the stochastic multiperiod location transportation problem

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The Stochastic Multiperiod Location TransportationProblemWalid Klibi, Francis Lasalle, Alain Martel, Soumia Ichoua,

To cite this article:Walid Klibi, Francis Lasalle, Alain Martel, Soumia Ichoua, (2010) The Stochastic Multiperiod Location Transportation Problem.Transportation Science 44(2):221-237. http://dx.doi.org/10.1287/trsc.1090.0307

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The Stochastic Multiperiod LocationTransportation Problem

Walid Klibi, Francis Lasalle, Alain MartelInteruniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT),

Université de Montréal, Montréal, Quebéc H3C 3J7, Canada, andFaculté des Sciences de l’Administration, Département Opérations et Systèmes de Décision,

Université Laval, Québec City, Québec G1V 0A6, Canada{[email protected], [email protected], [email protected]}

Soumia IchouaInteruniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT),

Université de Montréal, Montréal, Quebéc H3C 3J7, Canada, andDepartment of Computer Science and Engineering, Johnson C. Smith University, Charlotte, North Carolina 28216,

[email protected]

This paper studies a stochastic multiperiod location-transportation problem (SMLTP) characterized by mul-tiple transportation options, multiple demand periods, and a stochastic demand. We consider the determi-

nation of the number and location of the depots required to satisfy customer demand as well as the mission ofthese depots in terms of the subset of customers they must supply. The problem is formulated as a stochasticprogram with recourse, and a hierarchical heuristic solution approach is proposed. It incorporates a tabu searchprocedure, an approximate route length formula, and a modified procedure of Clarke and Wright (Clarke, G.,J. W. Wright. 1964. Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res. 12568–581). Three neighbourhood exploration strategies are proposed and compared with extensive experimentsbased on realistic problems.

Key words : location problem; transportation problem; stochastic customer order process; stochasticprogramming; Monte Carlo scenarios; tabu search

History : Received: June 2008; revisions received: December 2008, September 2009; accepted: September 2009.Published online in Articles in Advance March 5, 2010.

IntroductionDepot location decisions arise at the strategic plan-ning level of distribution networks. Fundamentally,in a business context, location-allocation problemsinvolve the determination of the number and locationof the depots required to satisfy customer demand aswell as the mission of these depots in terms of thesubset of customers they must supply. Deterministic,dynamic, and stochastic models have been proposedin the last decade to solve variants of the depot loca-tion problem. Comprehensive reviews of the litera-ture on these models are found in Owen and Daskin(1998) and Klose and Drexl (2005). Three formulationsof the problem under uncertainty were studied based,respectively, on stochastic programming (Birge andLouveaux 1997; Snyder and Daskin 2005), robust opti-mization (Kouvelis and Yu 1997), and queuing theory(Berman, Krass, and Xu 1995). In most of these for-mulations, the demand is the main random variableconsidered.When the distribution network designed is in

operation, on a daily basis, the depots must shipthe products ordered by their customers to specified

ship-to points. Moreover, nowadays, an increasingnumber of companies rely on external transportationresources to ship their products to customers and theydo not have their own vehicle fleet. In this context,depending on the size of the orders received, theymay be shipped in single-customer partial truckloads(STL) or full truckloads (FTL), on multidrop truck-load routes (MTL), or via less-than-truckload (LTL)transportation. The profits of the distribution networkdepend heavily on the efficiency of the transporta-tion decisions made. These transportation options aredefined more precisely in the following pages. Thevehicle routing problems (VRP) encountered for theMTL case were studied extensively in the literature(see Laporte and Osman 1995 for a review). Althoughexact solution methods were developed for determin-istic routing problems (Toth and Vigo 1998), theircomplexity led most researchers in this field to pro-pose heuristic solution methods (Laporte et al. 2000).A few stochastic versions of the VRP problem werealso studied (Laporte and Louveaux 1998). Despitethe fact that location and transportation problems areclearly interrelated, the large classical literature on

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Klibi et al.: The Stochastic Multiperiod Location Transportation Problem222 Transportation Science 44(2), pp. 221–237, © 2010 INFORMS

these problems assumes that they are independent.In the last few years, however, major efforts havebeen devoted to the development of integrated mod-els such as location-routing and inventory-routingmodels. A recent review of these integrated models isfound in Shen (2007).The first classification of location-routing problems

(LRPs) is found in Laporte (1988), who proposes var-ious deterministic formulations of the problem. Morerecent papers (Chien 1993; Tuzun and Burke 1999;Barreto et al. 2007; Prins et al. 2007) present heuris-tic methods to solve deterministic LRPs. A multieche-lon version with inventory is addressed in Ambrosinoand Scutella (2005). An uncapacitated LRP with dis-tance constraints is studied in Berger, Coullard, andDaskin (2007); they propose a set-partitioning formu-lation and a branch-and-price solution approach. Thehierarchical structure of the problem is stressed inNagy and Salhi (1996a, b), who develop an heuris-tic in which a routing phase is embedded into thelocation method. A dynamic LRP defined over a plan-ning horizon is examined in Laporte and Dejax (1989)and in Salhi and Nagy (1999). A stochastic LRP isstudied in Laporte, Louveaux, and Mercure (1989)and in Albareda-Sambola, Fernandez, and Laporte(2007). In the stochastic program considered, depotlocations and a priori routes must be specified in thefirst stage, and second-stage recourse decisions dealwith first-stage route failures. Recently, Shen (2007)proposed a stochastic LRP model based on routingcost estimations. An approach to solve continuousLRPs is presented in Salhi and Nagy (2009). Com-prehensive reviews of location-routing models and oftheir applications are provided in Min, Jayaraman,and Srivastava (1998) and Nagy and Salhi (2007).The LRP models found in the literature have two

main shortcomings when compared to the strategicneeds of distribution businesses using external trans-portation resources.First, most LRP models assume that the distributor

has its own vehicle fleet and that the transportationproblems encountered are purely VRPs. When com-mon or contract carriers are used, more options areavailable, namely, single-customer full (FTL) or partial(STL) truckload, multidrop truckload (MTL), and less-than-truckload (LTL) transportation. The FTL, STL,and MTL options are provided by truckload (TL) car-riers. The distinction comes from the way the truck isused by the shipper: FTL refers to the case where afull trailer is delivered to a single destination, STL tothe case where the trailer shipped is not loaded to fullcapacity, and MTL to the case where the truck deliv-ery route involves more than one destination. In thelast case, the route is elaborated by the shipper as if itwere its own truck. This leads to location-transportationproblems (LTP) instead of location-routing problems.

These problems must not be confused with the classof problems known as transportation-location problems,which are, in fact, transshipment-location problems aspointed out by Nagy and Salhi (2007).Second, most LRP models implicitly assume that

location decisions and routing decisions can be madesimultaneously for the planning horizon considered(a year, for example), i.e., that the routes do notchange on a daily basis and that customer demandis static. In the business context considered here,the location and mission of depots must be fixedfor the planning horizon but transportation decisionsare made on a daily basis in reaction to the cus-tomer orders received. This gives rise to what wecall stochastic multiperiod location-transportation prob-lems (SMLTP). Most LRP models do not consider thecustomer order arrival process explicitly. Salhi andNagy (1999) introduce a deterministic multiperiodlocation-routing problem to analyse the consistencyof the solutions provided by static location-routingmethods. Laporte and Dejax (1989) consider a relateddynamic location-routing problem but they do notrequire that the location and mission of the depots isfixed for the planning horizon. In our context, a ran-dom number of customers order a random amountof products on a daily basis, and these orders mustbe delivered the next day. The dynamic customerdemand pattern is therefore not known when theproblem has to be solved. We assume, however, thatthe demand process is stationary.The aim of this paper is to provide a more precise

definition of the SMLTP, to formulate it as a stochasticprogram with recourse, and to propose an heuristicmethod to solve it.Laporte (1988) has shown that the LRP is NP-hard.

For each time period, the transportation subproblemconsidered in the SMLTP is an open VRP having thesame NP-hardness property as a basic VRP (Sariklisand Powell 2000). Thus, the SMLTP is an NP-hardstochastic combinatorial optimization problem. Con-sequently, although exact solution approaches maybe used to solve small size instances of the SMLTPoptimally, they are not capable of handling realisticinstances. This justifies the development of heuristicsthat better achieve a good trade-off between compu-tation time and solution quality.The paper is organized as follows. The next sec-

tion provides a detailed description of the SMLTP andformulates it as a stochastic program with recourse.The following section shows how this stochastic pro-gram can be solved with a sample average approx-imation (SAA) mixed-integer program (MIP) basedon a Monte Carlo scenario sampling scheme. In thefollowing section, a solution approach is proposedto solve the SMLTP: It combines efficiently, in anested schema, a Monte Carlo scenario generation

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Klibi et al.: The Stochastic Multiperiod Location Transportation ProblemTransportation Science 44(2), pp. 221–237, © 2010 INFORMS 223

procedure, a transportation heuristic, and a location-allocation tabu search procedure. The section also pro-poses various neighbourhood exploration strategies.Finally, in the last section, experiments are designedto evaluate the various versions of the tabu heuristicproposed for problems with different realistic charac-teristics. Computational results are also presented anddiscussed.

