response time reduction in make-to-order and assemble-to-order supply chain design

IIE Transactions (2009) 41, 448466Copyright C IIEISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408170802382741

Response time reduction in make-to-orderand assemble-to-order supply chain design

NAVNEET VIDYARTHI1,, SAMIR ELHEDHLI2 and ELIZABETH JEWKES2

1Department of Decision Sciences and MIS, John Molson School of Business, Concordia University, Montreal, Quebec, Canada,H3G 1M8E-mail: [email protected] of Management Sciences, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1,E-mail: {elhedhli, emjewkes}@uwaterloo.ca

Received December 2006 and accepted June 2008.

Make-to-order and assemble-to-order systems are successful business strategies in managing responsive supply chains, characterizedby high product variety, highly variable customer demand and short product life cycles. These systems usually spell long customerresponse times due to congestion. Motivated by the strategic importance of response time reduction, this paper presents modelsfor designing make-to-order and assemble-to-order supply chains under Poisson customer demand arrivals and general service timedistributions. The make-to-order supply chain design model seeks to simultaneously determine the location and the capacity ofdistribution centers (DCs) and allocate stochastic customer demand to DCs by minimizing response time in addition to the xedcost of opening DCs and equipping them with sufcient assembly capacity and the variable cost of serving customers. The problemis setup as a network of spatially distributed M/G/1 queues, modeled as a non-linear mixed-integer program, and linearized usinga simple transformation and a piecewise linear approximation. An exact solution approach is presented that is based on the cuttingplane method. Then, the problem of designing a two-echelon assemble-to-order supply chain comprising of plants and DCs servinga set of customers is considered. A Lagrangean heuristic is proposed that exploits the echelon structure of the problem and uses thesolution methodology for the make-to-order problem. Computational results and managerial insights are provided. It is empiricallyshown that substantial reduction in response times can be achieved with minimal increase in total costs in the design of responsivesupply chains. Furthermore, a supply chain conguration that considers congestion is proposed and its effect on the response timecan be very different from the traditional conguration that ignores congestion.

Keywords: Response time, make to order, assemble to order, supply chain design, cutting plane method, Lagrangean relaxation

1. Introduction

Make-to-Order (MTO) systems are successful businessstrategies to manage responsive supply chains that are char-acterized by high product variety, highly variable customerdemand and short product life cycles. Because of mass cus-tomization and competition on product variety, many rmsadopt an MTO strategy to offer a variety of products anddeal with product proliferation. Dells manufacturing anddistribution of Personal Computers (PCs) is an excellentexample of an MTO supply chain (Margretta, 1998; Dell,2000). Dell typically offers several lines of product, witheach allowing at least dozens of features from which cus-tomers can select when placing an orderdifferent combi-nations of CPU, hard drive, memory and other peripherals.In Dells supply chain, multiple components are procured

Corresponding author

and kept in inventory at various assembly facilities, fromwhich they are assembled into a wide variety of nishedproducts in response to customer orders. Whereas each ofthese components takes a substantial lead time to man-ufacture, the time to assemble all these components intoa PC is low, provided there is sufcient assembly capac-ity and the components are available. In traditional Make-To-Stock (MTS) supply chains, the customer orders aremet from stocks of an inventory of nished products thatare kept at various points of the network. This is doneto reduce the delay in fullling customer orders, increasesales and avoid stockouts. However, the problems asso-ciated with holding inventory of nished products mayoutweigh the benets, especially when those products be-come obsolete as technology advances or fashion changes.While an MTO strategy eliminates nished goods inven-tories and reduces a rms exposure to the risk of ob-solescence, it usually spells long customer response time(Gupta and Benjaafar, 2004).

0740-817X C 2009 IIE

Response time reduction in supply chain design 449

In order to reconcile the dual needs of a quick responsetime and high product variety, many rms such as GeneralElectric, American Standard, Compaq, IBM, BMW andNational Bicycle use a hybrid strategy (i.e., mix of MTOand MTS) called the Assemble-To-Order (ATO) strategy,in which a subassembly, or a number of common subassem-blies used in several products, are assembled and placed ininventory until an order is received for the nished prod-uct (Song and Zipkin, 2003). This allows the rm to cus-tomize the orders by having the product ready using theMTO strategy, while taking advantage of the economiesof scale using the MTS strategy. Also, investment in thesemi-nished product inventory is smaller compared to theoption of maintaining a similar amount of nished goodsinventory. Furthermore, demand pooling benets can berealized. Although, maintaining a semi-nished productinventory in ATO systems lowers the customer responsetime as compared to a pure MTO system, it can be furtherreduced by minimizing congestion at the point of differen-tiation. Naturally, the response time to deliver the productis critical and forms the basis for competition. Consumerswillingness to pay a premium for a shorter response timeprovides further incentives for rms to reduce response timein MTO and ATO supply chains.

Although various integrated models of supply chain de-sign have been proposed in recent years to support leadtime reduction, these models have continued to be largelyguided by more traditional concerns of efciency and cost inMTS settings, where the primary focus is on minimizing thexed cost of facility location and the variable transporta-tion cost under fairly stable and deterministic customer de-mand settings. This approach is personied by the work ofDogan and Goetschalckx (1999), Vidal and Goetschalckx(2000), Teo and Shu (2004), Shen (2005), Eskigun et al.(2005) and Elhedhli and Gzara (2008). For example, Vidaland Goetschalckx (2000) present a model that captures theeffect of change in transportation lead time and demand onthe optimal conguration of the global supply chain net-work, assuming that the demand is deterministic. Eskigunet al. (2005) incorporate delivery lead time and the choice oftransportation mode in the design of a supply chain under adeterministic demand setting. These models tend to ignorecongestion at the facilities and its effect on response time.Their solutions prescribe locating facilities whose capacityutilization is very high, resulting in an excessively long re-sponse time when subjected to variability in service timesand randomness in customer orders. Reviews by Vidal andGoetschalckx (1997), Erenguc et al. (1999) and Sarmientoand Nagi (1999) also point out that most of the existingsupply chain design models do not consider measures ofcustomer service such as response time in making loca-tion/allocation decisions. Also, refer to the recent reviewby Klose and Drexl (2005). This is not surprising given thecomplexity of the model and the interplay of locational andqueueing aspects of the problem. To the best of our knowl-edge, Huang et al. (2005) is one of the rst to model the effect

of congestion in the design of distribution networks. Theymodel capacity using the mean and variance of the Dis-tribution Centers (DCs) as continuous variables, whereasour model considers capacity as a set of discrete optionswith known means and variances. They propose solutionprocedures based on outer approximation and Lagrangeanrelaxation, and tested on small instances of the problem.

Another growing body of literature that is related to ourwork and accounts for congestion and its effect on responsetime in strategic planning is models for facility locationwith immobile servers, stochastic demand and congestion(such as location of emergency medical facilities, re sta-tions, telecommunication network design, automated tellermachines or internet mirror site location). For an exten-sive review, refer to Berman and Krass (2002). Due to thecomplexity of the underlying problem, most papers in thisarea make very strong assumptions: (i) either the numberor capacity of the facilities (or both) are assumed to bexed; (ii) the facilities are assumed to be identical; (iii) thedemand arrival process is assumed to be Poisson; and (iv)the service process is usually assumed to be exponential(see, Amiri (1997), Marianov and Serra (2002), Wang et al.(2003) and Elhedhli (2006) and references therein). Despitethat, most of the techniques proposed to date to solve theseproblems, with the exception of Elhedhli (2006), are eitherapproximate or heuristic based. Our work is also similarin spirit to models for capacity planning with congestioneffects, for which only heuristic solution procedures havebeen reported; see Rajagopalan and Yu (2001) and refer-ences therein.

The objective of this paper is to model the effect ofcongestion on the response time and analyze the trade-off among response time costs, facility location and capac-ity acquisition costs, and outbound transportation costs inthe design of supply chain networks. More specically, wepresent a model to determine the conguration of an MTOsupply chain, where the emphasis is on minimizing the cus-tomer response time through the acquisition of sufcientassembly capacity and the optimal allocation of workloadto the assembly facilities (DCs) under stochastic customerdemand settings. The DCs are modeled as spatially dis-tributed queues with Poission arrivals and general servicetimes to capture the dynamics of the response time. Themodel is formulated as a non-linear Mixed-Integer Pro-gramming (MIP) problem and is linearized using piecewiselinear functions. We present a cutting plane algorithm thatprovides the optimal solution to the problem. Furthermore,we present a Lagrangean relaxation heuristic procedurefor solving large-scale instances of such integrated models.Then, we present a model for the two-echelon ATO sup-ply chain design problem, where a set of plants and DCsare to be established to distribute various nished prod-ucts to a set of customers with stochastic demand. DCs actas assembly facilities, where semi-nished products, pro-cured from plants are held in inventories, from which theyare assembled into a wide variety of nished products in

450 Vidyarthi et al.

response to customer demands. We propose a Lagrangeanrelaxation heuristic that exploits the echelon structure ofthe problem and uses the solution methodology proposedabove for the MTO problem. Explicit consideration of con-gestion effects and their impact on response time in makinglocation, capacity and allocation decisions in supply chainsdistinguishes this work from most other supply chain designmodels.

The rest of the paper is organized as follows. Section 2provides a non-linear MIP formulation of the MTO sup-ply chain design problem, a piecewise linearization andan exact solution approach based on the cutting planemethod. The simplications resulting from assuming ex-ponentially distributed service times (M/M/1 case) anddeterministic service times (M/D/1 case) are also explic-itly described. In Section 3, we present the formulation ofthe two-echelon ATO supply chain design problem and aLagrangean heuristic. Computational results and manage-rial insights are reported in Section 4. Finally, Section 5concludes with some directions for future research.

