Combined make-to-order/make-to-stock supply chains

Transcription

1 IIE Tranaction , Copyright C IIE ISSN: X print / online DOI: / Cobined ake-to-order/ake-to-tock upply chain PHILIP KAMINSKY 1, and ONUR KAYA 2 1 Departent of Indutrial Engineering and Operation Reearch, Univerity of California, Berkeley, Berkeley, CA 94720, USA E-ail: kainky@ieor.berkeley.edu 2 Departent of Indutrial Engineering, Koc Univerity, Eng 206, Ruelifeneri Yolu, Sariyer, Itanbul 34450, Turkey Received October 2006 and accepted Septeber 2007 Aulti-ite anufacturer erved by a ingle upplier in a tochatic environent i conidered. The anufacturer and the upplier have to decide which ite to produce to tock and which to produce to order. The anufacturer alo ha to quote due date to arriving cutoer for ake-to-order product. The anufacturer i penalized for long lead tie, iing the quoted lead tie and high inventory level. Several variation of thi proble are conidered and effective heuritic for the ake-to-order/ake-to tock deciion are deigned to find the appropriate inventory level for ake-to-tock ite. Scheduling and lead tie quotation algorith for centralized and decentralized verion of the odel are alo developed. Extenive coputational teting i perfored to ae the effectivene of the propoed algorith, and the centralized and decentralized odel are copared in order to quantify the value of centralized control in thi upply chain. A centralized control i not alway practical or cot-effective, the value of liited inforation exchange for thi yte i explored. [Suppleentary aterial are available for thi article. Go to the publiher online edition of IIE Tranaction for the following free uppleental reource: Online appendix including additional coputational analyi and proof.] Keyword: Inventory, lead tie, cheduling, upply chain, MTS/MTO 1. Introduction Inventory cot ake up a large portion of total cot in any upply chain, o effective inventory anageent i one of the ot iportant iue facing upply chain anager. Indeed, in a ulti-facility upply chain, a critical tactical deciion involve the identification of interediate tocking point in the upply chain, and the deterination of appropriate inventory trategie at thee tocking point. Manager ut deterine, in other word, which product or coponent of product will be ade to tock, and which will be ade to order. For thoe product or coponent that are ade to tock, appropriate inventory trategie ut be deterined, and for thoe product that are ade to order, approache for cutoer lead tie quotation ut be developed. Furtherore, if different product or coponent utilize coon production reource, operating trategie for thoe reource will ipact yte perforance. Traditionally, ot copanie utilized a puh or Make-To-Stock MTS yte, holding inventory at the end of the upply chain. However, in an MTS yte, fir need to be able to etiate deand to deterine how uch Correponding author to produce and tock, and o thee yte rely heavily on forecat, which in any cae are not very accurate. Thu, any progreive copanie have hifted to pull or Make-To-Order MTO yte, holding no inventory at all and producing to order. In thee yte, copanie produce baed on actual cutoer deand intead of forecat. Inventorie are eliinated, but cutoer ut now wait for delivery, perhap leading to lo of copetitivene on the part of the fir. The deciion to ue either a puh trategy or a pull trategy for a particular product therefore depend heavily on the characteritic of the yte. Indeed, in a upply chain, uing a puh trategy for oe product or coponent, and a pull trategy for other, ight be uch ore effective then uing either yte excluively. Becaue of thi, fir are beginning to eploy a hybrid approach, a puh pull trategy, or cobined MTO MTS yte, holding inventory of oe coponent, and producing other to order. In addition to inventory deciion, the cheduling of the production of order and the approach to lead tie quotation to cutoer alo have ignificant ipact on the perforance of upply chain, particularly in MTO upply chain. Copanie need to quote hort and reliable lead tie to their cutoer to reain copetitive in the arket and to increae their profit. For a copany that produce ultiple product with different characteritic, the deciion on X C 2009 IIE

2 104 Kainky and Kaya when to produce each order affect the copletion tie of anufacturing and thu the lead tie for that product. Thu, fir need a policy or approach that will help the to decide which ite or coponent to produce to tock and which ite or coponent to produce to order, what inventory level to aintain for ake-to-tock ite, how to quote lead tie to cutoer and how to equence order to efficiently ue liited reource. In thi paper, we explore approache to thee four interrelated deciion for atwo-tage upply chain. To the bet of our knowledge, thee four interrelated deciion have never been explored iultaneouly in the context of upply chain. However, ot of the coponent of thee deciion have been conidered eparately or in pair. MTO/MTS odel have generally been tudied for ingletage yte. Willia 1984, Federgruen and Katalan 1999, Carr and Duenya 2000 and Youef et al aue that deciion regarding which ite to produce to order and which to tock are ade exogenouly fro the odel, and they try to find the bet way to operate the yte given thee deciion. In contrat, Li 1992, Arreola- Ria and DeCroix 1998 and Rajagopalan 2002 all deign odel that focu on thi deciion a well a on aociated inventory level deciion. Reearcher have introduced a variety of odel in an attept to undertand effective due date quotation and equencing. Cheng and Gupta 1989, Kainky and Hochbau 2004 and Kekinocak and Tayur 2004 urvey due date quotation odel in detail. The ajority of earlier paper on due date quotation are iulation baed. For intance, Eilon and Chowdhury 1976, Week 1979, Baker and Bertrand 1981, Miyazaki 1981 and Bertrand 1983 conider variou due date aignent and equencing policie, and in general deontrate that policie that ue etiate of hop congetion and job content inforation lead to better hop perforance than policie baed olely on job content. Soe analytical reult do exit for liited verion of thee odel. Priarily, thee conit of deterinitic, coon due date odel, where a ingle due date ut be aigned for all job, and tatic odel, where all job are available at tie 0. For thee iplified odel, a variety of polynoial algorith have been developed ee, for exaple, Seidann et al. 1981, Panwalkar et al. 1982, Hall and Poner 1991, Chand and Chhajed 1992, Kahlbacher 1992 and Brucker 1998; however, thee reult do not extend in an obviou way to ore coplex odel. A erie of paper doe ue a variety of different approache to conider ore coplex odel. Kainky and Lee 2008 derive analytical reult for a odel in which a erie of job arrive at the yte over tie and in which all order ut be accepted. In other odel, not all potential job ut be accepted. Kekinocak et al conider a odel in which the fir can decide whether or not to accept an order for the cae in which revenue decreae with quoted lead tie. Soe reearcher approach lead tie quotation odel within a queuing theoretic fraework. For exaple, Wein 1991 conider a ulti-cla M/G/1 queuing yte under the objective of iniizing the weighted average lead tie ubject to the contraint of the axiu fraction of tardy job and the axiu average tardine, Duenya 1995 and Duenya and Hopp 1995 conider odel in which a cutoer probability of placing an order decreae with increaing lead tie, and Savaaneril et al alo conider the poibility of holding inventory to reduce quoted lead tie, thu increaing the probability that cutoer will place an order. In Kainky and Kaya 2006, we analyze pure MTO upply chain and deign effective cheduling and due date quotation algorith for the centralized and decentralized verion of thoe yte. We how that thee algorith are probabilitically ayptotically optial i.e., the relative error of thee heuritic when copared to the optial olution goe to zero a n for the objective of iniizing n = n cd d i + c T T i where d i i the quoted due date for job i, T i = C i d i + i the tardine of job i and c d and c T are the unit due date and tardine cot for the odel. In thi paper we integrate due date quotation with cobined MTO MTS deciion aking, conider everal different approache for equencing job and focu on two-tage upply chain odel. Our analyi provide guidance for deciding when to ue MTS and when to ue MTO approache in upply chain, and how to effectively operate the yte to iniize yte-wide cot. We alo quantify the value of centralization and inforation in thee yte by building both decentralized and centralized odel, obtaining good olution to both of thee clae of odel, and deigning coputational experient to explore the effectivene of our algorith and to copare the centralized and decentralized yte. We focu on finding anwer to the following quetion: 1. Which ite hould be produced MTO and which one MTS at each tep of the upply chain and what are the optial level of inventory for MTS ite? 2. Which ite hould be produced next when a facility becoe available for production? 3. What due date hould be quoted to each cutoer at the tie of arrival? 4. What i the benefit of a centralized upply chain a oppoed to a decentralized yte? 5. How uch gain can be achieved through inforation exchange between upply chain eber? 2. Model We conider a erial upply chain with two tage, a upplier and a anufacturer. A trea of cutoer order arrive at the anufacturer over tie and the anufacturer quote a lead tie d j for each order j when it arrive. In thi context,

