Reza Zanjirani Farahani a,*, Nasrin Asgari b

objective functions are formed. Constraints imposed on these two objectives cover all centers, which must be supportedby the DCs. Using Multiple Objective Decision Making techniques, the locations of DCs are determined. In the nal

* Corresponding author. Tel.: +98 21 66413034/66497; fax: +98 21 66413025.E-mail address: farahan@aut.ac.ir (R.Z. Farahani).

European Journal of Operational Research 176 (2007) 18391858

www.elsevier.com/locate/ejor0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved.phase, we use a simple set partitioning model to assign each supported center to only one of the located DCs. 2006 Elsevier B.V. All rights reserved.

Keywords: Location; MADM; Set covering; MODM; Set partitioning

1. Introduction

In this paper, a real-world case study about locating supportive centers in a military logistics system willbe investigated. First, we give a summary of the functions of this system; then the scope of the problem willbe presented.a Department of Industrial Engineering, Amirkabir University of Technology, Tehran, Iranb Supply Chain Management Research Group, Tehran, Iran

Received 27 July 2004; accepted 27 October 2005Available online 19 January 2006

Abstract

In this paper, locating some warehouses as distribution centers (DCs) in a real-world military logistics system will beinvestigated. There are two objectives: nding the least number of DCs and locating them in the best possible locations.The rst objective implies the minimum cost of locating the facilities and the latter expresses the quality of the DCslocations, which is evaluated by studying the value of appropriate attributes aecting the quality of a location. Qualityof a location depends on a number of attributes; so the value of each location is determined by using Multi AttributeDecision Making models, by considering the feasible alternatives, the related attributes and their weights according todecision makers (DM) point of view. Then, regarding the obtained values and the minimum number of DCs, the twoO.R. Applications

Combination of MCDM and covering techniques ina hierarchical model for facility location: A case studydoi:10.1016/j.ejor.2005.10.039

Our supply chain is composed of three types of facilities:

origins; supportive centers; supported centers.

These facilities are depicted in Fig. 1. The arrows show the direction of items ow. Now we give a shortdescription about the facilities.

1840 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 183918581.1. Origins (sources for supplying all needed goods and materials)

Facilities, which are referred to origins, are physically xed and their locations and characteristics areknown. Origins include harbors, airports and factories. Foreign goods are imported via harbors and air-ports, and other goods (goods manufactured by national factories) are shipped from manufacturer facto-ries. We can put these national factories into two categories:

Ministry of Defense subsidiaries; Independent factories (independent from Ministry of Defense).

When factories from both categories can supply what is needed, we assume they all have equal priorityand the selected origin will be the one that can best assist in achieving logistics goals; therefore, dierentorigins could be thought as competitors.

1.2. Supportive centers (warehouses, repair shops and terminals for receiving, storing and shipping all needed

goods and materials to supported centers)

There exist a xed number of supportive centers and possible location; but their number and locationsare currently unknown and are supposed be determined here. They receive the required items from the ori-gins and ship them to the supported centers; so they can be referred to as Central Warehouses (CW).There are no limitations on the relationship between the origins and supportive centers; i.e. requirementsof each supportive center can be supplied by any of the origins.

Logistics Department is the procurement unit of a military organization and plays an important role inprocurement of needed goods, materials and equipments and keeping them in ready-to-be-used state. It is asystem of great importance for achieving goals, fullling missions and increasing eciency and eectivenessof a military organization, and just like other operational systems assist the organization in choosing thebest organizational goals and strategies. Logistics Department system is made of a number of sub-systemssuch as warehousing, repair and maintenance, transportation, standardization, estimation, procurement,

Sources for supplying all needed goods and materials

Supported centers that

consume

shipped goods and materials

Origins

Supportive centers that serve as intermediate

nodes

Supportive centers Supported centersFig. 1. The three types of facilities serving in a military logistics system.

distribution, recycling, and assets control systems. In this case, the supportive centers are composed of threetypes of physical entities including warehouses, repair shops and transportation terminals.

