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597

Design of Stochastic Distribution Networks Using Lagrangian Relaxation Guy-Aimé Tanonkou, Lyes Benyoucef, and Xiaolan Xie

Abstract—This paper addresses the design of single commodity stochastic distribution networks. The distribution network under consideration consists of a single supplier serving a set of retailers through a set of distribution centers (DCs). The number and location of DCs are decision variables and they are chosen from the set of retailer locations. To manage inventory at DCs, the economic order quantity (EOQ) policy is used by each DC, and a safety stock level is kept to ensure a given retailer service level. Each retailer faces a random demand of a single commodity and the supply lead time from the supplier to each DC is random. The goal is to minimize the total location, shipment, and inventory costs, while ensuring a given retailer service level. The introduction of inventory costs and safety stock costs leads to a nonlinear NP-hard optimization problem. A Lagrangian relaxation approach is proposed. Computational results are presented and analyzed showing the effectiveness of the proposed approach. Note to Practitioners—The distribution network design problem considered in this paper was motivated by industrial applications encountered in an EU-funded project GROWTHONE. These applications, from automotive and textile industries, showed that supply chain design decisions are strongly linked to operational decisions and cannot be made without taking into account operational performances such as inventory costs and customer service levels. A simulation-based multiobjective optimization approach was proposed in ONE for supply chain design. The main drawback of the ONE approach is the excessive computational time. The goal of this paper is to propose an analytical distribution network design model that captures key operational performances yet remains tractable. The analytical model integrates the following decisions that are traditionally treated separately at different levels: the number and locations of DCs, the assignment of retailers to DCs, and the inventory level to keep at each DC. A Lagrangian relaxation-based approach is proposed to determine in a few minutes a nearoptimal distribution network. Index Terms—Facility location, Lagrangian relaxation, optimization, supply chain design.

Manuscript received February 12, 2007; revised August 23, 2007. First published March 12, 2008; current version published October 1, 2008. This work was supported in part by the Luxembourg Ministry of High Education and Research—Department of Science under Grant 03/88 and in part by the European Community Research Program under the Network of Excellence I*Proms “Innovative PROduction Machines and Systems.” This paper was recommended for publication by Associate Editor L. Shi and Editor N. Viswanadham upon evaluation of the reviewers’ comments. G.-A. Tanonkou and L. Benyoucef are with INRIA COSTEAM Project, ISGMP, Bàt. A, Ile du Saulcy, Metz 57000, France (e-mail: [email protected]; [email protected]). X. Xie is with Ecole Nationale Supérieure des Mines de Saint-Etienne (ENSM.SE), Engineering and Heath Division, 42023 Saint-Etienne cedex 2 France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASE.2008.917156

I. MOTIVATIONS

A

SUPPLY CHAIN is a dynamic, stochastic, and complex system, in which the performance of any given entity depends to a large extent on the behavior of other entities. To improve the overall performance of a supply chain, it is necessary to view the system as a whole. Among various supply chain related activities, the procurement and distribution of goods are playing an increasingly important role as a result of the globalization of the economy. Companies face complex decisions regarding the design or redesign of their distribution networks. These decisions include distribution center (DC) locations and inventory control, and have significant impacts on the network’s long-term performances. They are by nature complex, costly, and difficult to reverse. Facility location problems have been widely addressed in the literature. An early study [1] reviewed a number of mathematical models on facility location problem including models with or without capacity constraints, with single or multiple stages, etc. It concluded that it is very important to develop multistage and multicommodity models taking into account uncertainties and dynamics. A review [2] on global supply chain design problem, in which half of the reviewed models are dedicated to the design of distribution networks, emphasized on the lack of consideration of uncertainties in most existing models. Geoffrion and Powers [3] analyzed the evolution of distribution network design over the 20 years before 1995. They identified a number of elements, which have significantly contributed to the evolution of distribution networks, including the logistics functionalities, optimization algorithms, information systems, etc. They claimed that customer service and client requests will remain as the most fundamental aspects for research. Vidal and Goetschalckx [4] provided a review with emphasis on mathematical models. Problem formulations, solution methods, and computational experiences are identified and analyzed. The authors indicated that “very few models consider important stochastic aspects of a global supply chain, such as customer service level and lead times.” Klose and Drexel [5] presented an extensive state-of-the-art dedicated to facility location models for design of distribution systems. The model formulations and solution approaches vary widely in terms of fundamental assumptions, mathematical complexity, and computational performances. They focused in particular on continuous location models, network location models, mixed-integer programming models, and applications. Revelle and Eiselt [6] surveyed a number of important decisions problems in facility location. They stated that “the field is very active with many interesting problems still being investigated, both from a problem statement/formulation standpoint and from an algorithmic point of view. Although the field is active from a research

