A linear relaxation-based heuristic approach for logistics network design

Computers & Industrial Engineering 59 (2010) 964–975

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Computers & Industrial Engineering

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A linear relaxation-based heuristic approach for logistics network design

Phuong Nga Thanh a, Olivier Péton a,⇑, Nathalie Bostel b

a École des Mines de Nantes, IRCCyN 4 rue Alfred Kastler, B.P. 20722, 44307 Nantes Cedex 3, Franceb IUT de Saint-Nazaire, IRCCyN 58 rue Michel Ange, B.P. 420, 44606 Saint-Nazaire Cedex, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 February 2009Received in revised form 7 September 2010Accepted 8 September 2010Available online 19 September 2010

Keywords:Heuristic methodsLogistics network designMixed integer linear programmingLinear relaxationRounding procedures

0360-8352/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.cie.2010.09.007

⇑ Corresponding author. Tel.: +33 251 858 313.E-mail addresses: [email protected] (P.N.

(O. Péton), [email protected] (N. Bostel).

We address the problem of designing and planning a multi-period, multi-echelon, multi-commoditylogistics network with deterministic demands. This consists of making strategic and tactical decisions:opening, closing or expanding facilities, selecting suppliers and defining the product flows. We use a heu-ristic approach based on the linear relaxation of the original mixed integer linear problem (MILP). Themain idea is to solve a sequence of linear relaxations of the original MILP, and to fix as many binary vari-ables as possible at every iteration. This simple process is coupled with several rounding procedures forsome key decision variables. The number of binary decision variables in the resulting MILP is smallenough for it to be solved with a solver. The main benefit of this approach is that it provides feasible solu-tions of good quality within an affordable computation time.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Logistics network design is concerned with many strategic andtactical decisions. We consider a mixed integer linear programme(MILP) for designing and planning a complex supply chain over ahorizon of a few periods (typically 5 years). An optimal configura-tion must enable products to be produced and delivered to the cus-tomers at the lowest cost while satisfying a required service level.

It is widely considered as unrealistic to handle universal modelsincluding concerns about logistics, risk management, finance, so-cial and environmental issues, etc. As pointed out in Min and Zhou(2002), one has to know which essential components must bemanaged and then establish specific supply chain goals. In thiswork, we focus on the logistics issues and, more particularly, onthe location and allocation decision variables: location of ware-houses and production plants, allocation of warehouses to produc-tion plants, allocation of customer demand points to warehouses,definition of the material flows between the nodes of a complexlogistics network. The MILP under consideration can be viewedas a facility location problem, with multiple facilities, periodsand commodities, and additional constraints. Although a few tacti-cal issues are addressed by the model, the problem is clearly a stra-tegic one.

To make sure that the proposed solutions are compatible withthe industrial context and robust toward uncertainties in some

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Thanh), [email protected]

parameters (costs, demands), it is strongly advised to run severalcomputations of the model, with various network configurationsor logistics scenarios. Thus, the model is likely to be run repeat-edly until a strategic decision is taken. One prerequisite is thatthe underlying optimisation methods provide very good approxi-mations of the optimal solution within an acceptable amount oftime.

We propose a heuristic algorithm based on successive linearrelaxations of location decision variables, followed by correctionprocedures. The main contribution of this method is to yield feasi-ble solutions of good quality within a limited computation time.The paper is organised as follows. In Section 2, we recall the mainconcepts of supply chain design and planning and introduce theoptimisation problem. In Section 3, we present the linear relaxa-tion-based algorithm. In the computation experiments of Section4, our goal is to evaluate the performance as well as the limitationsof the approach.

2. A dynamic model for logistics network design

2.1. A brief review of the literature

The multi-period planning of a supply chain is an NP-hardproblem that has been addressed by many authors in the recentand abundant literature. There is such a large variety of enter-prise logistics networks that completely generic models wouldprobably not fit any company. The most recent models includemany features with the idea of reflecting some real cases orfocusing on some particular aspects of the location problems.

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Among the most widespread characteristics in the recent modelsare:

– a supply chain with multiple echelons and multiple products orfamilies of products,

– dynamic models where the data and variables may change atevery period,

– complex product flows, with an exchange of products betweenplants or warehouses, direct deliveries to some customers,reverse logistics, re-manufacturing, etc.

– a variety of constraints: competition or budget constraints, etc.– complex cost structures: fixed and variable costs, linear or non-

linear costs,– hybrid strategic/tactical models with inventories: average,

safety or cyclic inventories.

Dias, Captivo, and Climaco (2006) worked on the re-engineeringof a network composed of facilities and customers. The authorssuppose that facilities can be open or closed and can reopen morethan once during the planning horizon. The model is solved by pri-mal–dual heuristic methods. Melo, Nickel, and Saldanha da Gama(2006) aimed at relocating the network with expansion/reductioncapacity scenarios. Capacity can be exchanged between an existingfacility and a new one, or between two existing facilities undersome conditions. Vila, Martel, and Beauregard (2006) proposed adynamic model in a much more specialised context. They consid-ered an application in the lumber industry, but their model canbe applied to other sectors.

The methods for solving logistics network planning and designuse the classical tools of operations research. The exact methodsinclude branch-and-bound, Benders decomposition or the use ofa commercial solver (Bidhandi, Yusuff, Ahmad, & Bakar, 2009;Canel, Khumawala, Law, & Loh, 2001; Hamer-Lavoie & Cordeau,2006; Martel, 2005; Melo et al., 2006).

As emphasised by Melo, Nickel, and Saldanha da Gama (2009),when the number of discrete variables is large, realistically sizedproblems can only be solved with a heuristic method. Our workfalls into this category of problem. Most of the metaheuristic ap-proaches are proposed for basic models like static models for sim-ple network planning. Filho and Galvpo (1998) used tabu search tosolve a concentrator location problem. Syam (2002) mixed simu-lated annealing and Lagrangean relaxation to solve a 3-echelonproblem. Other authors have used genetic algorithms (Gong, Gen,Yamazaki, & Xu, 1997; Jaramillo, Bhadury, & Batta, 2002).