2. MTO supply chain design

Consider the problem of designing an MTO supply chain,where a set of DCs are to be established and equipped withsufcient capacity to serve a set of customers. Sufcientcapacity here implies being able to obtain service withoutwaiting for an excessively long time after the order is placed.The DCs maintain inventory of multiple components andfacilitate the assembly and shipment of a wide variety of n-ished products in a timely fashion without carrying expen-sive nished-goods inventory and incurring a long responsetime. Response time refers to the interval between the plac-ing of an order and receipt of the ordered product. In MTOsupply chains, because a customer order triggers the assem-bly of nished product from components, the response timeconsists of the assembly lead time and the delivery lead time.The delivery time between individual DCs and customersis relatively constant compared to the order fullment timeat DCs in such settings. Moreover, it can further be reduced(using alternative transportation modes or expedited deliv-ery services) to respond quickly to customer orders on ashort-term basis. However, the assembly lead time is highlydependent on the DC capacity and the allocated workloadand is difcult to change (on a short-term basis) once theDC is established.

2.1. Model formulation

We consider the setting depicted in Fig. 1. We assume thatthe demand for each product from each customer is inde-pendent and occurs according to a Poisson process. Oncethe demand for a product is realized at the customers end,the order is placed at the DCs. DCs will act as assembly facil-

Fig. 1. An MTO supply chain network.

ities and the customers orders arriving at the DCs are meton a First-Come First-Serve (FCFS) basis. We assume thateach DC operates as a single exible-capacity server withinnite buffers to accommodate customer orders waitingfor service.

Under these assumptions, the MTO supply chain is mod-eled as a network of independent M/G/1 queues in whichthe DCs are treated as servers with service rates propor-tional to their capacity levels, where the capacity levels arediscrete. We also assume that there is an unlimited supply ofcomponents and their inventory holding costs at the DCsare insignicant. Hence, the model formulated below si-multaneously determines the location and capacity of DCsand the assignment of customer to DCs by minimizing theresponse time costs in addition to the xed location andcapacity acquisition costs, the assembly and transportationcosts from DCs to customers. Besides capacity restrictions(steady-state conditions) at the DCs, and the demand re-quirements, there are constraints which ensure that at mostone capacity level is selected at the DCs. To model thisproblem, we dene the following notation.

Indices and parameters:

i = index for customers, i = 1, 2, . . . , I ;j = index for potential DCs, j = 1, 2, . . . , J;k = index for potential capacity level at DCs, k =

1, 2, . . . , K;fjk = xed cost of opening DC j and acquiring capacity

level k ($/period);cij = unit cost of serving customer i from DC j ($/unit);t = mean response time cost per unit time per customer

($/period/customer);i = mean demand rate for the product from customer

i (units/period);jk = mean service rate at DC j, if it is allocated capacity

level k (units/period); 2jk = variance of service times at DC j, if it is allocated

capacity level k.


Decision variables:xij = fraction of customer is demand served by DC j (0

xij 1);

yjk =1, if DC j is opened and capacity level kis acquired

0, otherwise.

Let the demand for the product at customer location ibe an independent random variable that follows a Poissonprocess with mean i. If xij is the fraction of customer isdemand served by DC j, then the aggregate demand ar-rival rate at DC j is also a random variable that follows aPoisson process with mean j =

Ii=1 ixij, due to the su-

perposition of Poisson processes. If the service times at eachDC follow a general distribution and each DC is modeledas an M/G/1 queue, then the mean service rate of DC j, ifit is allocated capacity level k, is given by j =

Kk=1 jkyjk

and the variance in service times is 2j =K

k=1 2jkyjk. This

service rate reects the server capacity or essentially thenumber of MTO products a DC can assemble and ship in agiven time period. Let j represent the mean service time atDC j (j = 1/j), j be the utilization of DC j (j = j/j)and CV2j be the squared coefcient of variation of ser-vice times (CV2j = 2j / 2j ). Under steady-state conditions(j < j) and FCFS queuing discipline, the expected aver-age waiting time (including the service time) at DC j is givenby the PollaczekKhintchine (PK) formula:

E[Wj(M/G/1)] =(

1 + CV2j2

)jj

1 j + j

=(

1 + CV2j2

)j

j(j j) +1j

j,and the expected total waiting time for DC j is obtainedby multiplying the waiting time at DC j by the expecteddemand as(

1 + CV2j2

)2j

j(j j) +j

jj.

The expected total waiting time for the entire system isgiven by

E[W (M/G/1)] =J

j=1

[(1 + CV2j

2

)2j

j(j j) +j

j

],

and can be written as:

12

Jj=1

[(1 + CV2j

) jj j +

(1 CV2j

) jj

].

This is equivalent to

12

Jj=1

[(1 +

Kk=1

CV2jkyjk

) Ii=1 ixijK

k=1 jkyjk I

i=1 ixij

+(

1 K

k=1CV2jkyjk

) Ii=1 ixijK

k=1 jkyjk

]. (1)

The resulting non-linear MIP formulation is

(PN) : minJ

j=1

Kk=1

fjkyjk+I

i=1

Jj=1

cijixij+tE[W (M/G/1)],(2)

subject to

Ii=1

ixij K

k=1jkyjk j, (3)

Kk=1

yjk 1 j, (4)J

j=1xij = 1 i, (5)

0 xij 1, yjk {0, 1} i, j, k. (6)The rst term in the objective function (2) represents the

xed cost (amortized over the planning period) of locatingDCs and equipping them with adequate assembly capacity.The second term accounts for the variable cost of assem-bly and shipment of products from DCs to customers. Thethird term is the expected total response time cost or the lostsales due to excessive response times between DCs and cus-tomers. The expected total response time cost is expressedas a product of average response time cost per unit time thatone of its customers spends in the system and expected totalwaiting time in the system. In this paper, for simplicity andtractability reasons, we assume that the cost of the responsetime is linearly proportional to the waiting time. However,the model can be extended to other cost functions such asa piecewise linear function or a cost function proportionalto max{0, E[W (M/G/1)] W0}, where W0 is the maximumtolerated waiting time for a customer. Moreover, the averageresponse time cost t may vary from customer to customer(tij) if desired, but we assume for simplicity that it is thesame across customers. It can be interpreted as a penaltyfunction that reects the true cost of not fullling customerorders in the committed lead time. In practice, determiningthe values of average response time cost can be challenging,however, one can rely on techniques outlined in Rao et al.(2000). To accurately reect lost sales due to unacceptableresponse time, Rao et al. (2000) surveyed Caterpillar dealersto determine the percent of customers who would renege ifa product was not available immediately, after 2 weeks, andafter 4 weeks. A lower bound on the response time cost canbe provided by the inventory holding costs (Eskigun et al.,2005). Furthermore, the problems can be solved iterativelywith different values of t to obtain a trade-off curve fromwhich decision makers may choose a solution based on theirpreference between location and capacity acquisition cost,transportation cost, and response time costs.

Constraints (3) ensure that the steady-state conditions(j j) at the DCs are met. Constraint set (4) ensuresthat at most one capacity level is selected at a DC, whereasconstraint set (5) ensures that the total demand is met.


Constraints (6) are non-negativity and binary restrictions.The formulation can easily handle single-sourcing require-ments by imposing binary restrictions on xij. This wouldrestrict the assignment of customer is demand to one andonly one DC j.

The non-linearity in (PN) arises due to the expressionof the total waiting time at the DCs, E[W (M/G/1)]. Itcan be shown that E[W (M/G/1)] is convex in aggregatearrival rate j, for a xed value of j and convex in ser-vice rate j, for a xed value of j, where j =

Ii=1 ixij

and j =K

k=1 jkyjk. Intuitively, one would expect thatthe waiting time increases with increasing marginal returnsas the arrival rate increases and decreases with decreasingmarginal returns as the service rate increases. In the nextsection, we deal with the non-linearity due to the expres-sion of the total average waiting time using a linearizationbased on a simple transformation and a piecewise linear ap-proximation. We also present an exact solution procedurebased on the cutting plane algorithm. Our cutting planealgorithm is similar to the outer-approximation algorithm(Duran and Grossman, 1986).

2.2. Linearization and cutting plane method

In order to linearize Equation (1), let us dene non-negativeauxiliary variables Rj, such that:

Rj = jj j =

Ii=1 ixijK

k=1 jkyjk I

i=1 ixijj

which implies thatI

i=1ixij = Rj1 + Rj

Kk=1

jkyjk

= jK

k=1jkyjk =

Kk=1

jkzjk (7)

where

zjk ={

0 if yjk = 0j if yjk = 1 j, k (8)

and j = Rj/(1 + Rj) is the server (DC) utilization.Since there is at most one k with yjk = 1 while yjk = 0 for

all other k = k, the expression zjk = jyjk can be ensuredby adding the following constraints:

zjk yjk j, k,K

k=1zjk = j j.

The function j = Rj/(1 + Rj) is concave. Given a set ofpoints Rhj indexed by H, j can be approximated by aninnite set of piecewise linear functions that are tangent toj at points Rhj (as shown in Fig. 2) that is

j = minhH

{1(

1 + Rhj)2 Rj +

(Rhj

)2(1 + Rhj

)2}

j

or

j 1(1 + Rhj

)2 Rj +(Rhj

)2(1 + Rhj

)2 j, h H.

Fig. 2. A piecewise linear approximation of Rj/(1 + Rj).


The expression for E[W (M/G/1)] reduces to

E[W (M/G/1)] = 12

Jj=1

{(1 +

Kk=1

CV2jkyjk

)Rj

+(

1 K

k=1CV2jkyjk

)j

}

= 12

Jj=1

(Rj +

Kk=1

CV2jkwjk + j K

k=1CV2jkzjk

)

where

wjk ={

0 if yjk = 0Rj if yjk = 1 and zjk =

{0 if yjk = 0j if yjk = 1 j, k.

