 6 mes he cos of he opmal soluon. However, hs case requres furher deal o show ha he gher bound holds. In parcular, one more way of combnng orders needs o be nroduced: Combne : Move Q uns from perod k o perod k, and Sep2 of he ransformaon procedure s replaced by he Sep 2 descrbed below. Le K Q K H k + K0 Q 0 + k H k + B k Q k K Q K 2 K 0 Q 0 + k B k Q k 454 Managemen Scence/Vol. 48, No., November 2002

10 Zero-Invenory-Orderng Polces H H k + K0 Q 0 K 0 Q 0 D k Q + k Q k Q D k Q D Proof. Please refer o he echncal appendx for deals. Lemma.0. If he curren soluon S sasfes (4), hen connues o hold afer Sep 2 has been execued. Proof. Ths s proven separaely for each of he hree possble combne seps. Please refer o he echncal appendx for deals. Sep 2. Combnng sep k a realer. If 4.6 K 2 H K + K 2, execue combne. Else f 4.6 K K 2 K + K 2 + K 2, execue combne 2. Else execue combne. Thus, n he remander of hs secon we consder a ransformaon procedure dencal o ha n. excep ha Sep 2 s replaced by Sep 2 and he soluon consdered a each eraon sasfes he condons n Lemma.8 raher han hose n Lemma.. For any nvenory orderng polcy S, le Z S be he sysemwde cos assocaed wh polcy S and Z S k be he oal cos assocaed wh sasfyng he demands of realer a perods k hrough T and all he demands (from perod o T ) of realers + hrough n. Le k be he ndex of he earles orderng perod a realer n whch an order s placed before all nvenory from he prevous order has been depleed. Observe ha o prove Theorem 2., suffces o show ha a any eraon of he ransformaon procedure he curren on-hand soluon S sasfes Z S Z S k Z Z S k (4) Obvously, hs condon holds for S = S. The followng lemmas show ha f he curren on-hand soluon sasfes (4), hen also holds afer execung Sep 2 and Sep. Ths mples ha he curren soluon a any eraon of he ransformaon procedure sasfes (4). Fnally, observe ha he soluon S generaed a he las eraon sasfes Z S k = 0, snce s a ZIO polcy, and hus Theorem 2. follows drecly from (4). Lemma.9. If he curren soluon S sasfes (4), hen connues o hold afer Sep has been execued. 4. Opmal Zero-Invenory- Orderng Polcy In hs secon we show ha when he number of realers s fxed, we can fnd he bes ZIO polcy by formulang an assocaed shores-pah problem, n me whch s polynomal n T and exponenal n he number of realers n. As we have seen, hs ZIO polcy has a cos whn a facor of 4/ from he opmal cos. Le = 2 T + be he se of dfferen me perods, where T + s used for noaonal convenence. Le N = 2 n be he se of realers. Consruc an acyclc graph G = V A, where V = ū = u u n u = n = } {{} n mes A = { u u n v v n v u for all ; here s a leas one componen such ha u <v ; for every wh u <v we have u = mn j= 2 n u j u (.e., all he componens ha changed had he same value, u } Gven an arc ū v, where ū = u u n and v = v v n, le k be he number of componens ha are dfferen n ū and v, and I = 2 k be he se of ndces of hose componens; ha s, for l = 2 k, l s such ha u l <v l. Observe ha k and by consrucon u = u 2 = =u k = u. The arc ū v represens orderng a perod u o sasfy demands of each realer l, l = 2 k, from perod u hrough v l. Thus, he cos assocaed wh hs arc s he cos of orderng hose uns a perod u and Managemen Scence/Vol. 48, No., November