1.3. Supported centers (centers that consume the shipped goods and materials)

These facilities are usually of xed locations. Items supplied by origins are shipped to supportive centersand are eventually distributed among supported centers. However, some items are transferred directly fromorigins to supported centers (direct shipment). Supported centers can be considered as regional warehouses(RW) in a supply chain. There are limitations for the relationship between the supportive centers and sup-ported centers; each supportive center covers a certain number of supported centers and each supportedcenter is assigned to only one supportive center (except for critical periods of time when some supportivecenters might help other supportive centers which need assistance).

If we take all components of each entity in Fig. 1 into account, Fig. 2 will be resulted. The network struc-ture of this gure could be thought as a supply chain, considering each node as a xed entity of supplychain. In Fig. 2, the arrows show the direction of material and information ow between the existing facil-ities. We have already briey explained the functionality of each facility.

When planning such systems, two phases can be considered: rst, nding the location of supportive cen-ters so that they cover all the supported centers; second, nding a shipment strategy for the ow of itemsbetween these three types of facilities by solving the relevant multi-commodity transshipment problem. Thescope of our problem is limited to the rst phase.

Direct shipment Supported centers

Supportive centers Origins

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1841.

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.Fig. 2. Network structure of Logistics Department system including its components.

Finding the number and locations of the supportive centers is the problem discussed in this paper. Inaddition, after nding the location of supportive centers, allocating the supported centers to the supportivecenters in a way that each supported center is assigned to only one supportive center, is also included as thenal step.

In this paper, we want to partition all supported centers in a minimum number of groups so that eachgroup is assigned to only one supportive center; the entities of each group receive their needed items fromtheir own supportive center, which is assigned to them. Note that, in a military location problem, the qual-ity of supportive centers location is an important issue that will be also investigated here. In summary, theproblem is to nd the number of needed supportive centers, locations of supportive centers and partitioningof supported centers in a way that each supported centre is assigned to only one supportive centre. Theobjective functions and the constraints of the problem are as follows:

Objective function 1: Maximizing the utility of the selected locations. Utility of a potential point dependson 23 attributes that will be explained later.

Objective function 2: Minimizing the number of supportive centers.Constraint 1: All of the supported centers must be covered (coverage criteria will be presented later)

by supportive centers.Constraint 2: Each of the supported centers must be supported by one and only one of the located

supportive centers.

In a problem relatively similar to that of this paper, but for health facilities in a new city, Berghmanset al. (1984) using techniques of location theory in network nd the smallest number of centers that willensure that all inhabitants are located within the critical distance. In their problem, a new city is under con-struction in a developing country. The future population projection of town is established and the ratio ofmedical personnel to inhabitants is xed according to health policy criteria. The primary care system shouldbe composed of a set of health centers, which are identical with regard to equipment and personnel. Theproblem is to determine the number and thus the size of the health centers, and their location. The solutiondepends on two opposing factors: the total construction cost, which is increasing with the number of cen-ters, and the walking distance for the patient, which is decreasing with the number of centers. They developa Fortran program in which the sensitivity of the solution is studied as a function of the given criticaldistance.

The case discussed in this paper is the problem concerns location theory, specially set covering problems.These problems have attracted signicant research eort since the beginning of 1960s. Since then, numerousproblem types have been identied and methodologies developed to solve these problems have been used tomake decisions pertaining to the location of facilities in many practical applications. Brandeau and Chui(1989) present a survey of representative problems that have been studied in location problems (see alsoGhosh and Rushton (1987) and Daskin (1995) for detailed discussions on location models).

Multicriteria analysis of location problems has received considerable attention within the scope of con-tinuous and network models in the recent years; there are several problems that are accepted as classicalones: the point-objective problem (Wendell and Hurter, 1973; Hansen et al., 1980; Pelegrin and Fernandez,1988; Carrizosa et al., 1993), the continuous multicriteria min-sum facility location problem (Hamacherand Nickel, 1996; Puerto and Fernandez, 1999), and the network multicriteria median location problem(Hamacher et al., 1998; Wendell et al., 1977), among others.

The case of this research is a multicriteria (including multiattribute and multiobjective) set coveringproblem. So far, multicriteria analysis of discrete Location Problems has attracted less attention. Severalauthors have dealt with problems and applications of multicriteria decision analysis in this eld. Forinstance, Ross and Soland (1980) worked on multiactivitymultifacility problems and proposed an interac-

1842 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858tive solution method to compute non-dominated solutions to compare and choose from. In Lee et al.

