Production/distribution system design with inventory considerations

Production/Distribution System Design with Inventory Considerations

Burcu B. Keskin,1 Halit Üster2

1 Department of Information Systems, Statistics and Management Science, University of Alabama, Tuscaloosa, Alabama 35487

2 Department of Industrial and Systems Engineering, Texas A&M University, College Station, Texas 77843-3131

Received 5 August 2009; revised 29 September 2011; accepted 6 January 2012DOI 10.1002/nav.21482

Published online in Wiley Online Library (wileyonlinelibrary.com).

Abstract: This study addresses the design of a three-stage production/distribution system where the first stage includes the set ofestablished retailers and the second and third stages include the sets of potential distribution centers (DCs) and potential capacitatedsuppliers, respectively. In this problem, in addition to the fixed location/operating costs associated with locating DCs and suppliers,we consider the coordinated inventory replenishment decisions at the located DCs and retailers along with the appropriate inventorycosts explicitly. In particular, we account for the replenishment and holding costs at the retailers and selected DCs, and the fixedplus distance-based transportation costs between the selected plants and their assigned DCs, and between the selected DCs andtheir respective retailers, explicitly. The resulting formulation is a challenging mixed-integer nonlinear programming model forwhich we propose efficient heuristic solution approaches. Our computational results demonstrate the performance of the heuristicapproaches as well as the value of integrated decision-making by verifying that significant cost savings are realizable when theinventory decisions and costs are incorporated in the production distribution system design. © 2012 Wiley Periodicals, Inc. NavalResearch Logistics 59: 172–195, 2012

Keywords: production/distribution; inventory; mixed integer nonlinear programming; heuristics

1. INTRODUCTION

Production-distribution system design (PDSD) problemsaddress strategic and tactical decisions regarding the designand operation of supply chains. A typical PDSD model simul-taneously considers the decisions regarding plant locations,distribution center (DC) locations, DC-plant and retailer-DCassignments as well as product flows from plants to retailersthrough DCs. The objective of the PDSD problem is to mini-mize the variable distribution costs, as well as the fixed costsof opening, equipping, and managing plants and DCs.

Several companies including Elkem Silicone [47], GEPlastics [46], the Kellogg Company [5], Frito-Lay [9],Digital Equipment Corporation [1], Ault Foods Limited[28], Libbey-Owens-Ford [20], and DowBrands [31] haveachieved substantial cost savings through the optimizationof production-distribution systems. Review papers on PDSD[8, 10, 11, 35] summarize the benefits and challenges associ-ated with integrating the overall decision processes in thesesystems while emphasizing the need for practical analyti-cal models and efficient solution methods. In this article,we consider an integrated location and inventory problem in

Correspondence to: H. Üster ([email protected])

the context of PDSD. Our problem generalizes the previousPDSD literature by taking inventory decisions into accountand explicitly modeling the interaction between inventoryand location decisions explicitly through transportation costs.

Specifically, we consider a three-stage PDSD problemwhere the first stage corresponds to existing retailers (cus-tomers) and the second and third stages consist of DCs andcapacitated suppliers, respectively, whose locations are to bedetermined. Opening (or selecting) a DC or supplier results ina facility-specific fixed operational cost. Each retailer facesa deterministic and stationary retailer-specific demand andkeeps inventory that is replenished from a single DC. EachDC, on the other hand, replenishes its inventory from a sin-gle capacitated supplier. That is, we impose single-sourcingrestrictions in both retailer-to-DC and DC-to-supplier assign-ments. Although single-sourcing requirements make thelocation problem much harder to solve due to capacity restric-tions at the suppliers, they help to ease control of the inventorydecisions throughout the supply chain. Since the inventory iskept at both the retailer and DC levels, we incorporate intoour model the coordination aspects of inventory decisionsas addressed in multiechelon inventory literature. Specifi-cally, as in [33], we assume that the inventory system of aselected DC and its associated retailers is operated under the

© 2012 Wiley Periodicals, Inc.

Keskin and Üster: Production/Distribution System Design 173

assumptions of the classical single-warehouse multiretailer(SWMR) problem in which a power-of-two policy is used.Under this overall setting, our problem concerns the deter-mination of the optimum (i) locations for the DCs and thesuppliers, (ii) assignments of the retailer to the DCs as wellas the DCs to the selected suppliers, and finally (iii) coordi-nated inventory decisions for the retailers and the DCs. Theoverall objective is the minimization of the total cost in thesystem which includes (i) the fixed operational costs of theDCs and suppliers, (ii) the transportation costs between theDCs and the retailers and between the suppliers and the DCs,and (iii) the inventory replenishment and holding costs of theretailers and the DCs.

This work generalizes two closely related studies in theintegrated location-inventory literature by Teo and Shu [45]and Üster et al. [48]. Both of these papers include jointlocation-inventory models where the inventory control at boththe DCs and retailers was investigated. Teo and Shu [45] con-sider a two-stage setting with inventory decisions and addressthe issue of coordinating replenishment activities between thewarehouses (DCs) and the retailers. We generalize this settingin mainly two directions.

First, Teo and Shu [45] assume that the DCs are suppliedby a single uncapacitated supplier whose location is fixed.The replenishment cost of a DC from the supplier is assumedto be constant, independent of the location of the DC. Inour case, we determine capacitated supplier locations and,at the same time, express the inventory replenishment costfor a DC as dependent (in addition to a fixed component) onthe location of the DC and its assigned supplier. As statedby Benton [4], a single supplier in a PDSD may be justifiedwhen (1) lower total cost results from a much higher volume(economies of scale); (2) the buying firm obtains more influ-ence with the supplier; and (3) lower costs are incurred tosource, process, expedite, and inspect by the use of a singlesupplier. However, in many real-life PDSD problems, as wellas in the literature [1,22,25,26,47], having multiple supplierswould be more appropriate due to (1) the suppliers’ capacityissues; (2) the need for maintaining competition among sup-pliers; and (3) the requirements for keeping a backup sourceas a protection from shortages, strikes, and other disruptionsof the suppliers. In our problem, the limitation of capacityat the suppliers is the main reason to consider multiple sup-pliers where suppliers have an aggregate production capacitylimit that they cannot exceed regardless of the inventory pol-icy. Additionally, during a design or redesign of a PDSD,many problems call for considering opening and/or closingplants for the best configuration along with the location ofwarehouses or DCs [14,15,26,27,30,43]. Main reasons mayinclude (i) a reconfiguration of the supply chain to respond tomarket changes as in Elkem Silicone [47] or GE Plastics [46]or (ii) eliminating or consolidating duplicated elements of asupply chain due to company mergers or acquisitions [22].

Replacing the potential suppliers with a set of a potentialplant locations, our model and solutions would be applicablein these settings. As the models in [45] do not consider anycapacity limitations and do not include supplier selection (andthe fixed cost of supplier selection), their model and resultsare not applicable for this problem. As selection of suppli-ers is a major issue for many distribution network design orredesign problems, it needs to be addressed via quantitativeOR tools.

Second, contrary to the approach in [45], which incorpo-rates inventory costs into the model by including an annualcost term for shipping the entire annual demand of a retailerfrom its assigned DC, we consider trip costs and frequenciesin expressing the transportation costs, which include fixedand distance-based variable components. The latter compo-nent directly relates to the dependence of fixed ordering costs,and in turn, inventory policy decisions, on location decisionsthrough transportation costs.

Our work also generalizes the study by Üster et al. [48] thatconsiders a three-stage integrated location-inventory model,with existing retailers at the first stage and a single supplierat the third stage. They develop efficient algorithms to deter-mine the location of a single DC (specifically, the coordinatesof the DC in a continuous location setting) serving retailersand inventory decisions under a power-of-two policy for thewarehouse and retailers simultaneously. We extend their workby considering multiple capacitated suppliers as well as mul-tiple DCs whose locations are to be determined in a discretesetting.

In summary, the main contributions of this article arethreefold. First, we create a formal model that provides anintegrated view of strategic facility location decisions andoperational transportation and inventory decisions. Second,we develop efficient construction and local search improve-ment heuristics as well as a simulated annealing based meta-heuristic to find near optimal solutions for the problem.Finally, by comparing the cost of the PDSD problem withinventory considerations with three benchmark models, wequantify the value of integrated decision-making, multiplesuppliers, and the inventory considerations at the retailers.Through this comparison, we also identify the importantproblem parameters that contribute the most to this value.

The remainder of the article is organized as follows: Thenext section discusses the existing literature on PDSD prob-lems and integrated location and inventory models. In Section3, we introduce the notation and the PDSD model withinventory considerations. Next, in Section 4, we discuss theheuristic solution approaches to this problem depending onthe problem characteristics. Section 5 presents a simulatedannealing based metaheuristic to improve on the heuris-tic solutions. In Section 6, we describe the details of thebenchmark model. Section 7 presents the numerical resultsregarding the performance of the solution approaches and the

Naval Research Logistics DOI 10.1002/nav

174 Naval Research Logistics, Vol. 59 (2012)

value of integrated decision-making. Finally, in Section 8, wesummarize our findings and conclusions.

2. LITERATURE REVIEW

Our article contributes to two streams of research. The firststream is the literature on PDSD problems focusing on dis-crete models for the location of plants and warehouses and thesecond stream includes the literature on integrated locationand inventory models.

There exists a large body of research on the modeling anddesign of various components of PDSD problems. In thisarea, several comprehensive reviews have been published[8, 10, 11, 35]. For an updated review and detailed classifi-cation of PDSD problems, see [16]. One important emphasisin all these review papers is the complexity of PDSD prob-lems, regarding different levels of logistics planning. Eventhough it is recognized that the four major dimensions oflogistics planning involving customer service levels, facilitylocation, inventory, and transportation decisions are highlyinterrelated [2,12], there is a lack of research integrating thesedecisions, investigating their interrelations, and planning thesupply chain as a whole.

A recent line of work in integrated location-inventory the-ory presents a remedy to this deficiency by placing particularemphasis on the inclusion of inventory costs in networkdesign problems. An even better approach for eliminating thisdeficiency is to consider joint optimization of facility loca-tion and inventory decisions. Our article contributes to theintegrated location-inventory theory literature in this context.We discuss these two areas of integrated location inventoryliterature next.

In the former area, papers, including [3, 6, 7, 9, 14, 21, 23,24, 36, 37, 39, 41, 44], consider inventory costs (e.g., order,holding, backlog, and shortage costs) in the context of distri-bution system design, while ignoring the effects of inven-tory ordering policies on optimal network design. In oneearly article in this context, Barahona and Jensen [3] con-siders a two-stage location model with fixed inventory costsand developed a solution method based on Dantzig-WolfeDecomposition. For a three-stage network design model,Jayaraman [14] models in-transit inventory and linear cyclestock costs, together with fixed facility location costs andunit-based transportation costs under deterministic demand,to determine the number and location of plants and dis-tribution centers (DCs). Even though their problem settingis similar to ours, they do not explicitly model inventorydecisions.

Considering stochastic demand, Nozick and Turnquist [23]analyze the impact of integrating inventory costs into a two-stage fixed-charge facility location model. A basic premise ofthe work by Nozick and Turnquist [23] is the considerationof safety stock costs to provide a desired level of service,

together with other fixed location and transportation costs,in determining the optimal number of DCs and their loca-tions. This article provides a linear approximation for safetystock costs as a function of the number of DCs. Croxton andZinn [6] extend the analysis by Nozick and Turnquist [23] toconsider a three-stage multiproduct network design problemwhere the number and location of DCs are determined whileminimizing total transportation, fixed location, and safetystock costs. This model is tested with data from a nationalretailer, and due to explicit consideration of inventory costs,an immediate result is a reduced number of DCs.

Daskin et al. [7], Shen et al. [36], Shu et al. [39], and Vid-yarthi et al. [49] also study network design problems understochastic demand. Daskin et al. [7], motivated to study thedistribution of perishable and expensive blood products tolocal hospitals, locate regional centers for blood platelets inthe first stage and assign these centers to local hospitals inthe second stage, by considering fixed locations and trans-portation costs as well as safety stock costs. They propose aLagrangian relaxation algorithm [7]. For this specific applica-tion, Shen et al. [36] and Shu et al. [39] present a set-coveringproblem with branch-and-price approach. The main result,other than the theoretical and algorithmic contributions inthese papers, is that integrating facility location decisionswith the cost of inventory risk-pooling explicitly has animpact on the number of regional centers located. Vidyarthiet al. [49], on the other hand, consider the design of a threestage production/distribution system where the locations ofcapacitated plants and DCs are determined, similar to our arti-cle. However, they do not consider any inventory decisionsneither at the DC nor at the retailer level and they do not dealwith the coordination issues that arise from keeping invento-ries at two levels. Nevertheless, they consider the inclusionof safety stock costs at the open DCs for all the retailers thatobserve stochastic demand and served by that DC.

In this article, our focus is not only on incorporating inven-tory costs in the PDSD problem, but also on determininginventory policy parameters together with facility locationand assignment decisions. Hence, our article contributes tothe second area of integrated location inventory literature byconsidering joint optimization of facility location and inven-tory decisions. The limited existing research in this area byRomeijn et al. [32], Teo and Shu [45], Üster et al. [48], andShu [38] is closely related to our research problems. Con-sidering the discrete facility location problem setting, Teoand Shu [45] study a two-stage, warehouse-retailer networkdesign problem that incorporates transportation and inven-tory cost functions under deterministic stationary demandover an infinite planning horizon. Given an unlimited exter-nal supplier, their goal is to determine how many warehousesto set up, where to locate them, how to serve the retailersusing these (uncapacitated) warehouses, as well as determineoptimal inventory policies for the warehouses and the retailers



Figure 1. Underlying structure of the PDSD-INV problem. [Colorfigure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

so that the total transportation, fixed facility, inventory replen-ishment, and holding costs are minimized. In their model, thetransportation costs are per unit per mile costs. They do notexplicitly consider the impact of trip distances and frequencyon transportation costs, and hence, they do not explicitlyaccount for the interdependency of inventory and the facilitylocation problem. They show that the network design prob-lem can be modeled approximately (to within 98% accuracy)as a set-partitioning problem that can be solved using thecolumn generation method. Romeijn et al. [32] extend thework by Teo and Shu [45] to consider demand variabilityand capacity congestion by including safety stock and con-gestion costs in the model. They also formulate the problemas a set-partitioning problem and solve it using a columngeneration approach. For the problem studied by Teo andShu [45], Shu [38] admits that the proposed column genera-tion approaches fail to handle large-scale network problemsand presents an efficient greedy algorithm that adds set ofretailers to the network iteratively until all the retailers havebeen assigned to certain warehouses.