11 Zero-Invenory-Orderng Polces holdng hem n nvenory unl her consumpon. Specfcally, he cos of hs arc s K 0 u d uv + d 2 uv2 + +d k uvk + c uv + c 2 uv2 + +c k uvk where duv s he oal demand faced by realer from perod u o v Ku 0 d uv + d 2 uv2 + +d k uvk s he cos of shppng d uv + d 2 uv2 + +d k uvk uns from he suppler o he warehouse a me u; and, cuv s he oal shppng and holdng cos for realer f we order a perod u o cover he demands n perods u u + v ;.e., v 2 c uv = K u d uv + h j d j+v I s easy o see ha he shores pah from o T + T+ T+ n G = V A corresponds o fndng he bes ZIO polcy. Usng a Fbonacc heapmplemenaon of Djksra s algorhm (Cormen e al. 999), he shores pah can be found n me O V log V + A, where V =O T + n, A =O T 2n, and n s he number of realers. The algorhm grows o be compuaonally expensve as he number of realers ncreases. Thus, he nex seps o developa heursc ha fnds a good ZIO polcy n polynomal me. Ths s he objecve of he nex secon. 5. Lnear-Programmng-Based Algorhm In hs secon we nroduce a lnear-programmngbased heursc ha generaes close-o-opmal ZIO polces and, hus, effecve soluons o he snglewarehouse mulrealer problem. We sar by formulang he problem of fndng an opmal ZIO polcy as an neger program. The algorhm s based on solvng he lnear programmng relaxaon of he resulng model and ransformng he fraconal soluon obaned no an neger soluon. Our mehod s smlar o he mehod presened n Chan e al. (999) for mulcommody nework flows wh pecewse lnear concave coss. The pecewse lnear concave coss assocaed wh shpmens from he suppler o he warehouse are j=u modeled as follows. Consder any me perod and he assocaed warehouse orderng cos funcon, K 0. Le R be he number of dfferen slopes n he cos funcon, whch we assume, whou loss of generaly, s he same for all me perods o avod cumbersome noaon. Le M r, M r, for r = R, denoe he lower and upper lms, respecvely, on he nerval correspondng o he rh slope of he pecewse lnear cos funcon. Noe ha M 0 = 0 and M R can be se o he oal quany ha may be shpped a perod, n l = d l. We assocae wh each of hese nervals, say r, a varable cos per un, denoed by r, equal o he slope of he correspondng lne segmen, and a fxed cos, f r, defned as he y-nercep of he lnear prolongaon of ha segmen. See Fgure 5 for a graphcal represenaon. Observe ha he cos ncurred by shppng a quany on a ceran range s he sum of s assocaed fxed cos plus he cos of shppng all uns a s correspondng lnear cos. Tha s, f we le Q 0 denoe he warehouse order a me, we can express he assocaed ransporaon cos, K 0 Q0,asK0 Q0 = f r + r Q0 where r s such ha Q0 r M M r Fnally, we defne he followng varables. For each = 2 T and r = 2 R, le Fgure 5 Cos f f 2 f M 0 U r = { f Q 0 M r M r, 0 oherwse. Pecewse Lnear Concave Funcon slope α slope α 2 slope α M M 2 M Quany 456 Managemen Scence/Vol. 48, No., November 2002

12 Zero-Invenory-Orderng Polces For each realer = 2 n, and perods k T, le Z = quany ordered by realer a me o sasfy demand a perod k and { Z = f Q 0 r M M r, 0 oherwse, Z r for each r = 2 R. In wha follows we refer o he U varables as nerval varables and o he Z varables as quany varables. To model orderng and nvenory coss a he realer level, we consder a dummy perod T + and defne, as n 4, for each realer = 2 n and perods <k T +, c = oal cos of orderng a perod o sasfy demand for perods hrough k and holdng he uns n nvenory unl her consumpon. Tha s, c = K ( k ) d j j= k 2 + j= h j ( k d l l=j+ Observe ha a ZIO polcy for realer can be nerpreed as a pah from o T + on a nework wh nodes 2 T + and arcs k, for <k T +, wh assocaed cos c. In wha follows, we refer o hs nework as he h realer s nework or G. Thus, o calculae orderng and nvenory coss a realer we formulae a shores-pah model on G usng he varables f an order s placed by realer a me X = o sasfy demands for perods hrough k, 0 oherwse, and flow conservaon consrans. We refer o X = X as he vecor of pah flows. The bes ZIO polcy can be found by solvng he