Impact on inventory costs with consolidation of distribution centers

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1 IIE Tansactons (2) 33, 99± Imact on nventoy costs wth consoldaton of dstbuton centes CHUNG PIAW TEO, JIHONG OU and MARK GOH Deatment of Decson Scences, Faculty of Busness Admnstaton, Natonal Unvesty of Sngaoe, 5 Law Lnk, Sngaoe 759 E-mal: fbateoc@nus.edu.sg Receved Mach 999 and acceted Febuay 2 The consoldaton of Dstbuton Centes (DCs) s a new tend n global logstcs management, wth a educton n nventoy costs often beng cted as one of the man bene ts. Ths ae uses an analytcal modelng aoach to study the mact on faclty nvestment and nventoy costs when seveal DCs ae consoldated nto a cental DC. In atcula, ou model suggests that consoldaton leads to lowe total faclty nvestment and nventoy costs f the demands ae dentcally and ndeendently dstbuted, o when they follow ndeendent but ossbly nondentcal Posson ocesses. Ths agees wth the concluson of the classcal EO and newsvendo models. Howeve, we show by an examle that, fo geneal stochastc demand ocesses, the total faclty nvestment and nventoy costs of a consoldated system can be n ntely wose o than that of a decentalzed system. Ths ases manly because the ode elenshment xed cost yelds a cost comonent ootonal to the squae oot of the mean value of the demand, whle the demand uncetanty yelds a cost comonent ootonal to the standad devaton of the demand. Whethe consoldaton s cost e ectve o not deends on the tade-o between these two comonents, as ndcated by an extensve numecal study. We also oose an algothm that solves fo a dstbuton system wth the total faclty nvestment and nventoy costs wthn 2 of the otmal.. Intoducton and ovevew As quoted by Rheem (997), ``Consoldaton and centalzaton ae ovng to be an excellent move fo comanes that know how to manage logstcs e ectvely''. Thee ae aleady many case eots on comanes steamlnng the dstbuton netwoks by consoldatng and centalzng the dstbuton oeatons. Fo nstance, Stales Inc., one of the bggest o ce oducts etales n the US, has ecently develoed a stategy to centalze and ugade ts US dstbuton netwok (Gouley, 997). Also, Dsney Stoes Inc. uses a dedcated Cental DC (CDC) n Memhs to suly moe than d eent tye of oducts to 36 stoes acoss the US, Canada and Pueto Rco (Jedd, 996). Some comanes go even futhe to centalze the woldwde dstbuton netwok, such as Benetton, the Italan sotswea comany. Benetton uses one CDC n Ponzano, Italy to seve ove 6 stoes n 83 countes aound the wold (Daan, 992). Poe and Rete (996) also eot that one mao ntenatonal comute ats comany stated o wth ve DCs n England, Mane, Calfona, Jaan and Hong Kong, but has snce estuctued ts global logstcs ocess and establshed Hong Kong as the sngle global DC. The followng easons have often been cted fo adotng the consoldaton stategy: Reduced faclty nvestment costs. A lage CDC s moe cost e cent to buld and oeate comaed to havng many smalle egonal centes. Inceased sevce qualty. Centalzed nventoy ensues bette qualty contol and vsblty of stocks wthn the system. Also at a moden CDC, moe value-added sevces can be ovded at lowe cost. Lowe total nventoy costs. By oolng demands togethe, the equed amount of nventoy to seve them s educed, and bene ts fom nceased economes of scale n uchasng and tansotaton oeatons can be acheved. Ths ae uses an analytcal modelng aoach to quantfy the e ects of consoldaton. In atcula, we study the mact on the faclty nvestment and nventoy costs when seveal Regonal Dstbuton Centes (RDCs) ae consoldated nto one CDC. We focus on the common ntuton that consoldaton educes total nventoy costs due to the sk oolng e ect. The model n ths ae s also atally motvated by a common queston faced by logstcs manages n South-East Asa, who often have to atonalze the set-u of the dstbuton system Ó 2 ``IIE''

2 Teo et al. stuctue, esecally afte some acquston o mege execses. The common queston osed s: ``Is thee a need to mantan moe than one DC n ths egon to suot egonal demand?'' The oblem s comlcated by the fact that the exstng DCs may not have dentcal nventoy-elated cost stuctues, due to the choce of d eent technologes and equment tyes. On the othe hand, the oblem s sml ed by the fact that customes fom d eent locatons n the egon a ected by the consoldaton execse ae nd eent about whee the odes ae shed fom, as elenshment lead-tmes fom d eent otental stes of DCs ae moe o less dentcal (snce they ae all based n South-East Asa, wth customes based manly n Noth Asa). To fomalze ou model, consde a m wth one oducton lant oducng a sngle oduct fo a collecton of aggegated demand onts wth andom demands. Wthout any loss of genealty, we assume that the m has selected a set of ossble locatons to set u RDCs to seve these demands. The man oblem the m faces s to choose the RDC locatons so that the total faclty nvestment and nventoy costs wll be mnmzed. We assume that the customes seved by d eent DCs ae nd eent about whee the odes ae shed fom, and also that the outbound tansotaton costs wll not change dastcally deendng on whee the odes ae shed fom. The man eason fo dong so s that we want to solate the e ect of DC odeng cost stuctue