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perspective, when it comes to applications, there appears to be a significant deficit, at least as compared to other, similar, fields.” Although location-allocation decisions are strongly linked to operational decisions, facility locations decisions are traditionally made without taking into account the operational performances of the related supply chain. Hence, there is a need for realistic yet tractable facility decision models. The rest of this paper is organized as follows. Section II reviews the existing literature dedicated to deterministic and stochastic facility location problems and summarizes the main contributions of this paper. Section III describes the stochastic inventory-location problem under consideration. Section IV presents the proposed Lagrangian relaxation-based approach. Section V discusses the computational experiments and sensitivity analyses. Section VI concludes this paper with some directions for future research. II. LITERATURE REVIEW The literature dedicated to deterministic and stochastic location problems is very rich. Classical models include the fixed charge facility location (FCFL) problem and the capacitated fixed charge location (CFLP) problem. The first model forms the basis of many of the location models that have been used in supply chain design. All parameters are deterministic and the problem is to find the locations of the facilities and the shipment pattern between the facilities and the retailers in order to minimize the combined facility location and shipment cost. A number of solution approaches have been proposed for FCFL. Simple heuristics typically begins by constructing a feasible solution and then greedily adding or dropping facilities from the solution until no further improvement can be obtained. A Tabu search method was proposed in [7] to solve the FCFL problem and it was tested successfully on small and moderate sized problems. A variable neighborhood search algorithm was proposed in [8] to solve the FCFL problem and the -median problem. Geoffrion [9] showed that when embedded in branch and bound, Lagrangian relaxation is powerful for identifying the optimal solutions of the FCFL model. Geoffrion and Graves [10] extended the traditional FCFL problem to include shipments from plant to DC and multiple commodities. Daskin [11] and Galvao [12] reviewed Lagrangian relaxation approaches for deterministic location problems. The literature on facility location theory ([11], [13], [14] and [15]) usually uses oversimplified deterministic facility location models that neglect important operational costs such as inventory holding costs. Snyder [16] presented a rich state-of-the-art on existing stochastic models for facility location problems. Many of these models try to minimize the expected cost or maximize the expected profit of the system. Others take a probabilistic approach, for example, maximizing the probability that the solution is in some sense “good.” Some models are solved using algorithms designed specifically for the problem, where others are solved using more general stochastic programming techniques. Stochastic versions of the capacitated -median problem (CPMP) and CFLP were presented in [17]. Customer demands, production costs, transportation costs, and selling prices are random variables. The goal is to determine facility locations, their capacities, and the allocation of customers to facilities in order to maximize the expected utility of profit.

In [18], a facility location model with random throughput costs at the DCs is considered. The objective is to minimize the deterministic transportation cost (plant to DC and DC to customer) plus the expected throughput cost at the DCs. The authors initially consider the network flow aspect of the problem (assuming the DC locations are given). Then they embed the expected cost model into a nonlinear integer program. This model is solved heuristically, since for each candidate solution to the location problem, a Lagrangian problem must be solved to compute the expected flows. For the joint transportation-location problem, Benders decomposition was used in [19] to solve a problem that is a combination of the CFLP and the stochastic transportation problem with random customer demands. While the contribution of inventory cost has been recognized for many years, only recently has been addressed the so-called inventory-location model by incorporating inventory decisions in facility locations models ([20]–[24], [25]). Barahona and Jensen [20] modified the uncapacitated facility location problem to implicitly consider limited inventory levels. Nozick and Turnquist [21] approximated inventory costs as part of the fixed facility costs assuming a linear relationship between inventory and the number of open facilities, and proposed a model that takes into account a given service coverage. Nozick and Turnquist [22] extended this model and included demand coverage in the objective function. In [23], a nonlinear integer inventory-location model was presented. The customers’ demands are uniform and rectilinear distances are used to represent the distances between the locations (plants to DCs and to customers). Each DC operates under a continuous review inventory system, and the location and the capacity of each plant are known and fixed. The problem is to determine the number of DCs and their locations, as well as the customers they serve, in order to minimize the fixed costs of operating the DCs, total DCs inventory holding costs, and total transportation costs. Since the general version of the problem is NP-Hard, they developed analytical models and proposed heuristic procedures for special cases obtained under some simplification assumptions. Stochastic versions of the joint inventory-location model were presented in [24]–[28] and [29]. They developed a location model with risk pooling that explicitly considers expected inventory cost when making facility location decisions, thus combining strategic and tactical decisions into a single model. This leads to a difficult nonlinear combinatorial optimization problem. Daskin et al. [25] presented a stochastic inventory-location model that incorporates working and safety stock inventory costs at the DCs, as well as the economies of scale that exist in the transport costs from the supplier to DCs. The model is identical to that proposed by Shen [24] and Shen et al. [26]. In both papers [24] and [26], the authors solved the problem by first recasting it as a set partitioning problem and then solving the resulting model using column generation. In [25], a Lagrangian solution algorithm is proposed for the case in which the variance-to-mean of the demand is the same for all retailers and the supply lead times are constants. Furthermore, a number of improvement heuristics were outlined for the problem. The algorithm was tested on two datasets consisting of 88 and 150 retailers, respectively. Computational results