Problem Description and FormulationThe Business ContextTo start with, let us examine the business contextof the SMLTP more closely. A company purchases(or manufactures) a family of similar products, con-sidered as a single product, from a number of supplysources. This product is sold to customers located in alarge geographical area and, hence, must be shippedto a large number of ship-to points. To provide goodservice, the company cannot satisfy customer ordersdirectly from the supply sources and must operatea number of uncapacitated distribution centers (alsoreferred to as depots). Customers order a varyingquantity of product on a daily basis and the com-pany wants to provide next-day delivery using com-mon or contract carriers. For a given day t, at eachdepot, when all the orders are in, the company plansits transportation for the next day and requests fromits carriers the trucks required to deliver products toship-to points through FTL, STL, MTL, or LTL trans-portation. When the orders are delivered, the trucksneed not return to the depot. Let L be the set of depotsconsidered, P the set of ship-to points where orderscan be delivered, Lp ⊂ L the subset of depots that areable to provide next-day delivery service to ship-topoint p ∈ P , and, conversely, Pl ⊂ P the subset of ship-to points that could be served by depot l. Also, forday t, let KTL

lt be the set of feasible TL (STL or MTL)delivery routes from depot l ∈ L, considering ship-to point demands, service requirements, and vehiclecharacteristics. Let Pk ⊂ P be the ordered set of ship-to points in route k ∈ KTL

lt . The TL tariff wk chargedby the carriers to assign a specific truck to route k isbased on the following formula:

wk =maxrkmk�TLl + al�Pk� − 1 �where

mk = total mileage of route k,rk = transportation cost rate per mile for the vehicle

associated to route k,TLl =minimum transportation charge for any TL

route starting at depot l,al = drop charge for any additional stop on a route

starting at depot l.

Note that the charge wk must be paid to the car-rier independently of the load shipped in the vehicle.When a customer orders more than a truckload on agiven day, we assume that the depot ships as muchas possible in full truckloads. Note also that severalroutes k ∈ KTL

lt could have identical ship-to point setsPk if vehicle types with different capacities and ratescan be used. However, if this occurs, there is no rea-son not to use the route with the lowest charge wk.We therefore assume, in what follows, that the setsKTLlt contain only nondominated TL routes.If the vehicle on route k has a load close to its

capacity, then multidrop TL transportation is usu-ally cheaper than LTL transportation. For the ordersreceived on a given day, if it is not possible to con-struct routes with a good load or capacity ratio for allorders, then it may be cheaper to send some ordersby LTL transportation. In particular, if a TL route ismore expensive than LTL shipments, then it shouldnot be used. More precisely, a route k ∈KTL

lt would notbe used if wk >

∑p∈Pk LTLl� p�dpt , where dpt is the

load to be dropped at ship-to point p ∈ Pk on day t,and LTLl� p�d is the LTL charge for the shipmentof a load d from depot l to ship-to point p. If itsalternative LTL routes dominate such a TL route, itcan be removed from the set of possible routes a pri-ori. We assume from now on that all the routes inthe set KTL

lt are also nondominated by their alterna-tive LTL routes. Conversely, on a day t, an LTL ship-ment on lane l� p would be considered only if it ischeaper than the least-cost STL shipment kl� p on thislane, i.e., only if LTLl� p�dpt <maxrkl� p mkl�p

�TLl .This implies that in the daily transportation planningprocess at depot l, in addition to the nondominatedfeasible TL routes k ∈ KTL

lt , all the economic depot toship-to point LTL routes k ∈KLTL

lt must be considered,i.e., the set of routes to consider is Klt = KTL

lt ∪ KLTLlt .

The LTL tariff paid for a single destination route k ∈KLTLlt is given by wk = LTLl� pk�dpkt , where pk is the

ship-to point of route k.Given the distribution network user context

described previously, the strategic decisions to makehere involve the selection of a subset of the depotsL∗ ⊂ L to operate during the planning horizon T con-sidered as well as the assignment of ship-to pointsP ∗l ⊂ Pl, l ∈ L∗, to these depots in order to maximizetotal expected profits. Note that, in what follows,the notation L is used to represent a subset of opendepots in L and L̄ = L\L is its complement. Simi-larly, Pl ⊂ Pl is used to represent the subset of ship-topoints assigned to depot l ∈ L. An important aspectof the problem is that the mission of the selecteddepots, defined by their customer sets P ∗

l , l ∈ L∗, mustremain the same for each day t ∈ T of the plan-ning horizon. When a depot l ∈ L is used, a fixedoperating cost Al is incurred and the unit value of

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xlp

Sources

Ship-topoints

Potential depotslocations

xl

p ∈PDays

MTL route STL route FTL shipment LTL shipment

• Location decisions• Allocation decisions

• Daily transportation decisions

Design level

User level

Compound demand process

l ∈ L

yklt, t ∈T

…

Figure 1 Multiperiod Location Transportation Network

products shipped from that depot is vl. The value vltakes into account the product production and pro-curement costs, inbound shipment costs, warehous-ing costs, and inventory holding costs. The unit priceof products sold to ship-to point p is up. The struc-ture of the multiperiod location-transportation net-work examined is illustrated in Figure 1, where thedesign and user problems are separated according tothe temporal hierarchy between location and trans-portation decisions.

The Distribution Network User ProblemAs previously explained, for a given distribution net-work with depots set L⊂ L and mission Pl, l ∈ L, on adaily basis, the depots l ∈ L receive orders from theircustomers p ∈ Pl and they make shipping decisionsfor the next day. It is assumed that the demands ofthe ship-to points p ∈ P follow a compound stationarystochastic process with a random order interarrivaltime qp and a random order size op. The cumulativedistribution functions of interarrival times and ordersizes are denoted, respectively, by F

qp · and F op · .

A possible realization of these compound stochasticprocesses over planning horizon T is illustrated inFigure 2 for exponential interarrival times and nor-mal order sizes. As can be seen, on a given day, onlya subset of the ship-to points may require products.Such realizations constitute demand scenarios andthe set of all demand scenarios associated with thecompound demand processes considered is denotedby �. The probability that demand scenario � ∈ �will eventually be observed is denoted by �� . For agiven scenario � ∈�, the set of ship-to points order-ing products on day t is denoted by Pt� , and theshipments to make on day t ∈ T at depot l ∈ L aredefined by the loads dpt� , p ∈ Plt� , where Plt� =Pl∩Pt� is the set of depot l ship-to points that orderproducts on day t.

Given the loads dpt� , p ∈ Plt� , l ∈ L, to deliveron day t, shipping decisions are made by the networkdepots users in two steps. First, for the loads that arelarger than a truckload, a decision is made to shipas much as possible in full truckloads. Let KFTL

plt � be the set of vehicle types (routes) selected to makefull truckload shipments to point p. Then, the residualloads to be inserted in the STL, MTL, or LTL ship-ments of depot l on day t are

d̄pt� = dpt� −∑

k∈KFTLplt �

bkyFTLklt � � p ∈ Plt� � (1)

where yFTLklt � is the number of truckloads shipped topoint p from depot l on route k, and bk is the capacityof route k vehicles. We assume that yFTLklt � is selectedto ensure that d̄pt� is nonnegative.Next, the best delivery routes must be constructed.

LetKlt� = set of nondominated feasible delivery

routes (i.e., such that Pk ⊂ Plt� and∑

p∈Pk d̄pt� ≤ bk,k ∈Klt� from depot l on day t under scenario �;�kp = binary coefficient taking the value one if

ship-to point p is covered by route k (i.e., if p ∈ Pk),and zero otherwise;yklt� = binary decision variable equal to one if

route k is used from depot l on day t under scenario�, and equals to zero otherwise.For demand scenario �, the best routes are obtainedat depot l on day t by solving the following trans-portation subproblem:

Cult� =Min

y

∑k∈Klt �

wkyklt� (2)

subject to ∑k∈Klt �

�kpyklt� = 1 p ∈ Plt� � (3)

yklt� ∈ !0�1" k ∈Klt� � (4)

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Ship-topoint 1

Ship-topoint 2

.

.

.

Ordered quantitiesTime between

consumer orders

T

Day t

Multiperiod demand scenario

Ship-topoint|P|

.

.

.

Figure 2 Ship-to Points Stochastic Demand Process

where y denotes the vector of all the routing decisionsand Cu

lt� is the cost of the optimal shipments madeby depot l on day t under scenario �. This model issimilar to the classical set partitioning formulation ofthe deterministic VRP (Toth and Vigo 1998).Furthermore, the shipments made on a daily basis

generate sales revenues. Taking these into accountas well as depot production and procurement, ware-housing, inventory holding, and customer shipmentcosts, the net revenues Ru� generated at the distri-bution network user level for demand scenario � aregiven by

Ru� = ∑l∈L

∑t∈T

[ ∑p∈Plt �

$up−vl dpt� %

− ∑p∈Plt �

∑k∈KFTLplt �

wkyFTLklt � −Cu

lt�

]& (5)

Because up is the unit price of products sold to ship-topoint p and vl is the unit value of products shippedfrom depot l, up−vl provides the margin of productsshipped from depot l to ship-to point p. These netrevenues are important elements to take into accountin the distribution network design model.