(Krarup and Pruzan, 1983). This ensures that the multiobjective formulation is not solvable in polynomialtime. In this context, when time and eciency become real issues, dierent alternatives can be used to approx-

imate the Pareto optimal set. One of them is the use of general-purpose MOCO heuristics (Gandibleux et al.,2000). Another possibility is designing ad hoc methods based on one of the following strategies:

Computing the supported non-dominated solutions; Performing a partial enumeration of the solutions space.

Obviously, the second strategy does not guarantee the non-dominated character of all the generatedsolutions since we only consider the solutions obtained during the partial search. Nevertheless the reductionin computation time can be remarkable.

2. The data

We present our methodology in ve phases. In each phase, we explain the type and the size of inputs,process and outputs.

Phase 1. Determining all attributes that inuence the utility of a location. These attributes are divided intothree groups as only pure non-compensatory (PN), pure compensatory (PC) or both (NC). The list of attri-butes aecting on the utility of a supportive location is as follows:

Climate: Temperature (PC) Rain (PC) Humidity (PC) Sunshine (PC)

Geological: Earthquake intensity (PN) Flood history (PN)(1981), an application of integer goal programming for facility location with multiple competing objectivesis studied. Solanki (1991) applies an approximation scheme to generate a set of non-dominated solutions toa bi-objective location problem. Recently, Ogryczak (1999) looks for symmetrically ecient location pat-terns in a multicriteria discrete location problem. In general, none of the above papers, focuses on the com-plete determination of the whole set of non-dominated solutions. The only exception is the paper by Rossand Soland (1980) that gives a theoretical characterization but do not exploit its algorithmic possibilities.

Nowadays, multiobjective combinatorial optimization (MOCO) (see Ehrgott and Gandibleux, 2000;Ulungu and Teghem, 1994) provides an adequate framework to tackle various types of discrete multicriteriaproblems.Within this emergent research area several methods are known to handle dierent problems such asdynamic programming enumeration (Villarreal and Karwan, 1981; for a methodological description; Klam-roth andWiecek, 2000; for a recent application to knapsack problems) and implicit enumeration (Zionts andWallenius, 1980; Zionts, 1979; Klein and Hannan, 1982; Rasmussen, 1986; Ramesh et al., 1986). Anotherapproach based on labeling algorithms can be seen in Captivo et al. (2000). It is worth noting that most ofMOCO problems are NP-hard and intractable (Ehrgott and Gandibleux, 2000; for further details). Even inmost of the cases where the single-objective problem is polynomially solvable the multiobjective versionbecomes NP-hard. This is the case of spanning tree problems and min-cost ow problems, among others.In the case of uncapacitated plant location problem (UPLP), the single-objective version is already NP-hard

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1843 Interruption of earth (PN)

Military: Active defensive (NC) Non-active defensive (PC) Frontier threats (NC) Internal threats (NC) Access to supported echelons (NC) Density of supported echelons (NC)

Economical:

1844 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858groups (Fig. 3):

Non-compensatory attributes: These are eight attributes which must be satised at a minimum standardlevel; otherwise the corresponding alternatives are not feasible and will be removed. For example, aboutFrontier threats there is a standard level about 90 km distance from the country frontier; it means thateach alternative having the distance less than 90 km from the frontier is not suitable for locating a sup-portive center.

Compensatory attributes: These are 21 very important attributes which determine the desirability of apotential location. However, no standard level is dened for them; all we know is the direction of theirdesirability. Note that some of the attributes like Frontier threats can be both non-compensatory andcompensatory. For example, we do not want to locate our supportive centers closer than 90 km to fron-tier. In addition, 110 km distance from the frontier is better than 100 km; i.e. the distance from frontier isan attribute with a dual aspect; so it is also compensatory.