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compared favorably with those of the set partitioning approach proposed by Shen [24] and Shen et al. [26]. A nonlinear integer-programming model for the multicommodity supply chain design problem was presented in Shen [30]. The model determines the location of facilities and the assignment of customers to the facilities in order to minimize a nonlinear objective function that includes the economies of scale costs at the facilities. It is the first multicommodity supply chain design model that incorporates supply chain costs exhibiting economies of scale. It generalizes many well-studied models. A Lagrangian relaxation solution algorithm is proposed, and compared with existing algorithms for different special cases of the proposed model. The main assumption is related to the nonlinear objective function to optimize, which is a summation of two concaves and nondecreasing functions. The inventory-location model considered in this paper is an extension of the inventory-location model proposed in [24]–[26]. We relax the assumption of identical variance-tomean ratio and further assume random supply lead times. The relaxation of identical variance-to-mean ration and the introduction of random supply lead times lead to a cost function that is no longer concave. Therefore, Shen [24], Daskin et al. [25], Shen et al. [26], and Shen [30] algorithms do not apply. The main contributions of this paper can be summarized as follows: (i) an analytical inventory-location model which takes into account both random demands and random supply lead-times. It takes into accounts the inventory holding cost of an economic order quantity (EOQ) policy and the cost of a safety stock to cope with randomness of both demands and supply lead-times; (ii) an original Lagrangian relaxation-based approach that is able to produce good feasible solutions and tight lower bounds; (iii) important relations between the structure of the optimal distribution networks and key design parameters established by an extensive numerical experiment; (iv) an original solution technique of relaxed problems that can be adapted to solve similar problems such as the models of [24]–[26] without imposing the assumption of identical variance-to-mean ratio. Note that part of the results of this paper appeared in a preliminary study Tanonkou et al. [31], which also considers both random demands and random supply lead-times. To reduce the problem complexity, the inventory-location model of [31] assumes that each DC keeps two separate safety stock, one to cope with the random demand and another to cope with the random supply lead-time. This assumption leads to relaxed problems of submodular cost function that are easier to solve. Unfortunately, separate safety stocks lead to excessive inventory and, in practice, a single safety stock is used to cope with all sources of randomness. This is taken into account in the inventory-location model of this paper. III. PROBLEM SETTING AND FORMULATION A. Problem This paper addresses the design of a stochastic distribution network in which a single supplier ships products (single product type) to a set of retailers via a set of DCs to locate. The number and location of the DCs are not given a priori. They are chosen

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Fig. 1. Structure of the distribution network.

from a set of retailer locations. We assume that the demand at and each retailer is normally distributed with a daily mean variance . The retailers are connected each others. Each retailer location can be selected to host a DC. The retailers that are selected to operate as DC order inventory from the supplier using an EOQ policy. The frequency of orders and the order quantity at each DC depend on the mean demand served by the DC, which, in turn, is a function of the assignment of retailers to the DC. To this working inventory, each DC keeps a safety stock to protect against the possibility of stockouts during the supply lead time. The supply lead time (in days) for deliveries from the supplier to the DC located at retailer is a random variable with mean and variance . The transportation times from DCs to retailers are not considered in this study. Fig. 1 illustrates the network structure of the supply chain under consideration. Thus, the problem considered in this paper is the following: given a set of retailers each facing an independent uncertain demand, we must decide how many DCs to locate, where to locate them, which retailers to assign to each DC in other to minimize the total location, shipment, working-inventory, and safety stock inventory costs. B. Mathematical Model The following notations are used to define the stochastic inventory-location problem under consideration. set of retailers indexed by ; DC located at retailer ; mean daily demand of retailer ; variance of daily demand of retailer ; fixed annual cost of locating a ; per-unit shipment cost from a to retailer ; fixed cost per order placed to the supplier by a ; fixed cost per shipment from the supplier to a ; per-unit shipment cost from the supplier to a ; inventory holding cost per unit per year in a ; mean lead-time in days from the supplier to a ; variance of the lead-time from the supplier to a ; number of working days per year; desired percentage of not stocking out at a DC during a retailer lead time; standard normal variate such that .