On the other hand, as mentioned earlier, Üster et al. [48]consider a continuous facility location problem in a three-stage setting. They locate the warehouse, given the locationsof the supplier and existing retailers, and determine the inven-tory policy parameters of the warehouse and the retailerssimultaneously. Their article integrates the continuous facil-ity problem [19] and the SWMR problem with power-of-twopolicy [33] to provide efficient solution algorithms for theintegrated inventory-location problem. As discussed in detaillater, we utilize their solution algorithms while solving thesubproblems in this study.

All of the aforementioned articles in the integratedlocation-inventory literature have assumed an uncapacitated,single supplier. In other words, none of the referencesreviewed above has considered the impact of supplier selec-tion (or opening/closing plants) on the network design deci-sions. In many real situations, however, especially in a globalsupply chain environment, there is a need to evaluate supplychain designs with a number of suppliers/plants. Therefore,

this article proposes a three-level production/distributionnetwork design model to attempt to fill this gap.

3. MODEL FORMULATION

To model our three-stage integrated PDSD and inventorycontrol problem PDSD-INV, depicted in Fig. 1, we first intro-duce the sets I = {i : 1, . . . , m}, J = {j : 1, . . . , n}, andK = {k : 1, . . . , K} which denote the sets of retailers (firststage), candidate DC locations (second stage), and candidatesupplier locations (third stage), respectively. We present thecomplete notation for the input parameters in Table 1.

In this model, we use six sets of decision variables. Thefirst four sets of decision variables, given below, are relatedto the selection of DC and supplier locations as well asthe assignment of retailers to the DCs and the DCs to thesuppliers,

Zk ={

1, if supplier k is selected,0, otherwise.

Xj ={

1, if DC j is selected,0, otherwise.

Yij ={

1, if retailer i is assigned to DC j ,0, otherwise.

Vjk ={

1, if DC j is assigned to supplier k,0, otherwise.

The last two sets of decision variables address the inventoryrelated decisions which are T R

i , reorder interval at retailer i,and T DC

j , reorder interval at DC j .

Table 1. Notation for input parameters.

Di Demand at retailer iCk Annual throughput capacity at supplier kfj Fixed (annual) cost of selecting candidate DC jgk Fixed (annual) cost of selecting candidate supplier k

KDCj Fixed ordering cost of DC j

KRi Fixed ordering cost of retailer i

hRi Inventory holding cost rate at retailer i

hDCj Inventory holding cost rate at DC j , hDC

j ≤ hRi , ∀i ∈ I

HDCj Echelon holding cost rate at the DC j , HDC

j = hDCj

HRij Echelon holding cost rate at retailer i, HR

ij = hRi − HDC

j

Tb Fixed base period (set a priori)pDC

jk Fixed cost of transportation (releasing a shipment) to DC jfrom supplier k

pRij Fixed cost of transportation (releasing a shipment) to

retailer i from DC j

rDCjk Per mile transportation cost from supplier k to candidate

DC j

rRij Per mile transportation cost from candidate DC j to

retailer i

dDCjk Distance between a candidate DC location j and a candidate

supplier location k

dRij Distance between a retailer i and a candidate DC location j



Before introducing the model in detail, let us summarizethe underlying modeling assumptions regarding inventorydecisions for DCs and retailers which closely follow the onesfor SWMR lot sizing problem [33]. Given a fixed DC j ∈ J ,its fixed supplier k ∈ K, and a set of retailers Ij assignedto this DC j , i.e., Ij = {i ∈ I : Yij = 1}, the operationcharacteristics include the following:

• Deliveries from supplier k to DC j , and from DC j toretailer i ∈ Ij are direct and instantaneous. The sizesand frequencies of the deliveries are determined bythe inventory policy parameters.

• Each delivery between two facilities results in atransportation cost which consists of a fixed (load-ing/unloading) cost (pDC

jk , pRij ) and a variable mileage

cost (rDCjk , rR

ij ). In other words, each delivery betweensupplier k and DC j would result in a transportationcost of pDC

jk + rDCjk dDC

jk . Since, on average, there willbe 1/T DC

j shipments annually, the total transportationcost from selected suppliers to selected DCs is givenas

∑j∈J

∑k∈K

(pDC

jk + rDCjk dDC

jk

)Vjk

T DCj

.

Similarly, each delivery between DC j and retailer i

would result in a transportation cost of

∑i∈I

∑j∈J

(pR

ij + rRij d

Rij

)Yij

T Ri

.

• DC j ’s inventory is replenished at successive reorderintervals of T DC

j , incurring a total cost of trans-portation and ordering per replenishment given bypDC

jk + rDCjk dDC

jk + KDCj .

• The inventory of retailer i ∈ Ij is replenished atsuccessive reorder intervals of T R

i incurring a costof pR

ij + rRij dR

ij + KRi , which represents the total

transportation and ordering costs per replenishment.• Echelon holding costs accumulate at a rate of HDC

j atDC j over T DC

j and at a rate of HRij at retailer i over

T Ri .

• The reorder intervals, T DCj and T R

i , i ∈ Ij , are power-of-two multiples of a fixed base period, Tb. [33]shows that, while the optimal replenishment strategyis intractable, a power-of-two constrained inventorypolicy is proven to come within 6% of a lower boundon the minimum achievable costs. For a selected DCj and the set of retailers Ij served by this DC, theinventory holding costs are given as:∑

i∈Ij

1

2HR

ij DiTRi +

∑i∈Ij

1

2HDC

j Di max{T R

i , T DCj

}

using the echelon holding cost rates (see [40], p.98).For a detailed discussion of power-of-two policies anda single warehouse, multiretailer system, we refer thereaders to [33], [40], and [48].

The PDSD-INV can now be formulated as the followingnonlinear integer program:

Min∑j∈J

fjXj +∑k∈K

gkZk +∑i∈I

∑j∈J

(pR

ij + rRij d

Rij

)Yij

T Ri

+∑j∈J

∑k∈K

(pDC

jk + rDCjk dDC

jk

)Vjk

T DCj

+∑j∈J

KDC

j

T DCj

+∑i∈Ij

KRi

T Ri

+∑i∈Ij

1

2HR

ij DiTRi

+∑i∈Ij

1

2HDC

j Di max{T R

i , T DCj

} Xj

subject to∑j∈J

Yij = 1, ∀ i ∈ I (1)

∑k∈K

Vjk = 1, ∀ j ∈ J (2)

Yij ≤ Xj , ∀ i ∈ I, j ∈ J . (3)

Vjk ≤ Zk , ∀ j ∈ J , k ∈ K. (4)

Vjk ≤ Xj , ∀ j ∈ J , k ∈ K. (5)∑i∈I

∑j∈J

DiYijVjk ≤ CkZk , ∀ k ∈ K. (6)

T DCj =

{2νj Tb, νj ∈ Z, if Xj = 1,n/a, otherwise

∀ j ∈ J . (7)

T Ri = 2υi Tb and υi ∈ Z, ∀ i ∈ I. (8)

Xj ∈ {0, 1}, Yij ∈ {0, 1}, ∀ i ∈ I, j ∈ J . (9)

Zk ∈ {0, 1}, Vjk ∈ {0, 1}, ∀ j ∈ J , k ∈ K. (10)

The objective function of PDSD-INV minimizes the totalcost of designing and operating the distribution system, thatis, the sum of (i) the fixed cost of locating the DCs, (ii) thefixed cost of locating the suppliers, (iii) the transportationcost from selected DCs to the retailers, (iv) the transporta-tion cost from selected suppliers to selected DCs, and (v) theannual ordering and holding costs at the DCs and the retail-ers, respectively. Constraints (1) ensure that each retailer isserved by exactly one DC. Constraints (2) stipulate that eachDC is assigned to exactly one supplier. These sets of con-straints, together with the integrality constraints (9) and (10),state that there is single-sourcing between the first and second



stage and between the second and third stage. Constraints (3),(4), and (5) are the assignment constraints to ensure that onlyselected facilities (DCs and suppliers) are utilized. In particu-lar, constraints (3) state that the retailers can be assigned onlyto the selected candidate DCs. Constraints (4) ensure that aDC can be assigned only to a selected supplier. Constraints(5) ensure that a supplier can be assigned only to a selectedcandidate DC. Constraints (6) ensure capacity restrictions atthe suppliers. Note that the capacity restriction at a supplieris based on the annual throughput, and hence, the total annualdemand of the retailers assigned to that particular supplier.This capacity is based on an aggregate production capacitylimit, regardless of the inventory policy. Therefore, it limitsthe size of the subsets of retailers that are served by one DCand on the number of DCs that are served by that supplier.On the other hand, DCs do not have explicit capacity con-straints, assuming that the physical size of the facilities andthe inventory turns are large enough. Constraints (7) and (8)state the power-of-two restrictions on the reorder intervals ofthe selected DCs and the retailers. Finally, constraints (9) and(10) are the standard integrality constraints.

Note that this formulation is complicated, not only dueto the nonlinearities in the objective function, but also dueto the nonlinearities in constraints (6), (7), and (8). We mayeliminate the nonlinearity in the capacity constraint (6) byintroducing a new variable that keeps track of the retailersassigned to a supplier. In particular, for retailer a i ∈ I, a DCj ∈ J , and a supplier k ∈ K, we define

Wijk =

1, if retailer i is assigned to supplierk through DC j ,

0, otherwise.(11)

Then, we can replace constraint (6) with the followingconstraints

Yij + Vjk − 1 ≤ Wijk , ∀ i ∈ I, j ∈ J , k ∈ K. (12)∑i∈I

∑j∈J

Di Wijk ≤ Ck Zk , ∀ k ∈ K, (13)

thereby eliminating nonlinearity in the constraints (6) at theexpense of additional variables and constraints. The nonlin-earity in the objective function and in the constraints (7) and(8) that are due to inventory considerations still complicatesthe problem. Hence, heuristic and metaheuristic approachesare important to obtain good solutions; we discuss them next.

4. HEURISTIC SOLUTION APPROACHES

We first devise an efficient construction heuristic whichprovides a feasible solution to PDSD-INV. This feasible solu-tion is later used as the initial solution in an improvement

heuristic we develop. The improvement heuristic utilizescombinations of neighborhood searches to modify the initialsolution. In the context of cost evaluation of a given solu-tion, we describe a continuous location based (CL-based)approach, which utilizes the results from [48], to determinethe assignments between subsets of retailers and sets ofopen DCs and suppliers. We start by describing the solutionrepresentation and evaluation approach used in our algo-rithms. Then, we describe the construction heuristic, theimprovement heuristic, and their components in detail.

4.1. Solution Representation and Evaluation

Based on the problem characteristics, any feasible solutionto PDSD-INV can be represented as mutually exclusive andcollectively exhaustive subsets of retailers and sets of openDCs and suppliers. We let K′ ⊂ K and J ′ ⊂ J be the set ofselected suppliers and the set of open DCs, respectively, andS denote the set of retailer subsets in which each S ∈ S isserved by a unique DC j ∈ J .

We define cjkS , the cost of serving the retailers in S ⊂ S viaDC j ∈ J ′ and supplier k ∈ K′, to have three components,including:

• supplier-specific fixed selection cost, gk ,• DC specific fixed location cost, fj , and• system-wide inventory replenishment, inventory

holding, and transportation costs, denoted byIT (j , k, S). This cost is estimated by solving aSWMR lot-sizing problem given as

IT (j , k, S)

= min

{∑i∈S

(pR

ij + rRij d

Rij

)T R

i

+(pDC

jk + rDCjk dDC

jk

)T DC

j

+ KDCj

T DCj

+∑i∈S

KRi

T Ri

+∑i∈S

1

2HR

ij DiTRi

+∑i∈S

1

2HDC

j Di max{T R

i , T DCj

}}. (14)

subject to

T DCj = 2νj Tb and νj ∈ Z. (15)

T Ri = 2υi Tb and υi ∈ Z, ∀ i ∈ S. (16)

T DCj ∈ R+ and T R

i ∈ R+, ∀ i ∈ S. (17)

Note that this SWMR lot-sizing problem takes theinteraction of inventory decision variables and trans-portation costs into account. In particular, the trans-portation cost between supplier k and DC j has anexplicit impact on the reorder interval of the DC by



influencing the total cost per replenishment from thesupplier. Similarly, the transportation cost betweenDC j and the retailers in S influences the reorderinterval of the retailers. For fixed supplier k, DC j ,and a retailer set S, IT (j , k, S) is solved using theapproach provided in [33].

Hence, cjkS is given by gk +fj + IT (j , k, S), for all j ∈ J ′,k ∈ K′, and S ⊂ S.

In general, the quality of a feasible solution is representedby the value of the objective function, i.e., the cost that itimplies. To evaluate this cost for given J ′, K′, and S, weuse the following representation of the objective function ofPDSD-INV, i.e.,

Z(J ′, K′, S) =∑k∈K′

∑j∈J ′

∑S⊂S

cjkS =∑k∈K′

gk +∑j∈J ′

fj

+∑j∈J ′

∑k∈K′

∑S∈S

IT (j , k, S) xjkS

where xjkS is a binary assignment variable denoting whethersubset S ∈ S is assigned to supplier k ∈ K′ through DCj ∈ J ′. Our algorithms are designed to determine the con-tents of J ′, K′, and S as well as xjkS for j ∈ J ′, k ∈ K′, andS ∈ S.

Note that a feasible solution implies that the capacity con-straints at the selected suppliers are satisfied. That is, in afeasible solution

∑j∈J

∑i∈S;S∈S DixjkS ≤ CkZk for each

selected supplier k ∈ K′. This constraint dictates the size ofthe retailer sets as well as how many DCs a supplier can serve.However, it does not restrict the use of the power-of-two basedinventory policies.