The Distribution Network Design ProblemThe SMLTP is a hierarchical decision problem dueto the temporal hierarchy between the location deci-sions and the transportation decisions. At the strategiclevel, the only decisions made here and now are theselection of the subset of facilities L∗ ⊂ L to use dur-ing the planning horizon T considered and the mis-sion P ∗

l l ∈ L∗ of these facilities. After a deploymentlead time, on a daily basis, the transportation deci-sions discussed previously are made by the network

users. However, the network design decisions mustbe considered when making daily transportation deci-sions and, conversely, adequate network design deci-sions cannot be made without anticipating the netrevenues (5) generated by daily sales and transporta-tion decisions for a given distribution network duringthe network usage horizon T . The best possible antic-ipation involves the explicit inclusion, in the designmodel, of the transportation model (2–4) and of thenet revenue expression (5) but with the informationavailable at the time the network design decisionsare made. Because the ship-to point demands for thehorizon T are not known when the design decisionsare made, this information takes the form of the setof potential demand scenarios � previously defined.This leads to the formulation of the SMLTP as a two-stage stochastic program with recourse (Ruszczynskiand Shapiro 2003), where the first stage deals withdepot location and mission decisions and the secondstage deals with daily transportation decisions. Thefollowing first-stage decision variables are required toformulate the model:

xl = binary variable equal to one if depot l is opened,and to zero otherwise;

xlp = binary variable equal to one if ship-to point p isassigned to depot l, and to zero otherwise.

The notation x is used to denote the vector of all thesedecision variables and xl is used to denote the vectorof depot l ship-to point assignment variables. Notethat a given binary vector x specifies unique designsets L and Pl, namely, L = !l � xl = 1", L̄ = !l � xl = 0",and Pl = !p � xlp = 1", l ∈ L.The stochastic programming model to solve is the

following:

R∗ =MaxxRx = ∑

�∈�� Rdux�� −∑

l∈LAlxl (6)

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subject to

∑l∈Lp

xlp = 1 p ∈ P� (7)

xlp ≤ xl l ∈ L� p ∈ Pl� (8)

xl� xlp ∈ !0�1" l ∈ L� p ∈ Pl� (9)

where, based on (2–5), the optimal value Rdux�� ofthe second-stage program for design x and scenario� is given by

Rdux�� =∑l∈L

∑t∈T

{ ∑p∈Pt�

[up− vl dpt�

− ∑k∈KFTLplt �

wkyFTLklt �

]xlp−Cdu

lt xl�� }

(10)

withCdult xl�� =min

y

∑k∈Klt �

wkyklt� (11)

subject to

∑k∈Klt �

�kpyklt� =xlp p∈Pt� � (12)

yklt� ∈ !0�1" k ∈Klt� & (13)

In the first term of objective function (6), expectednet revenues are calculated, and in the second term,depot fixed costs are subtracted to get expected prof-its. Constraints (7) in the first-stage program enforcesingle depot assignments for ship-to points and con-straints (8) limit ship-to point assignments to openeddepots. Constraints (12) in the second-stage programare coupling relations, ensuring that daily route selec-tions respect depot mission decisions.Notwithstanding the inherent combinatorial com-

plexity of this model, it would be virtually impossi-ble to solve because the set of demand scenarios � isusually extremely large. In fact, when the interarrivaltimes and order size distribution functions F qp · andF op · are continuous, there is an infinite number ofpossible demand scenarios. This is the case, for exam-ple, when interarrival times are exponential and ordersizes are log normal, a frequent case in practice. Thisdifficulty can be alleviated, however, through the useof Monte Carlo scenario sampling methods.

Sample Average Approximation ModelThe approach proposed to reduce the stochastic com-plexity of our problem is based on the Monte Carlosampling methods presented in Shapiro (2003) andapplied to the VRP in Verweij et al. (2003) and Rei,Gendreau, and Soriano (2007) and to supply chainnetwork design problems in Santoso et al. (2005) and

Vila, Martel, and Beauregard (2007). A random sam-ple of scenarios is generated outside the optimiza-tion procedure and then a sample average approxi-mation (SAA) program is constructed and solved. Theidea is to first generate an independent sample ofn equiprobable scenarios !�1� & & & ��n"=�n ⊂� fromthe initial probability distributions of order interar-rival times and order sizes, which also removes thenecessity of explicitly computing the scenario proba-bilities �� . Then, based on (6–13), the SAA programobtained is the following:

�Rn = Maxx�y

1n

∑�∈�n

∑l∈L

∑t∈T

{ ∑p∈Pt�

[up−vl dpt�

− ∑k∈KFTLplt �

wkyFTLklt �

]xlp

− ∑k∈Klt �

wkyklt�

}−∑

l∈LAlxl (14)

subject to∑l∈Lp

xlp = 1 p ∈ P� (15)

xlp ≤ xl l ∈ L� p ∈ Pl� (16)∑k∈Klt �

�kpyklt� = xlp�

� ∈�n� l ∈ L� t ∈ T � p ∈ Pt� � (17)

xl� xlp� yklt� ∈ !0�1"� ∈�n� l ∈ L� p ∈ P� t ∈ T � k ∈Klt� & (18)

The scenarios in sample �n ⊂� used in the modelare generated directly from the cumulative distribu-tion functions of interarrival times and order sizesFqp · and F op · , p ∈ P . Assuming that the customerorders are independent of each other, to sample a sce-nario � ∈�, we generate independent pseudorandomnumbers uq and uo uniformly distributed on the inter-val $0�1% and we compute the inverse, F q−1p uq andF o−1p uo , of the distributions of interarrival times andorder sizes. The MonteCarlo procedure used to gen-erate the daily demands dpt� , p ∈ P , t ∈ T of theship-to points for scenario � is presented in Figure 3.In this procedure, the continuous variable ) is usedto denote order-arrival times. Order arrivals are gen-erated in the interval %0� �T �% and mapped onto thecorresponding planning periods t ∈ T . More than oneorder can arrive in a given planning period. Repeatingthis Monte Carlo sampling procedure n times yieldsthe required sample of scenarios �n. Note that all theProcedures presented in this paper use the followingsyntax:

Procedure(input_variable1� & & & �procedure_parameter1� & & & �output_variable1� & & &).

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MonteCarloF qp · � F op · � p ∈ P �T �dpt� � p ∈ P� t ∈ T For all p ∈ P , do:) = 0; dpt� = 0, t ∈ TWhile ) ≤ �T �, do:Generate the Uniform $0�1% random numbers uq and uoCompute the next order arrival time ) = ) + F q−1p uq and t = �)�

Compute the planning period t demand dpt� =dpt� + F o−1p uo

End whileEnd do

Figure 3 Procedure Monte Carlo for the Generation of Scenario �

Clearly, the quality of the solution obtained withthis approach improves as the size n of the sample ofscenarios used increases. The SAA model above has astructure similar to the deterministic location-routingproblem (Berger, Coullard, and Daskin 2007), but itseparates explicitly assignment and routing decisionsand it is much larger. It uses an exponential num-ber of binary decision variables for route selection.A preestablished set of nondominated routes can begenerated as input to the model, but only small prob-lem instances can be solved to optimality this waywith commercial solvers. A better approach is to usecolumn generation, which avoids the explicit consid-eration of all the possible routes. When �T � ∗ ��n� islarge, however, this optimal approach can be usedonly for relatively small problems. Our aim in thenext section is to propose an heuristic method to solverealistic problem instances.

Solution ApproachThe General SchemeThe SMLTP is a hierarchical decision problem forwhich a hierarchical heuristic solution approach is anatural fit. Nagy and Salhi (2007) present a reviewof sequential, clustering, iterative, and hierarchicalheuristic approaches to solve the LRP. The differencebetween these heuristics relates mainly to how thesolution method treats the relationship between thelocation and the routing subproblems. The heuristicsolution approach proposed in this section is a nestedmethod that integrates location-allocation and trans-portation decisions in a hierarchical manner. In syncwith the bilevel problem definition provided in Fig-ure 1, the solution approach proposed builds on auser level transportation heuristic and a design levellocation-allocation heuristic combined into an efficientnested procedure. For the design problem, the tabusearch heuristic proposed locates depots and assignsship-to points to the opened depots. It is a local searchprocedure that explores neighbouring depot config-urations and perturbs the ship-to point assignmentsusing a set of restricted moves. For the user trans-portation problem, a modified and extended Clarke

and Wright (1964) savings heuristic is proposed. Thisuser heuristic is also used to evaluate potential movesin the design heuristic.Clearly, several hundred moves may be considered

during the solution process and an exact evaluationof each potential solution, using the user heuristic,would be too time-consuming. This is particularlytrue given the stochastic nature of our problem. Tosolve the SAA model (14–18), the evaluation of thesolutions considered must be based on an estima-tion of their expected value for the scenario sample�n generated. To reduce the calculation effort, wepropose a fast approximate move evaluation proce-dure based on a route cost estimation formula. Inthe taxonomy proposed by Talbi (2002), our solutionapproach could be classified as a low-level hybridheuristic. The following sections present our user levelheuristic and our design level heuristic.