Surprisingly enough, few heuristic methods have been reportedin the specialised literature, although they seem to be efficient forthe most complex problems. The most common heuristic methodsuse some ‘‘add” and ‘‘drop” procedures (Saldanha da Gama & Captiv-o, 1998), or rely on primal–dual methods (Dias et al., 2006) orLagrangean relaxation (Hinojosa, Kalcsics, Nickel, Puerto, & Velten,2008; Hinojosa, Puerto, & Fernández, 2000; Pirkul & Jayaraman,1998). Heuristic methods based on simple neighbourhood struc-tures and local improvements are seldom used because of the heter-ogeneity of the entities considered (suppliers, plants, warehouses,transport, customers, machines). More efficiency might be achievedby recent metaheuristic methods that combine various neighbour-hoods within adaptive algorithms.

2.2. Description of the problem

We consider a network composed of four layers, like the one de-picted in Fig. 1.

The first layer consists of a set of potential first tier suppliersthat provide the company with raw materials. Production stepscomposed of plants make up the second layer. Product storageand distribution steps, carried out by warehouses, are the main

constituents of the third layer. Finally, the fourth layer is composedof retailers or final customers. We adopt a flexible network struc-ture: products can be transferred between plants or delivered di-rectly from plants to important customers. This approach enablesus to model various practical situations. For example, we can dealwith the second tier suppliers once the first tier suppliers havebeen selected. In this case, the first tier suppliers should be consid-ered as production centres in the second layer. Some other changesin the network configuration would require a modification of thecurrent mathematical model, but the main conclusions of this pa-per remain valid for many location problems in the context of acomplex supply chain.

A complete description of the problem and the mathematicalmodel are presented in Thanh, Bostel, and PTton (2008) and canbe found in the Appendix. In this paper, we only recall the maincharacteristics of the model.

We consider a multi-commodity supply chain where everyproduct has its own bill of materials. The manufacture of theseproducts may be decomposed into different steps performed in dif-ferent production plants. Every plant or warehouse has a limitedcapacity along with lower and upper limits for the level of utilisa-tion (running a plant at 1% or at 100% of its capacity is not author-ised). We also model the possibility of increasing the capacity of anexisting facility by building some physical extensions, calledcapacity options.

Our purpose is to make strategic decisions over a multi-periodplanning horizon. These decisions concern the selection of suppli-ers, the opening or closing of facilities, capacity planning for openfacilities, and production and distribution management. Produc-tion management consists of allocating manufactured productsto open plants and managing subcontracted products (we supposethat the company can subcontract a part of its production). Distri-bution management consists of allocating the customers to open orrented warehouses. The model does not include any single-sourc-ing constraint. Some inventory management aspects are also in-cluded as they influence strategic decisions such as capacityplanning. We suppose that the inventories are planned only inwarehouses, not in plants.

The mathematical model includes four types of binary decisionvariables, most of which are related to the location of facilities. Letus define the binary variables xt

i that have the value 1 if a facility iis active at period t, and 0 otherwise. Depending on the context, avariable xt

i stands for a supplier, a plant or a warehouse. A sup-plier is said to be active if it delivers at least one raw material.A plant or a warehouse is active if it is open in the correspondingperiod.

As shown in Thanh et al. (2008), variables xti concerning plant

and warehouse locations are the core of the model. They have adirect influence on the largest part of the objective function.Other binary variables concern the capacity options and theselection of suppliers. The MILP also includes continuous decisionvariables that model the product flows along the network (quan-tities of products transferred from the suppliers to the plants, thewarehouses and the customers). Since the remainder of thepaper only uses variables xt

i , we do not give more details aboutother variables. The interested reader will find them in theAppendix.

The cost structure consists of two parts. The fixed costs are re-lated to the supplier selection, the opening, the closure or theextension of facilities and, finally, to the running of existing facili-ties. The variable costs are associated with the main logistics oper-ations: production, storage, transportation between entities anddistribution to the customers. We consider one linear objectivefunction, which is the sum of all the fixed and variable costs. With-out loss of generality, this function is considered to be time-independent.

Fig. 1. The logistics network considered: (A) some facilities may not be used. (B) Suppliers can deliver to several manufacturing centres and manufacturing centres can bedelivered to by several suppliers. (C) Production of goods can be split among various manufacturing centres. In this case, there must be a link between the correspondingfacilities. (D) There can be direct deliveries from manufacturing centres to some important customers. (E) The customers can be delivered to by various distribution centres(no single-sourcing constraint).

966 P.N. Thanh et al. / Computers & Industrial Engineering 59 (2010) 964–975

The set of linear constraints can be divided into four categories:demand satisfaction constraints, capacity limitation constraints,coherence constraints (for example, only open facilities can pro-duce) and integrality and non-negativity constraints.

The MILP models many realistic hypotheses that have not beensimultaneously taken into account in the literature (Gebennini,Gamberini, & Manzini, 2009). The model is deterministic but,through the analysis of a set of discrete scenarios, we can handlethe uncertainty of some parameters and propose a robust solutionfor the strategic decisions. It focuses on strategic decisions but alsointegrates some tactical decisions or seasonal effects: maximumallowed level of inventory, maximum production capacity of plant,etc. This integration of tactical/operational decisions within strate-gic models is increasingly common in facility location (Melo et al.,2009). To conclude, the considered model is general enough to cov-er a wide variety of situations and provide answers to major stra-tegic decisions. It is particularly suited to companies with stable orpredictable activity.

3. Linear relaxation and rounding methods

3.1. LP-rounding

The linear relaxation of an MILP consists of replacing everyintegrality constraint by its continuous counterpart. In the con-text of minimisation, solving this relaxation gives a lower boundof the optimal value of the MILP. The idea of LP-rounding is tosolve the linear relaxation of some MILPs, and to round the frac-tional variables to recover integer feasible solutions. Despite itspoor formal guarantee of performance, LP-rounding is knownto yield good lower bounds for some assignment or locationproblems (Benders & van Nunen, 1983). It has been recently ap-plied to the general assignment problem (French & Wilson, 2007)and lot-sizing problems (Hardin, Nemhauser, & Savelsbergh,2007). In the field of facility location or logistics network plan-ning, linear relaxation-based methods have been used for thesingle facility location and more complex models (Levi, Shmoys,& Swamy, 2004). Cordeau, Pasin, and Solomon (2006) solve theLP relaxation and reintroduce integrality constraints later. Addi-

tional cuts are generated until an optimal integer solution isfound. Velásquez, Melo, and Nickel (2005) propose an LP-round-ing method for a supply chain design problem. Due to the pres-ence of budget constraints, their approach does not always yieldfeasible solutions.