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The decision variables are

For a generic probability distribution of the supply lead time to , the demand during the lead time is a random variable with mean (9) and variance

Before formally formulating the problem, we outline the different components of the objective function. Consider first the inventory cost, fixed ordering cost, and shipment cost related to . Let the expected annual demand be units and let each be the order quantity of . The total annual fixed ordering cost is given by

(10) The safety stock level at to ensure that stock-outs occur at supply delivery with a probability of or less is (11)

(1)

Consequently, the annual safety stock cost of all DCs is then (12)

The total annual shipment cost is given by (2) and the total inventory-holding cost is given by

Using (8) and (12), the DC-retailer transportation cost and the total fixed location cost , the nonlinear mixedinteger mathematical formulation of the problem is as follows: (13)

(3)

with

Then, the sum of order, shipment, and inventory-carrying is given by costs at the (4) By taking the derivative of (4) with respect to it to zero, we obtain

and setting

(5)

(14) subject to

By substituting cost becomes

into the cost function (4), the total annual (15) (6)

Note that the expected annual demand assigned to a regional is

(16) (17) where

(7) Hence, the total order, shipment, and inventory-carrying costs at the (6) depends on the assignment variable ( ) and is as follows: (8) Consider the safety stock cost of . For a given service level, determining the required safety stock level is related to the nature of the probability distribution of the demand during the supply lead time. We assume that the daily demands at retailer are independent and identically distributed random variables of normal distribution with mean and standard deviation .

The objective function (14) to minimize is the sum of the following costs: the first term corresponds to the fixed cost of locating facilities, the second term is the variable transportation cost from the supplier to the DCs, as well as the variable shipment cost from the DCs to the retailers, the third term represents the expected working inventory cost at the DCs (assuming that each DC uses an EOQ policy), and finally the last term represents the cost of holding safety stock at the DCs to maintain a service level of . Constraints (15) ensure that each retailer is

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assigned to exactly one DC. Constraints (16) state that retailers to opened DC . Concan only be assigned straints (17) are standard integrity constraints. The problem described by the above integer-programming model is nonlinear and the determination of exact solutions is a NP-hard problem [11]. Therefore, in this paper, we propose a Lagrangian relaxation-based methodology to approximate the exact solutions.

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the solutions of the relaxed problem that capture most important features of the optimal solution of the initial problem (13). The constraints (19) and (21) are relaxed and added to the objective function with Lagrangian multipliers and , respecis tively. The resulting relaxed problem

IV. SOLUTION METHODOLOGY Existing techniques of [24]–[26] and [31] for solving inventory-location models consist in relaxing the so-called single source constraints (15). Although the relaxation of these constraints lead to independent DC-level subproblems, unfortunately the cost functions of these DC-level subproblems are no longer concave or submodular functions and existing techniques of [24]–[26] and [31] do not apply. , we introduce the expected demand To solve problem at as a new decision variable. Using and the above is equivalent to notations, problem

(23) subject to constraints (20) and (22). The relaxation of constraint (19) makes the location decisions of different DCs independent. Consequently, the relaxed is equivalent to problem (24)

(18) where

is the subproblem defined by

with

(25)

with subject to (19) (20) (21) (22) In the following section, we propose a Lagrangian relaxationbased method to solve the above problem.

Property 1: a) b) For each c)

for all Lagrangian multiplier with , the

.

. of the relaxed problem d) If the solution is a feasible solution to the initial problem , then is an optimal solution to problem . Proof: Properties (a) and (d) are standard results of La, with and , grangian relaxation. If (25) implies

A. Lagrangian Relaxation The proposed Lagrangian relaxation-based method consists of: (i) relaxing constraints (19) and (21), and introducing the corresponding terms into the cost function; (ii) solving the relaxed problem for each setting of Lagrangian multipliers to obtain a lower bound; (iii) deriving a feasible solution to obtain an upper bound; and (iv) maximizing the lower bound using a subgradient method. The efficiency of the method is ensured by: (i) the tight lower bound and (ii) the feasible solutions derived from

.