Similarly, because there exists at most one k with yjk = 1while yjk = 0 for all other k = k, the expression wjk = Rjyjkcan be ensured by adding the following constraints:

wjk Myjk j, k,K

k=1wjk = Rj j,

where M is the usual Big-M.The resulting linear MIP formulation is

(PL(H)) : minJ

j=1

Kk=1

fjkyjk +I

i=1

Jj=1

cijixij

+ t2

Jj=1

{Rj + j +

Kk=1

CV2jk(wjk zjk)}

, (9)

subject to

Ii=1

ixij K

k=1jkzjk = 0 j, (10)

Kk=1

yjk 1 j, (11)J

j=1xij = 1 i, (12)

zjk yjk 0 j, k, (13)

j 1(1 + Rhj

)2 Rj (Rhj

)2(1 + Rhj

)2 j, h H, (14)j

Kk=1

zjk = 0 j, (15)

wjk Myjk 0 j, k, (16)K

k=1wjk Rj = 0 j, (17)

yjk {0, 1}; 0 xij, zjk 1; j, Rj, wjk 0; i, j, k.(18)

The steady-state conditions (j < j) translate into ca-pacity constraints, and are enforced by the constraints (10)and (13) and forced to


and varying batch sizes. For exponentially distributed ser-vice times at the DCs, the total expected waiting time forthe entire system is given by

E[W (M/M/1)] =J

j=1

j

j j

=J

j=1

( Ii=1 ixijK

k=1 jkyjk I

i=1 ixij

)=

Jj=1

Rj.

The resulting linear MIP model is

(PM/M/1L(H)

): min

Jj=1

Kk=1

fjkyjk+I

i=1

Jj=1

cijixij + tJ

j=1Rj

(19)

subject to Equations (10) to (15),

yjk {0, 1}; 0 xij, zjk 1; j, Rj 0 i, j, k.The model is structurally identical to the service sys-

tem design model presented in Amiri (1997) and Elhedhli(2006).

2.3.2. Systems with deterministic service times(M/D/1 case)

In many cases, the processing/assembly of nished prod-ucts at the DCs often involves repeated steps without muchvariation (Kim and Tang, 1997). This is particularly truefor MTO products with limited options and batch size ofone such as Dells PCs. For deterministic service times, theexpected waiting time for the entire system is given by

E[W (M/D/1)] = 12

Jj=1

(j

j j +j

j

)= 1

2

Jj=1

(Rj + j).

The resulting linear MIP model is as follows:

(PM/D/1L(H)

): min

Jj=1

Kk=1

fjkyjk

+I

i=1

Jj=1

cijixij + t2J

j=1(Rj + j), (20)

subject to Equations (10) to (15),

yjk {0, 1}; 0 xij, zjk 1; j, Rj 0 i, j, k.The cutting plane algorithm described above can be used tosolve these models to optimality. Alternatively, one can relyon the Lagrangean heuristics. Some computational resultsare provided in Section 4.

3. ATO supply chain design

In this section, we consider the design of an ATO supplychain, where we seek to locate a set of plants and DCs todistribute a product with a non-trivial bill of materials to

Fig. 3. An ATO supply chain network.

a set of customers with stochastic demand. The DCs willact as intermediate facilities between the plants and thecustomers and facilitate the shipment of products betweenthe two echelons, as shown in Fig. 3.

The semi-nished products are produced at the plantsand shipped to the DCs. Once the demand is realized at thecustomers end, the order is placed to the DC and the nalproduct is assembled and the demand is met. Hence, thesupply chain network is a combination of the MTS echelon(plant-DC echelon) and the MTO echelon (DC-customerechelon). The problem environment is characterized by astochastic customer demand that has to be satised froma set of DCs and where sufcient capacity has to be ac-quired in order to avoid long response times. To model thisproblem, we dene the following additional notation.

m = index for potential plants, m = 1, 2, . . . , M;n = index for semi-nished products, n = 1, 2, . . . , N;gm = xed cost of opening a plant at location m ($/pe-

riod);Pm = maximum available capacity of plant m (units);cjmn = unit production and transportation cost for semi-

nished product n from plant m to DC j;n = number of units of semi-nished product n required

to make one unit of nished product;um = decision variable that equals one, if plant m is

opened; zero, otherwise;vjmn = number of units of semi-nished product n pro-

duced from plant m and shipped to DC j.

Under the assumption that the demand at customer i isan independent random variable that follows a Poisson pro-cess with mean i and the service time at each DC followsa general distribution, each DC is modeled as an M/G/1queue, whose mean service rate, if it is allocated capacitylevel k, is given by j =

Kk=1 jkyjk and the variance in

service times is given by 2j =K

k=1 2jkyjk. Under steady-

state conditions (j < j) and FCFS queuing discipline, the


total average waiting time for the entire system (service plusqueuing time) is given by Equation (1). The resulting non-linear MIP formulation that simultaneously determines thelocation and capacity of plants and DCs, the shipment lev-els from plants to DCs, and allocation of customers to DCsby minimizing response time costs in addition to xed facil-ity location and capacity acquisition costs, production andtransportation costs between echelons is as follows:

(PATO) : minM

m=1gmum +

Jj=1

Kk=1

fjkyjk +J

j=1

Mm=1

Nn=1

cjmnvjmn

+I

i=1

Jj=1

cijixij + t2J

j=1

(1 +

Kk=1

CV2jkyjk

)

I

i=1 ixijKk=1 jkyjk

Ii=1 ixij

+ t2

Jj=1

(1

Kk=1

CV2jkyjk

) Ii=1 ixijK

k=1 jkyjk, (21)

subject to

Jj=1

Nn=1

vjmn Pmum m, (22)

Mm=1

vjmn =I

i=1nixij j, n, (23)

Ii=1

ixij K

k=1jkyjk j, (24)

Kk=1

yjk 1 j, (25)J

j=1xij = 1 i, (26)

0 xij 1, yjk, um {0, 1}, vjmn 0i, j, k, m, n. (27)

The objective function (21) consists of the xed cost ofopening plants, the xed cost of locating DCs and equip-ping DCs with the required capacity level, the variable costof producing and procuring semi-nished product, the vari-able cost of serving customers from DCs and the total wait-ing time costs at the DCs. Constraints (22) are capacityrestrictions on the opened plants and permit the use ofopened plants only. Constraints (23) are commodity owconservation equations at the DCs. Constraints (24) ensurethat the steady-state conditions at the DCs are met. Con-straints (25) ensure that at most one capacity level is selectedat a DC whereas constraints (26) ensure that the total de-mand is met. Constraints (27) are non-negativity and binaryconstraints.

3.1. Lagrangean relaxation

There are number of ways in which the model can be re-laxed in Lagrangean fashion (see Klose and Drexl (2005)and reference therein). In this paper, we exploit the eche-lon structure of the ATO supply chain using Lagrangeanrelaxation to decompose the model into two subproblems.Note that in (PATO), constraints (22) relate to the MTSechelon, and constraints (24) to (26) relate to the MTOechelon, whereas constraints (23) are the ow conservationconstraints that link the two echelons. Upon relaxing theow conservation constraints (23) with dual multipliers jn,the problem decomposes into two subproblems:

(SPMTS) : minM

m=1gmum +

Jj=1

Mm=1

Nn=1

(cjmn jn)vjmn, (28)

subject to

Jj=1

Nn=1

vjmn Pmum m, (29)

um {0, 1}, vjmn 0 j, m, n. (30)

(SPMTO) : minJ

j=1

Kk=1

fjkyjk +I

i=1

Jj=1

Nn=1

(cij + njn)ixij

+ t2

Jj=1

(1 +

Kk=1

CV2jkyjk

) Ii=1 ixijK

k=1 jkyjk I

i=1 ixij

+ t2

Jj=1

(1

Kk=1

CV2jkyjk

) Ii=1 ixijK

k=1 jkyjk, (31)

subject to

Ii=1

ixij K

k=1jkyjk, (32)

Jj=1

xij = 1 i, (33)

Kk=1

yjk 1 j, (34)

0 xij 1, yjk {0, 1} i, j, k. (35)

Subproblem (SPMTS) is a linear MIP model that deter-mines the location of plants and the ow of semi-nishedproducts into the DCs, whereas the subproblem (SPMTO) isa non-linear MIP model that provides the location and ca-pacity level of DCs and the allocation of customers to DCs.Note that the subproblem (SPMTO) is the MTO supply chaindesign model presented in the previous section and hencewe use the proposed cutting plane algorithm to solve it.

From model (PATO), we can derive some valid con-straints. For example, consider the following set of


constraints:

Mm=1

Pmum I

i=1i, (36)

Mm=1

vjmn n(

maxk

jk) j, n, (37)

Jj=1

Mm=1

vjmn nI

i=1i n. (38)

Constraints (36) are aggregate capacity constraints forthe MTS echelon. Constraints (37) are derived from Equa-tions (23) and (24) and constraints (38) follow from Equa-tions (23) and (26). Constraints (37) imply that the totalow of semi-nished products through a DC should notexceed the DCs maximum throughput capacity, whereasconstraints (38) ensure that the ow of every semi-nishedproduct from plants to DC is at least equal to the bill ofmaterials multiplied by the demand of that product fromall the customers. These constraints are redundant in theoriginal MIP formulation, but they improve the quality ofthe subproblem solutions in terms of the feasibility to theoriginal problem upon relaxing the ow conservation con-straints. This results in better heuristic solutions. Therefore,we add these set of constraints to (SPMTS) as follows:

(SPMTS) : minM

m=1gmum +

Jj=1

Mm=1

Nn=1

(cjmn jn)vjmn,

(39)subject to

Jj=1

Nn=1

vjmn Pmum m, (40)

Mm=1

Pmum I

i=1i, (41)

Mm=1

vjmn n(

maxk

jk) j, n, (42)

Jj=1

Mm=1

vjmn nI

i=1i n, (43)

um {0, 1}, vjmn 0 j, m, n. (44)

3.1.1. The lower boundThe Lagrangean lower bound is given by the solution of theLagrangean dual problem, max[v(SPMTS) + v(SPMTO)]which is equivalent to

max

{minhIu,v

Mm=1

gmuhm +J

j=1

Mm=1

Nn=1

(cjmn jn)vhjmn

+ minhIx,y

Jj=1

Kk=1

fjkyhjk +I

i=1

Jj=1

Nn=1

(cij + njn)ixhij

+ t2

Jj=1

(1 +

Kk=1

CV2jkyhjk

) Ii=1 ix

hijK

k=1 jkyhjk

Ii=1 ix

hij

+ t2

Jj=1

(1

Kk=1

CV2jkyhjk

) Ii=1 ixijK

k=1 jkyjk

}.