and elenshment lead-tmes on the cost e ectveness of the consoldaton stategy. When the DCs to be consoldated ae neaby (as n the Sngaoe context), these assumtons ae vald. Ou model also ales to the oblem of vendo/sule selecton. The consoldaton ssue studed hee can be tanslated to the ssue of consoldatng sule bases (.e., whethe to seve the customes by allocatng the demand to one o moe sules). Snce a d eent sule gves se to a d eent lead-tme (deendng on the locaton of sules) and odeng cost stuctue (e.g., EDI-lnked o not etc.), ou model essentally addesses the queston as to whethe t s bene cal to dvde the demand nto one o moe gous, to be seved by d eent sules. A susng concluson fom ths study s that t s sometmes cost-e ectve to seaate the mateal ow nto d eent gous, to exlot the cost economes o eed by d eent sules. In most evous wok on nventoy contol, the demand±suly system s assumed gven and the focus s on the detemnaton of the otmal nventoy contol olcy. On the othe hand, n the lteatue on faclty locaton oblems, the nventoy contol asect s seldom consdeed. In ou oblem, howeve, the faclty locaton decsons and the demand assgnments must take nto account the mact on nventoy costs at the DC. We call ths the locaton-nventoy oblem. Ou st concluson s that when the demands ae dentcally and ndeendently dstbuted (..d.), consoldaton s an otmal stategy fo faclty and nventoy costs mnmzaton. As a coollay, ths mles that consoldaton s otmal f the demands follow ndeendent but ossbly nondentcal Posson ocesses. Ths agees wth the concluson of the classcal EO and newsvendo models. Howeve, when the demands ae some geneal stochastc ocesses, the consoldaton stategy may not always lead to a educton n the total faclty nvestment and nventoy costs. In ths case, we wll see that the odeng xed cost gves se to a cost comonent ootonal to the squae oot of the mean value of demand and the demand uncetanty yelds a cost comonent ootonal to the standad devaton of the demand. Whethe consoldaton s cost e ectve o not deends on the tade-o of these two comonents, as ndcated by an extensve numecal study we conducted n the wok. The key to the soluton of the oblem les n the evaluaton of the otmal cost fo any gven dstbuton system stuctue. Fo that, we wll obtan tght bounds on the otmal cost, whch ae extensons of the bounds obtaned by Gallego (998). Whle Gallego takes the odeng cost as a constant, we wll nstead allow t to be a ece-wse lnea concave nceasng functon of the odeng sze. The bounds can be exlctly evaluated fo demands of nomal dstbutons. As noted n Tms and Goenevelt (984), ``fo most tems n nventoy contol the demand attens ae elatvely smooth so that one can n geneal safely use two-moment aoxmatons based on nomal demand denstes''. Unde the nomal demand assumton, and wth the hel of the bounds, we st constuct a wost case examle n whch the consoldaton stategy s n ntely wose o than the otmal dstbuton stategy (whch uses moe than one DC), and then oose an algothm that solves fo a dstbuton stategy wth the total nventoy costs wthn 2 of the otmal. The algothm s based on the wok esented n Chakavaty et al. (985) and uns n O n 2 m tme. The est of the ae s oganzed as follows. The lteatue evew n Secton 2 shows that ou wok falls n the ntesecton of the lteatue on faclty locaton and the lteatue on stochastc nventoy contol oblems. The geneal fomulaton of the model s esented n Secton 3. In Secton 4, we consde the..d. demand case, and n Secton 5, the geneal demand case. The numecal exement s esented n Secton 6. We conclude the ae n Secton Lteatue evew Tycally the lteatue on nventoy contol assumes a gven demand dstbuton, and seeks the otmal nventoy contol olces at the DC (see Gaves et al. (993) fo a thoough evew of ths eld, fo the sngle-echelon and mult-echelon nventoy systems). Thee ae seveal ecent

3 Consoldaton of dstbuton centes studes that ntegate nventoy decsons wth othe suly chan actvtes. Fo nstance, Fedeguen and Zkn (984a), Anly and Fedeguen (99), and Vswanathan and Mathu (997) ntegate tansotaton outng wth nventoy decsons. These woks study the sngle-day nventoy-outng oblem and catalze on many esults fom the vehcle-outng lteatue. Fo the ntegaton of oducton wth nventoy, whee multle comonents ae equed to oduce one at, the nteested eade s efeed to Lee and Bllngton (986), Cohen and Lee (988), and Ettl et al. (996) fo moe detals. These models ay close attenton to catung the ntedeendence of base-stock levels at d eent stoes and how ths ntedeendence a ects the oveall system efomance. In the lteatue on faclty locaton oblems, a standad fomulaton ncludes xed faclty setu costs and tansotaton costs (Geo on and Powe, 995 Revelle and Laote, 996), and usually the demands ae statc and detemnstc. The focus s manly on the tansotaton costs of the system, whle the nventoy costs at the DC ae often gnoed o sml ed to be ndeendent of the system stuctue. An exceton s n Baahona and Jensen (996) who studed a veson of the dstbuton netwok desgn oblem fo comute sae ats. The model nds the locaton of waehouses and the custome assgnment, mnmzng the total cost of buldng the waehouses and mantanng the nventoes at the vaous locatons. The focus, howeve, s