The User Problem HeuristicConsider the user problem for a given distributionnetwork with depots set L ⊂ L and mission Pl, l ∈ L.At the user level, under scenario �, for all the ship-topoint orders Plt� received by depot l on day t, theobjective of depot l is to select the FTL, STL, MTL, andLTL shipments and minimize its transportation costs.As explained previously, this is achieved with a two-step procedure. The first step determines the numberof full truckload shipments to make and the residualship-to point loads. The number of full trucks of eachtype k ∈KFTL

plt � to ship on day t from depot l is foundby solving the following simple integer programs byinspection:

$yFTLklt � %k∈KFTLplt �

= arg(maxy

∑k∈KFTLplt �

bkyk

∣∣∣∣ ∑k∈KFTLplt �

bkyk ≤ dpt� �

yk = 0�1� & & &)� p ∈ Plt� & (19)

Then, the residual loads d̄pt� , p ∈ Plt� to beshipped are computed with (1).The second step solves program (2–4), i.e., it finds

the best TL (STL or MTL) or LTL shipments to deliverthe residual loads. To solve this transportation prob-lem, we propose a modification of the simple and effi-cient VRP heuristic, based on perturbed Clarke andWright (CW) (1964) savings and 2-opt improvementsand developed by Girard, Renaud, and Boctor (2006).The main differences between our problem and a clas-sical VRP are that (i) two different direct deliverytransportation modes can be used (STL or LTL), and(ii) for all modes considered, the vehicle used doesnot return to the depot after its last drop. Clearly, thebest direct delivery mode for a given ship-to point can

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w(l, p)w(l, p′ )

w(l, p, p′)vs.

l

pp′

p

LTL

TL

l

oror

p′

Figure 4 Alternative Routes Considered in the Savings Calculationsfor Depot l

be determined a priori by comparing their respectivecosts, so that the cost of the best direct shipment fromdepot l to ship-to point p is

wl�p = min$LTLl� p� d̄pt� �maxrl� p ml�p �TLl %�

p ∈ Plt� & (20)Then, as illustrated in Figure 4, for two ship-to pointsp and p′, the savings associated with using a multi-drop TL route (l� p� p′ instead of the best direct ship-ments (l� p and (l� p′ can be calculated with thefollowing expression:

epp′ = wl�p +wl�p′ −wl�p�p′ �

p ∈ Plt� � p′ ∈ Plt� � \!p"� (21)

where wl�p�p′ =maxrl� p�p′ ml� p�p′ �TLl +al. The per-turbed CW heuristic works as the original CWalgorithm but uses the following modified savingsformula:

epp′ = wl�p +wl�p′ −+pp′wl�p�p′ �

p ∈ Plt� � p′ ∈ Plt� \!p"� (22)

where the weight +pp′ is randomly selected betweentwo predetermined limits +− and ++ for every pair(p�p′ . The 2-opt heuristic is then applied to improveeach route of the solution obtained. The procedureis repeated , times and the best solution found isretained. The total transportation cost of the best solu-tion found is denoted by �Cdu

lt Plt� . The net revenuegenerated by depot l ∈ L on day t ∈ T of scenario� ∈�n is given by

�Rdult Plt�

= ∑p∈Plt �

[up−vl dpt� −

∑k∈KFTLplt �

wkyFTLkt �

]

− �Cdult Plt� & (23)

The user problem heuristic proposed is summarisedin Figure 5, in the procedure User. This procedurecan be used to evaluate any given network designx under any given scenario � ∈ �. The total net

Userbk� k ∈KFTLplt � �dpt� � p ∈ Plt� �,�+−�++� �Rdu

lt Plt�

S1: (a) Solve (19) by inspection to obtain the FTL shipments

yFTLklt � � k ∈KFTLplt � � p ∈ Plt� &

(b) Compute the residual loads d̄pt� , p ∈ Plt� , with (1).S2: (a) Select the best direct delivery transportation modes

(LTL vs. STL), and compute their costs wl�p , p ∈ Plt� ,with (20).

(b) Solve the resulting open routing problem , timeswith the modified Clarke and Wright (1964) algorithmusing the savings epp′ , p ∈ Plt� , p′ ∈ Plt� \!p" computedwith (22), and retain the best transportation solution.

S3: Compute the net revenue �Rdult Plt� with (23).

Figure 5 Procedure User for Depot l in Period t Under Scenario �

revenues generated by design x for the scenario �considered is obtained simply by summing net rev-enues over all depots and days, i.e., by calculating�Rdux�� = ∑

l∈L∑

t∈T �Rdult Plt� . When a sample �n

of n Monte Carlo scenarios is used, an estimate �Rnx of the expected value of the design considered is thusgiven by

�Rnx =1n

∑�∈�n

∑l∈L

∑t∈T

�Rdult Plt� −

∑l∈LAl& (24)

Tabu Search Heuristic for the DistributionNetwork Design ProblemThe heuristic proposed to solve the SAA program(14–18) is based on tabu search, a local iterativeapproach that is able to escape from local optimaby allowing a degradation of the objective functionas opposed to pure descent methods. Because non-improving moves are allowed, solutions previouslyencountered during the search may be revisited. Toavoid this cycling phenomenon, a short-term memorycalled a tabu list keeps track of the most recent moves.The interested reader will find more details about thisapproach in Glover and Laguna (1997).At each iteration of the tabu heuristic, the current

design xc available is improved. Potential nontabusolutions in the neighbourhood of xc are evaluatedwith a fast but approximate route cost estimation for-mula. The best potential solution found is then eval-uated more precisely with User to determine if it isbetter than the best solution x∗ found to date. The fol-lowing paragraphs describe the main features of thetabu search heuristic proposed as well as three alter-native neighbourhood exploration strategies.

Neighbourhood Structure. At each iteration, anew design xm is generated from the current design xc

using one of the following three moves: (i) drop (closean opened distribution center and reassign its ship-topoints); (ii) add (open a closed distribution center andassign some ship-to points to it); or (iii) shift (closean opened distribution center and open a closed onewhile modifying some ship-to point assignments).

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LetMc be the set of all possible moves from the cur-rent design xc at an iteration of the algorithm. Then,using the operator ⊕ to denote a move, the generalneighbourhood of design xc is defined by the feasi-ble solution set GNxc = !xm � xm = xc ⊕m� m ∈Mc".Unfortunately, the size of GNxc increases rapidlywith the number of ship-to points and potential loca-tions, and it would be too time-consuming to assessall the potential solutions in GNxc . We thereforerestrict the search for a better design to moves asso-ciated to dropping, adding, or shifting depots com-bined with a greedy reassignment of ship-to pointsbased on a unitary net revenue maximisation rule.Also, using the concept of area of influence intro-duced by Nagy and Salhi (1996a), the moves consid-ered are restricted to adjacent depots. Two depots areneighbours if there exists at least one ship-to pointfor which they are, respectively, the nearest and thesecond-nearest depots. Let � l be the set of neigh-bours of depot l. Then, for each open depot l ∈ Lcspecified by the current design xc, we define a region�l including depot l, the depots in � l , and theship-to points Pcl′ , l

′ ∈ !l"∪ � l ∩Lc assigned to thesedepots. In our algorithm, only drop, add, or shiftmoves within regions are considered. These featureslimit the search for a better design to a restricted neigh-bourhood RNxc ⊂ GNxc defined by limited movesubsets �l ⊂Mc, l ∈ Lc. For a given region �l , thefeasible moves considered are:(i) Drop move mdropl : If � l ∩ Lc �= �, close

depot l and assign each ship-to point p ∈�l ∩ P todepot lp = argmaxl′∈� l ∩Lcp up− vl′ −wl′� p .

(ii) Add moves �addl : If � l ∩ L̄c �= �, forall l′ ∈ � l ∩ L̄c, open depot l′ and assigneach ship-to point p ∈ �l ∩ P to depot lp =argmaxl′′∈�l ∩Lcp ∪!l′" up− vl′′ −wl′′� p .

(iii) Shift moves (�shiftl : If � l ∩ L̄c �= �, for alll′ ∈� l ∩ L̄c, open depot l′ and close depot l simulta-neously and assign each ship-to point p ∈�l ∩ P todepot lp = argmaxl′′∈� l ∩Lcp ∪!l′"up− vl′′ −wl′′� p .In what follows, the following sets of possible

moves are used:

�drop = !mdropl "l∈Lc � �add =⋃l∈Lc

�addl �

�shift =⋃l∈Lc

�shiftl �=�drop ∪�add ∪�shift�

�l = !mdropl "∪�addl ∪�shiftl &

Note that only the moves leading to a feasible solu-tion are considered. For example, if a drop move cre-ates an isolated ship-to point, i.e., a point too far fromother depots to permit next-day delivery, then it is notconsidered. Note also that during the neighbourhoodexploration, the moves in � are evaluated only if theyare not tabu.