Thus can be explicitly written as

(MP) : max

1 + 2,

subject to

1 +J

j=1

Mm=1

Nn=1

vhjmnjn M

m=1gmuhm

+J

j=1

Mm=1

Nn=1

cjmnvhjmn h Iu,v,

2 I

i=1

Jj=1

Nn=1

(nixhij

)jn

Jj=1

Kk=1

fjkyhjk

+I

i=1

Jj=1

cijixhij

+ t2

Jj=1

(1 +

Kk=1

CV2jkyhjk

) Ii=1 ix

hijK

k=1 jkyhjk

Ii=1 ix

hij

+ t2

Jj=1

(1

Kk=1

CV2jkyhjk

) Ii=1 ixijK

k=1 jkyjkh Ix,y,

where Iu,v is the index set of feasible points of the set:{(um, vjmn): Equations (40) to (43); um {0, 1}; vjmn

0, j, m, n},and Ix,y is the index set of feasible points of the set:

{(xij, yjk): Equations (32) to (34); xij 0; yjk {0, 1}, i, j, k}.

We use Kelleys cutting plane method, in which the point is the solution of the relaxed master problem (RMP), de-ned on subsets Iu,v Iu,v and Ix,y Ix,y (Kelley, 1960).This from (RMP) is used to solve the subproblems(SPMTS) and (SPMTO), and generate two cuts of the form:

1 +J

j=1

Mm=1

Nn=1

vijmnjn M

m=1gmuim +

Jj=1

Mm=1

Nn=1

cjmnvijmn,

(45)

2 I

i=1

Jj=1

Nn=1

(nixi

ij

)jn

Jj=1

Kk=1

fjkyi

jk

+I

i=1

Jj=1

cijixi

ij +t2

Jj=1

(1 +

Kk=1

CV2jkyijk

)

I

i=1 ixiijK

k=1 jkyijk

Ii=1 ix

iij

+ t2

Jj=1

(1

Kk=1

CV2jkyijk

)

I

i=1 ixiijK

k=1 jkyijk

. (46)


The index sets Iu,v and Ix,y are updated as Iu,v {i} andIx,y {i}, respectively, as the algorithm proceeds throughthe iterations.

3.1.2. The heuristic: nding a feasible solutionThe rst subproblem (SPMTS) provides the location ofplants (um) and the ow of semi-nished products into theDCs (vjmn), whereas the second subproblem (SPMTO) pro-vides the location and the capacity decisions of the DCs(yjk), the assignment of customers to DCs (xij). Note thatthe link between the two subproblems is the ow balance ofproducts in and out of DCs. Hence, a feasible solution toproblem (PATO) can be constructed by solving (SPMTS) withthe additional set of constraints

Mm=1 vjmn =

Ii=1 nixij,

where xij is obtained from the solution of (SPMTO). Theoverall procedure is shown in Fig. 4. Computational resultsare provided next.

4. Computational results and insights

In this section, we report our computational experienceswith the proposed solution methodologies and presentsome insights. All the proposed solution procedures were

coded in C and the MIP problems were solved using ILOGCPLEX 10.1 (using the Callable Library) on a Sun Blade2500 workstation with 1.6-GHz UltraSPARC IIIi proces-sors. In the implementation of the iterative cutting planealgorithm and after the solution of the relaxed MIP, we usethe procedure CPXaddrows() to append the cuts generatedand exploit warm starting.

4.1. Test problems

The test problems are derived from the 2000 census dataconsisting of 150 largest cities in the continental UnitedStates (see Daskin (2004)). We generate nine sets of testproblems by setting the number of customers (I) to the 50,100 and 150 largest cities, and the potential DC locations(J) to the ve, ten and 20 most populated cities. The meancustomer demand rates i are obtained by dividing the pop-ulation of those cities by 103. The unit transportation costscij are obtained by dividing the great-circle distance betweenthe customer i and the potential DC location j by 100. Theservice rate of DC j equipped with capacity level k, is set tojk = k

i i (where k = 0.15, 0.20, 0.45 for I = 50; k =

0.10, 0.20, 0.30 for I = 100; k = 0.10, 0.15, 0.20, 0.30, 0.45for I = 150). The xed costs, fjk are set to 100 jk to

Fig. 4. Solution procedure for two-echelon ATO supply chain design.


reect economies of scale. For the M/M/1 case, the vari-ance of service times 2jk are set to the mean service rates,jk whereas for the M/G/1, the 2jk is obtained by settingcoefcient of variation (CV ) to 1.5. The average responsetime cost t is set to (i j(icij)/(I J)), where isthe response time cost multiplier and icij denotes the totalproduction and transportation cost associated with the or-der from the ith customer served by the jth DC. In order toexplore the sensitivity of the solution to different levels ofresponse time costs, the multiplier is tested for 0.1, 1, 5, 10,50, 100 and 200 (higher value of models the situation inwhich losing a customer order due to high expected waitingtime is extremely costly). For the ATO supply chains, theinstances are generated by setting the capacities of plantsto: Pm = U [0.1, 0.5]

Ii=1 i whereas their xed cost are

set to: gm = U [1000, 2000]

(Pm). In order to comparethe performance of the cutting plane method and the La-grangean heuristic for different values of the ratio of plantcapacities to total demand (r ), the instances are generatedby setting the capacities of plants to: Pm = (r

Ii=1 i)/M

and the xed costs to: gm = U [100, 110]

(Pm). The pro-duction coefcients (bill of materials) n were randomlygenerated in the range U [1, 5] and rounded up to the near-est integer value.

4.2. An illustrative case study

Using the rst set of test problems (where I = 50, J = 5 andK = 3), we illustrate that the MTO supply chain congu-ration that considers congestion and its effect on responsetime can be different from the conguration that ignorescongestion. Furthermore, we show empirically that sub-stantial reduction in response times can be achieved withminimal increase in total costs in the design of responsivesupply chains.

The rst set of test problems consists of 50 customers,ve potential DC locations (j = 1, New York, NY; j = 2,Los Angeles, CA; j = 3, Chicago, IL; j = 4, Houston, TX;j = 5, Philadelphia, PA) that can be equipped with threecapacity levels (k = 1, small; k = 2, medium; k = 3, large).The problem is solved to optimality (with a gap of 106) us-ing the cutting plane algorithm. Table 1 summarizes the re-sults for different values of the response time cost by setting to 0, 0.1, 1, 10, 100 and 1000 under four different cases:M/G/1 case with CV = 1.5, M/M/1 case, M/G/1 casewith CV = 0.5, and M/D/1 case. The table shows the totalobjective function value (TC), xed cost (FC), variable cost(VC), total response time cost (RC), total expected wait-ing time (E(W )), average DC utilization (), DCs openedand their capacity levels (Open DCs (yjk)), expected wait-ing time at the DCs (Wj), DC utilization (j), the numberof cuts generated (CUT), the number of iterations required(ITR) and the CPU time in seconds (CPU(s)) under dif-ferent scenarios. The supply chain network congurationfor two extreme values of response time cost ( = 0 versus

= 1000) are shown in Fig. 5. Figure 6(a) shows the effectof changing response time cost on the total expected waitingtime (E(W )), and Fig. 6(b) shows the effect of changing thetotal expected waiting time E(W ) on the sum of the xedand transportation costs. The observations are as follows.

1. Figure 5 shows that the supply chain conguration thatignores congestion opens four DCs (medium size DCsin New York, Los Angeles and Chicago and a smallDC in Houston) whereas the conguration that consid-ers congestion opens ve DCs, all with capacity level 3.Also, there is substantial reallocation of customer de-mand among the DCs in order to balance the work-load to reduce the DC utilization and overall responsetime in the system. Hence, the supply chain congurationthat considers congestion and its effect on response timecan be different from the traditional conguration thatignores congestion. However, we observe that at veryhigh values of , the congurations (location, capacityand demand allocations) are not signicantly differentamong the M/G/1, M/M/1 and M/D/1 cases.

2. As we see from Table 1, even with very small valuesof average response time cost, t = 0.1 meani,j(icij)or t = 1 meani,j(icij), substantial improvement in thetotal expected waiting time (E[W ]) can be achievedover t = 0. For example, for the M/G/1 case (CV =1.5), E[W ] decreases from 77.41 units to 48.37 units for = 0.1 and 1 respectively. This is due to the even distri-bution of demand among DCs. Figure 6(a) also showsthat the substantial reduction in response time can beachieved with a small value of the response time costs.This is because, as we increase the magnitude of the re-sponse time cost, DCs with higher capacity are usedand/or the number of DCs opened increases, averageDC utilization decreases, thereby reducing congestionand improving average response time. From Fig. 6(b),we see that the left portion of the curves are quite at,indicating that substantial improvement (decrease) in re-sponse time can be achieved with a small increase in xedand transportation costs.