3 Cobined ake-to-order/ake-to-tock upply chain 105 the lead tie repreent the tie until the order i expected to coplete proceing. An order can be for one of K ditinct job type, each with different aociated cot. Order arrive at rate λ and each arriving order i for job type i with probability δ i, i = 1, 2,...,K.Tofacilitate our analyi, we aue exponentially ditributed, tationary and independent inter-arrival tie, o each order for job type i arrive at rate λ i = λδ i.totart proceing each order, the anufacturer require a coponent pecific to that particular job type anufactured by the upplier. The coponent for job type i require a rando proceing tie with ean µ i at the upplier, and the order require a rando proceing tie with ean µ i at the anufacturer. In general, inventory of oe or all of the job type can be held by the anufacturer, and inventory of oe or all of the coponent type can be held by the upplier. We aue no delivery lead tie between the upplier and the anufacturer, o if the anufacturer need to proce a particular type and no inventory of the coponent of that type exit at the upplier, the anufacturer can begin proceing iediately after the upplier ha proceed the neceary coponent. Our objective i to iniize the total expected inventory, lead tie, and tardine cot in thi yte. Thu = K { hi E[I i ] + ci d E[d i] + ci T E[W i d i ] +}, 1 where h i i the unit inventory holding cot of job type i, ci d i the unit lead tie cot of job type i and ci T i the unit tardine cot for job type i. E[I i ] denote the ean aount of inventory of job type i, E[d i ] denote the ean lead tie of job i and E[W i d i ] + denote the ean tardine of job type i. In general, the optial inventory policy ight be tate dependent and quite coplex. However, to facilitate our analyi, and conitent with traditional analyi of related odel ee, for exaple, Arreola-Ria and DeCroix 1998 and the reference cited therein, we aue that a bae tock policy i ued for inventory control of MTS ite, o that inventory of job type i at the anufacturer i initially R i, and whenever an order of type i arrive at the anufacturer, a production order i ent to replenih inventory. If R i = 0, a MTO production yte i eployed for job type i. Ifdeand i higher than the inventory level, then the extra deand i backlogged and atified later when it i produced. We alo aue that order arrive ufficiently lowly that the yte i table, that i we aue that for the anufacturer: K λ i/µ i < 1, and for the upplier: K λ i/µ i < 1. We conider three verion of thi odel, which we briefly decribe here and dicu in ore detail in ubequent paragraph. We analyze thee three odel verion in order to develop inight into their effective operation, and copare thee three odel to generate inight into the value of centralization. In the centralized verion of the odel, the entire yte i operated by a ingle anager, who i aware of the inventory level and proceing tie at both the upplier and the anufacturer. Thi anager decide on the inventory level for each cla of job, a well a the production chedule and the lead tie that hould be quoted to each cutoer. In the decentralized full inforation odel, the anufacturer and the upplier are aued to work independently, but the anufacturer ha full inforation about both hi/her procee and thoe of the upplier. Both partie attept to iniize their own cot in a equential anner. The upplier act firt to deterine hi/her optial inventory level and then the anufacturer act to iniize hi/her own cot uing the optial inventory level for the upplier. Finally, in the iple decentralized odel, the anufacturer and the upplier work independently fro each other, but the anufacturer ha very liited inforation about the upplier tatu. The objective of the anager or anager in each of thee yte i to iniize holding, lead tie and tardine cot, and to do thi, due date quotation, equencing and inventory anageent have to be coordinated. Ideally, thi would require iultaneou conideration of thee three iue, but thi i intractable. Thu, the approach we have elected to follow for thi odel and throughout the paper i lightly different. Oberve that if anageent were able to quote lead tie that preciely atched the waiting tie of order in the yte, the objective would iplify to iniizing K {h ie[i i ] + c d i E[W i]}. Although it i not poible to be preciely accurate with lead tie quotation in thi yte, thi obervation otivate our approach: we firt deterine a cheduling approach deigned to reduce the u of waiting tie, and then baed on that chedule, we find the optial inventory level to iniize K {h ie[i i ] + c d i E[W i]} and deign a due date quotation approach that preent due date that are generally cloe to the copletion tie uggeted by our cheduling approach. We note that thi type of approach wa applied in Kainky and Kaya 2008 for pure MTO verion of thi yte. In the next ection, we introduce a preliinary inglefacility odel and analyze thi odel. In Section 4, we preent our upply chain odel, algorith and reult in detail, and in Section 5, we preent our coputational analyi. 3. Single-facility odel 3.1. The odel Although our ultiate goal i to analyze ulti-facility yte, we begin with a preliinary analyi of a inglefacility yte. We focu on finding the optial inventory level for thi yte and the condition under which an MTO trategy or an MTS trategy i preferred for thi facility, a well a on deigning effective cheduling and due date quotation heuritic for thi yte. We focu on a ingle facility identical to the anufacturer upplier yte

4 106 Kainky and Kaya decribed above, except that the anufacturer doe not need to obtain a coponent fro the upplier. Thi odel can be conidered a pecial cae of the upply chain odel decribed above where the proceing tie and cot at the upplier are all zero, o that the coponent are available to the anufacturer a oon a an order arrive. In other word, a ingle anufacturer face a trea of cutoer order that arrive over tie and the anufacturer quote a lead tie d j for each order j when it arrive. There are K ditinct job type, each with different aociated cot. Order arrive at rate λ and each arriving order i for job type i with probability δ i, i, i = 1, 2,...,K. Inter-arrival tie are exponentially ditributed, tationary and independent, and each order of type i require a rando proceing tie not known until proceing i coplete with ean proceing tie µ i. The facility can utilize both MTS and MTO policie to iniize inventory and lead-tie-related cot. Our goal i to find an effective operating policy to iniize thee cot. An operating policy for thi proble conit of value for bae tock level R i for each job type a well a approache for equencing and lead tie quotation. Recall that we will ipleent a three-phae approach to thi proble. Firt, we will develop a equencing approach, and then we will develop an inventory etting approach and a due date quotation approach. To analyze thi yte, we divide job in proce and waiting to be proceed into two clae, cutoer job, repreenting actual order, and replenihent job, repreenting production order whether they exit to replenih inventory or to eet external deand. Whenever an order arrive at the yte, if that ite exit in the inventory, the deand i iediately atified fro the inventory. Since that order i iediately atified, we do not add a cutoer job to the yte, but we do add a replenihent job to the yte, ince it ut be anufactured to replenih inventory. On the other hand, when an order arrive and there i no inventory of that ite, a cutoer job i placed in the production queue, and a lead tie i quoted. Thu, at any tie the production queue ay contain cutoer job repreenting currently unatified order, and replenihent job that are going to be produced to replenih inventory. We collectively refer to cutoer and replenihent job in the production queue a job in the production queue. Oberve that the production queue operate a in a pure MTO yte becaue job are placed in the production queue to replenih inventory even though inventory i on hand to eet an order. Thu, the inventory level of an ite whether poitive or zero doe not ipact the production proce, but doe decreae the due date cot ince we atify thoe order iediately Analyi and reult A dicued above, our approach to thi proble tart with a equencing rule. Motivated by the effectivene of the SEPTA rule the rule of chooing the job with the Shortet Expected Proceing Tie aong Available job to iniize the total copletion tie for the pure MTO yte analyzed in Kainky and Kaya 2008, we equence job in the production queue according to SEPTA. Under thi rule, each tie a job coplete proceing, the hortet available job in the expected ene that ha yet not been proceed i elected for proceing. To quote a due date, we utilize a odified verion of the approach initially introduced in Kainky and Kaya When an order j for job type i arrive at the yte at tie r j, the following lead tie d j i quoted: { 0 ifii > 0atr j, d j = E[p i ] + E[M j ] + E[M j]λψ i τ i 1 λψ i τ i otherwie. Note that in general, if order j cannot be filled fro inventory, oe job in the production queue, job k, will be ued to atify that order. Thu, M j i the workload in front of job k at the tie of arrival that i, the total proceing tie of job to be proceed before job k, ψ i i the probability that an arriving job ha a proceing tie le than p i and τ i = E[p p < p i ]ithe expected proceing tie of a job given that it i le than p i. Thi lead tie quotation rule i deigned o that the following lea i true. Lea 1. For the odel decribed above, for every job cla i, the expected value of the quoted lead tie for that cla i i equal to the expected waiting tie of a cla i order. In other word, for all i, E[d i ] = E[W i ] where W i denote the actual waiting tie of order type i. We call thi equencing/due date quotation approach SEPTA-LTQ, and baed on the reulting chedule, we deterine the optial inventory level for each cla. We note, however, that our analyi i aenable to odification if different equence are conidered. In particular, we require the tationary ditribution of the nuber of job of each type in the yte, and thi depend on the equencing rule. To find the optial inventory level, we iniize the following objective function: K { hi E[I i ] + ci d E[W i] }, 2 where E[I i ]ithe expected inventory level and E[W i ]ithe expected waiting tie for job type i. There i an obviou trade-off between the inventory cot and the waiting tie cot in thi function. We can decreae the waiting tie of cla i job by holding additional inventory of that type but doing o will increae inventory cot. Oberve that holding additional inventory for an ite doe not affect the production queue. Thu, the waiting tie and the lead tie quotation procedure for the other clae are not ipacted by inventory deciion.