4.2. A Construction Heuristic

The main objective of the construction heuristic (CH) is toprovide an initial solution for PDSD-INV. Utilizing a greedyapproach outlined in Display 1, our CH provides a partitionof the set of retailers into mutually exclusive subsets and,for each subset S ∈ S, determines a feasible DC-supplierassignment, xjkS , j ∈ J ′, and k ∈ K′.

Initially, the open DC and supplier sets (J ′ and K′, respec-tively) as well as the set of retailer subsets (S) are empty.Next, we sort the set of available suppliers, aS, in decreasingorder, according to the ratio of their capacity to fixed facilityopening cost. Starting with supplier k1 at the beginning of thesorted list, among the available DCs, aDC, we determine theclosest DC j1 to this supplier. The supplier-DC pair (k1, j1)

will be serving the retailers in subset S, i.e., xj1,k1,S = 1. Todetermine the contents of S, we sort the not-yet-served (free)retailers with respect to their proximity to this selected DCj1. We greedily add free retailers to subset S in the deter-mined order as long as the capacity constraint of supplier

Display 1 Procedure Construction(K′, J ′, S, Z)

1: Initialize: freeRet = I, K′ = ∅, aS = K, J ′ = ∅,aDC = J , S = ∅, and Z = 0

2: Sort aS according to the ratio Ck/gk , k ∈ K, in adecreasing order

3: while |freeRet| > 0 do4: k1 = aS[1]5: S = ∅6: remainingCap = Ck1

7: j1 = arg minj∈aDC{(pDCj ,k1

+ rDCj ,k1

dDCj ,k1

)}8: Sort freeRet according to {(pR

i,j1+ rR

i,j1dR

i,j1)} in an

increasing order9: for i ∈ freeRet do

10: if Di < remainingCap then11: remainingCap = remainingCap − Di

12: S = S ∪ {i} and freeRet = freeRet \ {i}13: end if14: end for15: Z = Z + cj1,k1,S

16: S = S ∪ {S}17: K′ = K′ ∪ {k1}; aS = aS \ {k1}18: J ′ = J ′ ∪ {j1}; aDC = aDC \ {j1}19: end while20: Return K′, J ′, S and Z

k1 is not violated. If a retailer is added to S, it is removedfrom the set of free retailers. Once the contents of S is final-ized, S is included in S. Additionally, we mark k1 and j1 asthe selected supplier and open DC, respectively. With thisstep, we essentially include them in K′ and J ′, respectively,and remove them from the sets of available suppliers aS andDCs aDC. We compute cj1,k1,S and modify Z(J ′, K′, S) byincorporating this cost into the current value of Z(J ′, K′, S).These steps are repeated until all of the retailers are servedby a DC and a supplier. Finally, the CH returns the setsJ ′, K′, and S, and the objective value Z(J ′, K′, S) of thesolution.

4.3. Local Search Algorithm

Our overall local search (LS) algorithm modifies the initialfeasible solution provided by the CH using a combination ofneighborhood functions to obtain a better solution. Typically,a neighborhood function modifies the key attributes of a solu-tion to generate neighboring solutions in a heuristic searchframework. For our problem, we list the attributes of our solu-tion as mutually exclusive subsets of retailers (S ∈ S) andDC-supplier pairs (links) that serve these subsets.

To modify S ∈ S, we utilize a combination of sim-ple neighborhood functions: move neighborhood, exchangeneighborhood, and new set construction functions. The move



and exchange neighborhood functions help us modify thecontents of each subset of retailers for intensification pur-poses. The new set construction function helps us constructnew subsets of retailers for diversification purposes. Com-bining several different simple neighborhood functions helpsus search the solution more thoroughly and efficiently. Weexplain these components in detail below.

To modify DC-supplier links, we use a CL-based approachto utilize the results by [48] when determining DC-supplierassignments. Üster et al. [48] consider an integrated contin-uous location and inventory problem (ICLIP) for a supplier-multiretailer setting to determine the location of the DC andthe inventory decisions of the DC and the retailers. Given asubset of retailers with known locations, we determine thebest central DC location and supplier to serve this subset bysolving an ICLIP for each potential supplier that has enoughcapacity to serve this subset. We explain the details of theCL-based algorithm in the following section.

In Display 2, we present the overall LS algorithm whichstarts with the initial feasible solution obtained from the con-struction heuristic. We initialize the set of subsets of retailersS and the best objective value (Z∗) from the solution of theconstruction heuristic. If there is only one subset of retailers,we check to see if we can generate new subsets by invokingthe new subset construction function. If generating new sub-sets proves beneficial and reduces the overall cost, we updatethe set of subsets of retailers with the new set obtained fromthe new subset construction function. Otherwise, if S containsmore than one subset of retailers, we perform a subset contentimprovement (move and exchange) neighborhood search, asdescribed in Sections 4.5 and 4.6, respectively. Next, wecheck whether the number of subsets can be increased byinvoking the new subset construction module. We continuein this manner by updating S and DC-supplier assignmentsthrough a series of operations until the cost stops improving.In that case, we return the best solution and its associated costZ∗ as the solution to the overall algorithm.

Display 2 Procedure LS (K′, J ′, S, Z∗)1: S = SCH ; Set Z∗ = ZCH ; Set Z0 = ∞2: while Z∗ < Z0 do3: Z0 = Z∗4: if |S| = 1 then5: Perform NewSet(S, Z∗)6: else7: Perform Move[S, Z∗]8: Perform Exchange[S, Z∗]9: Perform NewSet[S, Z∗]

10: end if11: end while12: return Z∗, S, K′, and J ′

Note that the CL-based approach to determine theDC-supplier pairs is crucial to calculating the objectivevalue associated with a solution, and hence, evaluating thegoodness-of-fit of a solution. Whenever there is a change inthe contents of the subsets or assignments between DC andsuppliers, the best DC-supplier pairs are determined via solv-ing an integrated continuous location and inventory problemfor each subset and supplier combination, described next.

4.4. Continuous Location Based Assignment

We utilize the results in [48] when determining the assign-ments of subsets of retailers, S ∈ S, to a DC and a supplier.Recall that Üster et al. [48] consider an integrated continu-ous location and inventory problem (ICLIP) in a three stagesetting where transportation costs are a function of distance.The solution approach in their article, the Perturb Algorithm,determines the location of the central DC, given the locationsof the supplier and the retailers as well as the inventory policyparameters of the central DC and the retailers.

In our local search algorithm, given a subset of retailerswith known locations, we determine the best central DC loca-tion and supplier to serve this subset by solving an ICLIP foreach potential supplier that has enough capacity to serve thissubset. In other words, by solving the ICLIP at most |K|times, we obtain a continuous DC location associated witheach candidate supplier as well as the corresponding cost ofthis particular assignment. We pick the lowest cost solutionto determine which DC and supplier pair should be assignedto this subset of retailers. However, since we are restricted tocandidate DC locations, rather than a continuous DC loca-tion, we determine the three best candidate DC locations thatare closest to the continuous DC location with the lowest cost.We evaluate each one of these candidate DC locations andtheir associated supplier by estimating the cost of assign-ing it to the current subset of retailers. In other words, wecompare the cost of three DC-supplier pairs for the currentsubset of retailers. We assign the subset of retailers to theone with the lowest total cost. Before we move on to a newsubset of retailers, we update the remaining capacity of theselected supplier. We continue in this manner until all subsetsare served by a DC-supplier pair. We invoke this procedure todetermine the DC-supplier assignments whenever a subset’scontent is changed via move or exchange operations or a newsubset is created.

4.5. Move Neighborhood Search (Move[S, Z∗])Move neighborhood search modifies the content of a subset

S ∈ S by moving a retailer in S to another subset in the set ofsubsets S. In the beginning of the search, we initialize the bestcost Z∗ with the best cost obtained so far in the overall pro-cedure. Starting with a randomly selected subset S1 ∈ S, we



move each retailer i in S1 to another randomly selected subsetS2 ∈ S, S1 = S2. After each move, we calculate the objectivevalue of the new solution, Zmove. If this move improves Z∗,i.e., Zmove < Z∗, we modify S to include the updated S1 andS2. We update the value of Z∗ with Zmove, and start the moveneighborhood search over, taking into account the modifiedS. For this purpose, the move neighborhood searches for the“first-best-move.” We continue in this manner until we canno longer improve Z∗ by moving a retailer from S1 to anyother subset S2 ∈ S. At that point, we return Z∗ and the asso-ciated S as the new solution. We outline the pseudo-code ofthe move procedure in Display 3.

Display 3 Procedure Move[S, Z∗]1: Set Z∗ = Z(J ′, K′, S), ZMove = ∞2: repeat3: Select S1 ∈ S such that |S1| > 14: Select S2 ∈ S such that S1 = S2

5: for all i ∈ S1 do6: S1 = S1 \ {i}. S2 = S2 ∪ {i}. Update S7: Calculate ZMove(J ′, K′, S)

8: if ZMove < Z∗ then9: Z∗ = ZMove

10: return Z∗ and S11: end if12: end for13: until Z∗ < ZMove

14: return Z∗ and S

4.6. Exchange Neighborhood Search(Exchange[S, Z∗])

The exchange neighborhood search modifies the contentsof two subsets by swapping retailers between these twosubsets. Display 4 provides the pseudo-code of the algorithm.

In the beginning of the search, we initialize the best cost Z∗with the best cost Z(J ′, K′, S) obtained so far in the overallprocedure. We also initialize the cost of exchange neighbor-hood search, Zexchange, as infinity. For each pair of subsetsof S, we swap contents of these subsets one retailer at atime. More specifically, starting with two subsets S1 ∈ S andS2 ∈ S such that S1 = S2, we swap a retailer i in S1 with aretailer j in subset S2 ∈ S. After this exchange, the subsetsS1 and S2 are modified to S

temp1 and S

temp2 . Then, the set of

subsets, S, is modified to Stemp by eliminating the originalsubsets S1 and S2 and including the new subsets S

temp1 and

Stemp2 . To evaluate the new solution, we calculate its objec-

tive value Zexchange with the new Stemp. Similar to the stepin move neighborhood search, before calculating Zexchange,we need to determine the DC-supplier pairs that will servethe newly constructed subsets S

temp1 and S

temp2 , as well as the

other subsets in Stemp, via solving the integrated continuouslocation inventory problem, i.e., CL-based approach, for eachsubset and supplier combination. If this exchange improvesZ∗, we update the value of Z∗. We also update S with themodified Stemp and start the exchange neighborhood searchover using the modified S. For this purpose, the exchangeneighborhood searches for the “first-best-exchange.” If theexchange does not result in an improved objective value, analternate pair of retailers are swapped between subsets S1 andS2, as described in Step 5 of the exchange procedure given inDisplay 4. We continue in this manner until we can no longerimprove Z∗ and we return Z∗ and the associated S as the newsolution.

Display 4 Procedure Exchange[S, Z∗]1: Set Z∗ = Z(J ′, K′, S), ZExchange = ∞2: repeat3: for all S1 ∈ S do4: for all S2 ∈ S such that S1 = S2 do5: for all i ∈ S1 and j ∈ S2 do6: Stemp = S7: Stemp = S temp \ {S1, S2}8: S

temp1 = S1 \ {i}, S

temp1 = S1 ∪ {j}

9: Stemp2 = S2 \ {j}, S

temp2 = S2 ∪ {i}

10: Stemp = Stemp ∪ {S temp1 , S temp

2 }11: ZExchange = Z(J ′, K′, Stemp)

12: if ZExchange < Z∗ then13: Z∗ = ZExchange

14: S = Stemp

15: Go to Step 316: end if17: end for18: end for19: end for20: until Z∗ < ZExchange

21: return Z∗.

4.7. New Set Construction (NewSet[S, Z∗])We use a new set construction procedure to modify the con-

tents of S. In particular, via new set construction, we increasethe number of subsets in S by one. While generating this newsubset, we utilize the distances among retailers.

First, we initialize smallestSize to the smallest |S| suchthat S ∈ S. Let S∗ be the new set to be constructed. Let S1 bethe first set in S. We first find the two most distant retailers,i∗1 and i∗2 , in S1. We randomly pick and remove one of theretailers, say i∗1 , from S1, and add it to the new set S∗. Next,we loop through all the subsets S ∈ S \ S1 to determine theretailers closest to retailer i∗1 . We include these retailers in thenew subset S∗ and eliminate them from their original subsets



until the size of S∗ is at least as large as smallestSize. Then,we add S∗ to S. Finally, we determine the objective valueZnewSet(S). Display 5 summarizes this procedure.

Display 5 Procedure NewSet[S]1: Set smallestSize = minS∈S{|S|}; S∗ = ∅2: Pick S1 ∈ S randomly. (i∗1 , i∗2 ) = arg maxi1∈S1,i2∈S1{di1,i2}3: S1 = S1 \ {i∗1 }, S∗ = S∗ ∪ {i∗1 }4: while |S∗| < smallestSize do5: for all S ∈ S \ S1 do6: i∗ = arg mini∈S{di∗1 ,i}7: S = S \ {i∗}, S∗ = S∗ ∪ {i∗}8: end for9: end while

10: S = S ∪ S∗11: return ZnewSet(S)

5. SIMULATED ANNEALING ALGORITHM

Simulated annealing (SA), first proposed by Kirkpatricket al. [17], is one of the most well-developed and widely usediterative techniques for solving optimization problems [34].The books by [13, 18], and [42] provide an introduction toboth the theoretical and practical aspects of annealing aswell as example applications including traveling salesman,graph partitioning, quadratic assignment, matching, lineararrangement, and scheduling.

A strong feature of the SA metaheuristic is that it is botheffective and robust. Although notoriously difficult to ana-lyze, the usual justification for its empirical success [29] isthat it avoids getting stuck at a local optima. Regardless of thechoice of initial solution, it produces high quality solutions. Itis also relatively easy to implement. Due to these advantages,we devise an SA algorithm for our problem.