manly algothmc, and vey estctve assumtons ae mosed on the nventoy costs to make the model tactable. The wok n ths ae can be vewed as a faclty locaton model that catues the mact of the nventoy elated costs wth moe sohstcaton, and wth stochastc demands at the custome locatons. Also, nstead of beng uely algothmc, we deve stuctual nsghts nto the cost e ectveness of consoldaton. Ou oblem s comlcated by the fact that closed fom exessons fo the otmal nventoy costs, condtoned on the demand assgnment, ae not avalable fo most demand dstbutons. Ou analyss n ths ae s only made ossble by seveal ecent advances n the aea of stochastc nventoy theoy, due manly to the semnal woks of Zheng (992) and Gallego (998). These woks deved stuctual nsghts and bounds fo the otmal nventoy cost functons whch we wll utlze late n ths ae. Waehouse consoldaton has also been studed n the context of sk-oolng of nventoes. Een (979) ovdes the semnal ece on the otental bene ts of skoolng n ths settng. He demonstated that the exected cost of the decentalzed system gows lnealy wth the numbe of DC (m), wheeas the exected cost of the centalzed system gows wth m. Thus, we have the socalled ``sk-oolng ncentve'' fo centalzng the nventoes. Een assumed that the odeng, holdng and enalty-cost stuctues ae dentcal acoss all ossble DC locatons. When the cost stuctues ae not dentcal, he ndcated that t mght not be cost e ectve to centalze demand. Ou counte examle n Secton 5 valdates ths conectue. Seveal othe eseaches have also looked at the sk-oolng ssue fom d eent esectves. We efe the eades to Een and Schage (98), Fedeguen and Zkn (984b) and Schwaz (989) fo a detaled dscusson. Most of the eseach so fa, howeve, assume a gven netwok stuctue and seeks only to undestand the mact of sk-oolng n the nventoy context. 3. Model fomulaton In the locaton-nventoy oblem, thee ae a total of n demand onts, ndexed by ˆ 2... n. We assume that the demand fo the oduct at demand ont can be modeled by a contnuous stochastc ocess fd t : t g, wth mean ate l e unt tme and vaance 2. The ocesses fd t : t ˆ 2... ng ae ndeendent. We wll only consde dstbuton stateges that seve the demands at each demand ont exclusvely fom a sngle DC (.e., slttng of demand s not allowed). The m has selected m ossble DC locatons, ndexed by ˆ 2... m. If a DC s set u at locaton, a xed setu cost f s ncued, and the lead-tme to suly fom the lant to ths locaton s L. Fo the uose of ths model, we assume that f can be amotzed ove the tme eod of consdeaton.e., f s a e unt tme ``chage'' (to ay back a loan ove a vey long hozon). On ths bass, f can also be teated n the model as the dect vaable cost nvolved n unnng a DC and ths vaable cost s DC-sze deendent. The man esults n ths ae eman vald fo the case when f s a concave functon of the amount of demand assgned to DC. Fo ease of exoston, we wll assume that f s a constant. We futhe assume that the lant has unlmted oducton caacty. The nventoy costs at DC nclude an odeng cost k e ode, beng the odeng sze, and ootonal nventoy holdng (es. enalty) costs accumulatng at a constant ate h (es. ) e unt stock (es. backode) e unt tme. The odeng cost k s modeled as a ece-wse lnea concave nceasng functon of the odeng sze to account fo the economes of scale n uchasng and tansotaton oeatons. We now de ne y ˆ f a DC s to be set-u at locaton, othewse, and ( f demand ont s assgned to the x ˆ DC setu at locaton, othewse. Let Y ˆfy : ˆ... mg and ˆfx : ˆ... m, ˆ... ng. Thus, the aggegate demand ocess assgned to DC s P n ˆ x d t. Next, we denote the total demand

4 2 Teo et al. dung lead-tme L by D, whch deends on the demand assgnment. Unde mld assumtons on d t ˆ... n, the otmal long un aveage nventoy cost can be shown to be gven by (Zheng, 992 Gallego, 998) C k P n ˆ ˆ mn x l R G y dy whee s the otmal eode ont, the otmal odeng sze, and G y ˆE h y D D y : 2 The tem k P n ˆ x l = e ects the aveage odeng cost when the aveage demand s P n ˆ x l (whch deends on the demand assgnment decson). The tem G y dya e ects the aveage holdng and stockout cost of the system, as the nventoy level of the system can be shown to be unfomly dstbuted between and (Zheng, 992). The locaton-nventoy oblem studed n ths ae can thus be fomulated as follows: Poblem P subect to mn f y x ˆ fo ˆ... n x y fo ˆ... n x y 2f g fo all : C 4. Identcal and ndeendent demand ocesses In ths secton, we consde the secal case of..d. demand ocesses. To smlfy the notaton, we assume l ˆ, fo ˆ... n. Snce demand ocesses ae dentcally dstbuted, the lead-tme demand D at DC deends only on the numbe of demand onts assgned to t. If the latte s n, we use D n to denote the leadtme demand and also wte G y G y n ˆE h y D n D n y : We now de ne n ˆag mn Z G y n dy fo > : Then n coesonds to the otmal eode ont n the stochastc nventoy oblem when the ode quantty s xed at. Accodng to Zheng (992), we have G n n ˆG n n 8 < mn y G y n f ˆ, H n s convex n, : G n n othewse, and S n n Z n G y n dy ˆ Z H y n dy s also convex n. Hence, S a n as n fo a : 3 As the oofs to the above esults ae qute ntcate, we efe the eades to the ognal ae by Zheng (992) fo the devatons. Ou st esult shows that consoldaton s otmal n ths case: Theoem. The consoldaton stategy s otmal fo the locaton-nventoy oblem when the demand ocesses ae..d.. Poof. Let ˆfx g Y ˆfy g be an otmal soluton to Poblem (P), then n ˆ P l x ˆ P x and the otmal value of Poblem (P) s gven by m v n > f k n G y n dya A: ˆ 4 Now f all n demand onts ae consoldated to DC, the total cost would be Z f k n G y n dya : We consde the weghted aveage of the above values wth weght fo as n =n: Z A m f k n G y n dya A n ˆ m ˆ mn v n > k n Z n n G y n dya A: Note that f all n demand onts ae consoldated to DC, the total demand D n ˆd d 2 d n s the sum of n..d. andom vaables. Then n D n has n coes of d d 2 d n, whch can be eaanged and wtten as a sum of n gous of n..d. andom vaables. Fo examle, the st gou s d d 2 d n, the