3. As the response time cost becomes dominant comparedto other cost components, DCs with higher capacity lev-els are opened, and average DC utilization decreases,thereby improving (decreasing) the response time. Forexample, in Table 1, in the M/G/1 case (CV = 1.5),for = 1, four medium size DCs are opened, whereasfor = 1000, ve large size DCs are opened. The aver-age DC utilization decreases from 0.83 to 0.44 and theexpected waiting time decreases from 48.37 to 5.44.

4. As we increase the magnitude of the response time cost,the transportation cost decreases initially and then in-creases. For example, in Table 1, in the M/G/1 case (CV= 1.5), for = 0, TC = 103, 577; for = 1, TC = 101,849; and for = 1000, TC = 106, 029. This is due tothe reallocation of customer demand among DCs in anattempt to reduce the total expected waiting time.


Table 1. Comparison of the MTO supply chain network congurations for M/G/1, M/M/1 and M/D/1 cases: an illustrativeexample

M/G/1 (CV = 1.5) M/M/1 (CV = 1) M/G/1 (CV = 0.5) M/D/1 (CV = 0) = 0 TC 146, 965 146, 965 146, 965 146, 965

FC 43, 388 43, 388 43, 388 43, 388VC 103, 577 103, 577 103, 577 103, 577RC 0 0 0 0E(W ) 0.96 0.96 0.96 0.96Open DCs 1(2), 2(2), 3(2), 4(1) 1(2), 2(2), 3(2), 4(1) 1(2), 2(2), 3(2), 4(1) 1(2), 2(2), 3(2), 4(1)Wj [138.79, , 5.27, ] [138.79, , 5.27, ] [138.79, , 5.27, ] [138.79, , 5.27, ]j [0.99, 1.00, 0.84, 1.00] [0.99, 1.00, 0.84, 1.00] [0.99, 1.00, 0.84, 1.00] [0.99, 1.00, 0.84, 1.00]CUT 0 0 0 0ITR 1 1 1 1CPU(s) 0.14 0.14 0.14 0.14

= 0.1 TC 148, 567 148, 333 148, 042 146, 724FC 46, 816 43, 388 43, 388 43, 388VC 101, 494 104, 235 104, 093 104, 033RC 257 710 560 503E(W ) 77.41 214.05 169.00 152.40 0.83 0.96 0.96 0.96Open DCs 1(2), 2(2), 3(2), 4(2) 1(2), 2(2), 3(2), 4(1) 1(2), 2(2), 3(2), 4(1) 1(2), 2(2), 3(2), 4(1)Wj [10.09, 33.51, 2.49, 2.83] [33.93, 98.34, 7.24, 74.55] [42.56, 124.89, 6.66, 94.06] [48.12, 139.37, 6.43, 107.05]j [0.99, 0.97, 0.71, 0.74] [0.97, 0.99, 0.88, 0.99] [0.98, 0.99, 0.87, 0.99] [0.98, 0.99, 0.87, 0.99]CUT 4 20 20 20ITR 2 6 6 6CPU(s) 0.35 0.88 1.69 0.9

= 1 TC 150, 269 149, 683 149, 286 149, 139FC 46, 816 46, 816 46, 816 46, 816VC 101, 849 101, 744 101, 649 101, 609RC 1604 1,124 822 714E(W ) 48.37 33.9 24.79 21.55 0.83 0.83 0.83 0.83Open DCs 1(2), 2(2), 3(2), 4(2) 1(2), 2(2), 3(2), 4(2) 1(2), 2(2), 3(2), 4(2) 1(2), 2(2), 3(2), 4(2)Wj [10.09, 15.08, 2.49, 3.39] [10.09, 18.11, 2.49, 3.21] [10.09, 22.02, 2.49, 3.06] [10.09, 24.19, 2.49, 3.00]j [0.91, 0.94, 0.71, 0.77] [0.91, 0.95, 0.71, 0.76] [0.91, 0.96, 0.71, 0.75] [0.91, 0.96, 0.71, 0.75]CUT 4 8 16 12ITR 2 3 5 4CPU(s) 0.33 0.41 0.91 0.45

= 10 TC 157, 274 155, 674 154, 324 153, 874FC 52, 078 49, 447 49, 447 49, 447VC 101, 407 101, 640 101, 638 101, 616RC 3789 4,587 3,240 2811E(W ) 11.43 13.84 9.77 8.48 0.67 0.75 0.75 0.75Open DCs 1(3), 2(3), 3(2), 4(2) 1(2), 2(3), 3(2), 4(2) 1(2), 2(3), 3(2), 4(2) 1(2), 2(3), 3(2), 4(2)Wj [1.75, 2.27, 2.10, 1.94] [6.63, 2.27, 2.54, 2.39] [6.63, 2.27, 2.58, 2.36] [6.78, 2.27, 2.65, 2.28]j [0.64, 0.69, 0.68, 0.66] [0.87, 0.69, 0.72, 0.71] [0.87, 0.69, 0.72, 0.70] [0.87, 0.69, 0.73, 0.69]CUT 12 12 8 12ITR 4 4 3 4CPU(s) 0.65 0.62 0.49 0.60

= 100 TC 182, 451 176, 255 172, 486 170, 940FC 57, 340 57, 340 57, 340 54, 709VC 101, 987 101, 494 101, 494 101, 557RC 23, 124 17, 421 13, 650 14, 675E(W ) 6.98 5.26 4.12 4.43 0.56 0.56 0.56 0.61Open DCs 1(3), 2(3), 3(3), 4(3) 1(3), 2(3), 3(3), 4(3) 1(3), 2(3), 3(3), 4(3) 1(3), 2(3), 3(3), 4(2)Wj [1.45, 1.68, 0.96, 1.05] [1.54, 1.84, 0.91, 0.97] [1.54, 1.84, 0.91, 0.97] [1.75, 2.15, 1.00, 1.54]j [0.59, 0.63, 0.49, 0.51] [0.61, 0.65, 0.48, 0.49] [0.61, 0.65, 0.48, 0.49] [0.64, 0.68, 0.50, 0.61]CUT 16 4 4 16ITR 5 2 2 5CPU(s) 0.77 0.26 0.35 0.93

(Continued on next page)


Table 1. Comparison of the MTO supply chain network congurations for M/G/1, M/M/1 and M/D/1 cases: an illustrativeexample (Continued)

M/G/1 (CV = 1.5) M/M/1 (CV = 1) M/G/1 (CV = 0.5) M/D/1 (CV = 0) = 1000 TC 358, 156 316, 316 289, 863 280, 873

FC 71, 675 71, 675 71, 675 71, 675VC 106, 029 102, 974 99, 683 99, 460RC 180, 453 141, 666 118, 506 109, 738E(W ) 5.44 4.27 3.57 3.31 0.44 0.44 0.44 0.44Open DCs 1(3), 2(3), 3(3), 4(3), 5(3) 1(3), 2(3), 3(3), 4(3), 5(3) 1(3), 2(3), 3(3), 4(3), 5(3) 1(3), 2(3), 3(3), 4(3), 5(3)Wj [0.64, 1.39, 0.78, 0.84, 0.55] [0.66, 1.51, 0.77, 0.83, 0.50] [0.68, 1.67, 0.76, 0.84, 0.43] [0.72, 1.67, 0.76, 0.85, 0.41]j [0.39, 0.58, 0.44, 0.46, 0.35] [0.40, 0.60, 0.44, 0.45, 0.34] [0.40, 0.63, 0.43, 0.46, 0.30] [0.42, 0.63, 0.43, 0.46, 0.29]CUT 35 35 25 10ITR 8 8 6 3CPU(s) 1.66 1.45 1.2 0.39

4.3. Performance of the cutting plane method for MTOsupply chain design

Table 2 displays the performance of the cutting planemethod for MTO supply chain design problems underM/G/1 (CV = 1.5), M/M/1 (CV = 1.0) and M/D/1

(CV = 0) cases for varying problem sizes. The columnsmarked FC, VC and RC represent the xed costs, the vari-able production and transportation, and the response timecosts respectively, expressed as a percentage of total costs(TC). The columns marked E(W ) are the total average wait-ing time in the system, is the average DC utilization and

Fig. 5. Effect of changing response time cost on the supply chain network conguration: (a) = 0; (b) the M/D/1 case, = 1000; (c)the M/M/1 case, = 1000; and (d) the M/G/1 case (CV = 1.5), = 1000.


Fig. 6. (a) Effect of changing response time cost (t) on the total expected waiting time E[W ]; and (b) effect of changing expectedwaiting time E[W ] on the xed costs plus the transportation costs.

DC represents the number of DCs opened. The table alsodisplays the number of constraints generated (CUT), thenumber of iterations of the algorithm (ITR) and the to-tal CPU time in seconds required to obtain the optimalsolution.

The results show that the average CPU time for M/G/1,M/M/1 and M/D/1 cases are 48, 21 and 9 seconds re-spectively, whereas the average number of cuts requiredare 26, 25 and 25 respectively. Also, the maximum CPUtime for M/G/1, M/M/1 and M/D/1 cases are 1147,410 and 107 seconds respectively, whereas the maximumnumber of cuts required are 66, 60 and 50 respectively.The computation times reveal the stability and the ef-ciency of the cutting plane algorithm for different percent-ages of xed, variable and response time costs, whereasthe number of iterations imply that only a fraction of theconstraints in (PL(H)) is required. As the magnitude ofresponse time cost (t) increases, the percentage of responsetime cost becomes more signicant with respect to othercost components and the algorithm seems to require moreCPU time and iterations as large number of cuts are gen-erated. Furthermore, in almost all of the instances, theM/G/1 case requires more cuts, and hence more CPUtime to solve than the M/M/1 and M/D/1 cases. Thisis attributed to the non-linearity in the expression of ex-pected waiting time for M/G/1 queues. It is also worth-while noting that the computational times for the secondset of instances (I = 50, J = 20 and K = 3) are compara-tively higher than others because the optimal solution hashighly congested DCs. In the model, this corresponds to thevalue of R/(1 + R) approaching one. At the at portionof R/(1 + R), a higher number of cuts is needed to closethe gap.