The basic requirements of the SA algorithm are a neighbor-hood structure on the set of feasible solutions and a numberof parameters which govern the acceptance or rejection ofnew solutions generated during the search. SA is a ran-domized search method that tries to improve a solution bywalking randomly in the space of possible solutions andgradually adjusting a parameter called “temperature.” Thesequence of temperatures and the number of iterations forwhich they are maintained is called the annealing sched-ule. The quality of the solution is very sensitive to both ofthese factors. In our implementation, we experimented exten-sively to find an effective combination of temperatures andtemperature durations. Next, we describe the details of ourimplementation.

The SA algorithm starts with an initial solution obtainedfrom the construction heuristic (see Section 4.2) andexamines a local neighborhood for better solutions. As in the

improvement heuristics, the SA algorithm operates over thesubsets of retailers. Hence, one of the most crucial inputsis the initial set of subsets of retailers, S0. To assign thesubsets of retailers to DC-plant pairs, again, the CL-basedapproach is utilized. Display 6 depicts the simulated anneal-ing algorithm in detail. In addition to the initial solution, theSA algorithm requires an initial temperature, T0; a coolingrate, α; a time constant, β; the total allowed time for theannealing process, MaxTime; and, finally, the time until thenext parameter update, M [34, pages 53–55].

Display 6 Procedure SA(S0, L, T0, α, β, M , MaxTime)1: T = T0

2: Scurrent = S0; Zcurrent = Z(S)

3: Sbest = Scurrent; Zbest = Zcurrent

4: Time = 05: while Time ≤ MaxTime do6: Call Metropolis(Scurrent, Zcurrent, Sbest, Zbest, T , M)

7: Time = Time + M

8: T = α · T ; M = β · M

9: end while10: return Sbest, Zbest

The core of the SA algorithm is the Metropolis proce-dure, presented in Display 7. The Metropolis procedure, afterreceiving the current solution {Scurrent, Zcurrent }, the temper-ature, T , and the number of metropolis loops, M , as inputs,simulates the annealing process at a given temperature T . Inthe Metropolis procedure, as in the improvement heuristics,we utilize exchange and move neighborhoods to define a newsolution within the local neighborhood, as well as new subsetconstruction procedure to create new retailer subsets. How-ever, in the Metropolis procedure, as opposed to the localsearch algorithm where we find the “first-best-solution” inthe neighborhoods, we consider solutions randomly gener-ated within exchange and move neighborhoods (as in lines 2-3in Display 7). Furthermore, instead of considering “Move,”“Exchange,” and “New Subset Construction” sequentially,we consider these routines simultaneously. The new solutionis the solution that provides the lowest cost among these threeprocedures. The acceptance of this new solution depends onhow the cost of this solution Znew compares to the cost of thecurrent solution Zcurrent. In particular, we compute the vari-ation �Z, �Z = Zcurrent − Znew, in the objective functionproduced by the new solution. If �Z ≤ 0, the Metropolisprocedure selects the new solution and replaces it with theexisting solution. Furthermore, if the new solution is betterthan the best solution obtained so far, we replace the best solu-tion and its cost with the new solution. On the other hand, if�Z > 0, the Metropolis procedure selects the new solution,



which is worse than the current solution, with a probabilityP(Random < e(− �Z

T)) where Random is a random number

from a uniform distribution. At high temperatures (T → ∞),the above probability approaches 1. On the contrary, the prob-ability converges to zero as T → 0 and the Metropolisprocedure becomes highly conservative in accepting non-improving solutions. As typical in simulated annealing, theprobability of accepting a non-improving move decreases andthe time spent within the Metropolis procedure increases asthe number of iterations progresses. These behaviors of theSA algorithm are controlled by decreasing the temperatureand increasing the Metropolis loop time after every call ofthe Metropolis procedure.

As mentioned before, the performance of the SA algo-rithm depends on the values of algorithmic parameters T0,α, β, M , MaxTime. To find the best experimental setting,we performed a 2k factorial analysis to determine how theseparameters influenced the SA performance. After identify-ing the best experimental setting, we ran the algorithm fornumerous instances to establish the effectiveness of the SA insolving the integrated PDSD with inventory considerations.The results of the parametric analysis and the performanceof SA are presented in Section 7.

Display 7 Procedure Metropolis(Scurrent, Zcurrent, Sbest,Zbest, T , M):

1: while M > 0 do2: Perform MoveRandom on Scurrent to obtain

{Smove, Zmove}3: Perform ExchangeRandom on Scurrent to obtain

{Sexchange, Zexchange}4: Perform NewSetConstruction on Scurrent to obtain

{SnewSet, ZnewSet}5: Znew = min{Zmove, Zexchange, ZnewSet}6: Snew = arg min{Zmove(Smove), Zexchange(Sexchange),

ZnewSet(SnewSet)}7: �Z = Zcurrent − Znew

8: if �Z ≤ 0 then9: Scurrent = Snew; Zcurrent = Znew

10: if Znew ≤ Zbest then11: Sbest = Snew; Zbest = Znew

12: end if13: else14: if Random < exp(−�Z

T) then

15: Scurrent = Snew; Zcurrent = Znew

16: end if17: end if18: M = (M − 1)

19: end while20: return Sbest, Zbest

6. BENCHMARK MODELS

In this section, we describe three benchmark modelsto demonstrate the value of integrated decision making(Section 6.1), inventory keeping at retailers and multiecheloninventory coordination (Section 6.2), and multiple suppliers(Section 6.3). The computational results based on these mod-els reveal insights into the design and inner dynamics of theproduction system later in Section 7.

6.1. BM1: Value of Integrated Location andInventory Model

First, we describe a benchmark model for comparingthe solution of the integrated location and inventory modeland evaluating the effectiveness of the integrated modelPDSD-INV. The benchmark model (BM1) follows the typicalsequential framework in the literature where location deci-sions precede inventory decisions (see also [7, 48]). In manyof the integrated location-inventory problems, as the overallproblem is challenging, the idea is to partition the integratedproblem into two subproblems that can be solvable usingknown, or rather easily implementable, solution methods.Specifically, in this study, we first solve a two-stage PDSDproblem (BM1-PDSD) without inventory considerations todetermine the supplier and DC locations, the assignments ofthe selected suppliers to the open DCs, and the assignmentsof the open DCs to the retailers. Then, given these locationand assignment decisions, we determine the inventory pol-icy parameters at each open DC and their assigned retailers(BM1-INV ).

6.1.1. Modeling BM1-PDSD

Ignoring inventory decisions initially, the problem PDSD-INV reduces to a two-level PDSD problem, BM1-PDSD,where a number of DCs and capacitated suppliers are to belocated with respect to the retailer locations while minimizingthe total cost in the system. The total cost includes the fixedcost of locating DCs and suppliers, as well as the transporta-tion costs from the selected suppliers to the open DCs andfrom the open DCs to the retailers. These transportation costsare unit-based transportation costs and ignore the impact ofinventory decisions.

Using the notation defined in Section 3, we formulate theBM1-PDSD as follows:

Min∑j∈J

fjXj +∑j∈J

gkZk +∑i∈I

∑j∈J

∑k∈K

uijkDiWijk

subject to∑j∈J

Yij = 1, ∀ i ∈ I. (18)



Yij ≤ Xj , ∀ i ∈ I and ∀ j ∈ J . (19)

Wijk ≤ Zk , ∀ i ∈ I, ∀ j ∈ J and ∀ k ∈ K. (20)

Wijk ≤ Xj , ∀ i ∈ I, ∀ j ∈ J and ∀ k ∈ K. (21)∑k∈K

Wijk = Yij , ∀ i ∈ I and ∀ j ∈ J . (22)

∑i∈I

∑j∈J

DiWijk ≤ CkZk , ∀ k ∈ K. (23)

Xj ∈ {0, 1}, Yij ∈ {0, 1}, ∀ i ∈ I and ∀ j ∈ J . (24)

Zk ∈ {0, 1}, Wijk ∈ {0, 1}, ∀ i ∈ I, ∀ j ∈ J and ∀ k ∈ K.(25)

In this formulation, the objective function minimizes thesum of the fixed cost of locating the DCs, the fixed cost oflocating the suppliers, and the total transportation costs fromthe selected suppliers to the retailers through open DCs. Inthe objective function, uijk denotes the per-unit transporta-tion cost from supplier k to retailer i through DC j , for alli ∈ I, j ∈ J , k ∈ K. Due to lack of better information, weestimate uijk as θ ∗ (rR

ij + rDCjk ) for all i ∈ I, j ∈ J , k ∈ K,

where θ is a constant coefficient used to convert per-mileper shipment transportation cost (represented by [rR

ij + rDCjk ])

into a per-unit per-mile transportation cost. In explanation, aθ value implies a shipment size of 1/θ units. In our numericalexperiments, we selected 0.001, 0.005, 0.01, and 0.05 as thevalues of θ , i.e., lot sizes of 1000, 200, 100, and 20 units,respectively.

Constraints (18) ensure that each retailer is served byexactly one DC. Constraints (19), (20), and (21) are theassignment constraints stating that only selected facilities(DCs and suppliers) are utilized. Constraints (22) stipulatethat if a link between a customer and a DC exists, it can beserved by only one selected supplier. In other words, theseconstraints ensure that a selected DC can be assigned to onlyone selected supplier. Constraints (23) are the capacity con-straints for the suppliers. Finally, constraints (24) and (25)are the standard integrality constraints.

We solve BM1-PDSD to optimality using CPLEX 12.0and determine the optimal values of Xj , Zk , and Wijk for alli ∈ I, j ∈ J , k ∈ K.

6.1.2. Modeling BM1-INV

Once Wijk is determined by solving BM1-PDSD for alli ∈ I, j ∈ J , k ∈ K, we use this information to form sub-sets of retailers that are served by a single DC and a singlesupplier. Then, for each such subset S, the inventory deci-sions of each retailer in that set and the DC serving that setare addressed by solving the corresponding SWMR problemusing the approach in [33].

Let S be the set of subsets of retailers. For S ∈ S, thereis a unique DC-supplier (jS , kS) pair that serves S such that

jS ∈ J and kS ∈ K. Then, we formulate the BM1-INV asfollows:

Min∑S∈S

{∑i∈S

(pR

i,jS+ rR

i,jSdR

i,jS

)T R

i

+(pDC

jS ,kS+ rDC

jS ,kSdDC

jS ,kS

)T DC

jS

+ KDCjS

T DCjS

+∑i∈S

KRi

T Ri

+∑i∈S

1

2HR

i,jSDiT

Ri

+∑i∈S

1

2HDC

i Di max{T R

i , T DCjS

}}

subject to

T DCjS

= 2νjS Tb and νjS∈ Z, ∀jS ∈ J , S ∈ S. (26)

T Ri = 2υi Tb and υi ∈ Z, ∀i ∈ S, S ∈ S. (27)

T DCjS

∈ R+ and T Ri ∈ R+, ∀i ∈ S, ∀jS ∈ J , S ∈ S.

(28)

This formulation can be decomposed for each subset S ∈S, and the inventory decisions for each retailer i ∈ S, andDC jS , jS ∈ J , S ∈ S can be obtained by solving a cor-responding SWMR lot-sizing problem using the approachin [33].

After the location, assignment, and inventory decisions aredetermined through solutions of the BM1-PDSD and BM1-INV, we evaluate the cost of the benchmark model using theobjective function of the original formulation PDSD-INV. Wecompare this cost with the cost of the PDSD-INV solved bythe heuristic approaches to determine the value of integrateddecision-making.

6.2. BM2: Value of Inventory at Retailers

This particular benchmark determines the value of inven-tory keeping at the retailers and the value of inven-tory coordination between retailers and their correspondingDCs. In many of the integrated location-inventory models[7, 24, 32, 36, 37], the inventory at the retailers is ignoredand the risk is pooled at the DCs. Different from this lineof work, we consider inventory replenishment decisions notonly at distribution centers (DCs), but also inventory deci-sions at retailers as well as the coordination among them.To evaluate the impact of incorporating these two complicat-ing factors, we devise the second benchmark model (BM2).In this benchmark model, we first determine the location andassignment decisions regarding suppliers and DCs while con-sidering the inventory only at DCs and ignoring the inventoryat retailers and its coordination with the corresponding DCs.Next, based on the location and assignment decisions, thecoordinated inventory policies at open DCs and retailers are



determined. More specifically, at the first stage, we considerthe following model BM2-S1:

Min∑j∈J

fjXj +∑k∈K

gkZk

+∑j∈J

∑k∈K

(pDC

jk + rDCjk dDC

jk

)Vjk

T DCj

+∑j∈J

KDC

j

T DCj

+ 1

2HDC

j

∑

i∈Ij

Di

T DC

j

Xj

subject to

(1), (2), (3), (4), (5), (6), (9), (10),

T DCj ≥ 0, ∀ j ∈ J . (29)

We solve BM2-S1 via a modification of the algorithms wedeveloped for PDSD-INV. For BM2-S1, the solution repre-sentation stays the same as PDSD-INV. However, the defi-nition of IT (j , k, S) does not include any retailer inventoryterms or any inventory coordination constraints. Hence, fora given set of retailers S, the solution of IT (j , k, S) is foundby solving a modified EOQ problem where all of the demandfrom S is collected at DC j that is served by supplier k. Morespecifically, the modified IT (j , k, S) is given as:

Min∑j∈J

KDC

j

T DCj

+ 1

2HDC

j

∑

i∈Ij

Di

T DC

j

where the optimal inventory decision at DC j is found as:

T DCj =

√√√√2(KDC

j + pDCjk + rDC

jk dDCjk

)HDC

j

∑i∈S Di

.

Furthermore, the modified cost cjkS of serving retailers in setS via DC j and supplier k is calculated as

cijS = gk + fj +√

2(KDC

j + pDCjk + rDC

jk dDCjk

)HDC

j

∑i∈S

Di .

After obtaining the location and assignment variables (X,Y, V, and Z), at the second stage of BM2, we invoke BM1-Invto obtain the coordinated inventory decisions both at the DCsand retailers. As in BM1, we capture the cost of BM2 using theobjective function of the original formulation PDSD-INV. Wecompare this cost with the cost of the PDSD-INV to deter-mine the value of inventories at the retailers as well as thevalue of coordination.