5 Consoldaton of dstbuton centes 3 second gou can be d n d n 2 d 2n, etc. As each gou has the same dstbuton as D n, we denote them by D k n n kˆ. Theefoe, n n G y n ˆn n Eh y D n D n y ˆ n Eh n y n D n n D n n y ˆ n E h n kˆ n n y Dk n n D k n n n y kˆ n n E h n n y Dk n kˆ D k n n n y n ˆ E h n y D n D n n n y n ˆ G n y n : The nequalty follows fom the fact that x y x y. Thus, A m v n > f ˆ Z n k n G n y n dya A ˆ m ˆ mn ˆ m ˆ mn m ˆ mn v n > f k n n n v n > f k n n n v n > k n Z n =n Z n =n n Z =n C C G y n dya A C C G y n dya A G y n dya A: The last nequalty follows fom (3). As stated n (4), the ght-hand sde s the otmal value coesondng to soluton ˆfx g Y ˆfy g : Thus, we conclude that consoldatng to one of the m DCs wll gve se to the otmal cost. Remak. The above esult s a lttle susng, snce n geneal S n s not concave n n. In ealty, demand dstbutons ae usually non-dentcal. Howeve, when the demands ae Posson ocesses of the same mean ate, Theoem would aly. If the ates ae d eent, we can slt the ocesses nto..d. sums of a `basc unt' ocess when ossble. If no common `basc unt' s found, a contnuous aoxmaton scheme can then be aled. Thus, t can be seen that Theoem would aly f the demands ae ndeendent Posson ocesses. Notce that f demands ae geneated fom many ndeendent ndvdual onts, the Posson ocess s a good aoxmaton to the demands. Thus, Theoem has motant am catons n the desgn of the otmal dstbuton stategy. 5. Geneal ndeendent demand ocesses Fo geneal demand ocesses, the otmal nventoy cost functon s vey comlex. We st seek to bound the otmal nventoy cost wth moe manageable functons. Wth the hel of these bounds, we esent an examle to show that consoldaton can be n ntely wose o than the otmal decentalzed system. We wll also oose an algothm that solves fo a dstbuton stategy wth the total nventoy costs wthn 2 of the otmal. 5.. Bounds fo otmal nventoy cost Consde a sngle locaton, sngle oduct, contnuous evew stochastc nventoy system n whch all stockouts ae backodeed. The demand s stochastc wth mean ate l, and the lead-tme demand s D. Relenshments need a ostve lead-tme L fo delvey and ncu a xed cost of k whch s a ece-wse lnea concave nceasng functon of the odeng sze e ectng the economes of scale n the uchasng and tansotaton costs. The othe costs nclude ootonal nventoy holdng (enalty) costs accumulatng at a constant h e unt stock (backode) e unt tme. De ne G y ˆEh y D D y and denote y ˆ mn y G y. Let C d be the soluton value of the mnmum nventoy cost when the andom demand s aoxmated by ts mean value. Assumng k ˆK to be a constant ndeendent of, Gallego (998) (movng a esult of Zheng) obtans lowe and ue bounds fo Z C mn C k l G y dya A: We use the followng theoem fom Zheng (992) and Gallego (998)

6 4 Teo et al. q C d G y C C d 2 G y 2 : 5 G y s the bu e cost. The bounds ae tght when the demand s detemnstc. We now extend Gallego's bounds to the case whee k s a ece-wse lnea concave nceasng functon. Theoem 2. If k s a ece-wse lnea concave nceasng functon, then lk mn h G y C h 2 mn lk h h 2 G y 2 2 h h 6 whch mles that both the lowe and ue bounds ae wthn 2 of the otmal. The oof we used below s based on a ece-wse lnea convex aoxmaton to the otmal nventoy cost functon. The bound n Gallego (998) s deved usng a d eent agument. Poof. Let a ˆ E DD y Š and a 2 ˆ E DD > y Š, and de ne a two-ont dstbuton D e as follows: P D e ˆ a ˆP D y ˆ h P D e ˆ a 2 ˆP D > y ˆ h h : D e s elated to D though E D e ˆa h a h 2 h ˆ E DD y ŠP D y E DD > y ŠP D > y ˆ E D ˆlL: By Jensen's nequalty, we have G y ˆE h y D D y Š ˆ E h y D D y D y ŠP D y E h y D D y D > y ŠP D > y h y E DD y Š E DD y Š y ŠP D y h y E DD > y Š E DD > y Š y ŠP D > y ˆ h y a a y ŠP D y h y a 2 a 2 y ŠP D > y Let G e y E h y D e D e y Š ˆ h y a a y ŠP D y h y a 2 a 2 y ŠP D > y : It can be easly ve ed that: 8 ll yš y a, >< h G e y ˆ h a 2 a a < y < a 2, >: y > a h y llš 2. On the othe hand, G y ˆE h y D D y Š ˆ h y a h a h 2 y h ˆ h h a 2 a : Hence we have G e y ˆ Gd y fo y a and y a 2, G y fo a < y < a 2. Fgue dects the elatonshs of the thee functons, G, G e, and G d. Let Z Ce mn c e lk G e y dya A then Ce C : To evaluate Ce, we follow the ocedue taken by Zheng (992) to st x and nd the otmal and then nd the otmal e. Fom Fg., t can be seen that fo a 2 a, the otmal ˆa, and when ˆ a 2 a, we get the mnmal value 7 ˆ E h y D e D e y Š: Fg.. The elatonsh between G y G d y and G e y.