Neighbourhood Evaluation. The evaluation of adesign xm ∈ RNxc involves the estimation of itsexpected value for the sample �n of Monte Carloscenarios generated. This could be done by applyingthe User heuristic to all l�� t ∈ Lm × �n × T andthen computing �Rnxm with (24). However, evaluat-ing all possible moves using the User heuristic wouldbe too time-consuming. Possible moves must thus beevaluated using a fast approximation. The evalua-tion function proposed is based on a linear regres-sion route length estimator (introduced by Daganzo1984 and extended by Nagy and Salhi 1996b) modi-fied to account for LTL transportation. It estimates thetotal transportation cost �Cdu

lt Pmlt � of depot l on day

t under scenario � as a function of the set of ship-to points Pmlt � visited from depot l on that dayunder design xm. The function is obtained with amultiple regression model based on three explana-tory variables: (i) a linehaul variable 1lPmlt � /NSl ,where 1lP is the sum of the distances from depotl to all p ∈ P , and NSl the average number ofstops on the routes from depot l; (ii) a detour vari-able 1lPmlt � /

√�Pmlt � � ; and (iii) an LTL variable4LTLl 5lP

mlt � , where 5lP is the sum of the load

distances from depot l to all p ∈ P and is 4LTLl theproportion of load distances from depot l on LTLroutes. This leads to the following cost approximationformula:

�Cdult P

mlt � ≈ �Cdu

lt Pmlt �

= 6̂1

(1lP

mlt �

NSl

)+ 6̂2

(1lP

mlt � √�Pmlt � �

)

+ 6̂34LTLl 5lPmlt � � (25)

where 6̂1, 6̂2, and 6̂3 are regression coefficients associ-ated to the linehaul, detour, and LTL variables, respec-tively. These regression coefficients are estimatedusing a sample of historical daily delivery routes or asample of daily routes obtained with User for differ-ent network designs. The parameters NSl and 4LTLl areinitially estimated with the same route sample, butthey are updated in the algorithm every time the Userheuristic is used to evaluate a new design.An adequate approximation of �Rnxm is then ob-

tained by replacing �Cdult P

mlt � in (23) by �Cdu

lt Pmlt �

and then by substituting resulting expression in (24),to get:

�Rdult P

mlt � =

∑p∈Pmlt �

[up−vl dpt� −

∑k∈KFTLplt �

wkyFTLkt �

]

− �Cdult P

mlt � � (26)

�Rnxm = 1

n

∑�∈�n

∑l∈Lm

∑t∈T

�Rdult P

mlt � −

∑l∈Lm

Al& (27)

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Restricted Neighbourhood Exploration. Giventhe restricted neighbourhood structure previouslydefined, three different strategies are considered toexplore the neighbourhood RNxc of the currentsolution xc. These strategies specify the order in whichthe drop, add, and shift moves are made during eachiteration of the algorithm. These three strategies havethe following characteristics.Strategy 1 (S1): This strategy is inspired from the

tabu search approach proposed by Nagy and Salhi(1996a) to solve a deterministic location-routing prob-lem. It is a straightforward version of the tabu searchmethod. At each iteration, all the drop, add, and shiftmoves in� are considered to generate a new solution.Strategy 2 (S2): This strategy is an extension of

the three-phase hill-climbing method proposed byKuehn and Hamburger (1963) to solve a determinis-tic location-allocation problem. Assuming that a min-imum number of depots are initially opened, thefirst phase explores add moves �add only, the sec-ond phase considers drop moves �drop exclusively,and the third phase concentrates on shift moves �shift.In our implementation, only nontabu shift moves areconsidered.Strategy 3 (S3): This strategy starts with the initial

solution obtained by assigning each ship-to point tothe feasible depot, with the maximum marginal netrevenue. Clearly, in this solution, all the potentiallyinteresting depots are opened. Then, at each itera-tion, a drop move in �drop is performed, followed byshift moves in �shift. This process continues until thespecified maximum number of iterations is reached.It is worth noting that the shift moves made at eachiteration can be viewed as an intensification phasein the region of the search space that contains solu-tions having the same number of opened depots asthe current solution. In addition, following each shiftmove, a reassignment procedure for borderline pointsis applied. This intensification phase is a tabu searchbecause nonimproving moves are allowed.Note that the moves ml ∈� considered are based

on changes to the status of depots, followed by agreedy adjustment to ship-to point assignments toensure that the resulting design is feasible. There isno guarantee, however, that the assignments adjust-ment made is optimal. It may therefore be profitable,when all moves are performed, to refine the adjust-ments made to get the new solution xc. To do this, weelaborated a reassignment procedure for borderlinepoints, inspired from an heuristic proposed by Zain-uddin and Salhi (2007) to solve the capacitated Weberproblem. A point p assigned to depot l ∈ Lm (i.e., inPml is considered borderline if its distance from depotl exceeds a predefined target distance mmax, i.e., ifml�p > mmax. Let � be the set of borderline ship-topoints. A ship-to point p is reassigned from depot l to

Reassignxc�mmax�7max�x′� �Rnx′ Set x′ = xc and compute �Rnx′ with (27)Construct the borderline point set �For all 4max ∈7max, doxm = x′

For all p ∈�, do

lp = argminl′∈Lmp

!4′l =ml′� p /ml�p � 4′

l ≤ 4max"� xmlp = 0� xmlp�p = 1&

Compute �Rnxm with (27).If ( �Rnxm > �Rnx′ , then x′ = xm and �Rnx′ = �Rnxm

End do

Figure 6 Reassignment Procedure

depot lp ∈ Lmp if its distance ratio 4lp =mlp�p /ml�p is

the lowest among the eligible depots, provided that itdoes not exceed a predefined value 4max. When this isdone for all the borderline points p ∈� associated toxm, a new solution vector x′ results. Several alternativesolution vectors x′ can be generated by considering aset 7max of 4max values. The solutions thus generatedcan then be evaluated with (27) to determine whichone is the best. This gives rise to the reassignmentprocedure presented in Figure 6 and applied in thethree strategies.

Tabu Lists. During the search procedure, two tabulists of varying length are kept. When a depot l isadded or dropped in the new current solution, l isinserted into a drop or add list �1. On the other hand,when the new current solution is obtained by shiftingtwo depots l and l′, the pair l� l′ is inserted into ashift list �2. In both cases, the oldest element of the listis removed. Note that when a depot l is inserted in �1,all the shifts involving this depot are also insertedin �2. At each iteration, the length of the relevant listis randomly generated in the interval $81�L�/2� �L�/2%for �1 and interval $82�L��2�L�% for �2, where 81�82 ∈%0�1$ are predefined parameters. Note that these inter-val bounds have been adequately fixed after severalpreliminary tests and are close to those considered inNagy and Salhi (1996a).

Solution Algorithm Initialization. Before the tabusearch is started, the solution process must be initial-ized. This involves first fixing the various parametersrequired by the procedures used. In addition to theparameters already defined, the following two param-eters are required to control the tabu search:

MI=maximum number of iterations,MNI= the maximum number of iterations without

improvement.

Next, a sample of n demand scenarios d� =$dpt� %p∈P� t∈T , � ∈ �n must be generated using pro-cedure MonteCarlo n times, and the neighbour sets� l , l ∈ L must be created. Initial solutions must thenbe constructed for the neighbourhood exploration

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Initializestrategy�n�,�+−�++�81�82�mmax�7max�MI�MNI,d� �� ∈�n � � (l � l ∈ L �xo� �Rnxo

Set the heuristic parameters n�,�+−�++�81�82�mmax,7max�MI�MNI

For all � ∈�n, do MonteCarloF qp · � F op · � p ∈ P �T �dpt� � p ∈ P� t ∈ T

Obtain the neighbour sets � l , l ∈ LIf strategy= 2, thenSet L0 =�, �P = PWhile �P �= �, do

l′ = argmaxl∈L̄0

∑p∈ �P∩Pl

up − vl −wl�p

Set L0 = L0 ∪ !l′", Pl′ = �P ∩ Pl′ and �P = �P\Pl′End while

Else set L0 = LConstruct an initial design xo by assigning eachship-to point p ∈ P to the depot

lp = argmaxl∈L0p up − vl −wl�p

For all l�� t ∈ Lo ×�n × T , doUserbk� k ∈KFTL

plt � �dpt� � p ∈ Polt� �,�+−�++��Rdult P

olt�

Compute the expected value �Rnxo with (24)

Figure 7 Initialization Procedure

strategy considered. For Strategy 2, an initial solu-tion x0 is obtained by sequentially opening the closeddepot l ∈ L̄ maximizing the marginal net revenues ofthe ship-to points �P not yet allocated. For Strategies1 and 3, L0 is initially set to L. Then, for all cases,the missions are obtained by assigning each ship-topoint p ∈ P to the depot lp ∈ Lp, with the maximummarginal net revenues. The resulting design x0 is thenevaluated with the User heuristic and the expectedvalue function (24). Figure 7 presents the initializationprocedure thus obtained.

Tabu Search Procedure. The Tabu search proce-dure proposed appears in Figure 8. The iterationsof the algorithm are controlled by two parameters:iter, the current number of iterations, and iter_ni,the current number of iterations without improve-ment. In Figure 8, Step 2 examines different movesin the neighbourhood of the current solution xc.Step 3 applies the reassignment procedure to the cur-rent solution xc. Step 4 evaluates the best move andupdates the parameters of the transportation costsestimation function. Step 5 manages the tabu lists.Step 6 performs a shift move intensification when theparameter “Improve” equals one. Step 7 checks if thelast iteration improved the best solution to date x∗.Finally, Steps 8 and 9 control the iterations of the algo-rithm. If either of the iteration limits is not reached,the exploration continues from Step 2 with the cur-rent solution xc. If MNI iterations were made withoutimprovement, the exploration continues from Step 1with the best solution to date x∗. Otherwise, the algo-rithm stops. Note that to simplify the exposition, we

Tabuxo� �Rnxo ��MI�MNI� improve; x∗� �Rnx∗ Step 0. Set x∗ = xo and �Rnx∗ = �Rnxo and initialize the

tabu lists (�1 and �2 .Step 1. Set xc = x∗ and �Rnxc = �Rnx∗ Step 2. �Rnx′ = 0

For all l ∈ Lc , doConstruct the region �l For all ml ∈�l and ml not tabu, doSet xm = xc ⊕ml and compute �Rnxm with (27)If ( �Rnxm > �Rnx′ , then x′ = xm and �Rnx′ = �Rnxm

End doEnd doxc = x′

Step 3. Reassignxc�mmax�7max�x′� �Rnx′ and set xc = x′

Step 4. For all l�� t ∈ Lc ×�n × T , doUserbk� k ∈KFTL

plt � �dpt� � p ∈ Pclt� �,�+−�++� R̂du

lt Pclt�

Compute the expected value �Rnxc with (24)Update the means NSl and 4LTLl , l ∈ Lc , using theaugmented set of generated routes