followng neger program: Problem P Mn T R = r= + n [ f r U r + r T n T + c X = = k=+ T = k= Z r ) s R Z r d k U r r = 2 R = 2 n and k T (5) T Z r = + d k X l r= l=k+ k T = 2 n (6) X lj f l = X jl = f l = T + (7) j j>l j j<l 0 f <l T = 2 n Z r 0 r = 2 R = 2 n and k T U r 0 r = 2 R and = 2 T X 0 = 2 n and k T + The frs se of consrans, (5), specfes ha f some quany s ordered a me by any realer and shpped on nerval r of he ransporaon cos funcon, hen he assocaed nerval varable, U r, mus be. These, ogeher wh he negraly of he U varables, are he only consrans needed o model he pecewse lnear concave coss; see Chan e al. (999). Obvously, consrans (5) could be aggregaed for all k. However, hs would consderably weaken he lnear programmng relaxaon of Problem P. Equaon (6) guaranees ha f a posve amoun s shpped o realer a me o sasfy demand a he realer a perod k, hen he realer mus order a perod o cover demands for perods hrough some l k. Observe ha hese consrans lnk he supplerwarehouse model wh he realer model. Fnally, he flow conservaon consrans (7) correspond o fndng, for each realer, a pah from o T + onhe realer s nework, G. Unforunaely, solvng hs neger program s compuaonally nracable for all bu small-sze problems. To overcome hs dffculy, we focus on analyzng he behavor of s lnear programmng relaxaon and ake advanage of s srucural properes o developan effecve heursc ha consrucs Managemen Scence/Vol. 48, No., November

13 Zero-Invenory-Orderng Polces an neger soluon o Problem P from s opmal fraconal soluon. For ha purpose, we sar by fxng he fraconal pah flows and sudyng he behavor of he resulng lnear program. Le X = X be he vecor of pah flows n a feasble soluon o he lnear programmng relaxaon of Problem P. Wha s he cos ha he lnear program assocaes wh hs soluon? Wha are he values of he correspondng nerval and quany varables, U r and Z r? Observe ha gven he vecor of fraconal pah flows X, he amoun o be shpped from he suppler o he warehouse a any me perod s known and herefore he lnear programmng relaxaon of Problem P can be decomposed no mulple subproblems, one for every me perod. For each me perod, he oal shppng cos, as well as he correspondng varables Z r and U r, can be obaned by solvng he followng problem, whch we refer o as he fxed-flow subproblem a perod or Problem FFX : [ R Problem FF X Mn n T f r U r + r Z r s.. R r= = k= Z r d k U r k = T and r = R (8) T Z r = + d k X s r= s=k+ Z r 0 k = T and = n (9) k = T = n and r = R U r 0 r = R The followng lemma, presened n Chan e al. (999), explcly characerzes he soluon o he lnear program for any gven (fxed) vecor of pah flows X. Lemma 5.. For any gven perod of me and fxed vecor of pah flows X, le he proporon of he demand of realer a me k whch s shpped a perod be T + s=k+ X s for k and = 2 n Rank he proporons,,n nondecreasng order of her values. Followng ha order, assocae a sngle ndex l o each par k so ha l and 2 L where L = n T +. Smlarly, le d l d k for each l = 2 L and s correspondng par k. Then, he opmal soluon o he fxed-flow subproblem a me s L l= K 0 ( L d s ) l l where 0 = 0 (0) s=l We are now ready o descrbe a polynomal me heursc ha fnds an effecve ZIO polcy based on he soluon o he lnear programmng relaxaon of Problem P. Lemma 5. wll be exensvely used by he algorhm o compue he ncrease n coss n he soluon o he lnear program when he vecor X s modfed n he search for an neger soluon. Lnear-Programmng-Based Algorhm Sep. Solve he lnear programmng relaxaon of Problem P. Le X = X be he opmal soluon. Inalze =. Sep 2. For each arc k, <k T + n nework G compue a margnal