4.4. Performance of the Lagrangean heuristic for ATOsupply chain design

The computational performance of the Lagrangeanheuristic for the two-echelon ATO supply chain modelfor the M/G/1, M/M/1 and M/D/1 cases is shownin Table 3. The second subproblem pertaining to theMTO echelon was solved to optimality using the cuttingplane approach. In all these test problems, the heuristicis activated at the nal iteration of the Lagrangeanprocedure. The Lagrangean bound (LAG-H) is ex-pressed as the percentage of heuristic solution and thequality of the heuristic solution (GAP) is expressed as:100 (Heuristic Solution LAG-H)/LAG-H. The tableshows the computational time of the subproblems (SP),the master problem (MP) and the heuristic (H) expressedas a percentage of the total computational time (CPU)for various single (L = 1) and multiple (L = 3 and 5)product instances. From these results, it is evident that theproposed heuristic succeeds in nding feasible solutionsthat are within an average of 2.81, 2.58 and 2.99% of theLagrangean bound in reasonable computational time: 427,390 and 345 seconds for the M/G/1, M/M/1 and M/D/1cases respectively. The total computational time can be ashigh as 1038 seconds in some cases. In terms of the size ofthe test problems, the heuristic is able to solve problems withup to 35 plants, 20 DCs and 150 customers, ve products,ve capacity levels and ve semi-nished products within amaximum of a 6% gap from the optimal solution. Table 4shows that the solution of subproblem 2 accounts for mostof the computational time, 89.08, 89.50 and 89.93% forM/G/1, M/M/1 and M/D/1 cases respectively. The mas-ter problem accounts for 9.87, 9.36 and 9.03%, whereas the

Tab

le2.

Com

puta

tion

alpe

rfor

man

ceof

the

cutt

ing

plan

em

etho

d:M

TO

supp

lych

ain

desi

gn

M/G

/1ca

se(C

V=

1.5)

M/M

/1ca

seM

/D/1

case

No.

IJ

K

FC

(%)

VC

(%)

RC

(%)

TC

E(W

)

DC

CU

TIT

RC

PU

FC

(%)

VC

(%)

RC

(%)

TC

E(W

)

DC

CU

TIT

RC

PU

FC

(%)

VC

(%)

RC

(%)

TC

E(W

)

DC

CU

TIT

RC

PU

150

103

0.1

4060

013

3,71

322

10.

936

245

240

600

133,

594

152

0.93

618

41

4060

013

3,48

495

0.92

618

41

139

592

136,

202

198

0.93

630

62

3959

113

5,18

112

60.

936

245

140

601

134,

356

660.

936

245

15

3958

314

0,63

560

0.85

624

52

4058

213

9,14

841

0.85

618

42

4059

113

7,86

226

0.86

630

63

1040

563

143,

888

350.

776

245

339

574

141,

829

370.

856

306

340

582

139,

498

230.

856

184

250

4252

615

8,72

314

0.61

66

22

4053

815

3,35

517

0.70

618

43

3954

614

8,62

814

0.77

618

42

100

3949

1216

8,46

714

0.61

618

45

4051

916

2,91

710

0.61

66

22

3952

915

6,21

511

0.7

618

43

250

203

0.1

5941

012

2,77

541

90.

9211

445

1059

410

122,

625

296

0.92

1144

58

5941

012

2,48

318

80.

9211

445

71

5643

112

4,84

319

80.

9110

405

1056

431

124,

153

139

0.91

1040

57

5941

112

3,48

299

0.92

1144

57

557

394

129,

842

117

0.86

1155

624

5839

312

7,93

780

0.87

1155

612

5542

212

6,04

563

0.91

1040

58

1057

376

134,

288

890.

8311

556

4457

385

131,

372

740.

8611

556

2258

393

128,

366

450.

8711

445

1050

5137

1215

4,12

341

0.71

1050

654

153

3710

147,

618

330.

7410

506

281

5438

814

0,91

524

0.76

1040

550

100

5233

1617

0,68

929

0.64

1166

711

4749

3515

160,

786

270.

7110

607

410

5237

1215

0,93

220

0.74

1050

610

73

100

53

127

730

196,

048

510.

834

124

127

730

195,

869

370.

834

83

127

730

195,

713

210.

834

124

15

2772

119

7,59

441

0.83

44

20

2773

119

6,93

226

0.83

44

20

2773

019

6,38

916

0.83

412

41

1026

722

199,

377

370.

834

165

127

721

198,

142

260.

834

124

127

731

197,

085

150.

834

42

050

2869

320

6,96

412

0.67

416

51

2770

320

4,85

813

0.75

412

41

2671

320

2,21

113

0.83

48

31

100

2867

521

2,39

912

0.67

416

51

2868

420

9,16

88

0.67

412

41

2769

420

6,23

58

0.75

412

41

200

2965

622

0,24

97

0.56

416

51

2866

521

6,43

06

0.61

48

31

2868

521

1,63

76

0.67

412

41

410

010

31

4752

016

5,40

213

40.

748

325

834

661

178,

665

284

0.93

630

65

3466

017

8,13

515

30.

936

245

35

5347

016

6,56

432

0.65

918

36

3762

118

0,10

457

0.86

721

43

3862

017

9,57

638

0.86

721

43

1052

471

167,

152

310.

659

274

637

621

181,

077

490.

857

285

337

621

180,

216

320.

867

214

250

5146

317

1,45

328

0.65

936

59

3860

318

6,48

026

0.77

728

54

3761

218

4,15

525

0.85

728

53

100

5044

617

6,74

928

0.65

927

412

3758

519

1,08

424

0.77

728

55

3859

318

7,49

415

0.77

728

54

200

4846

618

4,57

214

0.55

88

212

3559

619

8,18

216

0.70

618

44

3461

519

2,88

213

0.77

624

54

510

020

31

4752

016

5,21

815

90.

748

325

947

520

165,

061

119

0.74

832

56

4852

016

4,90

379

0.74

824

45

547

521

166,

275

860.

748

325

947

520

165,

867

650.

748

244

547

520

165,

458

440.

748

325

610

5347

016

6,75

132

0.65

918

35

5347

016

6,50

323

0.65

918

34

4752

116

5,93

936

0.74

824

45

5052

462

169,

629

290.

659

365

752

471

168,

556

200.

659

274

552

471

167,

667

130.

659

274

510

051

454

173,

133

280.

659

365

1051

463

170,

995

200.

659

274

552

462

169,

281

130.

659

274

520

046

477

179,

780

250.

658

325

2150

456

175,

861

200.

659

365

951

464

172,

442

130.

659

274

56

150

55

124

760

225,

407

149

0.91

416

52

2476

022

5,12

411

40.

914

165

224

760

224,

830

810.

924

165

25

2475

122

7,40

096

0.91

48

31

2476

122

6,56

364

0.91

412

41

2476

022

5,80

742

0.91

416

52

1025

741

229,

437

380.

824

83

124

751

227,

963

590.

914

165

224

761

226,

646

340.

914

124

150

2573

223

4,23

619

0.75

416

52

2673

123

2,79

413

0.75

44

21

2574

123

0,86

813

0.82

416

52

100

2572

323

8,46

218

0.75

412

41

2572

223

5,75

813

0.75

416

52

2673

223

3,47

78

0.75

44

21

200

2670

424

3,90

212

0.67

412

42

2671

324

0,68

98

0.67

48

31

2572

323

7,12

38

0.75

416

52

715

010

51

3169

020

7,16

429

40.

926

245

631

690

206,

933

224

0.93

624

55

3169

020

6,70

714

40.

936

245

65

3268

020

8,30

787

0.88

66

23

3466

020

7,97

588

0.89

728

55

3169

020

7,47

480

0.92

624

55

1032

681

209,

108

810.

886

184

532

681

208,

534

550.

886

62

234

660

208,

034

470.

897

285

650

3464

221

3,57

154

0.83

728

510

3464

221

1,75

937

0.84

728

56

3564

121

0,22

923

0.84

721

44

100

3165

321

7,91

737

0.8

624

513

3463

321

5,17

035

0.83

721

47

3464

221

2,31

422

0.84

721

45

200

3264

422

3,72

225

0.73

618

421

3165

422

0,22

024

0.8

624

516

3463

421

6,27

121

0.83

721

47

815

020

51

4852

017

7,44

433

20.

910

405

1148

520

177,

277

245

0.9

1040

58

4852

017

7,11

316

20.

910

405

85

4554

117

8,54

316

90.

899

274

1145

540

178,

141

115

0.89

927

47

4852

017

7,69

399

0.9

1040

59

1045

541

179,

538

157

0.89

936

516

4554

117

8,81

810

30.

899

274

845

540

178,

190

610.

899

274

750

4652

218

4,59

362

0.81

927

438

4553

218

3,14

764

0.85

936

527

4553

218

0,89

351

0.88

936

514

100

4551

418

8,33

559

0.81

927

454

4552

318

5,82

741

0.81

927

427

4552

218

3,62

036

0.85

936

524

200

4947

519

4,47

637

0.71

1030

421

144

515

190,

699

390.

819

274

6845

523

186,

730

240.

819

274

25m

in24

330

122,

775

70.

554

42

024

350

122,

625

60.

614

42

024

370

122,

483

60.

654

42

0m

ax59

7616

243,

902

419

0.93

1166

711

4759

7615

240,

689

296

0.93

1160

741

059

7612

237,

123

188

0.93

1150

610

7m

ean

4156

318

3,02

282

0.77

726

448

3958

318

2,99

567

0.80

725

421

3959

218

6,51

646

0.82

725

49

462

Tab

le3.