From a theoretical point of view, the quality of the linear relax-ation depends mostly on how close the formulation is to the com-plete characterisation of the convex hull of the original problem(Barros, 1998). Robert, Dejax, and Péton (2006) solved a strategicsupply chain planning problem that was very similar to the presentone. They observed that more than 60% of the binary variableswere directly set at 0 or 1 in the optimal solution of the linearrelaxation. This finding paved the way for the present study.

3.2. Description of the algorithm

The proposed LP-rounding performs successive linear relax-ations and rounding steps until enough variables have been setat 0 or 1. The resulting MILP is eventually handled by an LP solver.Location binary variables xt

i play a key role, especially those relatedto the plant and warehouse locations. In the original MILP, the xt

i

are binary decision variables. They become real variables in the lin-ear relaxation. The algorithm aims to fix the relaxed variables, inother words to set value xt

i ¼ 0 or xti ¼ 1 for some facility i and

some period t. Once fixed, xti appears as a constant in the sequel

of the process. When no further relaxed variable can be fixed at0 or 1, the resulting MILP is likely to be reduced enough so that asolver can find an optimal solution. The level r of already fixed vari-ables is an important performance factor when solving this result-ing MILP. If it is not large enough, we resort to supplementaryoperations in order to reach a target level s of fixed variables.The procedure that increases the number of fixed variables is usedonly for plant and warehouse location variables. The main schemeof the heuristic method is depicted in Fig. 2.

During Steps 1 and 3, inappropriate variable fixings may resultin infeasible solutions. Thanh et al. (2008) noticed that the MILPsolver was not always able to find feasible solutions for the largestinstances: this is one of the main concerns of our algorithm. Wepoint out that at each iteration of Step 1, a new linear programme

Fig. 2. The linear relaxation-based heuristic. In Step 1, we solve the linearrelaxation of the current MILP and round all decision variables xt

i whose relaxedvalues have a strong tendency toward 0 or 1. In practice, we look for every xt

i suchthat 0 6 xt

i < �0 or �1 < xti 6 1, where �0 and �1 are some given threshold values

close to 0 and 1, respectively. In Step 2, we wish to increase the number of fixedbinary variables x in order to reach the target level s. The distinct procedures forwarehouse and plant location variables are detailed in Sections 3.4 and 3.5. In Step3, the resulting optimisation problem is solved with an MILP solver and then thesolution is corrected.


denoted (LP) is created by rounding some decision variables to 0 or1. The new (LP) is solved at the subsequent iteration. The feasibilitytest checks if the current solution is feasible for the original (LP) (itcan be skipped at iteration 1). When the feasibility test exhibitsinfeasible solutions, a correction procedure repairs the currentsolution by enforcing the fixing of location variables. Two distinctprocedures are developed for the location variables associated withthe warehouses and the plants.

3.3. Correction procedure (Steps 1 and 3)

The correction procedure is divided into two phases. As a pre-caution, we reduce �0 and increase �1 to avoid obtaining the samesolution in the next iterations. This also decreases the probabilityof obtaining an infeasible solution.

We resort to the feasibility pump method for mixed-integerproblems (Bertacco, Fischetti, & Lodi, 2007; Fischetti, Glover, &Lodi, 2005). Let P = {x: Ax P b} denote the polyhedron associatedwith the linear relaxation of a given MILP. Given an integer infea-sible solution ~x R P, the feasibility pump method seeks the nearestfeasible solution x 2 P regardless of whether it is integer or not. Weminimise the distance between two vectors:

Min d~xðxÞ ¼Xi2Ijxi � ~xij; x 2 P

Function d~xðxÞ is not linear. However, for special cases whenexi 2 f0; 1g, it can be linearised as follows:

d~xðxÞ ¼X

i

exið1� 2xiÞ þ xi

dexiðxiÞ ¼

xi if exi ¼ 01� xi if exi ¼ 1

�In practice, every variable is not rounded at the current iteration,the feasibility pump has to consider a subset of variables. We solve:

Min d~xðxÞ ¼Xi2Iðexið1� 2xiÞ þ xiÞIndðiÞ

where Ind(i) = 1 if the variable exi is fixed, and 0 otherwise. For theobtained feasible solution x, we fix all binary variables whose valuesare exactly 0 or 1.

3.4. Setting binary variables for warehouse location (used in Step 2)

The desired level s of fixed location variables before Step 3 is akey parameter of the scheme in the sense that it influences thecomputation time of Step 3. If the level s has not been reached afterStep 1, it is necessary to increase the level of already fixed vari-ables. We resort to a distinct procedure to increase the level offixed plant location variables and warehouse location variables.

Concerning binary variables for warehouse location, we use thefollowing two rules: round variables that are close to 0 or 1 (withlooser thresholds c0 and c1 such that 0 6 �0 6 c0 and c1 6 �1 6 1)and round the largest fractional value to 1. These rules are depictedin Algorithm 1.

Algorithm 1. Set binary variables for warehouse location

Arguments:c0: pre-defined threshold to set a binary variable at 0c1: pre-defined threshold to set a binary variable at 1r: current level of fixed warehouse location variabless: desired level of fixed warehouse location variables

while r < s dofor every warehouse location fractional variable xi do

if xi < c0 thenxi 0

end ifif xi P c1 then

xi 1end if

end forif some new variables are set at 0 or 1 then

Solve (LP)if (LP) is infeasible then

Relax all xi that were fixed at 0end if

end ifif no new variable is set at 0 or 1 then

Select the largest fractional value �xi and all relatedvariables �xj

�xi 1�xj 1

end ifSolve (LP)Update r

end while

As we want to fix more binary variables than in Step 1, c0 and c1

must be further from 0 and 1 than �0 and �1. If the obtained solu-tion is infeasible due to the lack of warehouse capacity, we relax allbinary variables that were fixed at 0 at the current iteration.


The second part of Algorithm 1 gives priority to the variablesthat are close to 1. The variable with the largest fractional valueis selected and set at 1. In the underlying logistics assumptions,the plants or the owned (private) distribution centres cannotchange their status more than once during the whole planninghorizon. Thus, if the corresponding location variable xt

i is changedfrom 0 to 1, every variable xt0

i with t0 > t is also automatically setat 1. This is the meaning of related variables in Algorithm 1.