6.3. BM3: Value of Multiple Suppliers

The closest studies in the network design literature thatalso consider inventories at both DCs and retailers only focuson having a single aggregated supplier [45, 48]. However,in many real life cases of the production/design problem,we face selecting multiple suppliers. Wondering if the exist-ing aggregated supplier models can be efficiently addressthe three stage network design problem, we devise a thirdbenchmark model (BM3).

In BM3, assuming a single (aggregated) supplier, we firstdecide on the locations of the DCs as well as the coordi-nated inventory policies at the DCs and retailers. Next, giventhe open DCs and their corresponding sets of retailers alongwith the inventory decisions, we determine the locations ofcapacitated suppliers and their assignment to the selectedDCs.

More specifically, in the first stage, we solve the followingmodel BM3-S1:

Min∑j∈J

fjXj +∑j∈J

KDC

j

T DCj

+∑i∈Ij

KRi

T Ri

+∑i∈Ij

1

2HR

ij DiTRi

+∑i∈Ij

1

2HDC

j Di max{T R

i , T DCj

} Xj

+∑i∈I

∑j∈J

(pR

ij + rRij d

Rij

)Yij

T Ri

subject to

(1), (3), (7), (8), (9).

Since the algorithms for PDSD-Inv are capable of handlingmultiple suppliers, we utilize these algorithms while using asingle (dummy) supplier.

In the second stage, given the DC location, DC-to-retailerassignment, and inventory decisions (X, Y, ˆTDC, and TR), wesolve for Z and V using the model BM3-S2 below:

Min∑k∈K

gkZk +∑j∈J

∑k∈K

(pDC

jk + rDCjk dDC

jk

)Vjk

T DCj

subject to

(2), (4), (10),

Vjk ≤ Xj , ∀ j ∈ J , k ∈ K. (30)∑i∈I

∑j∈J

DiYijVjk ≤ CkZk , ∀ k ∈ K. (31)



Table 2. Parameter values for numerical experiments.

Di U[350, 1400] hRi U[5, 10]

fj U[100,000, 150,000] hDCj U[1, 4]

gk U[200,000, 300,000] pRij U[100, 400]

Ck U[0.5, 1.0] *∑

i∈I Di pDCij U[425, 1700]

KRi U[75, 300] rR

ij U[0.75, 3]KDC

j U[400, 1600] rDCjk U[1, 4]

BM3-S2 is similar to a capacitated facility location problemand hence, we solve BM3-S2 to optimality using CPLEX 12.0and determine the optimal values of Z and V. Note that dueto capacity restriction at the suppliers and single sourcingassumption, a particular (X, Y) may result in an infeasibleversion of BM3-S2. In this case, we go back to the first stageand modify the sets of retailers until they do result in a feasibleinput for BM3-S2.

After obtaining the location and assignment variables (X,Y, V, and Z), as in other benchmark models, we utilize theobjective function of the original formulation PDSD-INV toestimate the cost of BM3. We compare this cost with thecost of the PDSD-INV to determine the value of multiplesuppliers.

7. COMPUTATIONAL STUDY

The purpose of our computational study is two-fold.First, we conduct numerical studies on a wide range of testinstances to examine the performance of our solution algo-rithms, local search and simulated annealing, by utilizingthe construction heuristic solutions as benchmark. Second,we compare the integrated PDSD-INV approach and threebenchmark models sequential approach, i.e., solving BM1,BM2, and BM3, under varying input parameter values todetermine the value of integrated decision-making. All of thenumerical results are obtained with algorithms implementedusing C++ and run on a Pentium IV 3.2 Ghz machine with1 GB memory.

7.1. Experimental Data Generation

To test the solution approaches, we generate eight differ-ent instance classes where the number of retailers, numberof potential DCs, and the number of potential suppliers havetwo alternatives. Each class consists of 50 or 100 retailers, 20or 40 potential DCs, and 5 or 10 potential suppliers. In eachclass, we have 50 problem instances, generated randomlyusing the uniform distributions given in Table 2, resultingin a total of 400 problem instances. The data is obtainedusing similar parameter ranges as in [48]. We randomly gen-erate retailer, potential DC and potential supplier locationcoordinates in a square of size 150 and calculate the dis-tances using the Euclidean norm. To generate the supplier

capacities, we use the average demand in the data instance.In particular, we randomly generate the supplier capacityusing the uniform distribution U[Lcap, Ucap] for each sup-

plier k ∈ K where Lcap = 25∗∑

i∈I Di

|I| ∗ |I|50 = 0.5∗∑

i∈I Di

and Ucap = 50 ∗∑

i∈I Di

|I||I|50 = ∑

i∈I Di . With this capacityrestriction, on average, each supplier is capable of serving arandom number of retailers between 25 and 50 if there are 50retailers, and between 50 and 100 if there are 100 retailers inthe system.

7.2. Performance of LS and SA Algorithms

In this section, we report computational results illustratingthe performance and duration of the improvement heuristics.We define the performance of an improvement heuristic asimprovement over the construction heuristic (IOC), and wemeasure it as

IOC(%) = Z(C) − Z(H)

Z(C)× 100, (32)

where Z(C) is the objective value of the construction heuris-tic, and Z(H) is the objective value of the heuristic solutionapproach, i.e., local search or simulated annealing.

Before presenting the computational results regarding theperformance of SA, we discuss the experimental setting uti-lized in SA runs. To provide best SA parameters in termsof initial temperature T0, cooling rate α, time constant β,metropolis loop time M , and maximum allowed total timeMaxTime, we conducted an experimental analysis of the per-formance of the SA on varying problems and network sizes.Specifically, we selected three instances in classes 1, 4, 5,and 8. We ran these instances with 32 different SA settingswhere each SA parameter has a low and a high value, i.e. 25

factorial design, as given in Table 3.The value of temperature should be comparable to the over-

all cost of the problem since it has an impact on the acceptanceprobability. For this purpose, we selected T0 values to varybetween 10 and 50% of the initial construction cost, Z(C).The alternative values for α and β were selected followingthe commonly used values in SA applications. Finally, sincethe Metropolis procedure operates on subsets of retailers,metropolis loop time M and maximum allowed total timeMaxTime were selected as a fraction of total number of retail-ers in the network. With this experiment, we observed thathighest average IOC for a total of 12 instances was achieved

Table 3. Bounds on SA parameters for the performance analysis.

Parameters T0 α β M MaxTime

Low value 0.1 ∗ Z(C) 0.8 1 0.02 ∗ |I| 0.2 ∗ |I|High value 0.5 ∗ Z(C) 0.9 2 0.1 ∗ |I| 0.4 ∗ |I|



Table 4. Comparison of solution approaches.

IOC (%) Runtime (Secs.)

Class (|I|, |J |, |K|) LS SA CH LS SA

Class 1 (50, 20, 5) Min 0.00 0.00 0.00 0.28 6.53Ave 23.19 22.00 0.00 44.50 8.44Max 52.04 59.00 0.02 88.11 10.14

Class 2 (50, 20, 10) Min 0.00 0.00 0.00 0.86 13.02Ave 22.73 20.24 0.00 94.60 17.19Max 50.93 48.78 0.02 151.53 20.64

Class 3 (50, 40, 5) Min 2.82 7.06 0.00 23.08 6.45Ave 25.92 23.63 0.01 47.58 7.59Max 46.62 51.32 0.02 84.47 9.08

Class 4 (50, 40, 10) Min 0.00 0.05 0.00 0.59 13.30Ave 22.96 20.71 0.01 92.29 15.70Max 50.47 51.27 0.02 229.92 18.39

Class 5 (100, 20, 5) Min 0.00 0.00 0.00 0.92 27.91Ave 17.95 22.43 0.01 223.00 44.48Max 37.98 42.37 0.02 517.08 59.66

Class 6 (100, 20, 10) Min 0.00 0.56 0.00 2.11 58.41Ave 16.20 18.50 0.01 242.13 94.60Max 38.79 39.44 0.02 494.33 122.06

Class 7 (100, 40, 5) Min 0.00 0.00 0.00 0.94 24.11Ave 16.56 18.64 0.01 203.80 41.43Max 33.48 39.46 0.02 487.25 58.77

Class 8 (100, 40, 10) Min 0.00 0.00 0.00 1.63 54.45Ave 17.77 17.99 0.01 374.57 89.73Max 36.71 40.90 0.02 637.61 118.22

within the lowest possible duration using T0 = 0.5 ∗ Z(C),α = 0.8, β = 2, M = 0.1 ∗ |I|, and MaxTime = 0.2 ∗ |I|.We utilized this parametric setting in all of the subsequentSA runs.

In Table 4, we present the minimum, average, and maxi-mum percentage IOC for both local search (LS) and simulatedannealing (SA) algorithms. We also present the correspond-ing minimum, average, and maximum runtimes for thesealgorithms. It is clear that the LS and SA algorithms per-form very similarly in terms of solution quality since theimprovements they provide over the construction heuristic(CH) solutions are very close. It appears that the performanceof SA can be slightly better for larger instances included inClasses 5-8. On the other hand, the SA algorithm is signifi-cantly more efficient in terms of solution times, as observedin the last two columns of Table 4.

We also analyze the results in terms of which cost compo-nents contribute the most to the improvement over CH. First,in Fig. 2, we present the distribution of the average cost com-ponents over all of the instances and data classes for differentsolution methods. We group the cost components under sixdifferent categories: the fixed cost of selecting suppliers (Sup-Fix); the fixed cost of opening DCs (DCFix); the transporta-tion cost from selected suppliers to open DCs (SupDC-Tr);the transportation cost from open DCs to the assigned retailers(DCRet-Tr); the total inventory costs at open DCs (DCInv);and the total inventory costs at the retailers (RetInv). From

this figure, we see that the fixed cost of supplier selection andinventory costs at the retailers constitute, on average, 63%of the total costs, regardless of the solution method. This isimportant as these components have been ignored by othermodels in the integrated location-inventory literature. Addi-tionally, Table 5 demonstrates the contributions of differentcosts to the improvement obtained by LS and SA over CH.For each data class, the contribution from a particular cost C

(SupFix, DCFix, SupDC-Tr, DCRet-Tr,DCInv, or RetInv) iscalculated as

% -Contribution of C

= Average of

[CC − CH

Z(C) − Z(H)

]over all data instances,

where CC is obtained using CH and CH is the value of thesame cost component obtained using one of the heuristics.Analyzing the results over all the data sets, using either LSor SA, we observe that the most important contributions tothe improvement come from the fixed cost of selecting sup-pliers (55.5% with LS and 49.5% with SA), the fixed costof opening DCs (18.9% with LS and 19.7% with SA), thetransportation cost from open DCs to the assigned retailers(13.7% with LS and 20.2% with SA), and the total inventorycosts at the retailers (10.8% with LS and 10.5% with SA).

Note that the transportation cost from open DCs to theassigned retailers are influenced by the inventory decisionsat the retailers. Hence, inventory consideration at the retailersimplicitly impacts more than 20% of the improvement overCH. We conclude that the solution approaches (LS and SA)are very effective in solving the integrated location-inventoryproblem and also shed light on important cost components ofthe problem.

7.3. Value of Integrated Decision-Making

In this section, we present computational results to demon-strate the value of integrated decision-making over sequential

Figure 2. Cost components of the total cost given by differentsolution approaches.



Table 5. Comparison of Contribution of Cost Component to the IOC.

DS1 DS2 DS3 DS4 DS5 DS6 DS7 DS8

LS IOC(%) 23.19 22.73 25.92 22.96 17.95 16.20 16.56 17.77SupFix 64.70 61.16 61.98 32.40 60.61 49.84 64.19 49.75DCFix 19.89 23.73 20.95 15.25 13.30 18.16 14.14 26.31SupDC-Tr −0.28 −0.16 −0.18 1.11 −0.71 −0.45 −0.51 1.22DCRet-Tr 6.38 6.91 7.87 28.51 11.55 12.02 13.70 22.58DCInv 1.32 1.18 1.82 0.60 2.79 4.81 −0.75 −4.11RetInv 7.99 7.17 7.55 22.13 12.46 15.63 9.24 4.25Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00

SA IOC(%) 22.00 20.24 23.68 20.71 22.43 18.50 18.64 17.99SupFix 53.14 55.95 58.67 21.98 48.08 52.29 55.21 50.84DCFix 17.04 26.13 20.75 20.58 18.69 18.46 14.47 21.75SupDC-Tr 0.20 0.43 0.16 1.58 0.29 0.84 2.50 2.17DCRet-Tr 16.47 6.51 7.54 29.01 13.68 16.20 47.85 24.32DCInv 9.02 2.82 3.10 0.41 3.75 0.48 −22.57 −4.91RetInv 4.13 8.16 9.77 26.44 15.52 11.73 2.54 5.83Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00

decision-making in terms of location and inventory decisions.We measure the value of integrated decision-making as

Percentage gain over BM1(%) = Z(BM1) − Z(H)

Z(BM1)× 100,

(33)

where Z(BM1) is the objective value of PDSD-INV evalu-ated with the decision variables obtained through the solutionof BM1, and Z(H) is the objective value of the appropriatesolution approach for PDSD-INV. Since the transportationcosts in BM1 are adjusted using a θ value, there is an asso-ciated Z(BM1) for each value of θ . First, as a base caseanalysis, we choose a value of θ = 0.05. Next, we generalizethis analysis to investigate the impact of problem parame-ter variations on the value of integrated decision making. Inparticular, we conduct four different types of analysis:

1. Impact of θ : We vary θ as {0.001, 0.005, 0.01, 0.05}.2. Impact of fixed cost: We reduce the fixed cost of

facilities to 10, 20, 40, or 80% of the value that issuggested by the original data.

3. Impact of demand: High and low values of demandat the retailers are tested to investigate the robustnessof results for varying demand rates.

4. Impact of holding cost: High and low values of hold-ing costs at the DCs and retailers are tested to inves-tigate the robustness of results to currency exchangerisks in a global PDSD.