cos, c, as follows. The margnal cos s he oal ncrease n cos n he soluon o he lnear program ncurred when augmenng he flow on ha arc from he fraconal X o. Tha s, c = W + X k c k where W s he ncrease n ransporaon cos o he warehouse resulng from modfyng flow n he lnear program from X o. Ths cos ncrease can be easly calculaed usng Lemma 5.. Sep. Deermne he orderng epochs of realer by fndng he mnmum cos pah from o T + on nework G wh edge coss equal o he margnal coss. Sep 4. Updae he amoun and coss of warehouse orders a each perod o accoun for realer s orderng sraegy. Coss are updaed usng Lemma 5.. Sep 5. Le = + and repea Seps (2) (5) unl = n Managemen Scence/Vol. 48, No., November 2002

14 Zero-Invenory-Orderng Polces Ths algorhm runs n polynomal me. I requres solvng a lnear program wh O T 2 Rn varables and O T 2 Rn consrans, and hen performng Seps 2 hrough 5 o oban an neger soluon. Observe ha when applyng Lemma 5. a any sep of he algorhm, he rankng of he proporons assocaed wh he flow on an edge from suppler o warehouse can be updaed easly n O nt because proporons are only changed o values of 0 or. The ransporaon cos on edge can hen be compued n O nt + R. Ths s done a mos O nt 2 mes over all eraons of Seps 2 and 4. Consequenly, he complexy of performng Seps 2 hrough 5 s no more han O n 2 T + nt 2 R. 6. Compuaonal Resuls In hs secon we es he performance of he lnear-programmng-based algorhm n erms of boh compuaonal me and relave devaon from he opmal ZIO polcy. For hs purpose, he lnearprogrammng-based algorhm was coded n C++ and a varey of es problems, descrbed below, were solved on a Sun Ulra SPARCsaon; CPLEX 5.0 was used o solve he lnear and neger programmng problems. We apply he algorhm o wo ypes of problems. The frs s he sngle-warehouse mulrealer problem wh realer orderng cos represened by he modfed all-un dscoun cos funcon. The second s he sngle-warehouse mulrealer problem wh concave orderng cos funcons for he realers. Of course, n boh ypes of problems, warehouse orderng cos s a pecewse lnear concave funcon. Observe ha here exss an opmal ZIO polcy for he second ype of problems. Type Insances. We consder four problem classes correspondng o 5, 25, 50, and 00 realers. The plannng horzon s 2 perods and demands for each realer are generaed from a normal dsrbuon wh mean 00 and sandard devaon 20. Holdng coss are randomly generaed n he nerval [0.,0.6. Suppler-warehouse ransporaon coss are descrbed by pecewse lnear concave funcons wh fve breakpons beween 0 and he maxmum amoun ha could possbly be ordered from he suppler o sasfy realer demands. We consder he breakpons fxed and randomly vary fxed coss and slopes (whn a ceran sensble range) over me. Smlarly, he warehouse-realer ransporaon cos funcon for a parcular realer s a modfed all-un dscoun funcon wh eher 6, 7, or 8 prce breaks and agan fxed coss and slopes are randomly vared over me. Table shows, for each problem class, he average compuaon me of he lnear-programmng-based algorhm over 200 nsances generaed. For hese moderae-sze nsances esed, he opmal ZIO can be calculaed by solvng he neger program, Problem P, and used o evaluae he performance of he heursc soluon. The assocaed average compuaon mes are gven n he fourh column of Table. The cos of he opmal ZIO polcy obaned s compared o he heursc soluon n he las wo columns: The frs column repors he average rao for cases n whch he soluon o he lnear programmng relaxaon of Problem P was no neger. The second repors he average over all problems esed. Type 2 Insances. Here we sudy he performance of he lnear-programmng-based