which proves (b). If

, (25) becomes

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Property (c) is proved by the above relation together with the following relations:

B. Solving the Lagrangian Dual Problem proAccording to Property 1(a), the relaxed problem , i.e., vides a lower bound to the original problem where is the optimal value of (13). The dual problem deto obtain the best termines the Lagrangian multipliers lower bound, i.e.,

As is a piece-wise differentiable concave function, the above Lagrangian dual problem (DP) is solved using the following subgradient method:

where is the step size and

C. Deriving Feasible Solutions At each iteration of the subgradient algorithm for solving the , a feasible solution and, consequently, an dual problem is derived. For each soupper bound of the original problem of , if (19) and (21) are satisfied, then the lution solution is optimal. Otherwise, the solution is modified as follows to obtain a feasible solution. Constraint (21) can be easily with . Two cases are conenforced by replacing sidered with respect to (19). (i.e., no DC is open). Then, we Case 1) If choose to open a unique DC and assign all retailers to it. The DC location is selected such that the total cost is minimized. , for some . Case 2) At least one DC is opened, i.e., Then, for each retailer , (retailer is assigned to more 1) If than one DC in the Lagrangian solution), we with assign retailer to a single such that the total cost of the modified solution is minimum. 2) If (retailer is not assigned in the Lagrangian solution), we assign retailer to an open DC such that the total cost of the modified solution is minimum. , we compute an For this new feasible solution upper bound

is the Lagrangian vector at iteration is the gradient defined by

Two step-size procedures are used to solve the dual problem. The first procedure uses the Armijo’s step size algorithm (see , and is the smallest integer such that [32]) with

(26) D. Lagrangian Relaxation Heuristic

with . The second procedure is the standard subgradient optimization procedure (see [33]) with

The use of the two step-size rules is motivated by our computational experiences. Armijo step-size allows large step-size at the beginning but quickly gets trapped at some nondifferentiable frontier. The standard subgradient step-size is more conservative but has better convergence. We start with the Armijo type step size rule and continue as improvement is large enough. long as the dual function improvement becomes small, we switch When the cost to the standard subgradient step-size with

where is the last Armijo step-size. As is unknown, we replace it by the best upper bound known during the search is reduced if no improvement of process and the parameter is observed after a certain number of iterations.

In this section, we present the structure of the main Lagrangian relaxation heuristic to solve the original problem . Main Lagrangian Relaxation Heuristic Initialization and let its • Determine an initial feasible solution cost be • Set (precision) Repeat Steps 1-7 Step 1. Solve all relaxed subproblems and , (the algorithm for solving this compute subproblem is given in the subsection ). using (24). Step 2. Compute Step 3. Derive a feasible solution and compute the related upper bound UB. ( , UB). Step 4. Update the best-so-far solution Step 5. Determine the step size . . Step 6. Update the Lagrangian multiplier . Step 7. Until

.

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TANONKOU et al.: DESIGN OF STOCHASTIC DISTRIBUTION NETWORKS USING LAGRANGIAN RELAXATION

In the following, we will mainly focus on the presentation of . the solution procedure to solve subproblems E. Solving Subproblem The main objective of this section is to propose a polynomial algorithm for solving subproblems and determining . According to Property 1(b), we only need to consider such that . For , one of the following cases is encountered. , constraint (20) implies that and If for all and since . (i.e., is open), constraint (20) becomes redunIf dant and the problem of selecting the retailers to be served by is obtained by solving the second subproblem

(27) where is replaced by (i.e., is fixed). To summarize, the solution of the sub problem —

is:

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Since problem (27) is a minimization problem, it is equivalent to problem (29), shown at the bottom of the page. Consider the problem

(30) Then

(31) As we mentioned earlier, solving (30) is not trivial since the are not equal to 0. Hence, instead of parameters using the optimal EOQ at determined in Section III, our . We solve basic idea is to use the fixed order quantity of then the related subproblem (30) by searching simultaneously and the optimal order size . optimal value of Consequently, the second term of (30) depending on order quantity , is the total inventory running cost and fixed order/ shipment cost that can be replaced by the following cost:

—

(32)

Consequently, (25) can be rewritten as (28) We will mainly focus on solving the second subproblem . If we neglect the decision variable , then, the second sub problem (27) can be transformed into a submodular function minimization problem and hence can be solved polynomially by general submodular function minimization algorithms. An efficient construction algorithm was proposed is not a decision in [25] in the case when parameter variable and when the objective function of (27) has only one square root. In our case, solving problem (27) is not trivial are not equal to zero and is since all parameters a decision variable. In the following, we propose a polynomial algorithm to comthat takes into account the special structures of pute problem (27).