7 Consoldaton of dstbuton centes lk Za a G e y dya a 2 a ˆlk h G y h G y whee we have used the fact that a 2 a ˆ G y h h : 8 Fo > a 2 a, accodng to Lemma 2 n Zheng (992), the otmal s to be solved fom the followng condton: G e ˆG e : 9 Agan fom Fg., we can see that to make Equaton (9) hold, a. Snce fo y a G e y ˆG d y then Equaton (9) s the same as G d ˆG d whose soluton s gven n Zheng (992) as d ˆlL h= h : Futhemoe, lk R d d G e y dy Z d lk G d y dy 2 G y a 2 a A d ˆ l k Z 2l G y a 2 a G d y dya : The otmal s clealy geate than the EO soluton s 2l k h 2l G y a 2 a h s ˆ 2lk h h a 2 a 2 whch s geate than a 2 a. The coesondng cost value s smalle than 2l k =2l G y a 2 a Š 2l k =2l G y a 2 a Š h = h s ˆ 2l k h 2l G y a 2 a h s ˆ 2l k 2l G y 2 h h h h d s ˆ 2lk h h G y 2 s lk h= h 2 G y : G y The above s thus smalle than the mnmal value n (8) obtaned fo the case a 2 a. Theefoe Ce ˆ mn lk h h 2 G y a 2 a 2 lk ˆ mn h h 2 G y 2 h 2 h s 2lk h h G y 2 : To ove the ue bound, we note that the functon G f y ˆG d y y ll G y stctly domnates G y, and hence the otmal cost s bounded above by lk G f y dya mn ˆ G y mn lk h : h 2 Let denote the batch sze that attans the mnmum n the ue bound. Then k ˆf c fo some f and c. The lowe bound s at least s h cl 2lf h G y 2 wth equalty only when the otmal batch sze s attaned at a tunng ont. On the othe hand, the ue bound s gven by mn lk h h 2 cl G y mn G y lf h h by concavty of K s h ˆ cl G y 2lf h s h cl 2 2lf h G y 2 : In geneal, G y s not easy to chaacteze n closed fom. If the demand D follows nomal dstbuton, howeve, t can be shown to be ootonal to the standad devaton. Theoem 2 can thus be sml ed as follows. 2

8 6 Teo et al. Coollay. If the demand e unt tme s..d. and nomally dstbuted wth mean l and standad devaton, then thee exsts a ostve numbe ˆ L mn E h z D ll z D ll L z L such that G y ˆ and lk mn h h 2 C mn lk h h h : h The oof of ths coollay follows fom standad aguments n nventoy theoy. Fo the sake of comleteness, we nclude the easy oof. Poof. Unde the above assumton, the lead-tme demand D N ll 2 L. Then the otmal newsvendo quantty y s gven by ll z L fo some ostve numbe z whch deends on h and, and coesondngly, G y ˆmn E h y D D y Š y ˆ mn E h ll z L D D ll z L z ˆ L mn E h z D ll z D ll L z L ˆ whee n the last equaton ˆ L mn E h z D ll z D ll L z L deends on h, and L. Note that D ll N L and z ˆ mn E h z D ll z D ll L z L whch can be obtaned by solvng a classcal newsvendo model Wost case examle Consde a scenao whee we have two ossble DC locatons and two demand onts. At the DC, the odeng costs ae k ˆK, ˆ 2. At the demand onts, the demands ae nomally dstbuted wth mean l and standad devaton ˆ 2. Fo the DC, we de ne A ˆ 2K h = h and B ˆ 2, ˆ 2, whee s the ostve numbe as de ned n Coollay. To solate the e ect due to odeng cost and demand uncetanty, we may gnoe the e ect on the aametes due to the holdng (h) and enalty ( ) cost, by oe nomalzaton. Thus A and B oughly ndcate the magntude of the odeng cost and the elenshment lead-tme at DC esectvely. By (6), f we consoldate the demands at DC, the total q nventoy cost s at least A l l 2 B = 2 f we consoldate at DC 2, the total nventoy q cost s at least A 2 l l 2 B = 2. Howeve, f demand s seved by DC and demand 2 by DC 2 esectvely, the total cost s q only at most A l B 2 q A 2 l 2 B Now we choose (l ˆ =e ˆ ) and (l 2 ˆ =e 4 2 ˆ e) fo some abtaly small ostve constant e. Also we can make A ˆ e 2 and A 2 ˆ e 6 by sutable choces of K and K 2. We can smultaneously make B ˆ e and B 2 ˆ =e wth cetan choces of L and L 2. Wth these choces of the aametes, t can be ve ed that the cost fo the decentalzed stuctue s at most H e, whle both of the consoldated solutons have cost of at least H = e. Hence, the consoldated solutons ae n ntely wose o than the otmal decentalzed stuctue. In the above examle, the odeng cost of DC 2 O e 6 s much smalle than that of DC O e 2, although the elenshment lead-tme O =e s much lage than that of DC O e. Futhemoe, a lage demand l 2 ˆ O =e 4 wth low vaablty 2 ˆ e s assgned to DC 2, wheeas a small volume wth elatve lage vaablty l ˆ =e, ˆ s assgned to DC. Note that such demand assgnment s deal fo the sec c cost stuctues of the esectve DC. If we consoldate the volume and seve all the customes usng only one of the two DCs, the above calculaton shows that the nal nventoy cost wll be excessve. Ths s the fundamental eason why consoldaton stategy may not wok fo all demand scenaos The 2 -aoxmaton algothm Suose we elace the otmal nventoy cost functon n (P) by ts ue bound. Consde the followng oblem: P mn f y P mn l x k h h 2 subect to x y fo all x ˆ fo all x y 2f g fo all : G