5 Cobined ake-to-order/ake-to-tock upply chain 107 By Little law, we can write E[W i ] = E[N i ]/λ i and the objective function 2 becoe: K { hi E[I i ] + ci d E[N } K i]/λ i = {h i E[I i ] + c i E[N i ]}. 3 Then, we tate the following lea regarding the objective function 2. Lea 2. Objective function 2 i equivalent to the following function: K { hi E[I i ] + ci d E[W i] } = K {h i E[I i ] + c i E[N i ]} K R i = h i R i xf i x + c i x R i f i x, 4 x=0 x=r i where f i x denote the probability of having x job of type i in the production queue, c i = ci d/λ i and E[N i ] i the expected nuber of order for job type i waiting for production or equivalently, the expected nuber of cutoer job of type i in the production queue. An expreion of the for of Equation 3 wa conidered in Arreola-Ria and DeCroix 1998 for a different odel, and they derive reult iilar to thoe preented below for a firt-coe firt erved FCFS queueing dicipline. Below, we rederive their reult for copletene and clarity of expoition uing the notation of our odel, and extend thi analyi to SEPTA chedule. Lea 3. Optial inventory level for each job type can be found by decopoing the proble into K ubproble, one for each type. The proble therefore reduce to iniizing h i Ri x=0 R i xf i x + c i x=r i x R i f i xfor each i. Theore 1. The optial level of inventory R i i the iniu value x 0 that atifie: F i x c i. c i + h i Alo, it follow that: Corollary 1. It i optial to produce ite i MTO if and only if F i 0 c i /c i + h i. Corollary 2. A job type inventory level decreae and thu it ove toward MTO if it unit lead tie cot, proceing tie or arrival rate decreae or it unit holding cot increae. Note that thee reult are not retricted to ingle-erver queue or the SEPTA chedule, and they are independent of the arrival or anufacturing proce. However, thee characteritic affect F i x, the tationary ditribution of the nuber of ite of type i in the yte. To ae the effectivene of our cheduling algorith and to explore how the cheduling approach ipact F i x and the objective function, we analyze two different chedule, SEPTA and FCFS, and preent the following two reult for thee chedule. We eploy thee reult in our coputational analyi in Section 5. Corollary 3. If the FCFS cheduling rule i ued in the production queue, and we have an M/G/1 queue, to iniize Equation 4 it i optial to produce product i MTO if and only if K δ j E[e λ iµ j ] 1 δ ir i, r i 1 ρδ i j=1 where r i = c i /c i + h i, δ i = λ i /λ and ρ = K λ iµ i. Corollary 4. For acla i, let a repreent the et of job type with expected proceing tie le than that of type i, and let brepreent the et of job type with expected proceing tie greater than that of type i. Then, if the SEPTA cheduling rule i ued in the production queue, it i optial to produce product type i to order if and only if 1 ρλ i + λ a λ a ν a λ i + λ b 1 γ b λ i + λ a λ a ν a λ i λ i γ i λ i + λ a λ a ν a λ i r i where r i = c i /c i + h i,ρ= K λ iµ i,λ a = j a λ j i the total arrival rate of cla a job,λ b = j b λ j i the total arrival rate of cla b job,γ i z = E[e zp i ] i the Laplace tranfor aociated with the proceing tie of cla i and ν a z i the Laplace tranfor aociated with the length of a buy period in which only cla a job are proceed which can be found by olving the equation ν a z = γ a z + λ a λ a ν a z. Siilarly, the optial inventory level for MTS ite can be obtained eploying Theore 1 and uing the pgf given in the proof of Corollarie 3 and 4 in the online uppleent. 4. Supply chain odel Building on our analyi in Section 3, we analyze the inventory deciion, cheduling and due date quotation iue for two-tage upply chain. A we did for the ingle-tage yte, we develop effective heuritic to deterine optial inventory level at both facilitie and deign effective algorith for cheduling and due date quotation for both the centralized and decentralized verion of thee yte. Thee algorith allow u to copare the value of centralization and inforation exchange in upply chain under a variety of different condition. Oberve that when a anufacturer i working with a upplier, required coponent obtained fro thi upplier ay not be iediately available to the anufacturer at the tie an order arrive at the anufacturer. If thi i the cae, the anufacturer ha to wait for thee

6 108 Kainky and Kaya coponent before initiating production of that order. Thu, the upplier anufacturer relationhip ipact inventory level, a well a cheduling and lead tie quotation deciion. We odel thi yte a a two-facility erial upply chain with a anufacturer and a upplier where both partie can chooe to tock oe of the ite and ue an MTO trategy for the other in a ulti-ite, tochatic environent. A before, we aue that the upplier and the anufacturer eploy a one-to-one replenihent trategy and a bae tock policy for inventory control of their ite. The anufacturer tart with an inventory of Ri unit of finihed good and the upplier tart with an inventory of Ri unit of ei-finihed good that the anufacturer need to coplete hi/her production. We again define production replenihent job a we did for the ingle-facility odel. When an order arrive, if that ite i in the anufacturer inventory, the order i iediately atified. At the ae tie, a replenihent job of that cla i tarted in the yte, although the order i not placed in the anufacturer production queue until the coponent required for that job i received fro the upplier. If the upplier ha an inventory of the appropriate coponent, it i ent iediately to the anufacturer, and o the replenihent job appear iediately in the anufacturer production queue. If the upplier doe not have a poitive inventory level of that coponent, the upplier ut proce the coponent before ending it to the anufacturer, at which point the coponent appear in the anufacturer production queue. Note that regardle of whether or not the coponent i in inventory, the upplier place that job in hi/her production queue, either to hip to the anufacturer or to replenih the upplier inventory. If that ite i neither in the anufacturer nor the upplier inventory, then a lead tie i quoted to the cutoer and a cutoer job order i ent to both facilitie to atify thi order. Thi cutoer job appear iediately in the upplier production queue, and after it i delivered fro the upplier, it appear in the anufacturer production queue. If the ite i in the upplier inventory but not in the anufacturer inventory, then a cutoer job iediately appear in the anufacturer production queue, and areplenihent job i placed in the upplier production queue. A horter lead tie i quoted in thi cae ince the cutoer only need to wait for the production at the anufacturer, and waiting tie at the upplier i zero. Let xi and xi denote the nuber of job of type i in the upplier and anufacturer production queue, repectively. Then, let Ni denote the nuber of job of cla i waiting in the upplier production queue that hould be delivered to the anufacturer iediately upon copletion, Ii denote the aount of coponent inventory of cla i, Ni denote the total nuber of cutoer of cla i waiting in the yte for their order to be filled and Ii denote the aount of finihed good inventory at the anufacturer of cla i. Then Ni = ax { xi Ri, 0} Ii = ax { Ri xi, 0} Ni = ax { xi + ax xi Ri, 0 Ri, 0 } Ii = ax { Ri xi ax xi Ri, 0, 0 }. 5 In the following ubection, we develop heuritic for the centralized and decentralized upply chain for cheduling, inventory and lead tie quotation. Conceptually, each of our heuritic ha three ain tep: Step 1. Deterine the cheduling rule to be ued in each facility and find the tationary ditribution of the nuber of job in the yte. Step 2. Baed on the tationary ditribution and the objective function to be iniized, deterine the baetock inventory level. Step 3. Quote lead tie to the arriving cutoer baed on the chedule ued, the inventory level and the current tate of the yte at the tie of the arrival The centralized upply chain odel The odel When the anufacturer and the upplier belong to the ae fir, the yte can be odeled with a central agent that ha coplete inforation about both tage and ake all the deciion about both tage clearly, thi type of centralized control will be ore effective in iniizing the total cot in the yte. In thi ection, we conider thi type of odel, in which we aue that the anufacturer and the upplier work a a ingle entity and they are both controlled by the ae agent that ha all the inforation about both ide. The deciion about the cheduling at both facilitie and lead tie quotation for the cutoer a well a the inventory level for each party are ade by thi agent. In the centralized odel, our objective function to iniize i K { h i E [ Ii ] + h i E [ Ii ] + c d i E[d i ] + ci T E[W i d i ] +}, 6 where hi i the unit holding cot of ei-finihed good at the upplier and hi i the unit holding cot of finihed good at the anufacturer Analyi and reult In thi yte, we ue an approach iilar to the one we ued for the ingle-facility cae. We firt find the optial inventory level for any chedule that i independent of the workload or inventory level in the yte, in order to iniize the objective function: R, R = K { h i E [ Ii ] + h i E [ Ii ] + c d i E[W i ] }. 7