Replacing (32) into the objective function of problem (30) leads to relation (33), shown at the bottom of the page, with . Consider the problem

(34) and constant , the related subproblem For a fixed is an unconstrained nonlinear optimization problem with discrete variables which is similar to the pricing subprobin the square root term of lems in [25] except the term (33). The following property which is similar to [25, Th. 4.2] provides the characterization of the optimal solution of problem (34) and can be proved by similar arguments as in [25].

(29)

(33)

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Property 2: There exists an optimal solution of of all vectors such that the set

where the set

in

is partitioned into three subsets

where without loss of generality, and the elements of

conRemark 1: From Property 2, it is clear that each set elements that can be determined polynotains at most to consider is equal to mially. Further, the number of sets the number of changes of the set which is upper bounded which is by plus the number of changes of ordering in as the ordering of any pair upper bounded by switches only once. As a result, the set contains at most elements that can be determined in polynomial time. Remark 2: From Property 2 and the above algorithm, it is and and is indeclear that the set depends only on pendent of . Remark 3: In Step 3, the next value of such that the set changes is , where

with are sorted as follows:

From Property 2, can be determined by: (i) screening for all possible value of the order quantity ; (ii) determining the set of possible solutions for each ; (iii) deterby using relation (33) but restricting to the set mining of identified in (ii), i.e., the order quantity is replaced by the optimal order quantity for each . Continuous screening of is not necessary as, when indetermined by Property 2 does not creasing , the set change as long as the partition of the set in Step 1 and the ordering in Step 2 do not change. The following algorithm determines the set that contains all possible optimal solutions for all possible .

with

and . The combination of the above results and relations (30), (31), and (33) implies that relation (35), shown at the bottom of the page, which from the finiteness of , leads to relation (36), shown at the bottom of the page. , the optimal and can For each given , as be determined as follows:

Algorithm to Compute Step 1. Initialize Step 2. Determine

(37)

. according to Property 2 and do

. Step 3. Compute the next value of by increasing such changes, i.e., the set that the set changes or the ordering of elements of the set changes. Step 4. If no such exists, then STOP Else go to Step 2.

(38) Replacing (37) and (38) in (36) leads to (39)

(35)

(36)

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TANONKOU et al.: DESIGN OF STOCHASTIC DISTRIBUTION NETWORKS USING LAGRANGIAN RELAXATION

where

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TABLE I DATA SET

(40) Remark 4: In general, the solution of the relaxed problem does not satisfy the relaxed relation (21). However, if is differentiable at the dual optimal, i.e., does not change in of the neighborhood of the dual optimum, then the demand is exactly . This can be proved by letting the a with respect to to be zero. partial derivative of , Remark 5: It can be shown that if . As a result, the order at least one retailer is assigned to , where quantity can be restricted to

To summarize the subproblem follows.

TABLE II PERFORMANCE OF SOLUTION PROCEDURE (NBE = 1)

can be solved as TABLE III PERFORMANCE OF SOLUTION PROCEDURE (NBE = 5)

Algorithm to Compute Step 1. Initialize

and .

according to Property 2. Step 2. Determine ; Step 3. For each Step 3a.Compute and using (38) and (40). Step 3b.If , and . Step 4. Compute the next value of by increasing such that the set ( ) changes, i.e., the set changes or the ordering of elements of the set changes. Step 5. If , then STOP Else go to Step 2. V. COMPUTATIONAL RESULTS This section first evaluates the performance of the Lagrangian relaxation heuristic, and then investigates how the optimal distribution network depends on different key parameters. The heuristic solution procedures described in the previous section were coded in C++ and the numerical experiments carried out using a Pentium IV, 2.80 GHz, and 512 MB of RAM. We tested our algorithms on networks up to 150 retailers. The same data set is used to generate all problem instances as follows: is, respectively, • The number of retailers’ location # 10, 20, 40, 50, 60, 70, 80, and 150 retailers (each retailer location can be selected to host a DC). • Retailer demand: For each retailer, the mean daily de. mand is randomly generated with : For each potential location, the • Supply lead times average supply lead time is randomly generated with . • The standard deviations of daily demand and supply lead time are randomly generated with and .

, transportation cost , • Fixed facility location cost are randomly generated with reand shipment cost , and spect to . • Standard normal variate (target service level %). Table I summarizes the remaining parameters. For each set of parameters, NBE problem instances are generated and the following are computed. #

Number of opened DCs; Lower bound of the Lagrangian relaxation; Upper bound (best feasible solution found); Duality gap defined as Computational time in seconds.