Step 5. Update the tabu lists (�1 and �2 Step 6. If (improve= 1 , then

Tabuxc� �Rnxc ��shift�MI�MNI�0�x′� �Rnx′ xc = x′ and �Rnxc = �Rnx′

Step 7. If ( �Rnxc > �Rnx∗ , then x∗ = xc , �Rnx∗ = �Rnxc and iter_ni= 0

Else iter_ni= iter_ni+ 1Step 8. iter= iter+ 1Step 9. If (iter<MI) and (iter_ni<MNI), then go to Step 2

Else if (iter<MI), then set iter_ni= 0 and go to Step 1Else stop

Figure 8 Tabu Procedure

used the parameters MI and MNI when calling Tabuin Step 6 as in the main procedure. In the implemen-tation, however, the iteration limits parameters thatare used in Step 6 are not the same as in the mainprocedure.Using the Initialize and Tabu procedures described

previously, the neighbourhood exploration strategiesconsidered are implemented as follows:

Strategy 1 S1 :Initialize1�n�,�+−�++�81�82�mmax�7max�MI,MNI� d� �� ∈�n � � l � l ∈ L �xo� �Rnxo

Tabuxo� �Rnxo ��MI�MNI�0�x∗� �Rnx∗ Strategy 2 S2 :Initialize2�n�,�+−�++�81�82�mmax�7max�MI�MNI� d� �� ∈�n � � l � l ∈ L �xo� �Rnxo

Tabuxo� �Rnxo ��add�MI�MNI�0�xadd� �Rnxadd Tabuxadd� �Rnxadd ��drop�MI�MNI�0�xdrop��Rnxdrop Tabuxdrop� �Rnxdrop ��shift�MI�MNI�0�x∗� �Rnx∗

Strategy 3 S3 :Initialize3�n�,�+−�++�81�82�mmax�7max�MI�MNI� d� �� ∈�n � � l � l ∈ L �xo� �Rnxo

Tabuxo� �Rnxo ��drop�MI�MNI�1�x∗� �Rnx∗ .

The next section evaluates and compares thesestrategies.

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Computational ResultsPlan of ExperimentsTo test the heuristic approach proposed to solve theSMLTP, several problem instances were generatedbased on the following four dimensions: problem size,cost structure, demand process, and network charac-teristics. The problem instances were generated ran-domly but they were based on realistic parametervalue ranges obtained partially from the “Usemore”case documented in Ballou (1992) and from the dataof a real case. The problems were defined over var-ious U.S. regions and all distances were calculatedwith PC ∗MILER (www.alk.com) for the current U.S.road network. For all cases, it was assumed that theorder interarrival times are exponentially distributedwith a mean inter-arrival time + and that order sizesare log normal with a mean 9 and a standard devi-ation : . A one-year planning horizon, assumed toinclude �T � = 200 working days, was used.Problems of three different sizes were tested—

small (P1), medium (P2 , and large (P3 —as defined inTable 1. In each case, the problem size varies in termsof the number of depots and ship-to points consid-ered, and in terms of the U.S. geographical regionscovered. Based on the realistic industrial problemsexamined, the number of ship-to points in the prob-lems is much larger than the number of potentialdepots, and the latter was fixed at about 3% �P �. Tocapture different cost structures, two levels of fixedand variable costs were considered (see Table 2). Thefixed operating costs Al and the unit value of prod-ucts vl were selected randomly in the interval speci-fied in Table 2. The product prices on the market upwas fixed equal to the value in Table 2 for all ship-topoints.Next, demand processes are associated to the geo-

graphical coordinates of the ship-to points in aproblem. These demand processes are calibrated torepresent large, medium, or small customers. Twotypes of network are generated: (1) those composedmainly of large- and medium-sized customers (LN),and (2) those including mainly small- and medium-sized customers (SN). Table 3 provides the propor-tion of each type of customer in LN and SN net-works, as well as the +�9�: parameter values rangeused to generate specific instances. Finally, two ran-dom replications DS1�DS2 are generated for eachnetwork structure considered.

Table 1 Test Problems Sizes

Problem Number of Number ofinstance Geographical area depots ship-to points

P1 Central Northeastern United States 7 206P2 Northeastern and Midwest United States 15 706P3 Northeastern and Midwest United States 28 1�206

Table 2 Test Problems Cost Structures

High product value and price Low product value and price

High fixed (a) �230K�250K�� 19�21��23 (b) �230K�250K�� 9�11��13costs

Low fixed (c) �130K�150K�� 19�21��23 (d) �130K�150K�� 9�11��13costs

Note. (Cost structure): [fixed cost (Al range]; [product value (vl range];product price (up.

The combination of these four dimensions yields48 problem instances. Each instance is denoted asfollows:

i� j� k� l i ∈ !P1�P2�P3"� j ∈ !a� b� c�d"�k ∈ !LN�SN"� l ∈ !DS1�DS2"&

The three neighbourhood exploration strategiesproposed in the previous section were tested for allof these problem instances and the numerical resultsobtained are presented in the next paragraphs.

Numerical ResultsThe heuristics proposed were implemented in VB.Net2005 and the experiments reported in this sectionwere performed on a 2 GHz Dual Core workstationwith 3 GB of RAM. This section starts with a discus-sion of the calibration of the several procedures usedin the heuristics: Preliminary tests on various prob-lem instances were performed to fix the algorithmparameters and to study the stochastic behaviourof the solutions obtained. We then provide a com-prehensive analysis of performances of the 3 neigh-bourhood exploration strategies considered for the48 problem instances. In addition, for the small prob-lem (P1 instances, the heuristic is compared to theoptimal solution of the SAA model (14–18) obtainedwith CPLEX-11 when using all possible nondomi-nated routes. These problems were solved on a 64-bitserver with a 2.5 GHz Intel XEON processor and 16GB of RAM.

Procedures Calibration. The solution approach isbased on several procedures including a number of

Table 3 Ship-to Point Demand Structure

Ship-to point sizesand characteristics Large Medium Small

Larger ship-to points 15 65 20network (LN) (%)

Smaller ship-to points 10 30 60network (SN) (%)

� (cwt) �480�580� �300�400� �120�220�� (% �) (%) 7 10 16� (days) �2�5�4�5� �5�5�15�5� �20�5�35�5�

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Table 4 Heuristic Parameter Values

Procedure User Reassign Tabu

Parameters � �− �+ mmax �max �1 �2 MI MNI

Values range �1�50� �0�5�1�5� �1�2� �50�400� �0�1�5� �0�1� �0�1� �20�200� �0�20�Selected value 10 1 1.2 250 �1�0�75�0�5� 0.25 0.25 100 10

parameters that were calibrated with a set of prelim-inary experiments. Using several P1 and P2 instances,the three strategies were executed alternatively to fixthe heuristic parameters specified in the initializationprocedure. Table 4 presents the range of values testedfor each parameter and provides the value selected.Note that these values were fixed for the three strate-gies and present the best trade-off in terms of algo-rithm search speed and solution accuracy.Another important parameter to calibrate is the

number n of scenarios to generate (with the Monte-Carlo procedure) to obtain good solutions. It isimportant to note that because our demand processis stationary, the user problem must be solved byeach depot on a daily basis, and the planning hori-zon includes 200 days, when n scenarios are used,200 n user model instances sampled from the sameprobability distributions are solved by each depot. Forthis reason, a relatively small number of scenarios isrequired to obtain good results. To determine the bestvalue of n, different sample sizes were tested andthe quality of the solutions obtained was evaluatedusing a statistical optimality gap, as is usually donewhen the SAA method is used to solve stochasticprograms. This calibration was done using problemP1 with larger ship-to points (LN), and it was basedon the optimal solution of the SAA model (14–18)obtained with CPLEX-11. Values of n = 1�2�4�6�8were tested. We were not able to solve larger prob-lems to optimality without truncating the set of non-dominated routes.To estimate the statistical optimality gap for a sam-

ple size n, several SAA models based on indepen-dent samples of size n must be solved. Let �Rj

n andx̂jn, j = 1� & & & �m be the optimal value and an opti-mal solution of the SAA model for the m samplesused. Well-known results in stochastic programmingare that Ɛ$ �Rn% ≥ R∗, where Ɛ$ % denotes the expectedvalue, �Rn is the optimal value of (14), and R∗ is theoptimal value of (6), and that �Rn�m =∑m

j=1 �Rjn/m is an

unbiased estimator of Ɛ$ �Rn% (Shapiro 2003). A statis-tical upper bound on the optimal value is thus pro-vided by �Rn�m ≥ R∗. Also, the true objective func-tion value Rx̂jn ≤ R∗ of the feasible solutions x̂jn, j =1� & & & �m can be estimated with

�Rn′x̂jn =

1n′

∑�∈�n′

Rdux̂jn�� −∑l∈LAlx̂l

jn� (28)

where �n′ ⊂� is a sample of n′ scenarios generatedindependently of the sample used to obtain x̂jn, andwith n′ � n. A statistical lower bound on the optimalvalue is thus provided by �Rn′x̂

jn ≤ R∗. Furthermore,

if the User heuristic is used to solve the second-stageprogram, its value Rdux̂jn�� in (28) must be replacedby the value �Rdux̂jn�� ≤ Rdux̂jn�� calculated withUser to get �Rn′x̂

jn ≤ �Rn′x̂

jn . An estimate of the opti-

mality gap of the solution x̂jn can thus be calculatedas follows: gapn�m�n′x̂

jn = �Rn�m − �Rn′x̂

jn . Because m

SAA models with a sample of size n are solved, anestimate of the average gap for a sample of size n isgiven by gapn�n′ =

∑mj=1 gapn�m�n′x̂

jn /m.