7 Cobined ake-to-order/ake-to-tock upply chain 109 Then, we preent an effective cheduling algorith conitent with thi odel with the goal of iniizing the total waiting tie of the job in the yte and a lead tie quotation algorith that atche thee waiting tie. Uing the definition 5 and the equation ci d = λc i and E[W i ] = E[Ni ]/λ i due to Little law, we rewrite the objective function in ter of inventory level and job in the production queue: R, R = + c i E [ N i + h i + [ R i 1 R i +R i y i =R i K Ri, K { R i = h i E [ Ii ] + h i E [ Ii { ]} K = R i yi =0 yi =0 Ri +R i yi y i =0 h i R i y i P x i = y i, x i = y i [ R i 1 + c i + yi =0 yi =Ri yi =R i yi =Ri +R i yi R i R i y i P x i = y i yi =0 P x i = y i, x i = y i R i + R i y i y i ] y i R i P x i = y i, x i = y i P x i = y i, x i = y i y i + y i R i R i ]}, 8 where R and R are the array of inventory level at the upplier and the anufacturer and xi and x i are the nuber of cla i job in the upplier and anufacturer production queue, repectively. Oberve that for any chedule independent of inventory level and workload, both the upplier and the anufacturer production queue operate independent of the value of R, and the upplier production queue i independent of R. Unfortunately, the tate of the anufacturer production queue, and thu the tationary ditribution of nuber of job at the anufacturer f x, i a function of R. Theore 2. For fixed inventory level Ri for each cla i at the upplier, the optial level of inventory for the anufacturer are the iniu Ri value that atify: P xi Ri, x i Ri + P x i > Ri, x i + xi Ri + Ri c i c i + hi. 9 Thu, if the upplier i MTO inventory zero or the upplier inventory level i otherwie known we can characterize the optial anufacturer inventory level. ] Corollary 5. If the anufacturer i working with a pure MTO upplier, then the anufacturer optial inventory level for each cla are the iniu Ri value that atify: P xi + xi Ri c i c i + hi. 10 If thi i not the cae, we need to tart to ake approxiating auption to begin to characterize the anufacturer inventory level: Lea 4. If we aue that the change in the tationary ditribution of the nuber of job at the anufacturer i negligible with repect to w.r.t. a unit increae in the aount of the inventory level at the upplier, the optial level of R i i non-increaing w.r.t. R i. Unfortunately, in general, finding optial inventory level i difficult, becaue a entioned above, f x, the ditribution of the nuber of job at the anufacturer production queue, depend on the inventory level R at the upplier. If for the centralized odel, hi hi for a product type i, wecan how that an MTO trategy for type i at the upplier i optial, but typically we would not expect thi condition to hold that i, typically hi < hi. Thu, we are otivated to approxiate the optial R value. To do thi, we aue that the change in the tationary ditribution of the nuber of job at the anufacturer i negligible w.r.t. a change in the aount of the inventory level at the upplier. In thi cae, we can divide the proble into K ubproble and analyze each cla eparately. In the reaining part of thi ection, we tate our reult under thi auption. Unfortunately we cannot prove a reult iilar to Theore 2 for the inventory value at the upplier, ince the objective function 8 doe not poe the convexity tructure in Ri for fixed Ri. That i to ay that Ri + 1, Ri Ri, R i inot non-decreaing in Ri for every Ri. In order to deterine the optial level of R,wetherefore eploy a one-dienional earch on Ri.For each R i, we calculate the optial value of Ri, calculate the total cot uing the objective function 8 and pick the pair with iniu cot for each cla i. However, uing the propertie of the objective function, we decreae the ize of the earch pace by finding an upper bound on the value of Ri a in the following theore. Theore 3. Let R i 0 be the iniu value that atifie: P xi R i c i c i + hi. Then, the optial level of inventory Ri R i. We conclude that there i no need to earch for Ri beyond R i. Alo, thi bound allow u to oberve that: Corollary 6. It i optial for the upplier to ue an MTO trategy to produce product type i if F i 0 c i/c i + h i.

8 110 Kainky and Kaya In general, for the fixed inventory value Ri at the anufacturer, if for every Ri : 2 Ri, R i 2 Ri 0, then the following theore hold. An exaple of thi cae occur when hi hi. Theore 4. For fixed inventory level Ri at the anufacturer, if 2 Ri, R i / 2 Ri 0, the optial level of inventory at the upplier are the iniu value Ri that atifie: h i + c i P x i Ri + h i + c i P x i > Ri, x i + xi Ri + Ri ci. 11 Once inventory level are deterined, we turn to equencing and lead tie quotation deciion. Once again, we generalize the approache in Kainky and Kaya 2008 to MTS/MTO yte. In particular, we aue that proceing tie at the upplier and the anufacturer are exchangeable, and we proce the job according to SEPTA p Shortet Expected Proceing Tie Available rule, baed on total expected proceing tie p i = pi + pi atthe upplier, and in FCFS order at the anufacturer 1. To quote a due date, we utilize a odified verion of the approach initially introduced in Kainky and Kaya Thi approach wa hown to be able to iniize the objective function E[d i ] + E[W i d i ] +. Thi lead tie quotation algorith i deigned auing that job are cheduled according to SEPTA p at the upplier and FCFS at the anufacturer, and uing the inventory value and tate of the yte at the tie of job arrival. When an order j for job type i arrive at the yte at tie r j, the following lead tie d j i quoted: 0 if Ii > 0atr j, E [ ] pj + t j if Ii = 0, Ii > 0atr j, d j = dj + E [ ] { pj + ax t j + tj +lack j dj, 0} otherwie, where dj = E [ pj ] [ ] E [ ] M + E M j λpr{p < pj }E{p p < p j } j + 1 λpr{p < p j }E{p p < p j }. Let k denote the job in the production queue ued to atify order j. Then, Mj i the workload at the upplier in front of job k at tie r j, tj = l A E[p l ]where A i the et of job in the upplier queue cheduled before job k and to be ent to the anufacturer iediately after being proceed at the upplier and tj = l B E[p l ] 1 In Kaya 2006, we conider cheduling and lead tie quotation if proceing tie at the upplier and the anufacturer are not exchangeable. In eence, we focu on the bottleneck facility and develop a cheduling algorith iilar to SEPTA p and a lead tie quotation algorith for that chedule. where B i the et of job in the anufacturer queue at tie r j, lack j = d j p j λ pr{p < p j}e{p p < p j }+ l L in{d j p j λ l, I l }E[p l ]where L i the et of job type that will be cheduled after k at the upplier. Note that if we have inventory of job type l L, thoe job will be ent to the anufacturer before job k even though they are longer than k and ay caue an increae in the copletion tie of k, thu we add the lat coponent in the lack calculation to account for thi effect. A in the ingle-tage cae, the chedule ued in the yte only ipact the tationary ditribution of the nuber of job in the yte i.e., f x. With the equencing rule we preented above, the upplier i uing the SEPTA rule with repect to total proceing tie of the job and the anufacturer i cheduling job FCFS. Since the production queue at the upplier i independent of R and R,it operate jut like the ingle facility decribed in Section 3, o we can find the tationary ditribution of the nuber of job in the upplier production queue uing the pgf given in the proof of Corollary 4 in the online uppleent. However, the inventory level at the upplier affect the procee at the anufacturer, aking exact deterination of the tationary ditribution analytically difficult. Thu, we are otivated to ue the uual decopoition approach to approxiate the tationary ditribution of the nuber of job at the anufacturer. Note that when the inter-arrival tie are exponentially ditributed in a queueing odel like the well-known Jackon network odel preented firt in Jackon 1963, the departure proce i Poion ditributed. Since the inter-arrival tie to our yte i exponentially ditributed, we approxiate the departure proce fro the upplier with a Poion ditribution and thu we aue that the arrival to the anufacturer are Poion. Thu, we treat the procee at the anufacturer a a ingle facility with ultiple clae with Poion arrival cheduled FCFS. The probability generating function for the nuber of cutoer at teady tate in thi yte i given in the proof of Corollary 3 in the online uppleent. The required probabilitie to olve thi odel can be obtained uing thi pgf The decentralized upply chain, full inforation odel While oe upply chain are relatively eay to control in a centralized fahion, often thi i not the cae. Even if the tage in a upply chain are owned by a ingle fir, inforation yte, control yte and local perforance incentive need to be deigned and ipleented in order to facilitate centralized control. In any cae, of coure, the upplier and anufacturer are independent fir, with relatively liited inforation about each other. Ipleenting centralized control in thee upply chain i typically even ore difficult and cotly, ince the fir need to coordinate their procee, agree on a contract, ipleent an inforation technology yte for their procee, etc. Thu, for

9 Cobined ake-to-order/ake-to-tock upply chain 111 either centrally owned or independent fir, centralization ight not be worth the effort if the gain fro centralization are not ubtantial. Although centralization in upply chain ay be difficult and cotly to ipleent, decentralized yte ay be ubtantially le profitable. In oe cae, rather than agreeing to centralized control, fir ay elect to hare inforation in order to hopefully increae profit. We are therefore otivated to explore the gain fro inforation exchange, a thee gain ay approach thoe of centralization. In thi odel, the anufacturer ha coplete inforation about the entire yte, but ha no control over the upplier deciion. For thi yte, we find the optial inventory level for the anufacturer and the upplier, a well a an effective cheduling and due date quotation algorith. We alo coputationally ae the difference between the centralized odel and thi decentralized odel in the next ection The odel In thi decentralized upply chain odel, we aue that the two partie work independently fro each other and ai to iniize their own cot. However, the anufacturer ha full inforation about the procee at the upplier a well a hi/her own procee. We aue that the upplier alo incur a unit lead tie cot of ci in addition to hi/her unit holding cot of hi. Thu, the upplier work a a ingle facility ee Section 3 where the anufacturer i the cutoer of the upplier. The anufacturer ha a unit lead tie cot of c i and a unit holding cot of hi per unit tie for hi/her inventorie. Since the upplier work independently fro the anufacturer and trie to iniize hi/her own cot, the reult fro the ingle-facility cae Section 3 apply for optial inventory level deterination and equencing at the upplier. Then, for that equence and et of inventorie at the upplier, we find the optial inventory level and deign an effective cheduling and lead tie quotation algorith for the anufacturer Analyi and reult A in Section 3, the upplier objective i to iniize K {h i E[I i ] + c i E[N i ]}. Thu, the optial inventory level for cla i job at the upplier, Ri,ithe iniu x 0 that atifie: F i x c i ci +. h i Uing the approach in Section 3, the optial inventory level for the upplier can be obtained. We can alo tate the following corollary of Theore 3. Corollary 7. The optial inventory level for the upplier in the decentralized, full inforation cae will be greater than or equal to the optial level in the centralized cae when ci = c i and if the ae chedule are ued for both cae. Auing that the upplier ue the optial inventory level for hi/her facility, the anufacturer trie to iniize hi/her own cot for thoe fixed R value. Thu, we attept to iniize the objective function: K { h i E [ Ii ] + ci E [ Ni ]}. Uing the definition 5 and writing the explicitly for fixed R,weget the objective function that the anufacturer wihe to iniize: K R, R { = h i E [ Ii ] + ci E [ Ni ]} = { K h i R i +R i + y i =R i [ R i 1 R i yi =0 yi =0 Ri +R i yi y i =0 Pvx i = y i, x i = y i [ R i 1 +c i + yi =0 yi =Ri yi =R i yi =Ri +R i yi R i y i R i + R i y i y i ] y i R i P x i = y i, x i = y i P x i = y i, x i = y i P x i = y i, x i = y i y i + y i R i R i ]}. 12 Oberve that both the upplier and the anufacturer production queue operate independent of R for fixed R. Then, for fixed inventory level Ri for each cla i at the upplier, the optial level of inventory Ri for the anufacturer can be found by eploying Theore 2. Alo, the optial inventory level for a anufacturer working with a pure MTO upplier atify Corollary 5. We alo tate the following Corollary uing Corollary 7 and Lea 4. Corollary 8. The optial inventory level for the anufacturer in the decentralized, full inforation cae will be le than or equal to the optial level in the centralized cae if ci = c i and the ae chedule are ued in both cae auing that the change in the tationary ditribution of the nuber of job at the anufacturer i negligible w.r.t. a unit increae in inventory level at the upplier. Oberve that the optial inventory level for cla i at the upplier decreae a ci decreae. If we can chooe the unit lead tie cot ci to charge the upplier for backlog of type i,wecan coordinate the decentralized upply chain by charging ci ci d uch that the upplier inventory aount will be decreaed and the anufacturer inventory level will be increaed to the level that they carry in the centralized odel for the ae chedule. In thi cae, the total cot of the centralized upply chain can be achieved in thi decentralized odel.