We present the results of the base case comparison of thesolution approaches with BM1 in Table 6. Table 6 has threeparts summarizing the comparisons of BM1 with CH, LS,and SA, respectively. The first row of each part in Table 6presents the average gains due to integrated decision-making

over BM1 using the CH, LS, and SA algorithms, respec-tively. Other rows present the percentage contributions ofeach cost component to the gains obtained from integrateddecision making. Note that some entries are positive, indi-cating that cost component contributed to the savings, that isCBM1 > CH . However, there are some entries that are nega-tive, indicating that the BM1 outperforms the rival heuristic.Overall, all of the cost percentage contributions add to 100%.

Analysis of the results in Table 6 reveals three importantresults. Firstly, all of the heuristics outperform the BM1 withaverage gains ranging from 39 to 63%. As the size of thenetwork in consideration grows, the gains from integrateddecision making also increase. Secondly, as expected, theaverage gains with both the LS and SA algorithms are bet-ter than the gains with CH. Finally, the largest component ofthe gains comes from the savings due to supplier selectionand opening DCs. The network designs by the integratedmodel for all of the data sets outperform the network designsby BM1. Even though this resulted in slightly higher inven-tory costs at the retailers and transportation costs from openDCs to respective retailers, this increase is dominated by thesavings from the network design.

7.3.1. Impact of θ

As explained before, the transportation costs in BM1 areadjusted using a θ value. Hence, for each value of θ , thereis a corresponding Z(BM). In this experiment, we chooseθ to vary as 0.001, 0.005, 0.01, 0.05 to reflect lot sizes of1000, 200, 100, and 20 units in BM1. In Table 11, we reportthe duration of each BM1 as well as the minimum, average,and maximum gains due to integrated decision making overeach BM1 using the CH, LS, and SA algorithms, respectively.Note that BM1 with θ = 0.05 is the same as the base casepresented above. We now provide additional analysis.



Table 6. Average gains (%) with the integrated-decision making in comparison with the base-case benchmark BM1.


CH Ave.Gains 39.01 42.62 39.60 41.03 53.26 53.90 51.93 54.82SupFix 22.3 41.0 19.1 45.8 35.9 49.8 25.5 41.5DCFix 82.1 63.0 84.3 79.0 66.5 54.5 78.1 63.9SupDC-Tr 2.7 2.3 2.6 2.2 2.6 2.2 2.3 2.1DCRet-Tr −4.0 −3.5 −3.6 −10.4 −3.4 −3.6 −3.5 −3.8DCInv 1.0 0.2 0.9 −3.2 1.2 0.4 1.0 −0.3RetInv −4.0 −3.0 −3.3 −13.3 −2.8 −3.4 −3.4 −3.4

LS Ave.Gains 53.46 56.24 55.36 55.08 61.75 61.39 60.03 62.93SupFix 34.5 46.6 31.1 42.3 39.4 49.9 30.2 0.43DCFix 64.6 52.9 67.1 58.4 59.2 50.3 68.2 0.59SupDC-Tr 1.9 1.6 1.8 1.6 2.1 1.9 1.9 0.02DCRet-Tr −1.2 −1.0 −0.8 −1.3 −1.4 −1.8 −1.2 −0.02DCInv 1.1 0.6 1.1 0.3 1.6 1.0 1.4 0.00RetInv −0.8 −0.7 −0.4 −1.3 −1.0 −1.3 −0.5 −0.02

SA Ave.Gains 52.85 54.76 54.05 53.75 63.90 62.50 61.04 63.02SupFix 32.2 44.3 29.4 40.5 37.9 50.0 30.2 42.5DCFix 65.1 54.6 68.0 59.8 58.6 49.5 68.2 58.4SupDC-Tr 2.0 1.8 1.9 1.7 2.2 2.0 1.9 1.8DCRet-Tr −0.7 −1.2 −0.8 −1.4 −0.6 −1.5 −1.2 −1.9DCInv 1.5 0.8 1.4 0.3 1.7 1.0 1.4 0.4RetInv −0.1 −0.4 0.0 −1.0 0.2 −0.9 −0.5 −1.2

All of the solution procedures, compared to any BM1 forvarying θ , perform well. The lowest gains are obtained whenθ is set to 0.001. Even then, average gains range from 3.83to 13.49% for CH; from 16.88 to 22.02% for LS; and from16.82 to 26.49% for SA. The maximum gains are frequentlyin the 40% range with LS and SA when compared to BM1with θ = 0.001. As the transportation rate θ increases, i.e., lotsize per trip assumed by BM1 decreases, the average gainsincrease for all of the solution procedures. For BM1 withθ = 0.05, the base case, the average gains range from 39.01to 54.82% for CH; from 53.46 to 62.93% for LS; and from52.85 to 63.02% for SA. As before, for any given θ , theaverage gains with both the LS and SA algorithms are bet-ter than the gains with CH. In Table 11, we also observethat as the data sets become larger, i.e., as the number ofcustomers, DCs, or plants increases, the average gains aretypically higher. These results underscore the importance ofintegrated decision making.

Similar to the base case analysis, most of the gains areobtained through better selection of suppliers and opening theright number of the DCs in the right places. The BM1 solutionapproaches unnecessarily select a higher number of suppli-ers and locate a higher number of DCs. Even though havingmore suppliers and DCs help offset the cost of inventories andtransportation (that is, the total inventory costs are lower withthe BM1 models), it does not pay off in terms of fixed facil-ity costs (that is, the total facility location/selection costs aremuch higher with the BM models). Hence, significant gainsare possible with the integrated model.

One final observation from Table 11 is regarding the per-formance of the BM1 models. The runtime of each BM1 is

sensitive to the levels of parameters. BM1 with θ = 0.001takes the longest (with average runtime ranging from 48.11to 1561.24 seconds), and as θ increases, the runtime of BM1decreases dramatically.

7.3.2. Lower Fixed Costs

As reported above, one of the main differences betweenthe solution approaches (CH, LS, and SA) for the integratedPDSD-INV model and the varying BM1 models is that fewerfacilities (suppliers and DCs) are selected with CH, LS, andSA. For instance, while the average number of selected facil-ities with PDSD-INV is around 3 to 5, the number of selectedfacilities with BM1 models is around 12 to 14. Coupled withthe high fixed facility location costs, this outcome results in ahigher cost difference in the solutions of the two models. Forthis purpose, we experiment with lower fixed costs to see howfixed costs impact the value of integrated decision making.

We generate test instances where the fixed costs of facili-ties (plants and DCs) are changed to 10, 20, 40, or 80% of thevalue that is suggested by the original data of Classes 2 (smallinstances) and 6 (large instances) to test the performance ofthe solution algorithms. Using these lower fixed facility costs,we first test how the LS and SA perform when compared toCH. Next, we measure the gain from the integrated decision-making by comparing the solution algorithms to BM1 modelswith varying θ values.

In Table 12, in the dark shaded part of the table, we presentminimum, average, and maximum percentage improvementover CH for both LS and SA algorithms. As in Section7.2, the IOC(%) is calculated using (32). For the lowest



fixed cost experiment, i.e., using 10% of the original fixedcosts, the improvements of LS and SA over the construc-tion heuristic are the lowest. For data set 2, the averageimprovements are 6.62 and 9.08% with the LS and SA,respectively, whereas they are only 2.10 and 5.90% for dataset 6. As fixed cost percentage increases, average improve-ments increase. Using 80% of the original fixed costs, fordata set 2, the average improvements with the LS and SAalgorithms are 16.32 and 21.49%, respectively. For this case,the maximum improvements approach 48%, reaching closeto the performance using the original data. Similarly, for dataset 6, the average improvements with the LS and SA algo-rithms are 17.35 and 17.47%, and the maximum improvementgoes up to almost 40%. These results can be explained bythe emphasis that the CH puts on the choice of locations.Recall that the CH first selects locations, then determinesassignments. With increased fixed costs, a poor choice oflocations by the CH leads to higher objective values, thusproviding more opportunity for LS and SA to provide bettersolutions.

Even though we did not explicitly report the runtime of theheuristics, as before, the LS is the most time intensive heuris-tic with the average duration ranging from 45 seconds to 4minutes. However, the SA improves this performance with anaverage runtime ranging from 20 seconds to 1 minute. Thesedurations are reasonable considering the problem sizes inClasses 2 and 6.

In the rest of Table 12, we present how different benchmarkmodels with different transportation rates perform againstsolution approaches with lower location costs. As in Section7.3, the value of integrated decision-making is measuredusing (33). The main observation from Table 12 regardingthe value of integrated decision-making is that the gains fromall of the solution approaches over the BM1 models are lessthan the gains obtained with the original data. In the lowestfixed cost case, i.e., using 10% of the original fixed costs,the average gains over all of the solution approaches rangebetween 9.51 and 31.36%. This is mainly due to the gainsthat come from facility location decisions, and as the costof locating/selecting facilities goes down, the gains are notas high as the base case analysis. However, these gains arestill significant, especially considering their potential mone-tary value. As the fixed cost modification rate increases, i.e.,the data instances approach the original data, the gains overthe benchmark models increase. In terms of the runtimes, wenote that the runtime of the benchmark models are signifi-cantly faster using the data with lower fixed facility locationcosts. Using the lowest fixed costs, the average runtime of thebenchmark models ranges from 1.48 to 7.25 seconds. How-ever, when using 80% of the original fixed costs, the averageruntimes with BM1 θ = 0.001 are 305.34 and 515.42 secondsfor Classes 2 and 6, respectively. Using this fixed cost set-ting, as θ increases, the average runtime for BM1 models

decreases. With BM1 θ = 0.05, the runtimes are 2.34 and4.51 seconds for Classes 2 and 6, respectively.

7.3.3. Impact of Demand

In this experiment, we test the impact of low and highdemand on the integrated location-inventory model. For thispurpose, in every data class, each data instance is generatedassuming a low (350) or a high (1400) demand value for eachretailer, i.e., for all i ∈ I, Di = 350 for the low demand caseand Di = 1400 for the high demand case. All of the otherproblem parameters are kept as the same as in the base caseanalysis.

The first half of Table 13 presents the minimum, average,and maximum gains within each data class, obtained withCH, LS, and SA compared to BM1 with θ = 0.05 for lowand high demand, respectively. For low demand, the averagegains range from 54.14 (obtained with CH in DS4) to 73.73%(obtained with LS in DS8); whereas, for high demand, theaverage gains range from 33.32 (obtained with CH in DS1)to 62.23% (obtained with LS in DS8). As before, LS and SAperform better than CH. In fact, for these experiments, theLS algorithm slightly outperforms SA.

In general, the variations of BM1 perform worse for lowdemand cases than those for high demand cases. The dif-ference in average gains can be as high as 27.95% with CH,12.86% with LS, and 14.98% with SA (all of them obtained inDS1). The main reason is that the inventory cost componentswhere these benchmark models perform good have a lowerimpact when demand is low. Based on the cost componentanalysis, even though these benchmark models outperformthe rival heuristics in retailer and DC inventory costs; withlow demand, these costs are not large enough to offset theinferior performance of the benchmark models at other costcomponents. Hence, the integrated decision making is morevaluable under low demand situations.

7.3.4. Impact of Holding Cost

Changes in the currency exchange rates as well inter-est rates are one of the main issues of the global supplychains. Assuming that the holding costs are proportional tothe changes in these rates, it is reasonable to expect that theinventory holding cost rates can fluctuate. In this experiment,to capture this fluctuation effect, we test the impact of lowand high inventory holding costs on the integrated location-inventory model. For this purpose, to capture the lower endof the spectrum, in each data instance hR

i = 5 and hDCj = 1

for all i ∈ I and j ∈ J . On the other hand, at the higher endof the spectrum, in each data instance hR

i = 10 and hDCj = 4

for all i ∈ I and j ∈ J . All of the other problem parametersare kept as the same as in the base case analysis.



The second half of Table 13 presents the minimum, aver-age, and maximum gains for low and high holding costs,respectively. For low holding cost, the average gains rangefrom 39.22 (obtained with CH in DS1) to 65.55% (obtainedwith SA in DS8). For high holding cost, the range of the aver-age gains is similar: from 39.20 (obtained with CH in DS1) to63.39% (obtained with SA in DS8). In general, there is a veryslight difference in how CH, LS, and SA perform comparedto BM1 for low and high holding costs. Similar to the lowdemand case, with the low inventory holding costs, the gainsare inconsiderably higher. The main reason for the indiffer-ence in gains between low and high holding costs, in spite ofthe different performances of BM1 and the heuristics, is thelow percentage of the inventory holding costs compared tothe total costs. In many instances, the inventory holding costs(without the inventory replenishment costs) only contributeto 5% of the total costs. Hence, the difference in holding costrate does not significantly alter the average gains.

7.4. Value of Inventory at the Retailers

In this section, we present the computational results todemonstrate the value of inventory at the retailers and thevalue of the inventory coordination between retailers and theircorresponding DCs. Similar to Equation 33, we measure thisvalue as

Percentage gain overBM2(%) = Z(BM2) − Z(H)

Z(H)× 100,

where Z(BM2) is the objective value of PDSD-INV evalu-ated with the decision variables obtained through the solutionof BM2 and Z(H) is the objective value of the appropriatesolution approach for PDSD-INV. We present the minimum,average, and maximum gains obtained with CH, LS, and SAwith eight data sets in Table 7. Due to the structure of the CH,the average gains with CH are not pronounced. However, withboth LS and SA, average gains are around 20%, where SAis slightly better than LS. In particular, for data sets with 50retailers (DS1 through DS4), the average gains are higherthan 21% with maximum gains going higher than 45%. Fordata sets with 100 retailers, the BM2 performs slightly better.Nevertheless, average gains with LS and SA are higher than18% and the maximum gains can go up to 43.10%.

To better understand the source of these gains, we breakdown the gains with respect to each cost component. In Table8, we have three parts corresponding to the cost analysis forCH, LS, and SA, respectively. The first row of each partpresents the average gains over BM2 and the other rowspresent the percentage contributions of each cost componentto the gains obtained. Note that some entries are positive, indi-cating that the cost component from BM2 CBM2 is larger thanthe cost component from PDSD-INV (CH ), whereas someentries are negative, indicating that CBM2 < CH . Analyzing

Table 7. Comparison of solution approaches with BM2.