Com

puta

tion

alpe

rfor

man

ceof

the

Lag

rang

ean

heur

isti

c:A

TO

supp

lych

ain

desi

gn

M/G

/1ca

se(C

V=

1.5)

M/M

/1ca

seM

/D/1

case

No.

IJ

KL

MN

LA

G-H

(%)

GA

P(%

)S

P(%

)M

P(%

)H (%

)C

PU

(s)

LA

G-H

(%)

GA

P(%

)S

P(%

)M

P(%

)H (%

)C

PU

(s)

LA

G-H

(%)

GA

P(%

)S

P(%

)M

P(%

)H (%

)C

PU

(s)

150

103

110

30.

199

.78

0.22

84.3

114

.36

1.33

108

96.8

33.

1786

.06

12.7

51.

1985

98.3

31.

6792

.96

6.29

0.75

561

95.8

44.

1684

.55

13.5

11.

9411

199

.59

0.41

85.2

112

.96

1.83

8594

.58

5.42

91.3

48.

060.

6062

594

.77

5.23

87.1

11.5

11.

3911

899

.97

0.03

86.1

312

.19

1.68

8698

.84

1.16

92.0

55.

942.

0169

1098

.33

1.67

8314

.98

2.02

121

95.7

74.

2393

.14

5.53

1.33

8896

.94

3.06

88.4

310

.90

0.67

7150

98.9

11.

0991

.52

6.84

1.64

122

94.1

25.

8889

.78.

821.

4810

199

.86

0.14

88.5

89.

591.

8373

100

97.2

02.

8088

.78

11.1

30.

0912

795

.40

4.6

91.7

37.

011.

2610

498

.65

1.35

90.9

17.

931.

1675

250

203

110

30.

197

.07

2.93

87.8

511

.17

0.98

191

99.6

60.

3485

.99

12.4

11.

617

698

.23

1.77

93.5

95.

440.

9713

11

98.2

41.

7685

.03

13.1

51.

8219

296

.36

3.64

91.1

78.

290.

5417

994

.79

5.21

91.8

76.

361.

7714

75

99.3

20.

6885

.22

13.0

41.

7419

999

.07

0.93

92.5

66.

141.

318

195

.92

4.08

94.0

95.

390.

5214

810

97.8

92.

1188

.68

10.7

30.

5920

999

.37

0.63

90.5

58.

590.

8618

599

.73

0.27

86.7

711

.58

1.65

155

5094

.06

5.94

94.6

65.

170.

1721

199

.02

0.98

87.1

812

.46

0.36

186

97.7

12.

2991

.02

6.97

2.01

158

100

94.6

55.

3589

.75

9.67

0.58

213

95.0

94.

9192

.35

7.59

0.06

189

97.0

52.

9593

.33

6.03

0.64

162

310

05

33

205

199

.97

0.03

93.7

85.

680.

5418

298

.35

1.65

89.1

810

.14

0.68

168

98.5

61.

4486

.09

12.0

91.

8215

05

96.5

83.

4285

.45

14.3

90.

1618

599

.75

0.25

92.9

6.7

0.4

169

95.3

34.

6788

.18

11.2

70.

5515

910

97.2

32.

7786

.79

12.6

40.

5718

794

.26

5.74

87.1

410

.91.

9617

195

.07

4.93

86.8

911

.50

1.61

159

5097

.41

2.59

93.2

25.

830.

9519

096

.71

3.29

92.6

65.

431.

9117

699

.89

0.11

86.5

12.7

20.

7816

110

099

.68

0.32

93.1

16.

640.

2519

499

.22

0.78

92.2

56.

421.

3318

098

.85

1.15

88.0

211

.86

0.12

165

200

95.6

44.

3693

.61

5.06

1.33

196

99.5

50.

4588

.63

10.3

51.

0217

594

.97

5.03

85.3

812

.94

1.68

165

410

010

33

205

194

.47

5.53

91.7

38.

090.

1823

894

.01

5.99

87.3

511

.95

0.7

217

95.4

04.

6091

.29

8.25

0.46

197

594

.51

5.49

90.3

18.

261.

4326

895

.93

4.07

92.1

7.29

0.61

218

94.2

15.

7986

.72

11.7

21.

5620

110

98.8

31.

1790

.63

9.29

0.08

277

98.7

71.

2387

.36

11.1

81.

4622

297

.99

2.01

92.5

15.

881.

6120

250

97.2

22.

7886

.47

11.8

71.

6628

495

.41

4.59

86.5

112

.94

0.55

232

99.5

80.

4287

.82

11.8

40.

3420

410

097

.40

2.60

91.1

77.

411.

4229

698

.45

1.55

90.2

19.

10.

6923

496

.62

3.38

85.7

112

.99

1.30

210

200

95.9

44.

0685

.513

.22

1.28

298

97.4

52.

5590

.95

7.79

1.26

234

94.8

05.

2091

.81

6.52

1.67

215

510

020

33

205

198

.97

1.03

85.8

112

.14

2.05

297

98.0

81.

9287

.23

10.9

21.

8528

395

.40

4.60

92.2

25.

791.

9925

15

97.6

12.

3992

.95.

911.

1929

894

.37

5.63

92.1

57.

570.

2828

598

.65

1.35

93.1

15.

431.

4626

810

95.8

24.

1887

.46

11.1

31.

4130

097

.12

2.88

87.5

410

.71

1.75

285

99.6

10.

3988

.22

11.1

00.

6826

950

95.5

54.

4591

.68

7.47

0.85

309

97.0

32.

9789

.47

9.11

1.42

286

94.9

25.

0886

.04

12.1

31.

8327

110

099

.12

0.88

85.5

912

.88

1.53

313

98.6

61.

3487

.43

11.7

90.

7828

994

.12

5.88

86.8

112

.00

1.19

277

200

95.4

64.

5489

.19.

641.

2634

896

.44

3.56

92.2

27.

550.

2329

396

.12

3.88

91.1

68.

600.

2427

86

150

55

535

51

98.6

71.

3389

.13

9.93

0.94

551

99.9

40.

0686

.61

12.2

61.

1348

695

.08

4.92

92.2

16.

980.

8142

35

98.3

51.

6585

.54

13.9

60.

5056

394

.75

5.25

86.2

11.9

11.

8949

995

.08

4.92

93.1

95.

551.

2642

710

94.7

35.

2793

.29

6.17

0.54

576

96.0

53.

9590

.43

9.19

0.38

513

98.6

91.

3187

.81

10.8

61.

3344

450

99.8

30.

1789

.76

9.06

1.18

593

98.6

61.

3486

.77

11.6

81.

5553

699

.16

0.84

90.3

39.

110.

5645

110

099

.01

0.99

85.8

113

.66

0.53

599

96.9

03.

192

.06

7.38

0.56

544

97.2

22.

7891

.59

7.71

0.70

484

200

99.8

70.

1387

.42

12.0

70.

5161

799

.44

0.56

92.7

35.

441.

8354

794

.26

5.74

88.2

910

.55

1.16

485

715

010

55

355

199

.23

0.77

90.3

99.

290.

3271

599

.91

0.09

89.1

89.

531.

2964

594

.90

5.10

93.5

46.

090.

3758

85

97.2

32.

7791

.57.

630.

8771

799

.00

189

92

690

96.4

73.

5390

.24

9.10

0.66

598

1095

.34

4.66

94.5

55.

320.

1372

698

.73

1.27

89.7

18.

641.

6569

199

.17

0.83

89.6

9.16

1.24

615

5097

.13

2.87

90.7

47.

411.

8574

198

.15

1.85

91.6

7.61

0.79

697

95.2

74.

7388

.82

10.4

70.

7162

410

095

.27

4.73

91.7

46.

661.

6074

398

.50

1.5

87.9

111

.34

0.75

698

99.9

40.

0687

.06

12.8

80.

0662

720

098

.12

1.88

91.4

47.

151.

4174

796

.97

3.03

91.4

57.

141.

4170

697

.41

2.59

92.4

96.

960.

5563

28

150

205

535

51

98.3

11.

6991

.96.

981.

1298

195

.47

4.53

87.5

811

.25

1.17

922

95.0

74.

9394

.15.

130.

7776

55

96.6

83.

3289

.86

8.58

1.56

1000

99.8

10.

1991

.22

7.21

1.57

935

95.9

04.

1085

.95

12.4

01.

6580

010

95.1

24.

8893

.25.

870.

9310

0496

.02

3.98

87.9

610

.64

1.4

945

98.3

51.

6592

.89

6.46

0.65

810

5095

.71

4.29

84.3

214

.16

1.52

1007

99.5

00.

591

.95

7.83

0.22

955

94.3

55.

6586

.913

.03

0.07

871

100

98.6

41.

3685

.82

12.3

51.

8310

1994

.38

5.62

89.9

69.

180.

8697

299

.94

0.06

90.3

29.

600.

0889

020

094

.38

5.62

84.7

214

.67

0.61

1038

94.0

95.

9188

.84

9.49

1.67

976

99.7

20.

2892

.09

6.17

1.74

904

min

94.0

60.

0383

.00

5.06

0.08

108

99.9

70.

0385

.21

5.43

0.06

8594

.12

0.06

85.3

85.

130.

0656

max

99.9

75.

9494

.66

14.9

82.

0510

3894

.01

5.99

93.1

412

.96

2.00

976

99.9

45.

8894

.10

13.0

32.

0190

4m

ean

97.1

92.

8189

.08

9.87

1.05

427

97.4

22.

5889

.50

9.36

1.14

390

97.0

12.

9989

.93

9.03

1.04

345

463

Tab

le4.