9 Consoldaton of dstbuton centes 7 G denotes the otmal bu e cost at DC unde the demand assgnment. Let P Y denote the obectve value of the soluton Y n oblem P. Poblems (P) and (P) ae elated by the followng theoem: Theoem 3. Suose ( ˆfx g, Y ˆfy g) s an otmal soluton to Poblem (P), then Y s a feasble soluton to Poblem (P). Let P Y be the value of ths soluton and P the otmal value, we have P Y 2 P. Poof. By Theoem 2, P Y 2 P Y fo any Y. Thus, P 2 P. Hence, P Y P Y ˆ P 2 P. When the demands ae nomally dstbuted, Coollay gves an exlct exesson fo G. Then, Poblem (P) s equvalent to the followng concave attonng oblem: Fnd a atton S S 2... S of the set 2... nš (.e., [ S ˆ 2... nš S \ S ˆfo 6ˆ ) to mnmze the functon m g l 2 2S 2S whee ˆ g a b ˆf d a mn ak h h 2 b and n d a ˆ f a >, othewse. Such oblems have been extensvely studed n the oeatons eseach lteatue. In atcula, Chakavaty et al. (985) showed that wheneve g a b s concave n both a and b fo all, and f l = 2 l 2= 2 2 l n= 2 n then thee s an otmal atton wth a leasng oety. The sets S n the otmal atton s consecutve,.e., f k < l ae both n S, then 2 S f k < < l. In geneal, such stuctual esults ae not stong enough to guaantee olynomal tme constucton of the otmal soluton, snce the cost functon of any consecutve subsets deends on whch locaton they ae assgned to (.e., deendent on ). In the secal case when the cost functon g a b s ndeendent of the ndex, Chakavaty et al. (985) showed that the otmal consecutve otmze can be constucted by a smle shotest ath comutaton. The oblem has also been extended n seveal ways by many othes, notably Anly and Fedeguen (99), and Gal and Klots (995). Although the cost functon n ou nstance does deend on the locaton (.e., ndex ), the followng ooston shows that the deendence can be emoved: Pooston. Let h a b ˆmn ˆ...m g a b. Then m mn g l 2 atton S...S m ˆ 2S 2S m ˆ mn h l 2 : atton S...S m ˆ 2S 2S Futhemoe, thee s a consecutve otmal atton soluton to the ght-hand sde. Hence, the otmal consecutve atton can be obtaned by ndng the shotest ath n the gah as shown n Fg. 2. In ths gah, the cost of the edge s gven by c ˆh l k ˆ kˆ mn lˆ...m g l 2 k kˆ l k 2 k kˆ kˆ Fg. 2. Shotest ath on ths gah gves se to consecutve atton.

10 8 Teo et al. whch can be comuted n O n 2 m tme. Snce the gah s acyclc, the standad dynamc ogammng ecuson ocedue can nd the shotest ath n O n 2 stes. In summay, the concave attonng oblem, and hence Poblem (P), can be solved n O n 2 m tme. Poof. Let S... S m be an otmal consecutve atton to the left-hand sde. Suose fo some l 2 A > h@ l 2 A 2S 2S 2S 2S fo some locaton k. l A g k 2S [S k ˆ f k d@ 2S [S k 2S [S k l s f k d@ 2 k 2 2S [S k 2S l 2 A mn A f k d@ 2S k s s 2 k 2 2S 2 k 2 2S k ˆ g P l 2S l 2S 2 2S l [S k K k A v u 2K k h k k t@ l h k 2 k 2 A k 2S 2S v u 2K k h k k l h k 2 k 2 A k 2S k 2S k A h k k h k k 2 < g 2 A g l 2 A: 2S 2S 2S 2S k k Thus the atton T... T m (whee T ˆ S f 6ˆ k T ˆ and T k ˆ S [ Sk ) wll be a stctly bette atton, esultng n a contadcton. 6. Numecal exement We consde the e ectveness of the consoldaton stategy on a system wth two egonal DCs. As stated n the ntoducton, we ae concened hee only wth the faclty nvestment and nventoy costs when evaluatng the consoldaton stategy. The wost case examle above shows that consoldaton can be ``n ntely bad''. In ou numecal exement, we want to study the lkelhood of such stuaton and to undestand the tade-o between the faclty nvestment and nventoy costs. To ths end, nstead of smulatng the stochastc behavo of the system (whch gves only nsghts to dstbuton-sec c nstances), we have oted to comae the ue bounds obtaned fom the nomal aoxmaton to detemne how lkely t s fo consoldaton to be moe than 2 tmes o otmum. The exement eveals that the lkelhood of consoldaton beng ``n ntely bad'' deends on the d eence between the atos l = 2 and l 2= 2 2. In the exement, we assume k ˆK (constant) fo ˆ 2 to smlfy