10 112 Kainky and Kaya However, note that it ight not be in the bet interet of the anufacturer to decreae c in the decentralized odel, becaue decreaing c will decreae the upplier cot by decreaing the anufacturer gain, and the benefit gained by the anufacturer fro the decreae in the total upply chain cot ight not be worth the decreae in hi/her gain. It would be poible though to deign contract between the upplier and the anufacturer that will coordinate the upply chain and achieve the iniu total cot in the upply chain by dividing the gain between the partie in an efficient anner or by aking the upplier pay a certain percentage of hi gain to the anufacturer uch that both partie will benefit fro thi coordination and will agree to take part in it. We leave the detail of thee contract for future reearch. Next, we get the following corollary by ubtituting R i = 0inTheore 2. Corollary 9. Uing an MTO trategy for cla i job at the anufacturer i optial if and only if P x i R i, x i c i 0 c i + hi, 13 where Ri i the optial inventory level for cla i at the upplier. Since the anufacturer i working independently fro the upplier, he/he alo ue an SEPTA chedule according to hi/her own proceing tie to equence hi/her job, and eploy a lead tie quotation algorith iilar to the centralized odel ince he/he ha full inforation about the upplier. Thu, he/he will quote lead tie for order j of job type i a follow: 0 if I i > 0atr i, d j = E [ ] pj + t j + t j λψj τj if I 1 λψj τj i = 0, Ii > 0atr j, dj + E [ ] pj + M j + lack j otherwie, where dj = E[pj ] + E[M j ] + lack j, lack j = E[Mj ] λψ j τ j / 1 λψj τ j and ψ j = pr{p < pj }, τ j = E{p p < pj }, ψj = pr{p < pj }, τj = E{p p < pj }. For the equation above, let k denote the job in the production queue ued to atify order j note that k and j are job of the ae type and thu have equal expected proceing tie, A denote the et of job l in upplier queue with E[pl ] < E[pk ] cheduled before job k and to be ent to anufacturer iediately after being proceed intead of being kept a inventory at the upplier, and B denote the et of job at anufacturer queue at tie r j with E[pl ] < E[pk ]. Then, M j i the etiated workload at the upplier in front of job k at tie r j, tj = l A E[p l ]i the etiated workload for the anufacturer becaue of the current job at the upplier queue in front of job k, and tj = l B E[p l ]ithe etiated workload for the anufacturer becaue of the current job at the anufacturer queue that are horter than job k. Let L be the et of job of cla l with E[pl ] < E[pk ] and E[pl ] E[p k ], and then l j = lackj + E[Mj ]λ pr{p < pj, p < pj }τj + l L in{lack j + E[Mj ]λ l, Il } E[p l ] i the etiated workload for the anufacturer becaue of job with horter proceing tie at the anufacturer than job k that have not arrived to the yte yet but are expected to arrive before job k i finihed proceing at the upplier and are expected to ove to the anufacturer before job k. Finally, Mj = ax{tj + tj + lj dj, 0} i the etiated workload at the anufacturer when job k i oved to the anufacturer fro the upplier to atify order j and lackj = Mj λψk τ k /1 λψ k τ k ithe etiated workload for the anufacturer becaue of future job with horter proceing tie at the anufacturer that will arrive at the anufacturer while job k i till in the anufacturer queue. In thi cae, the tationary ditribution of the nuber of job at the upplier can be found uing the pgf given in the proof of Corollary 4 in the online uppleent. The reult fro the ingle-facility cae hold for the upplier. The anufacturer i alo cheduling job according to SEPTA. A in the centralized cae, we ue the decopoition approach to approxiate the tationary ditribution of the nuber of job at the anufacturer, and aue that the arrival to the anufacturer are exponentially ditributed. Thu, the tationary ditribution of the nuber of job at the anufacturer can alo be found by the pgf given in the proof of Corollary 4 in the online uppleent. Note that ince the anufacturer and upplier have different proceing tie for each cla and thu different chedule, their tationary ditribution will alo be different although both are found through the ae pgf Siple decentralized odel In any real yte, the upplier and the anufacturer have very liited inforation about each other. The anufacturer i unaware of the procee at hi/her upplier and need to ake hi/her own deciion without any inforation fro the upplier. The anufacturer ay only be aware, for exaple, of the average tie it take for the upplier to proce an order and end it to hi/her. For our odel of thi kind of decentralized upply chain, we deterine optial inventory level for the anufacturer and an effective way to chedule the order and to quote due date to the cutoer with little inforation fro the upplier. In the next ection, we copare the perforance of thi iple decentralized odel to that of the centralized odel and the decentralized odel with inforation exchange to provide inight a to whether or not centralization and inforation exchange are worth the effort The odel In thi odel, we again aue that the anufacturer and the upplier work independently fro each other, but in

11 Cobined ake-to-order/ake-to-tock upply chain 113 thi cae, the anufacturer ha no inforation about the upplier, and thu he/he cannot deduce the production chedule or inventory level at the upplier. The upplier till behave a in the inforation-haring odel above, and thu thoe reult till hold. However, in thi cae, the anufacturer only know the average tie it take for the upplier to deliver a job type i, denoted by E[di ]. Thu, for each job, the anufacturer act a if each job of type i i going to be delivered to hi/her fro the upplier after E[di ] tie unit. Baed on thi auption, we deterine the optial inventory level for the anufacturer and deign an effective cheduling and lead tie quotation algorith Analyi and reult Since the upplier act a a ingle facility a in the previou ection, the optial inventory level for cla i job, Ri are the iniu value that atify: Fi R c i i ci +. h i However, in thi cae, ince the anufacturer only know the average tie it take for an order of type i to be delivered by the upplier, when an order arrive, he/he aue, that order will be delivered by the upplier to hi/her after E[di ] tie unit. We odel the upplier a an M/D/ queue with deterinitic proceing tie E[di ]for type i and without any inventorie, o each job of type i will take exactly E[di ] tie unit to be delivered fro the upplier to the anufacturer a aued by the anufacturer; we ue analyi iilar to that of previou ection, o the anufacturer attept to iniize the following function: K R { = h i E [ Ii ] + ci E [ Ni ]} K = { h i + c i [ [ R i y i +y i =0 y i +y i =R i R i y i y i y i + y i R i P x i + x i = y i + y i P x i + x i = y i + y i ] ]}. Uing Corollary 5, the optial level of inventory are the iniu Ri value that atify: P xi + xi Ri c i c i + hi. We find the tationary ditribution of the job at the upplier uing the M/D/ queue odel with proceing tie E[di ]. Since we aue that each order i delivered to the anufacturer at tie r i + E[di ]bythe upplier, the procee at the upplier do not affect the arrival proce to the anufacturer and the ditribution of the inter-arrival tie of job to the anufacturer follow the ae exponential ditribution a the inter-arrival tie of job to the yte. Thu, the arrival proce of the job to the anufacturer i alo Poion ditributed, o the tationary ditribution of the nuber of job at both facilitie are calculated uing the pgf given in the proof of Corollary 4 in the online uppleent. Thu, uing the relation given in Section 3 for a ingle facility, we find the tationary ditribution of the nuber of job and the inventory level for the anufacturer and the upplier. Since the anufacturer focue on hi/her own objective, he/he ue an SEPTA chedule according to hi/her own proceing tie. We aue that the anufacturer ha no inforation about the upplier and i unaware of the chedule or inventory level there, but fro hi/her previou experience, he/he can deduce an average delivery tie for each cla of product. Thu, he/he ue thi piece of inforation to quote due date to hi/her cutoer. We ue a iilar approach to the previou cae to quote due date, but now, uing only the available inforation that the anufacturer ha, we etiate oe of the value that were known by the anufacturer in the previou cae we conidered. Thu, we quote lead tie for order j of job type i a follow: 0 if Ii > 0atr j, d j = E [ ] pj + t j + t j λψj τj if I 1 λψj τj i = 0, Ii > 0atr j, E [ ] [ ] di + E p j + M j + lack j otherwie, where k i the job in the production queue ued to atify order j and E[di ]ithe average delivery tie of cla i product fro the upplier. We ue the ae notation a in the previou ection, but in thi cae we only have inforation about the anufacturer. B denote the et of job in the anufacturer queue at tie r j with E[pl ] < E[pk ] and tj = l B E[p l ]. lj = E[di ]ψ k τ k /E[p ]ithe etiated workload of the anufacturer becaue of the job that will arrive fro the upplier before job k and have horter proceing tie than pk at the anufacturer. Then, Mj = ax{tj + lj E[di ], 0} i the etiated workload that job k ee when it arrive at the anufacturer, and finally, lackj = Mj λψk τ k /1 λψ k τ k i the etiated workload at the anufacturer becaue of future job with horter proceing tie than pk at the anufacturer that will arrive at the anufacturer while job k i till in the anufacturer queue. 5. Coputational analyi Uing the approache for inventory value deterination, cheduling and lead tie quotation developed in the previou ection, we deigned everal coputational experient to ae the effectivene of our algorith, how the perforance of cobined MTS/MTO yte differ fro that of pure MTO or MTS yte, and the difference in perforance between centralized and decentralized upply chain. For the centralized yte, we aue that both partie cooperate and ue the heuritic explained in Section 4.1 for finding the inventory level, cheduling, and lead tie