CH LS SA

DS1 Min 0.06 0.27 0.64Ave 0.40 21.65 22.69Max 0.83 52.27 46.56

DS2 Min 0.07 0.12 0.88Ave 1.37 21.54 22.76Max 16.30 54.42 52.12

DS3 Min 0.12 0.21 1.74Ave 1.64 23.91 21.63Max 9.08 51.47 46.49

DS4 Min 0.06 0.16 2.66Ave 2.27 21.21 22.90Max 46.79 45.14 49.43

DS5 Min 0.15 0.18 7.00Ave 2.06 20.28 21.97Max 34.57 43.10 41.77

DS6 Min 0.20 1.40 4.32Ave 0.59 19.15 19.48Max 1.68 40.12 40.12

DS7 Min 0.15 0.37 4.63Ave 0.51 18.23 20.51Max 1.17 38.22 40.58

DS8 Min 0.11 0.38 5.35Ave 0.56 18.88 19.45Max 2.75 41.56 41.58

the results from Table 8, we observe that the inventory costs atthe retailers only contribute to 11.01% to 23.98% of the gainsin LS and 10.77 to 28.09% of the same in SA. Next, we real-ize that considering inventory decisions at the retailers andinventory coordination affect the network design decisionsin a positive way. More specifically, on average, 37.01% and31.33% of the gains in LS and SA, respectively, come fromthe supplier selection decisions. Furthermore, on average,22.25% and 25.77% of the gains in LS and SA, respectively,are due to determining DC locations. These changes also pos-itively influence the transportation costs between retailers andDCs. On average, 9.50% of the gains in LS and 11.87% ofthe gains in SA are due to these transportation costs.

7.5. Value of Multiple Suppliers

In this section, using BM3, we quantify the value ofexplicitly considering supplier selection decisions in the inte-grated PDSD-INV problem. Specifically, the value of supplierselection in the integrated problem is estimated as

Percentage gain over BM3(%) = Z(BM3) − Z(H)

Z(H)× 100,

where Z(BM3) is the objective value of PDSD-INV evalu-ated with the decision variables obtained through the solutionof BM3 and Z(H) is the objective value of the appropriatesolution approach for PDSD-INV as before. We report the



Table 8. Components of the average gain over BM2.


CH Ave.Gains 0.40 1.37 1.64 2.27 2.06 0.59 0.51 0.56SupFix 0.00 17.16 15.27 −39.22 29.56 0.00 0.00 0.00DCFix 0.00 −7.25 1.52 −16.62 −13.20 0.00 0.00 0.00SupDC-Tr −24.11 −24.34 −17.57 −11.37 −6.54 −4.98 −7.21 −9.42DCRet-Tr −116.92 −138.63 −79.75 −31.56 −79.59 −35.38 −38.22 −49.63DCInv 182.97 196.71 146.02 143.56 172.30 123.12 122.96 137.80RetInv 58.37 56.36 34.50 55.21 −2.54 17.30 22.51 21.25

LS Ave.Gains 21.65 21.54 23.91 21.21 20.28 19.15 18.23 18.88SupFix 54.39 35.75 41.49 42.98 32.03 33.25 25.65 30.52DCFix 25.64 33.03 6.40 25.43 22.56 18.90 21.56 24.52SupDC-Tr −1.50 0.23 0.74 0.58 0.12 0.43 0.44 0.12DCRet-Tr −6.36 7.97 14.55 7.20 13.30 14.01 14.18 11.16DCInv 15.52 12.02 17.98 12.00 12.07 9.43 15.19 14.59RetInv 12.31 11.01 18.85 11.81 19.92 23.98 22.98 19.09

SA Ave.Gains 22.69 22.76 21.63 22.90 21.97 19.48 20.51 19.45SupFix 51.49 36.17 36.32 37.24 11.31 32.44 21.61 24.09DCFix 25.98 31.12 27.15 30.29 30.13 18.46 21.52 21.49SupDC-Tr 0.64 0.88 0.60 0.91 0.83 1.05 0.80 1.38DCRet-Tr 4.64 7.70 9.72 7.95 17.68 14.32 16.48 16.46DCInv 6.48 10.12 7.31 7.75 13.11 9.03 11.50 10.83RetInv 10.77 14.01 18.89 15.86 26.93 24.69 28.09 25.75

minimum, average, and maximum gains over BM3 with CH,LS, and SA in Table 9. We observe that the LS and SA performvery similarly; average gains range from 26.40 to 37.96% forLS and from 29.13 to 34.1% for SA while the maximumgains can go up to 60%. Even with CH, the average gains arearound 9% and maximum gains can go up to 48.25%.

The break down of these gains with respect to cost com-ponents, similar to the analysis for BM2, is given in Table10. For LS and SA, ∼27 and 58% of the gains are due to thefixed costs of the suppliers and DCs, respectively. For CH,∼97% of the gains are due to determining DC locations. Infact, we recognize that ignoring the supplier selection fromthe overall network design problem, increases the numberof DCs selected and hence, the corresponding fixed costscreate a significant gap between the performances of BM3and PDSD-INV. Additionally, these DC locations influenceinventory and transportation costs. Due to the increased num-ber of DCs, the transportation cost do not increase as much,however, inventory costs both at the DCs and retailers go upwith BM3. These results emphasize the impact of supplierdecisions on the overall integrated problem.

8. CONCLUDING REMARKS

In this article, we analyze a three-stage PDSD problemwith inventory considerations where the first stage consistsof retailers and the second and third stages consist of candi-date locations of DCs and capacitated suppliers, respectively.Furthermore, we explicitly consider inventory decisions atthe retailer and the DC levels. Each retailer replenishes its

inventory from a specific established DC at the second stage,and each selected DC replenishes its inventory from a specificcapacitated supplier located at the third stage. This problemis a generalization of the problems considered in the otherjoint location-inventory papers. In particular,

Table 9. Comparison of solution approaches with BM3.

CH LS SA

DS1 Min 0.00 13.77 8.77Ave 13.73 33.59 34.11Max 44.20 57.99 60.55

DS2 Min 0.02 5.12 6.71Ave 8.07 32.00 32.35Max 48.25 60.47 56.88

DS3 Min 0.73 16.55 11.65Ave 11.70 37.96 35.82Max 37.24 59.37 59.20

DS4 Min 0.02 10.75 15.48Ave 10.30 32.99 33.69Max 42.20 56.61 58.24

DS5 Min 0.08 9.71 11.01Ave 8.92 30.28 31.28Max 36.57 49.27 51.32

DS6 Min 0.15 9.67 7.66Ave 7.67 26.40 29.13Max 21.74 52.65 46.65

DS7 Min 0.09 5.96 6.21Ave 9.01 31.09 30.89Max 40.16 57.20 55.54

DS8 Min 0.05 9.27 10.45Ave 8.09 29.26 30.37Max 40.86 52.68 53.99



Table 10. Components of the average gain over BM3.


CH Ave.Gains 13.73 8.07 11.70 10.30 8.92 7.67 9.01 8.09SupFix −18.46 −74.26 −20.76 −29.74 39.72 −48.70 −29.87 −91.53DCFix 109.11 155.35 108.34 96.82 −78.60 114.78 116.40 158.03SupDC-Tr 17.60 18.88 15.18 20.74 44.11 19.13 33.97 22.22DCRet-Tr −20.56 −7.81 −4.53 3.00 39.31 0.62 −30.37 9.47DCInv 19.79 −6.35 3.27 −9.52 −6.71 5.11 4.02 −7.95RetInv −7.47 14.19 −1.51 18.70 62.15 9.10 5.73 9.76

LS Ave.Gains 33.59 32.00 37.96 32.99 30.28 26.40 31.09 29.26SupFix 34.01 27.78 32.01 32.56 31.54 14.25 27.16 18.90DCFix 57.99 59.60 57.37 55.28 49.72 67.44 58.14 64.48SupDC-Tr 1.59 1.44 1.40 1.58 1.43 1.87 1.72 1.48DCRet-Tr −0.02 1.51 1.42 1.71 2.86 1.46 1.22 −0.16DCInv 2.82 3.02 2.86 2.76 5.90 5.18 5.25 4.99RetInv 3.62 6.65 4.94 6.10 8.55 9.05 6.50 10.31

SA Ave.Gains 34.11 32.35 35.82 33.69 31.28 29.13 30.89 30.37SupFix 32.51 28.96 32.46 27.12 26.16 25.31 25.68 17.47DCFix 55.07 58.06 55.08 61.31 52.20 55.70 53.51 65.00SupDC-Tr 1.72 1.80 1.64 1.90 1.88 2.15 1.86 1.83DCRet-Tr 1.33 1.46 1.41 0.88 4.12 2.91 3.09 0.46DCInv 4.15 3.22 3.79 2.40 5.85 4.29 6.02 5.30RetInv 5.22 6.49 5.63 6.38 9.80 9.64 9.84 9.94

• it generalizes the three-stage continuous locationand inventory model by Üster et al. [48] to adiscrete setting with multiple DCs and multiplesuppliers;

• it generalizes the two-stage discrete location andinventory model by Teo and Shu [45] due to theconsideration of multiple capacitated suppliers andadditional inventory decisions at the DC level.

Table 11. Comparison of BM1 models with PDSD-INV solution approaches.

θ = 0.001 θ = 0.005 θ = 0.01 θ = 0.05

Time CH LS SA Time CH LS SA Time CH LS SA Time CH LS SA

Class 1 min 19.27 0.89 3.19 1.40 8.50 0.71 5.50 3.76 0.69 1.09 11.78 18.44 0.50 11.62 26.34 26.35ave 48.11 8.04 21.03 19.81 31.02 9.41 25.79 25.07 7.13 16.37 36.49 35.55 2.18 39.01 53.46 52.85max 295.02 33.04 39.45 44.49 82.00 39.71 43.79 70.87 12.67 49.24 56.28 60.09 7.45 53.92 67.85 69.10

Class 2 min 21.72 1.09 1.01 0.38 7.00 1.44 7.82 8.79 0.39 2.17 14.56 11.34 0.39 23.28 43.08 41.09ave 410.92 12.50 19.48 16.82 34.72 11.81 27.54 25.14 2.32 20.66 39.13 37.02 0.84 42.62 56.24 54.76max 5885.53 45.18 45.18 45.18 126.94 52.55 52.55 52.55 6.11 62.66 62.66 65.86 2.00 65.43 67.84 65.43

Class 3 min 60.20 0.59 0.41 4.51 46.08 0.59 2.87 5.23 18.41 3.40 17.66 14.68 4.36 25.54 35.95 36.40ave 201.91 3.83 21.07 18.67 123.46 7.99 27.36 25.50 34.03 16.49 38.31 36.18 14.81 39.60 55.36 54.05max 2245.19 11.68 43.38 48.60 342.09 19.29 49.33 53.51 74.84 33.45 62.60 65.89 31.08 48.27 69.07 70.32

Class 4 min 48.84 0.22 1.32 1.17 30.61 0.30 4.11 4.77 5.80 0.32 12.57 13.17 3.98 23.42 35.81 39.19ave 697.01 13.49 19.78 17.36 150.15 12.14 28.26 26.12 15.98 19.66 38.79 36.92 7.92 41.03 55.08 53.75max 3898.70 44.74 49.98 44.96 391.79 48.78 56.59 56.23 44.88 62.62 62.71 62.72 12.52 68.40 69.86 70.18

Class 5 min 14.17 0.74 2.86 4.88 3.53 2.73 15.80 17.20 1.70 5.48 22.36 23.51 1.55 44.05 53.25 55.98ave 115.22 9.42 22.02 26.49 38.44 17.73 32.60 36.44 2.40 34.17 46.20 49.25 1.68 53.26 61.75 63.90max 3336.09 36.36 45.69 47.42 225.81 42.35 50.80 55.46 4.36 53.34 61.49 64.22 2.16 69.96 74.36 75.82

Class 6 min 17.35 0.29 3.73 4.27 9.05 2.19 5.42 6.19 1.08 13.38 27.90 28.74 1.06 40.13 46.64 47.46ave 549.41 6.72 16.88 19.18 46.21 14.93 27.45 29.55 4.52 35.82 46.44 47.85 1.60 53.90 61.39 62.50max 3723.73 14.68 38.97 39.62 106.78 26.93 47.62 48.18 10.92 50.23 60.74 61.68 3.97 66.21 73.48 73.76

Class 7 min 43.02 0.36 0.51 5.48 11.72 2.03 1.43 8.31 1.14 10.01 30.09 33.26 1.13 35.85 47.86 49.22ave 382.01 9.63 20.01 22.01 44.60 16.81 29.04 30.86 3.99 34.02 45.32 46.77 1.47 51.93 60.03 61.04max 5946.09 41.42 41.42 44.36 158.48 46.01 46.10 49.75 9.89 54.20 58.99 57.66 3.44 67.96 70.03 72.60

Class 8 min 127.91 0.61 3.82 4.89 49.46 0.75 16.44 16.75 3.20 14.06 30.08 32.06 2.64 44.52 52.99 54.04ave 1561.24 10.41 20.52 20.77 304.97 16.14 31.03 31.19 20.89 39.84 50.84 51.02 6.40 54.82 62.93 63.02max 8940.90 40.21 40.21 40.90 1911.60 48.63 49.59 52.92 51.63 59.94 61.91 62.55 14.13 69.69 75.02 75.52



Table 12. Comparison of solution approaches with BM1 for classes 2 and 6 with low fixed location costs.