To evaluate alternative sample sizes, we calculatedthis average gap with m= 4 sample replications, andwe evaluated (28) with the User heuristic using sce-nario samples of size n′ = 100. The average gap val-ues obtained for different sample sizes are providedin Table 5. These values are expressed as a percent-age of the objective function value of the best designfound. It can be seen that samples of six or eightscenarios provide very good results. When inspect-ing the design found with these sample sizes, wenoted that they were all very similar: They openedthe same depots and only one or two ship-to pointassignments were different. However, the solutiontime increases considerably with the sample size. Forthese reasons, a sample size of n = 6 scenarios hasbeen selected as the best trade-off to use in the exper-iments. This means that 1,200 user model instances,sampled from the same probability distributions, aresolved by each depot every time a design is evalu-ated. The gaps in Table 5 suggest that the approachproposed is extremely accurate. This is due partly tothe fact that for the problem instance considered, theprofits generated are relatively high and the objec-tive function value has low variability, which couldmean that the value of stochastic solution (Birge andLouveaux 1997) is relatively low.

Analysis of Results. This section discusses thequality of the supply network designs obtained with

Table 5 Statistical Optimality Gap Values

Sample size (n) 1 (%) 2 (%) 4 (%) 6 (%) 8 (%)

gapn�100 (in %) 2.32 3.42 2.21 0.67 0.07

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Table 6 Comparison with CPLEX-11 Solution of SAA Model for Problem P1

Problem (a, LN) (b, LN) (c, LN) (d, LN) (a, SN) (b, SN) (c, SN) (d, SN)

S1 2,689,489 2,493,083 4,204,796 4,145,389 1,430,919 1,306,351 2,396,894 2,360,981S2 2,689,162 2,492,898 4,204,796 4,145,389 1,430,919 1,306,142 2,396,894 2,360,981S3 2,632,805 2,439,453 4,203,591 4,144,280 1,430,919 1,306,142 2,396,894 2,360,981CPLEX-11 2,689,481 2,493,069 4,204,851 4,145,394 1,431,210 1,306,349 2,397,181 2,361,277%-difference (%) 0.000 −0.001 0.001 0.000 0.020 0.000 0.012 0.013

the three neighbourhood exploration strategies pro-posed as well as their respective solution times. First,to determine how close to optimum the solutionsobtained are, the SAA model (14–18) was solved withCPLEX-11 for problem P1 instances, using an MIP rel-ative tolerance of 0.001 and a sample size n= 6. Theseare the largest SAA problems we were able to solve tooptimality. The results obtained with our three explo-ration strategies and with CPLEX are provided inTable 6. The values shown in Table 6 are the estima-tion of the true objective function value Rx providedby �Rn′x with a sample size n′ = 100. The last rowprovides the %-difference between the value of theCPLEX solution and the best design obtained withour heuristic search strategies. It can be seen that thedifferences are very small. For problem (P1, b, LN), thedesign provided by S1 is slightly better than the solu-tion provided by CPLEX. This is possible because theSAA is solved to optimality with a sample size n= 6,which is an approximation. For a given sample size,our heuristic can therefore provide a better solutionof the original stochastic programming model (6–13)than the SAA model. The SAA models solved include2,501,456 binary variables for the larger ship-to pointnetworks (LN), and they were solved on average in939 seconds by CPLEX-11 on our 64-bit server. It tookon average 10 seconds to obtain the best solutionfound with our heuristics on a 32-bit workstation.Next, our three exploration strategies were com-

pared for all problem instances. The true objective

Table 7 Mean Design Values for All Problem Types

P1 (P1, a, . , . ) (P1, b, . , . ) (P1, c, . , . ) (P1, d, . , . ) (P1, . , LN, . ) (P1, . , SN, . ) (P1, . , . , . )

S1 2�119�638 1�957�886 2�426�406 2�884�254 3�255�870 1�908�016 2�581�943S2 2�119�556 1�957�788 2�426�062 2�883�577 3�255�103 1�907�990 2�581�546S3 2�048�156 1�927�984 2�412�939 2�883�854 3�204�709 1�907�974 2�556�341


S1 9�578�586 8�845�539 8�599�984 9�631�837 10�712�395 5�780�462 8�246�428S2 9�521�073 8�836�155 8�595�897 9�633�208 10�677�916 5�782�614 8�230�265S3 9�560�879 8�809�231 8�570�404 9�616�201 10�691�160 5�758�618 8�224�889


S1 20�752�448 15�848�841 16�485�836 17�554�079 23�447�483 13�701�596 18�574�540S2 20�645�080 15�971�828 16�540�209 17�547�339 23�437�516 13�726�087 18�581�802S3 20�640�702 15�663�930 16�389�856 17�530�314 23�325�136 13�638�200 18�481�668

function value of a design x obtained was estimatedwith �Rn′x using a sample of n′ = 100 scenarios. For agiven problem instance, to ensure that solution strate-gies are compared on the same basis, the sample ofn= 6 scenarios used in the heuristic and the sampleof n′ = 100 scenarios used to estimate the design valuewere the same for the three exploration strategies.The mean design value obtained for different subsetsof problem instances are presented in Table 7. Thedot in the problem subset labels in Table 7 denotesall instances corresponding to a problem attribute.(P1, a, . , . ), for example, is the subset of four prob-lems of size P1 with cost structure a. The value ofthe best search strategy for each problem subset ishighlighted. More detailed comparisons of the threestrategies’ design value and solution times are pro-vided in Figure 9. In the three first plots of this fig-ure, for P1, P2, and P3, respectively, the design value%-deviation from the best-known solution is given forall the instances solved. The fourth plot provides acomparison of average time (in seconds) required tofind the best solution by problem size.When looking at the detailed results, the first obser-

vation that, comes out is that even if none of theproposed strategies is completely dominated, searchstrategies S1 and S2 provide better results. Strategy 1provides the best solution for 65% of the probleminstances solved and it is within 0.5% of the best

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8 16

500

450

400

350

300

250

200

150

100

50

0P1 P2

Problems

Mea

n tim

e (s

ec.)

P31514131211109

P3 instances

P1 instances P2 instances

7654321

1615141312111098765432116151413121110987654321

–6.00

–5.00

–4.00

–3.00

–2.00

Des

ign

valu

e de

viat

ion

(%)

Des

ign

valu

e de

viat

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(%)

Des

ign

valu

e de

viat

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(%)

–1.00

0.00

–9.00

–8.00

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–6.00

–4.00

–3.00

–2.00

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–0.50

–1.00

–1.50

–2.00

0.00 0.00

S1

S2

S3

S1S2S3

S1S2S3

S1S2S3

Figure 9 Exploration Strategies Design Value and Solution Time Comparisons

solution found for 96% of the problems solved. Thisis mainly because this strategy converges quickly to asolution with a good number of depots and then per-forms several moves to improve this solution. It usu-ally finds the best solution during the initial searchiterations. Strategy 1 is also very fast: The averagesolution times for P1, P2, and P3 are 13, 43, and 292seconds, respectively. Strategy 2 provides very goodresults for P2 and P3� but it is relatively dependent onthe quality of its initial solution. It provides the bestsolution for 52% of the problem instances solved andit is within 0.5% of the best solution found for 90% ofthe problems solved. It gives excellent results for largeproblems (P3 and for instances with smaller ship-topoints (SN). It is also the fastest strategy. Strategy 3finds the best design for only 27% of the probleminstances. It gives better results for small networkswith smaller ship-to points (SN). Due to the intensifi-cation phase added in Strategy 3, the number of shiftmoves grows rapidly. For large problems, the solu-tion time is significantly larger than for the two otherstrategies.When the network density is low, the number of

borderline ship-to points is high and the Reassignprocedure improves the design obtained significantly.When fixed depot costs are low, the design obtainedincludes more depots to save on transportation costs.Conversely, when ship-to points are smaller, thedesign obtained includes fewer depots. The cost struc-ture does not have a marked impact on the perfor-mance of the search strategies. For large problems

(P3 , however, S1 performs better for instances withlarge fixed and variable costs a , and S2 performsbetter for problems with extreme fixed to variablecosts ratios (b� c .Our results also show that the route length estima-

tion formula (27) is very accurate. Using it to evalu-ate trial moves provides very good designs. It givesvalues between $99%�104%% of the exact transporta-tion costs, computed with the user model, in the caseof larger ship-to point networks (LN) and between$98%�108%% in the case of smaller ship-to point net-works (SN).