12 114 Kainky and Kaya quotation. For the decentralized yte, we aue that each facility act independently and ue the heuritic for etting inventory level, cheduling and lead tie quotation intended to iniize it own cot a explained in Section 4.2 and 4.3. We ipleented our heuritic in C++, and preent the reult in the following ubection Single facility We perfored a iulation tudy to ae the effectivene of our heuritic for ingle-facility odel, and to deterine the value of ipleenting a cobined MTS/MTO yte. We generated n = 1000 job for each iulation, and conidered everal cenario with different nuber of job type K and different ultiplier h, c d and c T for cot function to evaluate the effect of thee paraeter on our yte for thi coputational tudy, we ue the ae h, c d and c T value aong different job type. For each cenario, we conidered ten intance with different exponentially ditributed arrival and proceing rate. In Table 1, the ratio of the average of total cot = K {h ie[i i ] + ci de[d i] + ci TE[W i d i ] + } are hown, where E[I i ] denote the average inventory, E[d i ] denote the average quoted lead tie and E[W i d i ] + denote the average tardine in a iulation run for product type i The ipact of MTS/MTO and equencing deciion The firt and econd colun copare a cobined MTS/MTO yte uing the SEPTA-LTQ algorith with pure MTS and MTO yte, repectively. For the pure MTO yte, we aue that no inventory i held for any of the product type. For the pure MTS yte, we aue that the copany carrie enough inventory to atify at leat 95% of the cutoer fro inventory and we find the inventory level that atify thi contraint. Recall that for the cobined yte, we utilize Theore 1 to find the optial inventory level. For the pure MTS yte, uing an approach iilar to that of Theore 1, we find the iniu value of x that atify the relation F i x 0.95 for product type i intead of F i x c i /c i + h i and ue thee x value a the inventory level for the pure MTS yte. The cobined yte, on average, provide a 20% decreae in cot a oppoed to pure MTS yte and a 15% decreae a oppoed to the pure MTO yte. In the third colun, we copare the cot of applying the SEPTA-LTQ algorith for thi cobined yte with the cot of applying the appropriate lead tie quotation algorith with a FCFS chedule. We ee that the SEPTA-LTQ algorith lead to about 15% lower cot on average than the FCFS-LTQ algorith. Note that with the FCFS chedule, ince future arrival do not affect the delivery tie of the previou order, the lead tie quoted with FCFS chedule have error only due to the tochaticity of the proceing tie of job already at the queue and thu, in general, have le error than the SEPTA chedule. Thu, if anything, lead tie quotation decreae the gain achieved by a SEPTA chedule. In the fourth colun, we analyze the effectivene of our lead tie quotation algorith. We copare the objective function uing the cheduling and lead tie quotation algorith SEPTA-LTQ with the cot function Table 1. Coparion of a cobined MTO-MTS yte with pure yte and with different chedule for a ingle facility with c d = 2 and different cobination of h, c T and K h=0.5 c T =2.5 MTS MTO SEPTA LTQ FCFS LTQ h=1 c T =2.1 MTS MTO SEPTA LTQ FCFS LTQ K = K = K = K = K = K = h=1 c T =2.5 MTS MTO SEPTA LTQ FCFS LTQ h=1 c T =3 MTS MTO SEPTA LTQ FCFS LTQ K = K = K = K = K = K = h=2 c T =2.5 MTS MTO SEPTA LTQ FCFS LTQ h=1 c T =5 MTS MTO SEPTA LTQ FCFS LTQ K = K = K = K = K = K = h=5 c T =2.5 MTS MTO SEPTA LTQ FCFS LTQ h=1 c T =10 MTS MTO SEPTA LTQ FCFS LTQ K = K = K = K = K = K = Average

13 Cobined ake-to-order/ake-to-tock upply chain 115 = K {h ie[i i ] + c d i E[W i]} uing the ae chedule SEPTA but without quoting lead tie. Intead we ue the actual waiting tie, W,ofjob in the yte in the calculation of. Note that the optial off-line LTQ algorith quote lead tie that are exactly equal to the waiting tie of the job in the yte, and thu i a lower bound for the LTQ algorith with a SEPTA-baed chedule. We ee that the total cot with the lead tie quotation algorith,, iabout 4% ore than. Thu, we conclude that our lead tie quotation algorith i effective for iniizing the objective function. Alo, we ee that the perforance of the lead tie quotation algorith increae a h or c T decreae. When h i lower, ore inventory will be held o when an order arrive, it i ore likely that the order i atified fro the inventory and there i le error in lead tie quotation. When c T i lower, the ipact of tardine cot on total cot decreae, and thu i cloer to The ipact of yte paraeter We alo ee the effect of the paraeter on the yte perforance in Table 1. Unurpriingly, a the unit inventory holding cot increae while all other paraeter reain contant, the perforance of the cobined yte ove toward the perforance of a pure MTO yte and give uch better reult than pure MTS yte ince holding inventory becoe uch ore cotly a h increae. The yte i affected in the ae way a the unit due date cot, c d, decreae becaue in that cae lead tie becoe le iportant and an MTO yte becoe uch ore attractive a c d decreae. We alo ee that the SEPTA-LTQ algorith give uch better reult than the FCFS-LTQ algorith a h increae alternatively c d decreae becaue a h increae, the yte ove toward an MTO trategy and the chedule ha an iportant effect on the objective in MTO yte; eanwhile, for MTS yte, ince the order are atified fro the inventory, the effect of iniizing the total copletion tie of the job on the yte perforance i not too ignificant. On the other hand, a the unit tardine cot, c T, increae while all other paraeter reain contant, MTO yte perfor wore and pure MTS yte becoe uch ore attractive. Alo, we ee that the perforance difference between the SEPTA- LTQ algorith and FCFS-LTQ algorith decreae a c T increae and the FCFS-LTQ algorith tart to give better reult a c T becoe very high. Thi happen becaue, with the FCFS-LTQ algorith, the future arrival do not affect the copletion tie of the previou job, and thu we can quote the lead tie uch cloer to the actual copletion tie o the tardine decreae. However, with the SEPTA-LTQ algorith, tardine cot will be higher and a c T increae, the cot of tardine overcoe the gain in total copletion tie with SEPTA-LTQ algorith, and thu FCFS-LTQ lead to lower total cot than SEPTA-LTQ. The reult in Table 1 are averaged acro arrival and proceing rate, but the perforance of our algorith clearly depend on thee rate. We alo analyze the perforance of our algorith for varying arrival and proceing rate and preent the reult in the online uppleent Effect of inventory deciion and lead tie quotation on upply chain We next explore the effect of inventory deciion and lead tie quotation on our yte. In the firt four colun of Table 2,without conidering lead tie quotation and uing the SEPTA cheduling algorith, we copare the objective function = K {h i E[I i ] + h i E[Ii ] + ci de[w i]} to explore only the effect of inventory deciion in thi yte. Recall that we ade oe auption about the interaction between the upplier and the anufacturer regarding the tationary ditribution of the nuber of job at the anufacturer and we alo aued that inventory value Table 2. Analyi of inventory deciion on centralized and decentralized yte with c d = 2, c T = 2.5 and different cobination of h, h and K h =0.5 h =2 LB DFI SD DFI SD Cen h =0.5 h =0.6 LB DFI SD DFI SD Cen K = K = K = K = K = K = h =1 h =2 LB DFI SD DFI SD Cen h =0.5 Cen h =1 LB DFI SD DFI SD Cen K = K = K = K = K = K = h =1.9 h =2 LB DFI SD DFI SD Cen h =0.5 Cen h =5 LB DFI SD DFI SD Cen K = K = K = K = K = K = Average