Non-compensatory: PN & NC

(8) Attributes affecting the utilization of apotential location

(24) Compensatory: PC & NC

(21) Native expert labors (PC) Economic activities (PC)

Infra-structures: Access to roads (PC) Access to rails (PC) Access to airports (PC) Access to harbors (PC) Access to water resources (PC) Access to power lines (PC) Access to dispatching centers (PC) Access to fuel stations (PC) Personnel convenience (PC)

There are a total number of 24 attributes. Regarding the above list, we can divide the attributes into twoFig. 3. Determining all attributes that aect on the utility of a location.

o : (cities) which should beevaluated for locating the

ut of desirability directionMax: Less than this Min: More than this

List of 3 non-compensatoryattributes The most important thing we need to know about both types of the attributes is the direction of theirdesirability. The direction of desirability for an attribute can be Max, Min or being in an interval. Forexample, less Frontier threats is desirable (Min), better Access to airport from a location is desirable(Max) and Temperature is an attribute which its desirability is the best between 18 and 22 C (Interval).

supportive centersInterval: Out of interval The direction of attributes desirability

Fig. 4. Evaluating non-compensatory attributes for alternatives.

Table 1An example showing the process of changing an Interval attribute to a Min attribute

# of months with Temperaturemore than 22 C (in a year)

Mean of Temperature morethan 22 C (in a year)

Weight associatedwith cooling costs

Total penalty of cooling

5 27.66 5 (5) * (27.66 22) * (5) = 141.5# of months with Temperatureless than 18 C (in a year)

Mean of Temperature lessthan 18 C (in a year)

Weight associatedwith heating costs

Total penalty of heating

6 9.93 1 (6) * (18 9.93) * (1) = 48.42Total penalty 189.92Process Output Input

33 feasible alternatives Removing the alternatives290 cities as first initial

alternatives

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1845Phase 2. In this phase, we use all possible alternatives as initial inputs of our process. Based on the natureof the supportive centers and existence of needed data about the locations, we consider 290 cities in thecountry as initial alternatives. In this phase, we only need to have information about non-compensatoryattributes of the alternatives. Regarding the information and the direction of desirability and standard lev-els, we can remove most of weak alternatives. Implicitly, non-compensatory attributes are those having themost value or weight in evaluation process. After passing this phase, the feasible alternatives reduced to 33.Fig. 4 shows the process.

In the next phase, we use MADM techniques to assess the quality of feasible locations; so we must bearin mind that:

All attributes of the type (Interval) must change to either Max or Min. Table 1 shows an example. Tem-perature (Interval) is an attribute that its desirability is being between 18 and 22 C. When the Temper-ature is less than 18 C, there are heating costs and when the Temperature is more than 22 C, there arecooling costs. In our country, the costs of cooling are about 5 times more than the costs of heating. Basedon the information, we have dened a penalty for heating and another one for cooling. As shown in Table1, total penalty to have the Temperature in the desired interval is 189.92; thus, we have changed andInterval type attribute to a (Min) type attribute.

3. The method

In this section, a suitable MADM technique is used to assess the quality of the locations and MODMtechniques are used to consider two mentioned objective functions.

3.1. MADM

Phase 3. Using MADM techniques to assess the quality of the locations. Normally, MADM methods areused to select the best alternatives. In this phase with respect to compensatory alternatives, we only try todetermine the value of each of the 33 feasible alternatives. Fig. 5 shows this process.

Suppose all of the attributes are of the types Min or Max. Using each MADM model is not feasible.Based on the type of available information about the attributes and the alternatives, using LINMAP,SAW, TOPSIS, ELECTRE and AHP is admissible (Hwang and Yoon, 1981). But a more exact investiga-tion shows that some techniques like TOPSIS and ELECTRE are the best for this case because we haveboth desirable directions (Min and Max); also the dimension of the attributes are dierent and should

1846 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858where Ai = the ith alternative considered; xij = the numerical outcome of the ith alternative with respect tothe jth criterion.

TOPSIS assumes that each attribute in the decision matrix takes either monotonically increasing ormonotonically decreasing utility. For the sake of simplicity, the proposed method will be presented as aseries of successive steps.

Step 1: Constructing the normalized decision matrix: An element rij of the normalized decision matrix R canbe calculated as

Output Process Input Value of each of 33 feasible alternatives which are potentially

suitable for locating the supportive centers

A suitable MADMmethod and its

formula for evaluating feasible alternatives

33 feasible alternatives List of 22 compensatory attributes The direction of attributes

desirability Weights of attributes 1be normalized. However, due to simplicity of implementation and calculation eorts, we have used TOP-SIS. In some cases, using group decision making techniques will also be useful (Hwang and Lin, 1987).