ConclusionsThis paper defines and formulates an importantstrategic planning problem for distribution businessesusing external transportation resources: the stochas-tic multiperiod location-transportation problem (SMLTP).The problem is characterized as a hierarchical deci-sion problem involving a design level taking networklocation and allocation decisions and a user leveltaking transportation decisions, and it is formulatedas a two-stage stochastic program with recourse. Weshow how a sample of multiperiod demand scenar-ios can be generated from the stochastic demand pro-cesses of customers and used to solve the problem.Because the resulting sample average approximationMIP is extremely large, it cannot be solved to opti-mality with commercial solvers for realistic problems

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and a hierarchical heuristic solution approach is pro-posed to solve it. It is based on a user-level transporta-tion heuristic and a design-level location-allocationheuristic. Three different strategies based on drop,add, and shift moves were proposed to explore theneighbourhood of a solution. A revised route lengthapproximation formula is also proposed to speed upcalculations.To test the quality of the heuristic developed,

several industrial cases were examined and usedto construct 48 realistic test-problem instances. Theexperiments made showed that the solution approachproposed provides good results in terms of solu-tion quality and solution time. We found that neigh-bourhood search strategies S1 and S2 give the bestresults. The excellent performance of the nested solu-tion approach proposed is principally due to the goodquality of the anticipation of transportation costs inthe design heuristic and to the efficient combination ofseveral procedures and heuristics. We believe that theapproach as it stands is sufficiently evolved to solvemost practical cases efficiently and effectively. Someroom is left, however, for additional research. Follow-ing are some future research avenues that could yieldsignificant payoffs:• We considered a context where a ship-to point

must be allocated to the same depot for the entireplanning horizon. In other contexts, a flexible allo-cation of ship-to points to depots may be possible.The user problem would then become a multidepottransportation problem and would be more difficultto solve.• We assumed that the demand process is station-

ary. If it were nonstationary, the size n of the scenariosamples to use to obtain good statistical optimal-ity gaps would be much larger and the SAA modelwould be more difficult to solve.• We were able to solve the SAA model to opti-

mality with CPLEX-11 only for small problems (P1 ,and we did not develop a specialized decompo-sition method to try to solve larger problems tooptimality. This may be very difficult because, as indi-cated earlier, the number of binary variables for routeselection grows exponentially, but it deserves furtherinvestigation.• The three search strategies examined could be

refined and fine tuned and other search methods,such as variable neighbourhood search, could be usedto elaborate alternative heuristics.

AcknowledgmentsThis research was supported in part by Natural Sciencesand Engineering Research Council of Canada Grant DNDPJ335078-05; by Defence R&D Canada; and by Modellium Inc.

ReferencesAlbareda-Sambola, M., E. Fernandez, G. Laporte. 2007. Heuristic

and lower bound for a stochastic location-routing problem.Eur. J. Oper. Res. 179 940–955.

Ambrosino, D., M. G. Scutella. 2005. Distribution network design:New problems and related models. Eur. J. Oper. Res. 165610–624.

Ballou, R. H. 1992. Business Logistics Management, 3rd ed. PrenticeHall, Englewood Cliffs, NJ.

Barreto, S., C. Ferreira, J. Paixao, B. S. Santos. 2007. Using clusteringanalysis in a capacitated location-routing problem. Eur. J. Oper.Res. 179 968–977.

Berger, R. T., C. R. Coullard, M. S. Daskin. 2007. Location-routingproblems with distance constraints. Transportation Sci. 41(1)29–43.

Berman, O., D. Krass, C. W. Xu. 1995. Location discretionary servicefacilities based on probabilistic customer flows. TransportationSci. 29(3) 276–290.

Birge, J. R., F. Louveaux. 1997. Introduction to Stochastic Program-ming. Springer, New York.

Chien, T. W. 1993. Heuristic procedures for practical-sized uncapac-itated location-capacitated routing problems. Decision Sci. 24(5)995–1021.

Clarke, G., J. W. Wright. 1964. Scheduling of vehicles from a centraldepot to a number of delivery points. Oper. Res. 12 568–581.

Daganzo, C. F. 1984. The distance traveled to visit N points witha maximum of C stops per vehicle: An analytic model and anapplication. Transportation Sci. 18(4) 331–350.

Girard, S., J. Renaud, F. F. Boctor. 2006. Heuristiques rapides pourle problème de tournées de véhicules. Administrative Sci. Assoc.Canada 2006 Proc., Banff, Alberta, Canada.

Glover, F., M. Laguna. 1997. Tabu Search. Kluwer Academic Pub-lishers, Boston.

Klose, A., A. Drexl. 2005. Facility location models for distributionsystem design. Eur. J. Oper. Res. 162 4–29.

Kouvelis, P., G. Yu. 1997. Robust Discrete Optimization andIts Applications. Kluwer Academic Publishers, Dordrecht,The Netherlands.

Kuehn, A. A., M. J. Hamburger. 1963. A heuristic program for locat-ing warehouses. Management Sci. 9 643–666.

Laporte, G. 1988. Location-routing problems. B. L. Golden, A. A.Assad, eds. Vehicle Routing: Methods and Studies. North-Holland, Amsterdam, 163–198.

Laporte, G., P. J. Dejax. 1989. Dynamic location routing problems.J. Oper. Res. Soc. 40(5) 471–482.

Laporte, G., F. V. Louveaux. 1998. Solving stochastic routing prob-lems. T. G. Crainic, G. Laporte, eds. Fleet Management and Logis-tics. Kluwer Academic Publishers, Boston.

Laporte, G., I. H. Osman. 1995. Routing problems: A bibliography.Ann. Oper. Res. 61 227–262.

Laporte, G., F. V. Louveaux, H. Mercure. 1989. Models and exactsolutions for a class of stochastic location-routing problems.Eur. J. Oper. Res. 39 71–78.

Laporte, G., M. Gendreau, J.-Y. Potvin, F. Semet. 2000. Classical andmodern heuristics for the vehicle routing problem. Internat.Trans. Oper. Res. 7 285–300.

Min, H., V. Jayaraman, R. Srivastava. 1998. Combined location-routing problems: A synthesis and future research directions.Eur. J. Oper. Res. 108 1–15.

Nagy, G., S. Salhi. 1996a. Nested heuristic methods for the location-routing problem. J. Oper. Res. Soc. 47(9) 1166–1174.

Nagy, G., S. Salhi. 1996b. A nested location-routing heuristic usingroute length estimation. Stud. Locational Anal. 10 109–127.

Nagy, G., S. Salhi. 2007. Location-routing: Issues, models and meth-ods. Eur. J. Oper. Res. 177 649–672.

Dow

nloa

ded

from

info

rms.

org

by [

193.

140.

109.

10]

on 2

4 M

ay 2

014,

at 0

4:24

. Fo

r pe

rson

al u

se o

nly,

all

righ

ts r

eser

ved.


Owen, S. H., M. S. Daskin. 1998. Strategic facility location: A review.Eur. J. Oper. Res. 111 423–447.

Prins, C., P. Prodhon, A. Ruiz, P. Soriano, R. W. Calvo. 2007. Solv-ing the capacitated location-routing problem by a cooperativeLagrangian relaxation-granular tabu search heuristic. Trans-portation Sci. 41(4) 470–483.

Rei, W., M. Gendreau, P. Soriano. 2007. A hybrid Monte Carlo localbranching algorithm for the single vehicle routing problemwith stochastic demands. CIRRELT Research Document 2007–24, CIRRELT, Montreal.

Ruszczynski, A., A. Shapiro, eds. 2003. Stochastic Programming,Vol. 10. Handbooks in Operations Research and ManagementScience, Elsevier, Amsterdam.

Salhi, S., G. Nagy. 1999. Consistency and robustness in location-routing. Stud. Locational Anal. 13 3–19.

Salhi, S., G. Nagy. 2009. Local improvement in planar facility loca-tion using vehicle routing. Ann. Oper. Res. 167(1) 287–296.

Santoso, T., S. Ahmed, M. Goetschalckx, A. Shapiro. 2005.A stochastic programming approach for supply chain networkdesign under uncertainty. Eur. J. Oper. Res. 167 96–115.

Sariklis, D., S. Powell. 2000. A heuristic method for the open vehiclerouting problem. J. Oper. Res. Soc. 51 564–573.

Shapiro, A. 2003. Monte Carlo sampling methods. A. Ruszczyn-ski, A. Shapiro, eds. Stochastic Programming, Vol. 10, Hand-books in Operations Research and Management Science. Elsevier,Amsterdam.

Shen, Z.-J. 2007. Integrated supply chain design models: A surveyand future research directions. J. Indust. Management Optim.3(1) 1–27.

Snyder, L. V., M. S. Daskin. 2005. Reliability models for facilitylocation: The expected failure cost case. Transportation Sci. 39400–416.

Talbi, E. G. 2002. A taxonomy of hybrid metaheuristics. J. Heuristics8 541–564.

Toth, P., D. Vigo. 1998. Exact solution of the vehicle routing prob-lem. T. G. Crainic, G. Laporte, eds. Fleet Management and Logis-tics. Springer, Berlin.

Tuzun, D., L. I. Burke. 1999. A two-phase tabu search approachto the location routing problem. Eur. J. Oper. Res. 11687–99.

Verweij, B., S. Ahmed, A. J. Kleywegt, G. Nemhauser, A. Shapiro.2003. The sample average approximation method applied tostochastic routing problems: A computational study. Computa-tional Optim. Appl. 24 289–333.

Vila, D., A. Martel, R. Beauregard. 2007. Taking market forces intoaccount in the design of production-distribution networks: Apositioning by anticipation approach. J. Indust. ManagementOptim. 3(1) 29–50.

Zainuddin, Z. M., S. Salhi. 2007. A perturbation-based heuristic forthe capacitated multisource Weber problem. Eur. J. Oper. Res.179 1194–1207.

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193.

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4 M

ay 2

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the stochastic multiperiod location transportation problem

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