Fixed cost modification

10% 20% 40% 80%

Min Ave Max Min Ave Max Min Ave Max Min Ave Max

Class 2IOC LS 0.00 6.62 20.13 0.00 11.12 25.84 0.00 12.86 32.80 0.00 16.32 42.47(%) SA 0.00 9.08 23.41 0.00 14.05 33.40 0.00 18.37 41.14 0.00 21.49 47.58

BM1 time 1.67 4.11 7.77 1.91 8.34 19.49 6.64 20.04 38.25 18.42 305.34 7154.34Gain over CH 0.74 11.19 31.81 1.70 14.38 40.38 3.20 18.11 49.69 5.74 23.14 56.50BM1 (%) LS 3.95 17.18 36.82 11.89 24.01 51.52 13.88 28.47 50.89 23.23 35.72 56.50θ = 0.001 SA 4.13 19.31 38.34 14.29 26.55 50.99 18.53 33.26 50.89 24.26 39.77 57.76




Class 6IOC LS 0.13 2.10 13.41 1.22 8.87 18.75 0.92 13.66 30.08 0.61 17.35 39.67(%) SA 0.00 5.90 11.51 1.30 9.25 18.75 0.98 13.38 27.81 0.66 17.47 36.87





After presenting a formulation for PDSD-INV, we devel-oped efficient heuristic solution approaches using neighbor-hood search algorithms, including a CL-based local searchheuristic and an SA algorithm. The CL-based local search andSA utilize the solution of ICLIP to determine the DC-supplierassignments for each retailer set. Based on our computationalresults, the SA finds better quality solutions (i.e., with lowercost) with lower run times.

We devised three benchmark models to measure the prac-tical value of the PDSD-INV. The first benchmark model(BM1), measuring the value of integrated decision making,compares the PDSD-INV with a sequential model where thenetwork design decisions precede the inventory decisions.

The second benchmark model (BM2), measuring the valueof retailer inventory and inventory coordination, considers atwo-stage decision making process. In the first stage, supplierand DC locations as well as DC inventories are determined,and in the second stage, given these decisions, the inventorydecisions of the retailers are determined. The third bench-mark model (BM3), measuring the impact of multiple sup-pliers, considers another two-stage process. In the first stage,assuming an aggregated dummy supplier, DCs are locatedand the inventory decisions at both levels are determined. Inthe second stage, given the selected DCs, their correspond-ing retailers, and inventory decisions, capacitated suppliersare selected.



Table 13. Comparison of solution approaches with BM1 with varying demand and holding costs.

Demand Holding cost

Low High Low High

CH LS SA CH LS SA CH LS SA CH LS SA

DS1 min 44.24 48.30 46.44 13.19 23.39 27.01 22.68 41.56 34.39 23.84 35.70 34.78ave 61.27 66.63 65.34 33.32 53.77 50.37 39.22 57.39 54.59 39.20 56.02 53.65max 72.12 79.30 78.24 53.37 72.38 69.29 61.82 72.18 72.12 60.78 70.41 70.33

DS2 min 39.70 44.08 42.23 16.51 33.75 33.25 23.61 42.56 41.85 24.21 40.92 41.64ave 57.61 61.00 59.59 41.83 56.54 53.52 43.56 57.15 56.97 43.32 56.07 56.05max 70.14 76.10 74.95 56.48 75.88 70.40 66.83 72.31 70.28 65.33 70.72 70.62

DS3 min 40.01 47.86 46.31 15.33 40.90 32.35 29.61 40.12 41.27 30.44 39.18 40.93ave 60.29 64.72 63.37 37.13 58.64 53.87 39.17 55.00 55.74 39.19 55.13 54.54max 70.74 78.52 77.42 57.81 69.29 68.71 47.72 71.32 70.01 47.20 0.45 68.23

DS4 min 36.69 44.06 44.06 25.53 37.20 38.09 28.45 41.79 37.94 28.98 36.41 38.09ave 54.14 60.38 60.51 47.29 59.32 57.14 41.25 55.93 55.96 41.10 54.60 54.56max 66.51 71.38 71.54 63.28 75.80 72.71 68.54 71.81 69.43 67.23 69.97 67.81

DS5 min 60.72 63.81 63.23 15.89 33.28 35.90 32.64 44.56 50.98 32.72 43.78 43.52ave 69.10 71.54 71.87 39.40 54.65 52.48 53.57 64.42 64.03 51.96 62.52 61.18max 77.58 84.55 80.05 54.04 67.75 52.48 65.51 77.06 74.66 63.26 74.46 1.45

DS6 min 61.03 60.04 61.03 23.56 35.47 34.81 32.72 40.22 44.38 32.44 39.20 39.11ave 71.77 72.53 71.89 52.38 59.71 59.21 54.73 63.97 63.96 53.16 61.70 61.61max 80.71 84.93 80.71 67.23 73.47 68.79 64.73 75.59 75.59 63.03 72.83 73.46

DS7 min 46.93 55.36 55.54 22.29 39.36 35.54 41.46 51.80 51.87 40.57 50.00 50.06ave 63.37 69.76 70.59 47.27 57.88 57.34 53.50 63.99 64.69 51.97 61.37 62.30max 70.23 80.98 77.23 60.33 71.68 71.54 60.75 77.25 77.18 58.72 73.51 74.49

DS8 min 59.08 62.34 58.52 36.33 45.78 45.17 41.47 55.08 58.59 40.88 53.60 55.16ave 68.84 73.73 69.63 55.69 62.23 61.86 56.27 64.93 65.55 54.72 62.90 63.39max 74.95 87.37 78.59 67.04 74.63 72.52 75.16 75.54 75.57 72.83 73.57 72.35

Both the local search and the SA provide significant costsavings over these benchmark models. In particular, withBM1, depending on how transportation costs are estimated,the average gains can range between 20 and 60%. With BM2,the average gains are around 20%, and with BM3, the averagegains are over 30%. Based on the analysis of the cost com-ponents, incorporation of supplier selection and inventoryconsiderations at the retailers influence the savings.

In the future, this article can be extended in multiple direc-tions. One future research direction is to consider stochasticdemand and safety stock issues that arise in this context. Itwould be valuable to quantify the amount of safety stockrequirements while considering a three stage network designwith cycle inventory decisions. It would be also interesting toevaluate the impact of safety stocks on the selection of DCsand suppliers. Another future research direction is to considermultiple sourcing as opposed to single sourcing. Multiplesourcing is preferred by a number of companies due to manypractical reasons. It would be useful for these companies tohave a tool to evaluate their network design needs while con-sidering inventories with multisourcing. Finally, the suppliercompetition, especially on unit price (with or without quan-tity discounts), can be incorporated into the network designwith inventory decisions.

ACKNOWLEDGMENTS

The authors thank an anonymous area editor and twoanonymous referees for their helpful comments on earlierversions of this article.

REFERENCES

[1] B.C. Arntzen, G.G. Brown, T.P. Harrison, and L.L. Trafton,Global supply chain management at digital equipment corpo-ration, Interfaces 25 (1995), 69–93.

[2] R.H. Ballou, Business logistics management, Prentice Hall,New Jersey, 1998.

[3] F. Barahona and D. Jensen, Plant location with minimuminventory, Math Program 83 (1998), 101–111.

[4] W.C. Benton, Purhasing and supply management, McGraw-Hill Irwin, New York, NY, 2007.

[5] G. Brown, J. Keegan, B. Vigus, and K. Wood, The kelloggcompany optimizes production, inventory, and distribution,Interfaces 31 (2001), 1–15.

[6] K.L. Croxton and W. Zinn, Inventory considerations in net-work design, J Bus Logist 26 (2005), 149–168.

[7] M. Daskin, C. Coullard, and Z.J.M. Shen, An inventory-location model: Formulation, solution algorithm and compu-tational results, Ann Oper Res 110 (2002), 83–106.

[8] S.S. Erengüç, N.C. Simpson, and A.J. Vakharia, Integratedproduction/distribution planning in supply chains: An invitedreview, European J Oper Res 115 (1999), 219–236.



[9] S. Erlebacher and R.D. Meller, The interaction of locationand inventory in designing distribution systems, IIE Trans 32(2000), 155–166.

[10] A.M. Geoffrion and R.F. Powers, Twenty years of strate-gic distribution-system design - an evolutionary perspective,Interfaces 25 (1995), 105–127.

[11] M. Goetschalckx, C.J. Vidal, and K. Dogan, Modeling anddesign of global logistic systems: A review of integrated strate-gic and tactical models and design algorithms, Eur J Oper Res143 (2002), 1–18.

[12] J.L. Heskett, A missing link in physical distribution system,J Market 30 (1966), 37–42.

[13] J. Hromkovic, Algorithmics for Hard Problems, 2nd ed.,Springer Verlag, New York, 2003.

[14] V. Jayaraman, Transportation, facility location and inventoryissues in distribution network design, Int J Oper Prod Manage18 (1998), 471–494.

[15] L. Kaufman, M.V. Eede, and P. Hansen, A plant and warehouselocation problem, Oper Res Q 28 (1977), 547–554.

[16] B.B. Keskin and H. Üster, Meta-heuristic approacheswith memory and evolution for a multi-product produc-tion/distribution system design problem, Eur J Oper Res 182(2007), 663–682.

[17] S. Kirkpatrick, J. Gelatt C. D., and M.P. Vecchi, Optimizationby simulated annealing, Science 220 (1983), 671–680.

[18] P.J.M. Laarhoven and E.H.L. Aarts, Simulated annealing:theory and applications, Springer, New York, 1987.

[19] R.F. Love, J.G. Morris, and G.O. Wesolowsky, Facilitieslocation: Models and methods, North-Holland, Amsterdam,Netherlands, 1988.

[20] C.H. Martin, D.C. Dent, and J.C. Eckhart, Integrated pro-duction, distribution, and inventory planning at libbey-owens-ford, Interfaces 23 (1993), 68–78.

[21] P.A. Miranda and R.A. Garrido, Incorporating inventory con-trol decisions into a strategic distribution network designmodel with stochastic demand, Trans Res-E 40 (2004),183–207.

[22] A. Nagurney, A system-optimization perspective for supplychain network integration: the horizontal merger case, TransRes 45 (2009), 1–15.

[23] L.K. Nozick and M.A. Turnquist, Integrating inventoryimpacts into a fixed charge model for locating distributioncenters, Trans Res -E 34 (1998), 173–186.

[24] L.K. Nozick and M.A. Turnquist, A two-echelon inventoryallocation and distribution center location analysis, Trans Res-E 37 (2001), 425–441.

[25] M. Ouhimmou, S. D’Amoursa, R. Beauregard, D. Ait-Kadi,and S. Singh Chauhan, Furniture supply chain tactical plan-ning optimization using a time decomposition approach, EurJ Oper Res 189 (2008), 952–970.

[26] S. Park, T.-E. Lee, and C.S. Sung, A three-level supply chainnetwork design model with risk-pooling and lead times, TransRes Part E 46 (2010), 563–581.

[27] H. Pirkul and V. Jayaraman, Production, transportation, anddistribution planning in a multi-commodity tri-echelon sys-tem, Trans Sci 30 (1996), 291–302.

[28] J. Pooley, Integrated production and distribution facility plan-ning at ault foods, Interfaces 24 (1994), 113–121.

[29] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numer-ical recipes in C: The art and science of computing, 2nd ed.,Cambridge University Press, Cambridge, 1992.

[30] H. Ro and D. Tcha, A branch-and-bound algorithm for thetwo-level uncapacitated facility location problem, Eur J OperRes 18 (1984), 349–358.

[31] E.P. Robinson, L.L. Gao, and S.D. Muggenborg, Designingan integrated distribution system at DowBrands, Interfaces 23(1993), 107–117.

[32] H.E. Romeijn, J. Shu, and C.-P. Teo, Designing two-echelonsupply networks, Eur J Oper Res 178 (2007), 449–462.

[33] R.O. Roundy, 98-percent effective integer-ratio lot-sizing forone-warehouse, multi-retailer systems, Manage Sci 31 (1985),1416–1430.

[34] S.M. Sait and H. Youssef, Iterative computer algorithms withapplications in engineering: Solving combinatorial optimiza-tion problems, IEEE Computer Society, Los Alamitos, CA,1999.

[35] A.M. Sarmiento and R. Nagi, A review of integrated analy-sis of production-distribution systems, IIE Trans 31 (1999),1061–1074.

[36] Z.J.M. Shen, C. Coullard, and M.S. Daskin, A joint location-inventory model, Trans Sci 37 (2003), 40–55.

[37] Z.M. Shen and M.S. Daskin, Trade-offs between customer ser-vice and cost in integrated supply chain design, Manuf ServOper Manage 7 (2005), 188–207.

[38] J. Shu, An efficient greedy heuristic for warehouse–retailernetwork design optimization, Trans Sci 44 (2010), 183–192.

[39] J. Shu, C.-P. Teo, and Z.M. Shen, Stochastic transportation–inventory network design problem, Oper Res 53 (2005),48–60.

[40] D. Simchi-Levi, X. Chen, and J. Bramel, Logic of logistics:Theory, algorithms, and applications for logistics and supplychain management, Springer, New York, 2005.

[41] L.V. Snyder, M.S. Daskin, and C.P. Teo, The stochastic loca-tion model with risk pooling, Lehigh University, Bethlehem,Pennsylvania, 2003.

[42] J. Spall, Introduction to stochastic search and optimization,Wiley, New York, 2003.

[43] A. Syarif, Y. Yun, and M. Gen, Study on multi-stage logis-tic chain network: a spanning tree-based genetic algorithmapproach, Comput Ind Eng 43 (2002), 299–314.

[44] C.P. Teo, J. Ou, and M. Goh, Impact on inventory costs withconsolidation of distribution centers, IIE Trans 33 (2001),99–110.

[45] C.-P. Teo and J. Shu, Warehouse–retailer network designproblem, Oper Res 52 (2004), 396–408.

[46] R. Tyagi, P. Kalish, K. Akbay, and G. Munshaw, GE plas-tics optimizes the two-echelon global fulfillment network atits high performance polymers division, Interfaces 34 (2004),359–366.

[47] N.L. Ulstein, M. Christiansen, and R. Granhaig, Elkem usesoptimization in redesigning its supply chain, Interfaces 36(2006), 314–325.

[48] H. Üster, B.B. Keskin, and S. Çetinkaya, Integrated warehouselocation and inventory decisions in a three-tier distributionsystem, IIE Trans 40 (2008), 718–732.

[49] N. Vidyarthi, E. Çelebi, S. Elhedhli, and E. Jewkes, Inte-grated production-inventory-distribution system design withrisk pooling: Model formulation and heuristic solution, TransSci 41 (2007), 392–408.


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Production/distribution system design with inventory considerations