14 116 Kainky and Kaya of different type at the upplier do not affect the tationary ditribution of other type at the anufacturer. We explore how thee auption effect the optial inventory level and the objective function with thi coputational tudy. We alo copare the centralized and decentralized verion of the cobined MTS/MTO upply chain with thi objective function. In the lat colun of Table 2, we alo analyze the effectivene of the lead tie quotation algorith for the centralized cae by coparing our original objective function Cen which include lead tie quotation with Cen. Throughout Table 2, we conider different nuber of product type K and ue the ae paraeter value over each of the product type, c d = 2 and c T = 2.5 for both the upplier and the anufacturer, and different cobination of h and h. To explore the effectivene of our algorith, we copare the objective function uing our inventory level for the centralized odel with the iniu objective function for that cae. In the iulation, we find the optial olution by conidering all of the poible cobination of inventorie of each type for the upplier and the anufacturer and electing the bet one. We ue thee iniu objective value a lower bound in our iulation. However, thi proce take a ignificant aount of tie, epecially when there are a large nuber of type. Although we can find the optial inventorie by trying all poible olution for the cae analyzed in thi experient ince they are of relatively all ize, note that thi ethod becoe alot ipoible to apply a the proble ize get bigger. Forexaple conider the cae when there are ten different job and the inventory of each job type at the upplier and the anufacturer can take a value fro zero to four. In that cae we need to evaluate different poible olution for the trial-and-error ethod and if each of the take a nanoecond 10 9 econd, the whole proce take ore than a day. In our heuritic, however, there i no interaction between the different type and we conider the eparately. Alo, we do only a one-dienional earch over the upplier inventory level up to an upper bound and find the correponding inventory level for the anufacturer uing Theore 2. Thu, for the ae cae, we would only need to ake 5 10 coparion in the wort cae, which can be copleted alot intantaneouly, and thu thi heuritic approach can be eaily applied to even larger proble. We contruct the lower bound by uing the optial inventory value found by exhautive earch, and copare it with the objective value obtained by uing the inventory value found by our heuritic. The firt colun in Table 2 copare thee two objective function for the centralized upply chain odel. We ee that the inventory value found by our heuritic are very cloe to the optial inventory value and there i only a 5% difference, on average, between the iniu cot and the cot obtained by uing our inventory value. The difference i due to our auption that having inventory of type i at the upplier doe not affect the tationary ditribution of the nuber of job at the anufacturer. Coparing the centralized and decentralized odel, we ee that the cot of the centralized odel are, on average, 10% le than thoe of the decentralized odel with full inforation. The cot aving due to inventory deciion increae to ore than 15% when we copare the centralized odel with the iple decentralized odel. We alo ee that, without centralization, if the anufacturer ha full inforation about the upplier, he/he can decreae the cot by about 7% by effectively adjuting hi/her own inventory level without changing anything ele. We ee that if the anufacturer had control over the upplier, he/he could cut hi/her cot ignificantly. However, even if the anufacturer did not have control over the upplier but had full inforation about the whole yte, he/he could till cut hi/her cot and increae hi/her profit. In the fifth colun of Table 2, we alo analyze the effectivene of our lead tie quotation algorith for the centralized odel by coparing Cen,which exclude lead tie quotation, with the original cot function Cen. Note that the difference between Cen and Cen i due to error in lead tie quotation. Oberve that the difference between Cen and Cen i about 5%, deontrating that our lead tie quotation algorith i effective for iniizing the objective function Cen. We can alo oberve the ipact of the paraeter h, h, c d and K on yte perforance. A the inventory holding cot at the upplier, h, increae while everything ele reain the ae, our heuritic give reult cloer to the lower bound. Intuitively, a h increae, the optial inventory level at the upplier and their effect on the anufacturer decreae, a we aued in our approxiation. In addition, if the upplier ue a pure MTO trategy, then our heuritic find the optial olution ince our approxiation i exact for thi cae. We ee the ae effect a h decreae, becaue in thi cae it i better to carry inventorie at the anufacturer than at the upplier, and the inventory level at the upplier and their effect on the anufacturer decreae. Siilarly, a c d decreae, the inventory level at the upplier alo decreae, and our heuritic give reult cloer to thoe of the optial olution. When we copare upply chain odel, we ee that a h decreae, the decentralized odel give reult iilar to thoe of the centralized odel. Thi i becaue a h decreae, the optial inventory level at the upplier for the centralized odel becoe cloe to the upper bound we preented in Theore 3, which i alo the optial inventory level for the upplier in the decentralized odel auing that the ae chedule i ued in both cae. Thu, a h decreae, the inventory level at the upplier and the inventory level at the anufacturer for the centralized and decentralized odel, approach each other and the objective value for the decentralized odel becoe cloer to the centralized odel objective value. We can oberve the ae effect a h increae, becaue in the centralized odel

15 Cobined ake-to-order/ake-to-tock upply chain 117 Table 3. Coparion of the centralized odel with DFI odel with K = 2, h = 0.5, h = 1, c = 2, λ 1 = 1, λ 2 = 1, µ 1 = µ 1 = 2, µ 2 = µ 2 = 2.1 and changing value for c i c 1 c 2 R S 1 R S 2 R M 1 R M 2 DFI a h increae, the inventory level at the anufacturer decreae and the inventory level at the upplier ove cloer to the upper bound. Thu, the decentralized odel reult get cloer to thoe of the centralized odel. For the lead tie quotation algorith, oberve that the perforance of the algorith increae a h or h decreae. Thi i becaue a the inventory holding cot decreae, ore inventory will be held and when an order arrive, it i ore likely that the order i atified fro the inventory and there i le error in the lead tie quotation. In Table 2, we ue the ae unit lead tie cot c d for the upplier and the anufacturer. In thi cae, auing the ae chedule i ued in the centralized and decentralized odel, the upplier carrie ore inventory and the anufacturer carrie le inventory in the decentralized odel with full inforation DFI odel than in the centralized odel ee Corollary 7 and Corollary 8. If we have the option to chooe the unit lead tie cot ci to charge the upplier for backlog for each product type i, wecan coordinate the decentralized upply chain by charging ci ci d uch that the upplier inventory aount will decreae to the level carried in the centralized odel, and thu the total cot of the centralized upply chain can be achieved by the DFI odel. Thi can be een in Table 3, in which we conider changing ci and cd i value. For the centralized verion of the odel with paraeter decribed in the table caption, the optial inventory level are R1 = 27, R 2 = 1, R1 = 19 and R 2 = 3. Since le inventory i carried at the upplier and ore inventory i carried at the anufacturer a ci decreae, by carefully electing c1 and c 2,weachieve centralized inventory level and cot and thu coordinate the DFI yte. However, if the odel ue different chedule, we are not necearily able to achieve the total cot of the centralized yte by chooing appropriate ci value in the DFI yte. Depending on the yte characteritic, arrival and proceing rate, and the chedule ued, the decentralized odel ight give higher cot, but it can alo give lower cot than the centralized odel. For exaple, for the cae dicued above, by uing the SEPTA p SEPTA p chedule in the DFI yte and SEPTA p FCFS in the centralized yte, the optial inventory level are R1 = 27, R 2 = 1, R1 = 15 and R 2 = 6 with the total cot for the centralized odel, which i lightly larger than the iniu cot achieved in the coordinated DFI yte, We alo analyze how the optial inventory level change in the centralized and decentralized odel depending on the arrival and proceing rate for different nuber of clae and preent the reult in the online uppleent Effectivene of a cobined yte and efficiency of heuritic for upply chain We alo copleted a iilar tudy to the one dicued above, except that lead tie quotation i included in our analyi, o the objective function becoe = K {h i E[I i ] + h i E[Ii ] + ci de[d i] + ci TE[W i d i ] + }.Weue the ultiplier h = 0.5, h = 1, c d = 2 and c T = 2.5 for thi cae. We are coparing the cot for the entire yte for the centralized and decentralized odel. In the firt colun in Table 4, we copare the total cot of the centralized odel uing the algorith SEPTA p LTQ C with the total cot of cheduling job according to FCFS in both yte and quoting lead tie appropriately. We conclude that the chedule ued to produce the job ha an iportant effect on the total cot and we ee that, on average, our heuritic perfor about 20% better than the coonly ued chedule FCFS. Recall fro Table 1 that thi wa alo the cae for the ingle-facility cae. Alo, when we copare the value in the econd colun of Table 4 with the value in the econd colun of Table 2 for paraeter h = 0.5, h = 1, c d = 2 and c T = 2.5, which copare the centralized cae and the decentralized cae with full inforation, we ee that there i a all difference between thee value. Thi difference i due to the incluion of lead tie quotation in Table 4. When the anufacturer ha full inforation in the DFI cae, he/he can quote lead tie a effectively a in the centralized cae and we can conclude that including lead tie quotation doe not have uch effect on the cot ratio between the centralized cae and the decentralized cae with full inforation. However, when we copare the ratio in the third colun of Table 4 with the ratio in the third colun of Table 2 for Table 4. Coparion of centralized and decentralized upply chain for a cobined MTO-MTS yte with h = 0.5, h = 1, c d = 2 and c T = 2.5 SEPTAp LTQ C / K FCFS LTQ Cen / DFI Cen / SD DFI / SD Average