When fixing a variable, values 0 and 1 are not consideredevenly. Indeed, setting a variable at 0 reduces the capacity andcan lead more easily to infeasible solutions. However, setting at 1a variable whose value is very far from 1 increases the cost and fa-vours a reduced utilisation of the corresponding facility. For thisreason, we resort to this rounding only when no new variablecan be fixed with the help of thresholds c0 and c1.

3.5. Setting binary variables for plant location (used in Step 2)

Algorithm 1 does not apply to the plant variables because themodel includes the possibility of subcontracting a part of the pro-duction. Moreover, the capacity of subcontracted plants is sup-posed to be unlimited. Hence, Algorithm 1 would always find afeasible solution for plant capacities with no guarantee of minimis-ing the objective function. Algorithm 2 is a greedy algorithm thatbuilds a feasible solution iteratively by rounding the largest frac-tional value up to 1.

Algorithm 2. Set binary variables for plant location

Parameters:z�: objective function value of the previous iterationz+: objective function value of the current iteration

for every plant location fractional variable xi doxi 0end forSolve (LP) and save z+

Relax all variables xi that were fixed at 0repeat

z� = z+

Select the largest fractional value �xi and all related variables�xj

�xi 1�xj 1xk 0"k – i,jSolve (LP) and save z+

Set dz = z+ � z�

if dz 6 0Relax all variables xk that were fixed at 0Solve (LP)

end ifuntil dz P 0

The main objective in Algorithm 1 is to increase the number offixed variables while keeping the solution feasible. In Algorithm 2,feasibility is not an issue: the aim is to minimise the objectivefunction.

4. Computational tests

The algorithm is implemented in C language while the linearrelaxations and the resulting problem in Step 3 are solved withthe MILP solver Xpress-MP (2007). In addition, to evaluate the per-formance of the method, each instance is solved separately with

Xpress-MP. The experiments are performed on a Pentium IV,3.2 GHz processor with 1 GHz of RAM.

4.1. Data description

We generated a set of 450 test instances, organised into 15 fam-ilies with logistics networks of increasing size. Families S1–S5(small-size instances) consider between 15 and 21 facilities and100–160 customers; families M1–M5 (medium-size instances)consider between 21 and 27 facilities and about 200 customers;families L1–L5 (large-size instances) consider up to 35 facilitiesand 300 customers. Each family is divided into three sub-familiesof increasing difficulty due to the load/capacity ratio. Ten instanceswith homogeneous characteristics are generated for each sub-fam-ily. The details are given in Table 1. This table also indicates thenumber of corresponding binary variables in the MILP. In the lastcolumn, we distinguish the number of binary variables corre-sponding only to location decisions. For every instance, we con-sider a strategic time horizon of five periods (of typically 1 year).

When generating the data sets, we followed the ideas of variousauthors: the customers’ demands are derived from Melo et al.(2006); the existing arcs in the logistics network were generatedaccording to a random procedure described by Cordeau et al.(2006); the capacity of the potentials and optional facilities comesfrom Melkote and Daskin (2001); the idea of transition phases afteropening or before closing a facility as well as the variable costs areinspired from Melachrinoudis and Min (2000); finally the fixed costsfollow the same formats as in Cortinhal and Captivo (2003). Everyfamily has exactly 30 instances, which have been confirmed as beingfeasible. Every family comprises three sub-families with decreasingload/capacity ratios: 10 instances of each of Type 1 (tight capacity),Type 2 (average capacity) and Type 3 (loose capacity). For example,for family S1, these sub-families are denoted S11, S12 and S13 respec-tively. The complete details can be found in Thanh (2008).

4.2. Tuning the solver and the algorithm parameter values

We carried out some experimental tests to determine goodparameter values for the solver and the algorithm. The XpressOptimizer uses the branch-and-bound method to solve the MILPand the parameters concern the node selection and the enforcingprocedures. We conclude that the parameters ensuring a goodtrade-off between the computation time and the quality of thesolution are the default parameters except for the parameternamed CUTSTRATEGY. This is set at the default value for Type 1 in-stances (the cut strategy is automatically selected) and no cutstrategy is considered for Types 2 and 3 instances.

Concerning the parameters of the algorithm, we use �0 = 0.03,�1 = 0.95, c0 = 0.05, and c1 = 0.9. The desired level of fixed decisionvariables is set at s = 0.6 for the families S1–S5, s = 0.7 for the fam-ilies M1–M5 and s = 0.8 for the families L1–L5. All the numericalresults reported were obtained with these parameter values. Weobserved that the results were not much influenced by reasonablechanges in these parameters.

4.3. Numerical results with the MILP solver

The model was first solved with Xpress-MP with a limited com-putation time of 1 h for families S1–S5, 2 h for families M1–M5, and3 h for families L1–L5. We propose this limitation after observinghow the objective function decreases during the computation time.In the first minutes of computation, the objective function generallydecreases quickly and then stabilises. For the largest datasets, veryfew improvements are observed after 2–3 h of computation. More-over these improvements are highly unpredictable. Thus, wedecided to stop the computations after a predefined time limit.

Table 1Description of the test instances.

Family # of Suppliers # of Plants # of Warehouses # of Customers # of Products Binary variables Location variables

S1 15 10 5 100 10 750 75S2 15 10 5 130 10 750 75S3 17 12 5 130 12 1020 85S4 17 12 7 160 12 1060 95S5 19 14 7 160 14 1180 105M1 19 14 7 180 14 1275 105M2 19 14 7 180 16 1275 105M3 21 16 9 180 16 1445 125M4 21 16 9 210 16 1445 125M5 23 18 9 210 16 1575 135L1 23 18 9 240 16 1575 135L2 23 18 11 240 18 1845 145L3 25 20 11 240 18 1995 155L4 25 20 13 270 18 2035 165L5 27 22 13 270 18 2185 175

Column 1 indicates the family of the instance. S, M and L stand for small, medium and large-size, respectively. Each family includes 30 instances. Columns 2–6 describe themain characteristics of every family: number of suppliers, plants, warehouses, customers and products. Column 7 shows the number of binary variables in the correspondingMILP. Column 8 details the number of binary variables representing the location decisions.