calculatons. Then, we have mn P l x k h s h 2 ˆ A l x fo some A and by Coollay, s G x ˆB 2 x fo some B : Exement: Gven a set of aametes A B A 2 B 2, deved fom a system wth two DCs, how lkely s t that consoldatng the two DCs nto one DC wll be n ntely wose-o? To elmnate the tval cases, we wll always geneate A and B such that A B =A 2 ˆ B 2. Othewse, f A A 2 and B B 2 (o vce-vesa), then consoldaton s clealy otmal. We geneate two tyes of oblem nstances: n Tye, A B A 2 ae unfomly cked n [, ] and n Tye 2, A B ae unfomly cked n [, ] and A 2 ˆ A 3. In the latte, A 2 A and hence B B 2. We next dscuss how we geneate the aametes fo l 2. Suose the otmal assgnment attons the demand onts nto two gous. Let l and 2 denote the total demand ate and vaance assgned to DC ˆ 2. We choose l 2 ˆ a < l ˆ b and fo both tyes of oblems, we x a ˆ, and take b fom the set f g. Note that snce the otmal atton has the consecutve oety, then by (), a mn l = 2 and b max l = 2. Hence the ato of a and b s a lowe bound of the d eence n the atos fl = 2 gˆ2. Now we let l l ˆ c whch eesents the stuaton when the demands ae consoldated, and x 2 2 ˆ. In ths way, 2 ˆ

11 Consoldaton of dstbuton centes 9 b c = c a, and l l 2 can also be exessed as a functon of a b c. We st geneate the values of A B A 2 B 2 (fo Tye and Tye 2 oblems seaately), then fo each A B A 2 B 2, wth a b angng ove the set f,... g, we geneate c s unfomly fom a b. We check next how many nstances of a b c gve se to cases whee consoldaton s not otmal fo the gven set of aametes A B A 2 B 2. Fo the exements, we want to obseve the numbe of tmes (S) when the cost aametes A B A 2 B 2 gve se to oblem nstances fo whch consoldaton may a ect the nventoy cost advesely (ndcated by the ule that ue bound functon obtaned fom consoldaton n ethe DC s lage than the total cost wthout consoldaton). We obtan nsghts to ths by seachng ove vaous combnatons of a b c. Fo such cost aametes, we also tabulate the numbe of oblem nstances (P) out of choces of (a b c) fo whch consoldaton may a ect the nventoy cost advesely. Fo Tye oblems, the value of S s as lsted n Table. The esult shows that the consoldaton stategy s etty obust n these stuatons. Fo nstance, gven geneatons of the aametes A B A 2 B 2, when a b ˆ, we geneated d eent values of c between a and b but could not nd any nstances whee consoldaton may be nfeo. Futhemoe, fo the Tye oblems, at most 4 out of geneatons of A B A 2 B 2 gve se to oblem nstances fo whch consoldaton may have an advese e ect. Ths haens n the case when a b ˆ. When such stuatons ase, the dstbuton of the numbe P s shown n Fg. 3. The y-axs ndcates the numbe of c s (out of ) whee consoldaton s nfeo. The x-axs ndcates the numbe Table. The values of S used n the Tye oblems a b (,) (,) (, ) (, ) (, ) S of the exemental tals (out of ). The gah ndcates that when at tmes whee the aametes A B A 2 B 2 ae not favoable fo consoldaton, then almost any choce of c wll esult n a case whee consoldaton may be moe than 2 tmes o otmum. The negatve e ect of consoldaton s moe aaent when the d eences of A B A 2 B 2 ae lage. Ths s seen n Tye 2 oblems, fo whch the S values ae lsted n Table 2. The esult shows that consoldaton stategy may not be vey e ectve fo Tye 2 oblems, when b=a >, as sgn cantly moe of the geneatons of A B A 2 B 2 gve se to oblem nstances fo whch consoldaton may not be otmal. The dstbuton of the numbe P s shown n Fg. 4. We obseve that n ths case when the ato b=a s aound, no negatve examle has been geneated. Ths llustates that the consoldaton stategy could be useful fo modeate values of b=a. In both oblem tyes geneated, wheneve a b ˆ, we ae not able to geneate any examle whee consoldaton may be nfeo. Thus the numecal exements seem to suot the hyothess: Gven any set of aametes A B A 2 B 2, deved fom a system wth two DCs, t s lkely that consoldaton wll be a good stategy, wheneve the atos of the mean to vaance fo each demand locaton ae smla (not moe than tmes aat). 7. Concluson The locaton-nventoy oblem n ths ae s motvated by the new tend of consoldatng and centalzng Table 2. The S values assocated wth the Tye 2 oblem a b (,) (,) (, ) (, ) (, ) S Fg. 3. Numbe of nstances whee consoldaton s nfeo fo Tye oblems. Fg. 4. Numbe of nstances whee consoldaton s nfeo fo Tye 2 oblems.