16 118 Kainky and Kaya Table 5. Analyi of the effectivene of the lead tie quotation algorith for the centralized odel / Cen K = K = K = Average paraeter h = 0.5, h = 1, c d = 2 and c T = 2.5, which copare the iple decentralized odel with the centralized odel, we ee that there i a large difference. Since the anufacturer ha very little inforation about the upplier, he/he can no longer quote reliable due date, o cot increae draatically in the iple decentralized cae due to poor lead tie quotation. The cot of the iple decentralized odel are about 40% wore than thoe of the centralized odel. Oberve that although there wa not uch difference between the decentralized odel with full inforation and the iple decentralized odel in Table 2, in Table 4, thi difference increae to about 30% due to the difficulty of lead tie quotation. In Table 5, we copare the cot Cen which exclude lead tie quotation with the original cot function Cen for the centralized odel uing different nuber of job, n.recall that the difference between thee two cot i iply due to the error in lead tie quotation. For a all nuber of job, the difference i about 25% but the perforance of the algorith increae ignificantly a the nuber of job increae and there i only about 6% difference on average between thee two cot for n = 1000 job. Thi deontrate the effectivene of our lead tie quotation approach for iniizing, atleat for relatively large nuber of job. We conclude thi ubection with the obervation that having full inforation about the upplier i critical to the anufacturer for lead tie quotation. In addition to full inforation, if the anufacturer ha coplete control over the upplier, total cot can be ubtantially decreaed with better inventory allocation. 6. Concluion We have conidered a variety of tylized odel of a upply chain with a ingle anufacturer and a ingle upplier in order to find effective inventory value that hould be carried at each facility and to ae the ipact of anufacturer upplier relation on inventory deciion and effective lead tie quotation. In our odel, we conider everal variation of inventory, cheduling and lead tie quotation proble in cobined MTO MTS upply chain in order to iniize a function of the total inventory, lead tie and tardine. We derive the condition for both the centralized n and decentralized verion under which an MTO or MTS yte hould be ued for each product at each facility and we preent algorith to find the optial inventory level. We alo preent effective lead tie quotation and cheduling algorith for centralized and decentralized verion of thi odel. Coputational tet deontrate the effectivene of thee approache. Uing our algorith, we invetigate the value of coordination chee involving inforation haring between upply chain eber for thi yte. We ee that cot can be cut draatically by uing a cobined yte intead of pure MTO or MTS yte and that inforation exchange between the upplier and the anufacturer i critical for effective lead tie quotation. We alo dicover that if centralization i not poible, inforation exchange in the decentralized odel can iprove the level of perforance draatically particularly with repect to lead tie quotation, although not a ignificantly a centralized control. In addition, we oberve that when the anufacturer ha ignificant inforation about the tatu of the upplier, he/he can iprove the perforance of the decentralized yte, and indeed ove cloe to the perforance of a centralized yte, by electing the appropriate long lead tie penalty to charge the upplier. Of coure, thee are tylized odel, and real-world yte have any ore coplex characteritic that are not captured by thee odel. Neverthele, thi i to our knowledge one of the firt paper that analytically explore inventory deciion, cheduling and lead tie quotation together in the context of a upply chain, and that explore the ipact of the upplier anufacturer relationhip in thee yte. In the future, we hope to evaluate ore coplex upply chain yte and ae how upply chain architecture can ipact cheduling and due date quotation deciion. We alo intend to expand thi reearch to conider different function of lead tie in the objective function. In oe yte, the anufacturer doe not have to accept all order and ha the option to reject certain order. Pricing and capacity deciion can alo be incorporated into thee odel. In all of thee odel and variant, the anufacturer need to develop trategie for yte deign, and for cheduling and due date quotation. Acknowledgeent Thi aterial i baed upon work upported by the National Science Foundation under grant DMI and DMI Reference Arreola-Ria, A. and DeCroix, G.A Make-to-order veru aketo-tock in a production-inventory yte with general production tie. IIE Tranaction, 308,

17 Cobined ake-to-order/ake-to-tock upply chain 119 Baker, K.R. and Bertrand, J.W.M A coparion of due-date election rule. AIIE Tranaction, 13, Bertrand, J.W.M The effect of workload dependent due-date on job hop perforance. Manageent Science, 29, Brucker, P Scheduling Algorith, Springer, New York, NY. Carr, S. and Duenya, I Optial adiion control and equencing in a ake-to tock, ake-to-order production yte. Operation Reearch, 485, Chand, S. and Chhajed, D A ingle achine odel for deterination of optial due date and equence. Operation Reearch, 40, Cheng, T.C.E. and Gupta, M.C Survey of cheduling reearch involving due date deterination deciion. European Journal of Operational Reearch, 38, Duenya, I Single facility due date etting with ultiple cutoer clae. Manageent Science, 41, Duenya, I. and Hopp, W.J Quoting cutoer lead tie. Manageent Science, 41, Eilon, S. and Chowdhury, I.G Due date in job hop cheduling. International Journal of Production Reearch, 14, Federgruen, A. and Katalan, The ipact of adding a aketo-order ite to a ake-to-tock production yte. Manageent Science, 457, Hall, N.G. and Poner, M.E Earline-tardine cheduling proble, I: weighted deviation of copletion tie about a coon due date. Operation Reearch, 39, Jackon, J.R Job hop-like queuing yte. Manageent Science, 10, Kahlbacher, H Terin-und ablaufplanung ein analyticher zugang., Ph.D. thei, Univerity of Kaierlautern, cited in Brucker Kainky, P. and Hochbau, D Due date quotation odel and algorith. To be publihed a a chapter in the forthcoing book Handbook on Scheduling Algorith, Method and Model, Leung, J.Y. ed., Chapan Hall/CRC, New York. Kainky, P. and Kaya, O Scheduling and due-date quotation in a MTO upply chain. Naval Reearch Logitic, 55, Kainky, P. and Lee,.-H Effective on-line algorith for reliable due date quotation and large cale cheduling. Journal on Scheduling, 11, Kaya, O.2006 MTO-MTS production yte in upply chain. PhD thei, Univerity of California, Berkeley, USA. Kekinocak, P., Ravi, R. and Tayur, S Scheduling and reliable lead tie quotation for order with availability interval and lead tie enitive revenue. Manageent Science, 47, Kekinocak, P. and Tayur, S Due-date anageent policie in Handbook of Quantitative Supply Chain Analyi: Modeling in the E- Buine Era, Sichi-Levi, D., Wu, D. and Shen,.M. ed, Kluwer, Norwell, MA, pp Li, L The role of inventory in delivery tie copetition. Manageent Science, 382, Miyazaki, S Cobined cheduling yte for reducing job tardine in a job hop. International Journal of Production Reearch, 19, Panwalkar, S.S., Sith, M.L. and A. Seidann 1982 Coon due date aignent to iniize total penalty for the one achine cheduling proble. Operation Reearch, 30, Rajagopalan, S Make-to-order or ake-to-tock: odel and application. Manageent Science, 482, Savaaneril, S., Griffin, P.M. and Kekinocak, P Analyi of lead tie quotation policie for bae-tock inventory yte. Working Paper, Georgia Tech. Seidann, A., Panwalker, S.S. and Sith, M.L Optial aignent of due date for a ingle proceor cheduling proble. International Journal of Production Reearch, 19, Week, J.K A iulation tudy of predictable due-date. Manageent Science, 25, Wein, L.M Due-date etting and priority equencing in a ulticla M/G/1 queue. Manageent Science, 37, Willia, T.M Special product and uncertainty in production/inventory yte. European Journal of Operation Reearch, 15, Youef, K.H., Van Delft, C. and Dallery, Y Efficient cheduling rule in a cobined ake-to-tock and ake-to-order anufacturing yte. Annal of Operation Reearch, 126, Biographie Phil Kainky i an Aociate Profeor in the Indutrial Engineering and Operation Reearch Departent at the Univerity of California, Berkeley. He received hi Ph.D. in Indutrial Engineering and Manageent Science fro Northwetern Univerity in Prior to that, he worked in production engineering and control at Merck and Co. Hi current reearch focue on the analyi and developent of robut and efficient technique for the deign and operation of logitic yte and upply chain. He i a co-author of Deigning and Managing the Supply Chain: Concept, Strategie and Cae Studie McGraw-Hill, 1999, 2003, 2008 which won the Book-of-the-Year Award and Outtanding IIE Publication Award given in 2000, and co-author of Managing the Supply Chain: The Definitive Guide for the Buine Profeional McGraw-Hill, He ha conulted in the area of production and logitic yte control. Onur Kaya i currently an Aitant Profeor at Koc Univerity in Turkey. He hold an M.S. in Statitic and a Ph.D. in Indutrial Engineering and Operation Reearch fro the Univerity of California, Berkeley. Hi ajor reearch interet are in upply chain anageent, inventory theory and control, cheduling and tochatic odel and their application.

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