The solver was able to find feasible solutions for 446 out ofthe 450 instances. The four remaining instances can be foundin families L2, L3 and L5. We observed that for these instances,much more than 3 h would be required to obtain feasible solu-tions. Every instance of families S1–S5 could be solved to opti-mality with computation times varying from 65 to 1419 s.Seventy-seven instances (out of 150) of families M1–M5 weresolved to optimality with computation times varying between22 min and 2 h. An average optimality gap (relative gap to thelower bounds) for M instances increased from 0% (family M1)to 1.4% (family M5). Finally, only 1 instance in families L1–L5could be solved to optimality. The average optimality gap after3 h of computation was only 0.75% for family L1 but 3.67% forfamily L5 (and even 6% for the 10 instances of Type 3 in familyL5). The computation time grew slowly for families S1–S5 butquickly for families M1–M5. Except for the unique solved in-stance, computation was stopped after 3 h and we recordedthe best feasible solution found so far.

4.4. Numerical results with the LP-rounding algorithm

Tables 2–4 report the results of our LP-rounding algorithm forsmall, medium and large-size instances, respectively.

Table 2Results of the LP-rounding method for the small-size (S) instances.

Family tH Steps 1 and 2 (%) Opt. 3 (# out of 10)

S11 25 68 10S12 28 72 10S13 29 83 10S21 30 71 10S22 39 75 10S23 39 74 10S31 58 60 10S32 70 76 10S33 85 72 10S41 114 61 10S42 121 67 10S43 136 67 10S51 194 57 10S52 209 54 10S53 211 57 10

For each family and each type of sub-family, 10 instances are solved. Column 2 reports thethe percentage of this time spent on Steps 1 and 2. Column 4 shows the number of resullast four columns detail the use of the correction procedure in Steps 1 and 3. In each casenecessary (columns 5 and 7) and the average number of calls to the correction procedu

The method combines three ways of assigning the value 0 or 1to a binary variable:

– the value is set directly at 0 or 1 by the LP relaxation,– the value is set at 0 or 1 by rounding rules,– the value is finally set at 0 or 1 by an exact MILP method.

We consider that solving the LP relaxation is the most practi-cal and efficient way of setting a variable at 0 or 1, followed bythe rounding rules. Solving the MILP is the most demanding pro-cedure. The objective is to find a good trade-off between the opti-mality gap and the computation time. Thus, an ideal run wouldavoid performing many correction loops and spending too muchtime on Step 3. This motivates our choice for reporting the num-ber of calls to the correction procedures and the time dedicated toStep 3.

Fig. 3 shows the evolution of the computation time when theinstance size increases. We make the distinction between eachtype of instance (loose or tight capacity). This highlights the factthat instances with tight capacity constraints are slightly easierto solve.

The method requires, on average, less than 1000 s for small andmedium-size instances and less than 1 h for the largest instances,

Corrections Step 1 Corrections Step 3

(# out of 10) (# of calls) (# out of 10) (# of calls)

6 1.5 0 –1 2 0 –8 1.8 0 –2 1.5 0 –5 2 0 –4 1.75 1 13 1.7 0 –5 1.6 0 –6 1.7 2 10 – 0 –2 1 0 –3 1.6 0 –1 1 0 –0 – 0 –2 1.5 0 –

average computation time of the heuristic method (in seconds). Column 3 indicatesting problems (out of 10 instances) that could be solved to optimality at Step 3. Thewe first provide the number of instances (out of 10) for which a correction step wasre (columns 6 and 8).

Table 3Results of the LP-rounding method for the medium-size (M) instances.

Family tH Steps 1 and 2 (%) Opt. 3 (# out of 10) Corrections Step 1 Corrections Step 3


M11 398 79 10 3 1 0 –M12 420 76 10 2 1 0 –M13 458 77 10 5 1.2 0 –M21 497 64 10 1 1 0 –M22 533 71 10 4 1.5 1 1M23 589 79 10 5 1.2 0 –M31 556 63 10 1 2 0 –M32 615 76 10 5 1.6 0 –M33 626 74 10 6 1.5 3 1M41 717 79 10 7 1.42 1 1M42 735 83 10 7 2 2 1M43 753 73 10 7 1.42 1 1M51 733 81 10 2 1 0 –M52 785 81 10 7 2.4 0 –M53 855 67 10 5 1.7 1 1

Fig. 3. Computation time of the heuristic for tight, average and loose capacity instances. The horizontal axis represents each family of instances, increasing in size. The verticalaxis represents the computation time (in seconds). The dots represent the average results over the 10 instances of each category of capacity (tight, average or loose) and eachfamily of instances.

Table 4Results of the LP-rounding method for the large-size (L) instances.

Family tH Steps 1 and 2 (%) Opt. 3 (# out of 10) Corrections Step 1 Corrections Step 3


L11 972 87 10 7 1.3 0 –L12 1154 85 10 8 1.25 0 –L13 1203 83 10 7 1.7 0 –L21 1964 85 10 3 1.3 0 –L22 2111 85 10 8 1.2 2 1L23 2337 83 10 3 1.3 0 –L31 2323 82 10 6 1.5 0 –L32 2551 81 10 6 1.4 0 –L33 2619 78 10 6 1.6 0 –L41 2246 73 10 2 2 1 1L42 2614 77 9 2 2 1 1L43 2666 75 9 2 2 1 1L51 5141 68 7 5 2 2 1L52 7091 55 5 7 2 3 1L53 7154 59 5 7 2 3 1


Table 5Efficiency of the heuristic method for small-size (S) instances.

Family tS tH Dt (%) Dz (%)

S11 65 25 �44 0.80S12 89 28 �61 0.80S13 105 29 �68 3.80S21 74 30 �32 1.30S22 158 39 �67 2.60S23 349 39 �12 2.80S31 150 58 �56 0.40S32 219 70 �53 1.35S33 386 85 �41 3.30S41 473 114 �82 0.27S42 643 121 �84 1.37S43 796 136 �80 1.95S51 837 194 �64 0.41S52 1157 209 �71 0.19S53 1419 211 �80 1.00

Columns 2 and 3 show the computation time tS of the MILP solver (in seconds) andthe computation time tH of the heuristic method (in seconds), respectively. Therelative difference Dt ¼ ðtH � tSÞ=tS � 100 is reported in column 4. As the value ofthe objective function has no intrinsic significance, we report the relative differenceDz between both objective functions in column 5. It is calculated asDz ¼ ðzH � zSÞ=zS � 100, where zH is the value of the objective function found by theheuristic method and zS is the value of the objective function found by the MILPsolver.