algorhm for nsances n whch all he ransporaon coss are pecewse lnear and concave. We agan consder dfferen problem classes, wh normally and ndependenly, dencally dsrbued realer demands wh mean 00 and sandard devaon 20, and generae 0 nsances for each class. Holdng coss are se o 0.2 per un per perod. The pecewse lnear concave ransporaon coss consdered have hree prce breaks (.e., four segmens wh dfferen slope) n he range from 0 o he maxmum possble demand ha could be sasfed usng ha lnk. Assocaed fxed coss and varable coss are randomly generaed over me. Table 2 descrbes he sx problem classes esed and repors he average compuaon me and he average rao of heursc o opmal soluons over he fve nsances esed for each class. We observe ha n all he nsances esed, he soluon o he lnear programmng relaxaon concdes wh he opmal neger soluon. Managemen Scence/Vol. 48, No., November

15 Zero-Invenory-Orderng Polces Table Compuaonal Resuls for Modfed All-Un Dscoun Warehouse-Realer Coss Number CPU Tme Frequency of Z H /Z ZIO Z H /Z ZIO Problem of CPU me wh IP fraconal Fraconal All class realers (seconds) (seconds) soluon cases cases Class 5 0 4/ Class / Class / Class / Table 2 Compuaonal Resuls for Concave Transporaon Coss on All Lnks Problem Number of Number of CPU Tme Z H /Z ZIO class perods realers (seconds) = Z H /Z Class Class Class Class Class Class Conclusons and Exensons We should pon ou ha he wors-case resuls descrbed n hs paper hold under farly general sengs. Indeed, a careful nspecon of he proofs of Theorems 2.2 and 2. reveals ha hese resuls hold for any general warehouse-realer orderng funcons and realer holdng coss sasfyng he followng properes: () hey are nondecreasng funcons of he amouns shpped and held, respecvely, and () he assocaed cos per un s nonncreasng n hose amouns. Moreover, boh condons are necessary o oban a fne wors-case performance bound on he bes ZIO polcy. Tha s, f a leas one of hese condons does no hold, hen he bes ZIO polcy has an unbounded wors-case performance (see Chan e al. 2002). Fnally, he bounds on he performance of ZIO polces developed n hs paper can be easly exended o a more general dsrbuon problem wh cenral socks, n whch he warehouse s allowed o carry nvenory. For a proof, please refer o he echncal appendx o hs paper avalable on he Managemen Scence elecronc companon page a hp:\\mansc. pubs.nforms.org/ecompanon./hml. Acknowledgmens The auhors hank Mchael Deampel and Mchael Zemer from Schneder Logscs for nroducng hem o hs problem. Research suppored n par by ONR Conracs N and N , and by NSF Conracs DDM , DMI , and DMI References Arkn E., D. Joneja, R. Roundy Compuaonal complexy of uncapacaed mul-echelon producon plannng problems. Oper. Res. Le Arora, S., M. Sudan Improved low-degree esng and s applcaons. Proc. 29h Annual ACM Sympos. Theory Compu. El Paso, TX, Chan, L. M. A., A. Murel, D. Smch-Lev. 999a. Producon/dsrbuon plannng problems wh pece-wse lnear and concave cos srucures. Workng paper, Norhwesern Unversy, Evanson, IL.,, Z. J. Shen, D. Smch-Lev. 999b. On he effecveness of zero-nvenory-orderng polces for he economc lo szng problem wh a class of pece-wse lnear cos srucures. Oper. Res. Forhcomng. Cormen, T. H., C. E. Leserson, R. L. Rves Inroducon o Algorhms. MIT Press, Cambrdge, MA. Fege, U A hreshold of ln n for approxmang se cover. J. ACM 45(4) Joneja, D The jon replenshmen problem: New heurscs and wors case performance bounds. Oper. Res. 8(4) Acceped by Thomas M. Leblng; receved Aprl, 200. Ths paper was wh he auhors 2 monhs for revson. 460 Managemen Scence/Vol. 48, No., November 2002