12 Teo et al. dstbuton oeatons n suly chan management. We analyzed the mact on the total faclty nvestment and nventoy costs of the consoldaton stategy. The key nsght ganed though ths study s that unlke n the classcal EO and newsvendo models, consoldaton does not always lead to educton n the total faclty nvestment and nventoy costs, and the d eences n the atos l = 2 of the demand ocesses lay an motant ole on the e ectveness of the consoldaton stategy. When the d eences ae small (less than ), the chances of the consoldaton stategy beng e ectve ae vey good. In atcula, fo Posson demand ocesses whch have the same such atos of value one, the consoldaton stategy s actually otmal. It s also otmal fo..d. demands. The ae can be extended n many dectons. We can consde mult-echelon locaton-nventoy oblems. Lm et al. (998) have obtaned some esults n ths decton. Also, the oblem wth multle oducts would be both challengng and nteestng. Acknowledgements The authos wsh to thank the Assocate Edto and two efeees fo the suggestons that have made the ae moe focused and cleae. Ths eseach was atally suoted by a gant fom the Sngaoe-MIT Allance Fellowsh 999±2 and NUS Reseach Poect Refeences Anly, S. and Fedeguen, A. (99) Stuctued attonng oblems. Oeatons Reseach, 39, 3±49. Baahona, F. and Jensen, D. (998) Plant locaton wth mnmum nventoy. Math Pogammng, 83(), ±. Chakavaty, A.K., Oln, J.B. and Rothblum, U.G. (985) Consecutve otmzes fo a attonng oblem wth alcatons to otmal nventoy goungs fo ont elenshment. Oeatons Reseach, 33, 82±834. Cohen, M. and Lee, H. (988) Stategc analyss of ntegated oducton-dstbuton systems: models and methods. Oeatons Reseach, 36, 26±228. Daan, P. (992) Benetton ± global logstcs n acton. Asa-Pac c Intenatonal Jounal of Busness Logstcs, 5(3), 7±. Een, G.D. (979) E ects of centalzaton on exected costs n a mult-locaton newsboy oblem. Management Scence, 25, 498± 5. Een, G.D. and Schage, L. (98) Centalzed odeng olces n a mult-waehouse system wth lead tmes and andom demand, n Mult-Level Poducton/Inventoy Systems: Theoy and Pactce, Schwaz, L.B. (ed.), Noth-Holland, Amstedam. 5±67. Ettl, M., Fegn, G.E., Ln, G. and Yao, D. (996) A suly netwok model wth base-stock contol and sevce equements. IBM Reseach Reot Numbe, RC Fedeguen, A. and Zkn, P. (984a) A combned vehcle outng and nventoy allocaton oblem. Management Scence, 32, 9±36. Fedeguen, A. and Zkn, P. (984b) Aoxmatons of dynamc multlocaton oducton and nventoy oblems. Management Scence, 3, 69±84. Gal, S. and Klots, B. (995) Otmal attonng whch maxmzes the sum of the weghted aveages. Oeatons Reseach, 43, 5±58. Gallego, G. (998) New bounds and heustcs fo olces. Management Scence, 44, 29±233. Geo on, A.M. and Powe, R. (995) Twenty yeas of stategc dstbuton system desgn: an evolutonay esectve. Intefaces, 25(5), 5±27. Gaves, S.C., Rnnooy Kan, A.H.G. and Zkn, P.H. (993) Logstcs of Poducton and Inventoy, Elseve, Amstedam, The Nethelands. Gouley, C. (997) Sules and demand: Stales uts en to ae and centalzes ts DC. Dstbuton, 6(), 6±62. Jedd, M. (996) Walt Dsney's logstcal magc. Dstbuton, 6(), 64±66. Lee, H. and Bllngton, C. (986) Mateal management n decentalsed suly chans. Oeatons Reseach, 4, 835±847. Lm, W.S., Ou, J. and Teo, C.P. (998) Cost e ect of consoldatng seveal one-waehouse mult-etale systems. Pent. Poe and Rete (996) Suly Chan Otmzaton: buldng the stongest total busness netwok. Beett Koehle Publshes. Revelle, C.S. and Laote, G. (996) The lant locaton oblem ± new models and eseach osects. Oeatons Reseach, 44(6), 864±874. Rheem, H. (997) Logstcs: a tend contnues. Havad Busness Revew, 6(), 8±9. Schwaz, L.B. (989) A model fo assessng the value of waehouse sk-oolng: sk-oolng ove outsde-sule leadtmes. Management Scence, 35, 828±842. Tms, H.C. and Goenevelt, H. (984) Smle aoxmatons fo the eode ont n eodc and contnuous evew (s S) nventoy systems wth sevce level constants. Euoean Jounal of Oeatonal Reseach, 7, 75±9. Vswanathan, S. and Mathu, K. (997) Integatng outng and nventoy decsons n one-waehouse multetale multoduct dstbuton systems. Management Scence, 43, 294±32. Zheng, Y.S. (992) On oetes of stochastc nventoy systems. Management Scence, 38, 87±3. Bogahes Chung Paw Teo s an Assstant Pofesso n the Deatment of Decson Scences. Well gounded n methodology. D Teo has ublshed n OR and SODA on netwok otmzaton oblems n the boad aeas of nventoy and logstcs management. Ou Jhong s a Seno Lectue n the Deatment of Decson Scences. D Ou has a tack ecod n ublshng n ntenatonally efeeed ounals. He s cuently Assocate Edto of the Asa Pac c Jounal of Oeatonal Reseach. Hs cuent eseach otfolo ncludes suly chan modellng and stochastc otmsaton. Mak Goh s an Assocate Pofesso and s cuently sevng as the Logstcs Management Coodnato n the deatment. He has ublshed wdely n ntenatonally efeeed ounals. Hs cuent eseach actvtes nclude suly chan management and stategy. He was also the Vce Pesdent of the Sngaoe OR socety, and sts on the Edtoal Boad of Suly Chan Management. Contbuted by the Inventoy Deatment