Table 6Efficiency of the heuristic method for medium-size (M) instances.


M11 1335 398 �59 0.57M12 1617 420 �62 0.90M13 2133 458 �74 1.46M21 3481 497 �79 0.45M22 3872 533 �77 0.60M23 4682 589 �86 2.40M31 4349 556 �74 0.57M32 4533 615 �74 0.32M33 4641 626 �82 1.67M41 6289 717 �87 1.54M42 6379 735 �78 3.60M43 6442 753 �74 1.60M51 6967 733 �80 1.83M52 2h 785 �89 2.69M53 2h 855 �79 1.71

No instance of families M52 and M53 could be solved to optimality by the solver.The calculation was thus stopped after 2 h (written 2 h).

Table 7Efficiency of the heuristic method for large-size (L) instances.


L11 3h 972 �91 2.70L12 3h 1154 �80 3.00L13 3h 1203 �79 1.79L21 3h 1964 �78 1.12L22 3h 2111 �69 1.09L23 3h 2337 �69 1.06L31 3h 2323 �66 1.27L32 3h 2551 �69 2.27L33 3h 2619 �78 1.32L41 3h 2246 �73 1.90L42 3h 2614 �61 1.60L43 3h 2666 �75 1.20L51 3h 5141 �51 2.22L52 3h 7091 �34 3.45L53 3h 7154 �40 2.99

Only one instance in family L21 could be solved to optimality. The average CPU timefor this family is still very close to 3 h (written 3 h for simplicity).


except for family L5. Most of the computation time is spent onSteps 1 and 2. They represent about 75% of the CPU except for fam-ily L5 (for which much of the computing time is devoted to Step 3).Each iteration of the linear relaxation is fast but must be repeatedseveral times and the correction steps are also time-consuming.Step 3 is quite reduced because most of the binary variables are al-ready fixed. For only half of the instances of family L5, the remain-ing MILP in Step 3 can be solved to optimality within the allocatedcomputation time. The computation time in Step 3 grows consider-ably when the size of the MILP increases. Cordeau et al. (2006) ar-gue that solving the problem to optimality is rarely justified inpractice. They stop the computation as soon as they obtain an inte-ger solution with an optimality gap less than 1%. Adopting thesame approach would surely reduce the total computation timeat the price of a small degradation in the objective function.

Although this does not appear in the tables, there is a highvariability in the execution times within every family. This observa-tion applies to both the MILP solver and to the LP-roundingalgorithm.

The last four columns of Tables 2–4 are related to the use of thecorrection procedure for infeasible solutions. Neither the problemsize nor the type of instances seems to have a significant impacton infeasibility. On the other hand, we observe that the number ofdetected infeasible solutions is greater in Step 1 than in Step 3, be-cause the corrections in Step 1 have increased the capacity of thenetwork. However, this infeasibility is quite easily corrected. Onaverage, we have two iterations with infeasibility in Step 1 andone iteration in Step 3. The LP-rounding method always finds feasi-ble solutions eventually. This underlines the efficiency of the correc-tion procedure.

4.5. Comparison between the MILP solver and the LP-roundingalgorithm solutions

We compare the solver solutions and the LP-rounding solutionswith respect to the value of the objective function and the runningtime. We define the relative deviation of the objective function Dz

and the relative deviation of the computation time Dt as follows:

Dz ¼zH � zS

zS� 100; Dt ¼

tH � tS

tS� 100

where zH is the value of the objective function found by the LP-rounding, zS is value of the objective function found by the MILP sol-ver, tH is computation time of the LP-rounding (called CPU in Tables2–4), and tS is the computation time of the MILP solver.

Tables 5–7 report the average values of Dz and Dt for each fam-ily. The LP-rounding method runs faster than the solver for all theinstances. This improvement is balanced by a loss in the quality ofthe solutions: the relative deviation Dz of the objective function va-lue varies from 0.19% to 3.6%. We observe that, for families S1–S5and M1–M3, the worst ratio Dz is obtained for instances with aloose load/capacity ratio. For families M4–M5 and L1–L5, the sol-ver is generally unable to find optimal solutions, so that the ratioDz is perturbed by the variable quality of the bound zS.

The average gap Dt varies between �12% and �91%. In 78% ofthe cases, the value of Dt is comprised between �50% and �80%.The heuristic method is initially more and more efficient as the sizeof the instance grows. For large-size instances however, the evolu-tion is quite different: Dt decreases as the size increases. This iscompletely logical since the computation time tH for the heuristicmethod is increasing while the computation time tS for the solveris artificially stabilised at the 3 h limit.

The load/capacity ratio does not seem to influence the value ofDt.

The Fig. 4 is a graphical representation of Table 7. It details theperformance of the LP-rounding method for each of the 150 large-

size test instances (families L1–L5). The figure shows the reductionin computation time obtained by the heuristic method for every

Fig. 4. Performance of the heuristic method for large-size (L) instances. The ratio Dt (from 100% to 0% of saved time) is given on the horizontal axis while the ratio Dz appearson the vertical axis (a negative value means that the heuristic method outperforms the solver). Each small square represents one of the 150 instances of families L1–L5. Let usconsider the example of instance A: the value of the objective function obtained with the LP-rounding is 1.2% worse than that obtained with the solver, but the computationtime is five times shorter (�80%).


instance. Nevertheless, for a group of difficult instances, this reduc-tion is not satisfactory enough (between 20% and 35% of timesaved, with degradation of the objective function). On the contrary,some instances could be solved much faster and efficiently by theLP-rounding algorithm.

5. Conclusion and further research

We have proposed an LP-rounding heuristic method combinedwith correction procedures for the design and strategic planning ofcomplex logistics networks. The method relies on successive relax-ations of the original mixed integer linear programme. At each iter-ation, some binary variables are set at 0 or 1, either directly by thelinear relaxation or by some rounding procedures. The last step ofthe heuristic method consists of solving the resulting problem ex-actly by a solver.

We aimed to develop an efficient procedure by relaxing as fewfixed variables as possible. The method is adapted to the context ofthe study since it focuses on the most critical binary variables ofthe original model. We use two different specific rules to fix theseimportant variables: the largest fractional value rule and the gree-dy algorithm.