.

A. Performances of Solution Procedure Consider first the case with . Table II presents the (million of dollars). The results where costs are expressed in lower bounds of the Lagrangian relaxation are very tight. of reIn the second experiment, for each fixed number # tailers, five problem instances are randomly generated. Numerare ical results of the five problem instances for each fixed # reported in Table III, which gives the minimum and the maximum of LB, UB, GAP, and #DC.

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Fig. 2. GAP versus number of iterations (#RL = 80).

Fig. 4. Number of DCs versus transportation costs ( = 1).

Fig. 3. Evolutions of LB and UB (#RL = 80). TABLE IV PERFORMANCE OF SOLUTION PROCEDURE (#RL = 80)

From Tables II and III, the feasible solutions of the Lagrangian relaxation heuristic are near-optimal with duality gap within 2%. The number of located DCs increases with the number of retailers. The CPU time increases with the problem size. It takes about 10 min for the problem with # retailers. and # reConsider now the case with tailers to investigate the convergence rate of our Lagrangian relaxation heuristic. Figs. 2 and 3 illustrate the evolution of the duality gap (GAP), LB and UB versus the number of iterations (#Iters) of the algorithm. The algorithm converges after 152 itand an upper erations with a lower bound . bound Consider the case with # retailers and NBE varying from 5 to 20. The numerical results reported in Table IV demonstrate that the proposed Lagrangian relaxation approach is quite robust and the duality GAP remains within 1%. B. Sensitivity Analysis We now investigate how the optimal distribution network depends on key parameters. More specifically, we consider the im-

Fig. 5. Number of DCs versus inventory costs (

= 1).

pact of the transportation costs, inventory holding costs, and the variances of demands and supply lead times. Consider first the impacts of transportation costs and inventory holding costs. For this purpose, we introduce two parameters and , respectively: : weighting factor of transportation cost; : weighting factor of inventory cost; and replace transportation and inventory holding costs by, respectively, . For our experiments, we varied separately and . To investigate only the impacts of inventory costs, is set to 1 and varies from 0.1 to 5. To investigate the impacts of transportation costs, is set to 1 and varies from 0.1 to 2. Table V presents the results with . Fig. 4 shows the relation between #DCs and transportation costs. Fig. 5 demonstrates the relation between #DCs and inventory costs. Table V shows that when the transportation costs and the inventory costs increase, the total cost of the distribution network increases. Nevertheless, under our experimental setting, we realized that the total cost of the optimal distribution network is more sensitive to the transportation costs than inventory costs. Furthermore, it seems that the problem becomes harder and the duality gap increases when the transportation costs are small, i.e., , and when the inventory costs are large, i.e., . This shows that transportation cost is a major component in the design of such distribution networks. Fig. 4 demonstrates the following. 1) When the transportation cost increases, the number of opened DCs increases. Intuitively, this result is quite

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TANONKOU et al.: DESIGN OF STOCHASTIC DISTRIBUTION NETWORKS USING LAGRANGIAN RELAXATION

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TABLE V IMPACTS OF TRANSPORTATION COSTS AND INVENTORY COSTS

Fig. 6. Number of DCs versus variability of lead time and demand.

We introduce a weighting fact and replace the standard deviaby, respectively, and . Fig. 6 shows the tions and change in the number of DCs when varies from 0 to 2. 1) Increasing the variability of retailers’ demands and DCs supply lead times reduces the number of DCs. This is in line with the risk-pooling concept in supply chain management. When the variability is high, the gain of safety stock and running inventory costs thanks to the reduction of DC numbers is more important than the increase of transportation costs. 2) The relation between the number of DCs and the variability of retailers’ demands and DCs supply lead times is fairly complex. 3) The variability of retailers’ demands and DCs supply lead times has significant impacts on the distribution network configuration. This has actually been pointed out by several simulation study of supply chains and supply chain design surveys. VI. CONCLUSIONS AND PERSPECTIVES natural as, when the transportation cost increase, DCs should be located closer to retailers in order to reduce transportation costs of DCs to retailers. 2) The relation between the number of DCs and the transportation costs is nearly linear. 3) More DCs are needed when the number of retailers increases. This result seems quite natural as, when the number of retailers increases, the economy of scale justifies the opening of more DCs. Fig. 5 shows the following. 1) When the inventory costs increases, the number of DCs decreases. In fact, reducing the number of DCs results in the reduction of running inventory and safety stocks and allows better accommodate the increase of inventory costs. 2) The relation between the number of DCs and the inventory costs is rather complex. For example, in the case where # , #DCs decreases for [0.1, 0.7], it remains constant for [0.7, 1], and it increases again when . 3) As in Fig. 4, the number of DCs increases when the number of retailers increases. This can be explained as in the case of Fig. 4. We now analyze the impacts of random demands and random supply lead times on the optimal configuration of the network.