Com

pari

son

betw

een

the

cutt

ing

plan

em

etho

dan

dth

eL

agra

ngea

nhe

uris

tic

for

AT

Osu

pply

chai

nde

sign

for

I=

100,

J=

10,K

=3,

M=

20an

dN

=1

Cut

ting

plan

em

etho

dL

agra

ngea

nhe

uris

tic

FC

(%)

VC

(%)

RC

(%)

E(W

)

DC

sC

UT

ITR

CP

U(s

)F

C(%

)V

C(%

)R

C(%

)E

(W)

D

Cs

LR

(%)

GA

P(%

)C

PU

(s)

Tig

htca

paci

ties

,r=

30.

150

500

689.

30.

965

103

714

4159

068

9.3

0.96

595

.88

4.12

251

149

501

209.

20.

965

103

1029

4159

121

8.2

0.96

596

.78

3.22

268

548

502

169.

30.

955

103

1585

4357

119

5.9

0.94

595

.04

4.96

269

1051

491

32.5

0.69

510

321

8042

562

51.8

40.

765

95.8

14.

1927

150

5147

214

.51

0.6

55

210

0244

552

17.4

70.

625

96.7

23.

2817

710

051

472

8.39

0.54

55

281

043

543

11.4

70.

65

94.9

95.

0127

850

050

464

8.39

0.54

55

296

443

498

10.8

20.

655

96.4

3.6

423

1000

4944

76.

210.

445

52

1100

4046

149.

820.

445

96.8

83.

1242

720

0046

4113

5.99

0.44

55

211

4839

4121

4.75

0.43

697

.11

2.89

444

5000

3832

303.

950.

327

285

1699

3333

343.

980.

327

97.0

12.

9945

1M

oder

ate

capa

citi

es,r

=5

0.1

4654

026

1.9

0.84

612

315

0039

610

332.

30.

854

98.1

11.

8920

51

4654

098

.12

0.83

66

296

439

610

122.

70.

844

96.8

33.

1740

75

4158

134

.16

0.83

44

210

1938

611

40.7

20.

844

96.6

93.

3132

510

4158

134

.16

0.83

44

289

838

611

40.7

20.

844

95.5

24.

4828

050

4156

421

.84

0.76

44

214

6739

583

27.5

60.

794

97.2

72.

7332

210

043

552

6.81

0.56

44

214

9540

572

7.74

0.58

497

.11

2.89

333

500

4254

46.

810.

564

42

1122

3753

107.

740.

584

96.8

83.

1236

510

0040

5010

6.81

0.56

44

213

7034

4818

7.74

0.58

494

.01

5.99

291

2000

4144

155.

520.

445

103

2524

4038

226.

630.

476

95.6

4.4

589

5000

3029

414.

550.

376

123

1981

3029

406.

530.

446

97.6

72.

3323

2L

oose

capa

citi

es,r

=10

0.1

4060

081

.01

0.83

44

211

0340

600

81.6

60.

834

96.5

73.

4354

61

4060

062

.75

0.83

44

295

539

600

63.8

50.

834

96.6

53.

3543

55

4060

140

.27

0.83

44

290

639

601

42.2

30.

834

98.9

11.

0948

710

3959

140

.27

0.83

44

287

539

601

41.7

80.

834

94.4

15.

5932

750

4058

211

.71

0.67

44

297

940

582

12.9

70.

674

96.1

93.

8135

410

040

564

11.7

10.

674

42

1140

4057

311

.75

0.61

496

.99

3.01

654

500

4155

57.

020.

564

42

1080

3852

118.

960.

564

97.2

72.

7376

510

0038

5210

6.94

0.56

48

317

6534

4719

7.84

0.56

498

.11.

964

220

0034

4719

6.85

0.56

48

318

8441

3821

7.54

0.37

695

.59

4.41

822

5000

3229

394.

430.

376

123

2148

3129

394.

430.

376

96.3

73.

6398

4M

in30

290

3.95

0.32

44

271

430

290

3.98

0.32

494

.01

1.09

177

Max

5160

4168

9.3

0.96

728

525

2444

6140

689.

30.

967

98.9

15.

9998

4M

ean

4350

763

.38

0.66

57

213

1439

529

640.

645

97.5

13.

4942

1

464


heuristic accounts for 1.05, 1.14 and 1.04%, on average forM/G/1, M/M/1 and M/D/1 cases respectively.

In Table 4, we compare the performance of the cuttingplane algorithm and the Lagrangean heuristic and reportthe results for one problem set (I = 100, J = 10, K = 3,M = 20 and N = 1) for different values of the ratio of plantcapacities to total demand (r = m Pm/ i i). The resultsshow that the Lagrangean heuristic outperforms the cuttingplane method in terms of computational time. On average,the Lagrangean heuristic takes 421 seconds whereas thecutting plane method takes 1344 seconds.

5. Conclusions

In this paper, we modeled and analyzed the effect of re-sponse time consideration on the design of MTO and ATOsupply chain networks. We presented an MTO supply chaindesign model that captures the trade-off among responsetime, the xed cost of opening DCs and equipping themwith sufcient capacity, and the transportation cost asso-ciated with serving customers. Under the assumption thatthe customer demand follows a Poisson process and servicetimes follow general distribution, the DCs were modeled asa network of single-server queues, whose capacity levels andlocations are decision variables. We presented a non-linearMIP formulation, a linearization procedure and an exactsolution approach based on a cutting plane method. Ourcomputational results indicate that the cutting plane algo-rithm provides optimal solution for moderate instances ofthe problem in few iterations and reasonable computationtimes.

We also presented a model for designing two-echelonATO supply chain networks, that consists of plants andDCs serving a set of customers. Lagrangean relaxation wasapplied to decompose the problem by echelonone for theMTS echelon and the other for the MTO echelon. While La-grangean relaxation provides a lower bound, a heuristic isproposed that uses the solution of the subproblems to con-struct an overall feasible solution. Computational resultsreveal that the heuristic solution is on average within 6% ofits optimal. We used the models to demonstrate empiricallythat substantial improvement (decrease) in response timecan be achieved with a minimal increase in total cost asso-ciated with designing supply chains. Also, we showed thatthe supply chain conguration (DC location and capacity,and allocation of customers to DCs) obtained using themodel that considers congestion can be very different fromthose obtained using the traditional models that ignoresresponse time.

The focus of our ongoing research is to develop modelsfor the design of MTO and ATO supply chains under moregeneral settings: general demand arrivals, general servicetime distributions and multiple servers. Due to the lack ofan exact expression for the waiting time in these settings, weplan to explore two approaches to model the problem. Therst approach relies on approximations of expected waiting

times whereas the second approach integrates simulationwithin an MIP framework.

Acknowledgement

The authors would like to thank two anonymous refereesand the Associate Editor for helpful comments.

References

Amiri, A. (1997) Solution procedures for the service system design prob-lem. Computers and Operations Research, 24, 4960.

Berman, O. and Krass, D. (2002) Facility location problems with stochas-tic demands and congestion, in Facility Location: Applications andTheory, Drezner, Z. and Hamacher, H.W. (eds.), Springer-Verlag, NewYork, pp. 329369.

Daskin, M.S. (2004) SITATIONfacility location software. Depart-ment of Industrial Engineering and Management Sciences, North-western University, Evanston, IL. Available at http://users.iems.nwu.edu/mdaskin.

Dell, M. (2000) Direct from Dell: Stratagies that Revolutionized an Indus-try, Harper Collins, New York, NY.

Dogan, K. and Goetschalckx, M. (1999) A primal decomposition methodfor the integrated design of multi-period production-distribution sys-tems. IIE Transactions, 31(11), 10271036.

Duran, M.A. and Grossman, I.E. (1986) An outer approximation algo-rithm for a class of mixed-integer nonlinear programs. MathematicalProgramming, 36, 307339.

Elhedhli, S. (2006) Service system design with immobile servers, stochas-tic demand and congestion. Manufacturing and Service OperationsManagement, 8(1), 9297.

Elhedhli, S. and Gzara, F. (2008) Integrated design of supply chain net-works with three echelons, multiple commodities, and technology se-lection. IIE Transactions, 40(1), 3144.

Erenguc, S.S., Simpson, N.C. and Vakharia, A.J. (1999) Integrated pro-duction/distribution planning in supply chains: an invited review.European Journal of Operational Research, 115, 219236.

Eskigun, E., Uzsoy, R., Preckel, P.V., Beaujon, G., Krishnan, S. and Tew,J.D. (2005) Outbound supply chain network design with mode selec-tion, lead times, and capacitated vehicle distribution centers. EuropeanJournal of Operational Research, 165(1), 182206.

Gupta, D. and Benjaafar, S. (2004) Make-to-order, make-to-stock, ordelay product differentiation? A common framework for modellingand analysis. IIE Transactions, 36, 529546.

Huang, S., Batta, R. and Nagi, R. (2005) Distribution network design: se-lection and sizing of congested connections. Naval Research Logistics,52, 701712.

Kelley, J. E. (1960) The cutting plane method for solving convex programs.Journal of the SIAM, 8, 703712.

Kim, I. and Tang, C.S. (1997) Lead time and response time in a pullproduction control system. European Journal of Operational Research,101, 474485.

Klose, A. and Drexl, A. (2005) Facility location models for distributionsystem design. European Journal of Operational Research, 162, 429.

Margretta, J. (1998) The power of virtual integration: an interview withDell computers Michael Dell. Harvard Business Review, 76(2), 7384.

Marianov, V. and Serra, D. (2002) Location-allocation of multiple-serverservice centers with constrained queues or waiting times, Annals ofOperations Resear

response time reduction in make-to-order and assemble-to-order supply chain design

Documents

order supply chains

order systems

order problem

supply chain conguration

order mto systems

responsive supply chains

mto supply chain margretta

product proliferation