The main contribution of the proposed LP-rounding algorithm isto ensure feasible solutions for every instance thanks to efficientcorrection procedures. The numerical results show the efficiencyof the approach for medium-sized instances: we obtain optimalor near optimal solutions with a reasonable execution time. Theobserved average reduction in computation time is around 80%while the loss in the objective function is 1.5%. However, we ob-serve that for the largest instances, the resulting MILP at Step 3is still large, and the time reduction of the heuristic method is only34–51% for L5 instances. For these more efficient specific methodsneed to be developed. One way to improve the objective functionwould be to replace the linear relaxation by a Lagrangean relaxa-tion. By doing this, we hope to increase the lower bound. ThisLP-rounding could be used to propose initial solutions for a moresophisticated method. Another possibility is to add a supplemen-tary step (local search) to decrease the upper bound after restoring

feasibility. With these improvements, we believe that the largestinstances could be solved very efficiently.

Appendix A. Mathematical model

A.1. Data sets

Tðt 2TÞ
planning horizon Sðs 2SÞ set of suppliers Iði 2 IÞ set of plants Jðj 2 JÞ set of warehouses JP set of private warehouses JH set of public warehouses Cðc 2 CÞ set of customers Oðo 2 OÞ set of capacity options Pðp 2 PÞ set of products Pr set of raw materials Pm set of manufactured products Pf set of finished products Cþi set of successors of a node i C�i set of predecessors of a node i
For plants and warehouses, we define two subsets, noted by
subscripts c(Ic, JPc, JHc) and o(Io, JPo, JHo) for the set of facilities thatmay close and open respectively.
A.2. Decision variables

A.2.1. Binary variables

xti
= 1 if the entity i is active at t, 0 otherwiseði 2S [I [JÞ
yti;o
= 1 if the capacity option o is added to i at t, 0 otherwiseði 2 I [JÞ
zts;p
= 1 if the supplier s is selected for p at t, 0 otherwise
v ts
= 1 if the supplier s gives a discount at t, 0 otherwise


A.2.2. Continuous variables

f tp;i;j
quantity of product p transferred from node i to node j at t
ktp;i

quantity of product p subcontracted for plant i at t

gtp;i
quantity of product p produced in plant i at t
htp;j

quantity of product p held in warehouse j at thebeginning of t

A.3. Parameters

All these parameters are part of the datasets and are never mod-ified by the heuristic.

A.3.1. Suppliers

HDts;p

1 if the raw material p is available from s at t, 0otherwise

Ats;p

available capacity of supplier s for p at t

A.3.2. Plants

Ki
initial capacity of production at i
Ki
maximal installable production capacity at i
HAti;o

1 if we can add the option o to facility i at t, 0 otherwise

KTo
capacity of option o
Ui
minimal percentage of utilisation of facility i
(independent of t)
Ut
i
maximal percentage of utilisation of facility i at t
HPti;p

1 if the plant i can manufacture p at t, 0 otherwise

A.3.3. Warehouses

The warehouses have almost the same characteristics as theplants. The only differences are the following:

Sj
initial storage capacity at j
KSo
storage capacity of option o (if o is added to awarehouse)
HStj;p

1 if the warehouse j can store p at t, 0 otherwise

A.3.4. Customers

Dtc;p

demand of customer c for product p at t

A.3.5. Products

HCp
1 if the product p can be subcontracted Bp0 ;p quantity of p0 necessary to manufacture one unit of p
(bill of materials)
LTp,i workload for the treatment of a unit p at facility
i 2 I [J

LSp,j
workload for the storage of a unit p at warehouse j 2 J
Ki,j
number of deliveries from plant i to warehouse j in oneperiod
A.3.6. Costs

CFs
fixed cost for selecting a supplier CDs discount if at least two raw materials are ordered from
the supplier s

COi
fixed cost for opening a facility i at a potential location CCi fixed cost for closing an existing facility i CAi,o fixed cost for adding capacity option o to a facility i CUi,0 fixed cost for operating a facility i CUi,o fixed cost for operating capacity option o at facility i CPp,i treatment cost of a unit p at facility i 2 I [J
CPp,0
subcontracting cost of a unit of p CSp,j holding cost of a unit of p at warehouse j 2 J
CTp,i,j
transportation cost of a unit of p from i to j
A.4. Objective function

Minz¼Pt2T

Ps2S

CFs:xts�CDs:v t

s ðz1 : supplier selection fixed costsÞ

þPt2T

Pi2Io[JPo

COi � xtþ1i �xt

i

� �ðz2 : opening facility fixed costsÞ

þPt2T

Pi2Ic[JPc

CCi � xti �xtþ1

i

� �ðz3 : closing facility fixed costsÞ

þPt2T

Pi2I[JP

Po2O

CAi;o � ytþ1i;o �yt

i;o

� �ðz4 : adding capacity options costsÞ

þPt2T

Pi2I[JP

CUi;0 �xti þPo2O

CUi;o �yti;o ðz5 : operating facility fixed costsÞ

þPt2T

Pp2Pm

Pi2I

CPp;i �gtp;iþCPp;0 �kt

p;i ðz6 : production variable costsÞ

þPt2T

Pp2Pf

Pj2J

CStp;j � ht

p;jþP

i2C�j

1Ki;j�

f tp;i;j

2

0@ 1A ðz7 : storagevariablecostsÞ

þPt2T

Pp2Pf

Pj2J

Pc2Cþ

j

CPtp;j � f t

p;j;c ðz8 : distribution variable costsÞ

þPt2T

Pp2P

Pi2S[I[J

Pj2Cþ

i

CTtp;i;j � f t

p;i;j ðz9 : transportation variable costsÞ

ð1Þ

The components z1–z5 are fixed costs. Concerning the supplierselection fixed costs z1, the company has to pay them for each per-iod and for each selected supplier. Whenever possible, the discountCDs is removed from the cost. Other costs that are paid once a yearare operating fixed costs. Other fixed costs are paid only once dur-ing the whole planning horizon, like the opening costs z2, closingcosts z3 and capacity option costs z3. Indeed, these costs are consid-ered as an investment for a new site (opening and adding capacityoption costs) or as a cost for removing installed things and for pay-ing for dismissing people.