In this paper, we have presented a Lagrangian relaxation- based approach to solve a single-commodity distribution network design problem with random demands and random supply lead times. A nonlinear integer-programming model was proposed. The model determines the location of DCs and the allocation of retailers to the DCs. The goal is to minimize the total fixed DCs location costs, running inventory and safety stock costs at the DCs and transportation costs through the network, while ensuring a given retailer service level. The resulting problem is difficult since it incorporates nonlinear working-inventory costs and nonlinear safety stock inventory costs. Computational results were presented and analyzed showing the effectiveness of the proposed approach. Future research includes design of distribution networks operating under other inventory policies such as base stock, or , having multiple suppliers, multiple commodities and with limited capacities for both DCs and suppliers. Another interesting research is to take into account correlated retailer demands. Market demands at various locations are usually highly correlated, especially through promotion efforts. Taking into the correlation of demands in distribution network design leads to more realistic but very complicated inventory-location models. The proposed approach does not directly apply and new optimization methods are needed.

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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008

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Guy-Aimé Tanonkou received the M.S. degree in applied mathematics and the Ph.D. degree in automation from the University of Metz, Metz, France, in 2003 and 2007, respectively. He is a researcher at INRIA (The French National Institute for Research in Computer Science and Control), Metz, France, since December 2004. He is the President and founder of the Association of Scientific Collaboration between Africa and Luxembourg (ACSAL). His primary research interests are supply chain management, inventory-location problem, and supply chain reliability.

Lyes Benyoucef received the M.S. and Ph.D. degrees both in operations research from the National Polytechnic Institute of Grenoble, Grenoble, France, in 1997 and 2000, respectively. He is a Research Associate at INRIA (The French National Institute for Research in Computer Science and Control) since September 2001. With INRIA, in addition to the fundamental research, he is involved in the following European projects: Optimization Methodologies for Networked Enterprises (ONE), Thematic Network on Extended Enterprises (TNEE), Green Rail Freight Transport for Chemical Goods (GrailChem), and Innovative PROduction Machines and Systems (I*PROMS). His research interests are in the broad area of supply chain management. He is especially interested in modeling, performance evaluation, simulation and optimization of supply chains, and E-procurement.

Xiaolan Xie received the Ph.D. degree from the University of Nancy I, Nancy, France, in 1989, and the Habilitation à Diriger des Recherches degree from the University of Metz, Metz, France, in 1995. Currently, he is a Full Professor of Industrial Engineering and Head of the Department of Health Care Systems Operation, Ecole Nationale Supérieure des Mines de Saint Etienne (ENSM.SE), a 190-year-old top ten French graduate engineering school. Before joining ENSM.SE, he was a Research Director at the Institut National de Recherche en Informatique et en Automatique (INRIA) from September 2002 to March 2005, a Full Professor at the Ecole Nationale d’Ingénieurs de Metz (ENIM), a French engineering school from 1999 to 2002, and a Senior Research Scientist at INRIA from 1990 to 1999. His research interests include design, planning and scheduling, supply chain optimization, performance evaluation, and maintenance of manufacturing, and healthcare systems. He is author/coauthor of over 200 publications including over 70 journal articles in leading journals of related field and four books on Petri nets. He has rich industrial application experiences with European industries. He is the French-Leader for the European Project FP6-IST6 IWARD on Swarm Robots for Health Services, for FP6-NoE I*PROMS on Intelligent Machines and Production Systems, for the GROWTH-ONE Project for the Strategic Design of Supply Chain Networks, and for the GRWOTH Thematic Network TNEE on Extended Enterprises. Dr. Xie is an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING and the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and was an Associate Editor of the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION (2001–2004). He has served as a Guest Editor of a 2001 Special Issue for the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION on Semiconductor Manufacturing Systems, a 2001 Special Issue for the International Journal of Production Research on Modeling, Specification and Analysis of Manufacturing Systems, a 2004 Special Issue for the International Journal of Production Research on Modelling of Reactive Systems, and a 2005 Special Issue for IJCIM on Discrete Event System Techniques for CIM. He has been served on International Program Committees for many conferences.

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