The second part of the objective function consists of variablecosts that depend on the product flows in the network. The pro-duction costs z6 are the sum of the product costs for internalplants and the subcontracting costs. Storage variable costs z7

are calculated as a function of the average inventory level. Inour model, this is defined as half of the delivered quantity (fromplants) which is calculated by the quotient of the total trans-ported quantity divided by the number of deliveries in one per-iod. The distribution cost z8 is calculated as a function of thedelivered quantity from warehouses to customers. Finally, thetransportation costs z9 are related to the product flows betweeneach node in the network.

A.5. Constraints

The following constraints apply for all t 2T:The first four constraints (1)–(4) concern demand satisfac-

tions on each layer of the network. The constraints (1) guaranteethat all customer demands are met. Note that the expressionj 2 C�c includes the possibility of direct linkage from plants tocustomers. The constraints (2) represent the flow conservationat warehouses. The constraints (3) ensure that plants receive en-ough raw materials and semi-finished products in order to man-


ufacture the required quantity of finished products. The expres-sion i0 2 C�i includes the possibility of linkage from plants toplants. The constraints (4) require that the quantity of manufac-tured and subcontracted products at a plant is larger than itsdelivered quantity.Xj2C�c

f tp;j;c P Dt

c;p c 2 C; p 2 Pf ð1Þ

ht�1p;j þ

Xi2C�j

f tp;i;j ¼

Xc2Cþ

j

f tp;j;c þ ht

p;j j 2 J; p 2 Pf ð2Þ

Xi02C�i

f tp0 ;i0 ;i þ kt

p0 ;i PX

p2Pm

Bp0 ;p � gtp;i i 2 I; p0 2 Pr [Pm ð3Þ

gtp;i þ kt

p;i PXj2Cþ

i

f tp;i;j i 2 I; p 2 Pm ð4Þ

Xp2Pm

LTp;i � gtp;i P Ui � Ki � xt

i þXo2O

KTo � yti;o

!i 2 I ð5Þ

The constraints (5)–(10) concern the capacity (initial + options) ofthe plants and warehouses. We suppose that a facility cannot rununder its minimum rate of utilisation and cannot exceed the maxi-mum rate of utilisation of its installed capacity. The constraints (5)and (6) represent these conditions for the plants. The constraints(7) and (8) are similar but applied to the warehouses. Moreover,warehouses cannot store more than their storage capacity (9). Final-ly, the installed capacity at any facility must not exceed its maximalinstallable capacity (10).Xp2Pm

LTp;i � gtp;i 6 Ut

i � Ki � xti þXo2O

KTo � yti;o

!i 2 I ð6Þ

Xp2Pf

Xc2Cþ

j

LTp;j � f tp;j;c P Uj � Kj � xt

j þXo2O

KTo � ytj;o

!j 2 J ð7Þ

Xp2Pf

Xc2Cþ

j

LTp;j � f tp;j;c 6 Ut

j � Kj � xtj þXo2O

KTo � ytj;o

!j 2 J ð8Þ

Xp2Pf

LSp;j � htp;j þ

Xi2C�j

1Ki;j� f t

p;i;j

0@ 1A 6 Sj � xtj þXo2O

KSo � ytj;o j 2 J ð9Þ

Ki � xti þXo2O

KTo � yti;o 6 Ki i 2 I [J ð10Þ

The constraints (11) and (12) are valid inequalities thatstrengthen the linear relaxation of the original problem.Xt2T

Xi2I

Kti � HPt

i;p � xti P

Xt2T

Xc2C

Dtc;p p 2 Pf ð11ÞX

j2JKj � HSt

j;p � xtj P

Xc2C

Dtc;p p 2 Pf ð12Þ

The constraints (13) require that suppliers deliver a raw mate-rial if, and only if, they are selected for this raw material. Moreover,their delivery cannot exceed their capacity. The constraints (14)state that a supplier, whenever selected, can only deliver a subsetof raw materials. We suppose that a supplier offers a discount if itis selected for at least two raw materials (15). The constraints (16)propose a minimal amount of the delivered quantity of each sup-plier in order to avoid small artificial orders just to obtain the dis-count. This threshold a is arbitrary and has to be adjustedaccording to the context (0.1 or 0.2 are typical values). The con-straints (17) and (18) state that a plant/warehouse can only deliverproducts that it can manufacture/store.

Xi2Cþs

f tp;s;i 6 zt

s;p � Ats;p s 2 S; p 2 Pr ð13Þ

zts;p 6 xt

s � HDts;p s 2 S; p 2 Pr ð14Þ

v ts 6 1=2

Xp2Pr

zts;p

!s 2S ð15Þ

Xi2Cþs

f tp;s;i P a � At

s;p � zts;p s 2S; p 2 Pr ð16Þ

f tp;i;i0 6 HPt

i;p � Ki i 2 I; i0 2 Cþi ; p 2 Pm ð17Þ

f tp;j;c 6 HSt

j;p � Kj j 2 J; c 2 Cþj ; p 2 Pf ð18Þ

The constraints (19) require that only opened plants can havesubcontracted products. The constraints (20) ensure that only anopened facility can add available capacity options. The constraints(21) and (22) prevent the facilities from changing their status(open or closed) more than once. Closed facilities cannot reopen,and open facilities cannot be closed. The status of public ware-houses cannot be changed after at least two periods (23). Addition-ally, capacity options cannot be removed (24). Finally, a facilitywhere capacity options are added cannot be closed (25).

ktp;i 6 xt

i � HCp �M i 2 I; p 2 Pm ð19Þ

yti;o 6 xt

i � HAti;o i 2 I [JP; o 2 O ð20Þ

xti P xtþ1

i i 2 Ic [JPc ð21Þ

xti 6 xtþ1

i i 2 Io [JPo ð22Þ

� 1 6 2 � xtj � xt�1

j � xtþ1j 6 1 j 2 JH ð23Þ

yti;o 6 ytþ1

i;o i 2 I [JP; o 2 O ð24Þ

xti P yt�1

i;o i 2 I [JP; o 2 O ð25Þ

The constraints (26) and (27) define the decision variables.

xtm; y

ti;o; z

ts;p; v t

s 2 f0;1g m 2 I [J [S; i 2 I [J;

o 2 O; s 2 S; p 2 Pr ð26Þ

f tp;m;n; k

tp;i; g

tp;i; h

tp0 ;j P 0 m 2 S [I [J; n 2 Cþm; i 2 I;

j 2 J; p 2 Pm; p0 2 Pf ð27Þ

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Documents

A linear relaxation-based heuristic approach for logistics network design