Thus, we need a decision matrix, direction of desirability of attributes and weight of each attribute. Wehave generated the decision matrix, have calculated the direction of attributes and have used brain storm-ing technique to nd the weights of the attributes. Details on TOPSIS and on the way to obtain the valuesof each alternative can be expressed as follows:

TOPSIS works based upon the concept that the best alternative should have the shortest distance fromthe ideal solution and the farthest from the negative ideal solution. The TOPSIS method evaluates the fol-lowing decision matrix which contains m alternative associated with n attributes (or criteria).Fig. 5. Using a MADM technique to determine the value of each feasible alternative.

6 7 6 7

Step 3:

Step 4:

Step 5:

Step 6:

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1847ser to A*Ranking the preference order: A set of alternative can now be preference ranked according to thedecreasing order of ci. However, we do not use ranked alternatives in our approach and the valueis dened as

ci Si=Si Si; 0 < ci < 1; i 1; 2; . . . ;m. 7It is clear that ci = 1 if Ai = A* and ci = 0 if Ai A. As ci approaches 1, the alternative Ai gets clo-j1

Similarly, the separation from the negative ideal one is given by

Si Xnj1

vij vj 2vuut ; i 1; 2; . . . ;m 6

Calculating the relative closeness to the ideal solution: The relative closeness of Ai with respect to A*Si Xn

vij vj 2vuut ; i 1; 2; . . . ;m. 5preferable alternative (ideal solution) and the least preferable alternative (negative ideal solution),respectively.Calculating the separation measure: The separation between each alternative and the ideal alterna-tive can be measured by n-dimensional Euclidean distance. The separation of each alternative fromthe ideal one is then given byA maxi

vij j j 2 J; mini

vij j j 2 J 0 j i 1; 2; . . . ;m

v1; v2; . . . ; vj ; . . . ; vnn o

;

A mini

vij j j 2 J; maxi

vij j j 2 J 0 j i 1; 2; . . . ;m

v1;v2; . . . ;vj; . . . ;vn

;

4

where J = {j = 1,2, . . . ,n} is associated with benet criteria and J 0 = {j = 1,2, . . . ,n} is associatedwith cost criteria. Then it is certain that the two created alternatives, A* and A, indicate the mostvm1 vm2 vmj vmn rm1w1 rm2w2 rmjw2 rmnwnDetermining ideal and negative ideal solutions: Let the two articial alternatives A* and A be denedas64 75 64 75

V

v21 v22 v2j v2n

vi1 vi2 vij vin

666666777777

r21w1 r22w2 r2jw2 r2nwn

ri1w1 ri2w2 rijw2 rinwn

666666777777. 3malized decision matrix V equals to:

v11 v12 v1j v1n2 3 r11w1 r12w2 r1jwj r1nwn2 3

matrix in this step as w w1;w2; . . . ;wj; . . . ;wn; j1wj 1. This matrix can be calculated bymultiplying each column of the matrix R with its associated weight wj. Therefore, the weighted nor-rij xijXmi1

x2ij

s,2

Step 2: Constructing the weighted normalized decision matrix: A set of weights is applied to the decisionPnof relative closeness is enough to continue the procedure.

123456789The output of this phase is ci that is the value of locating a supportive center in location i. Using MADMmethods, usually cis are between 0 and 1, where 1 indicates the ideal location and 0 the worst. In this case,the obtained values are shown in Table 2.

3.2. MODM

Phase 4. Using a multiple objective set covering model to nd the best locations. Covering problems hold acentral place in location theory. In these problems, we are given a set of demand points and a set of poten-tial sites for locating facilities. A demand point is said to be covered by a facility if it lies within a pre-spec-ied distance of that facility. Using the classication proposed by Daskin (1995), covering problems aredivided into two main classes: set (total) covering problems which consists of covering all demand pointswith the minimum number of facilities; and maximal (partial) covering problems which consist of coveringa maximum number of demand points with a xed number of facilities (Church and ReVelle, 1974). Daskinet al. (1988) examine extensions to covering models, such as the concepts of multiple, excess, backup, andexpected coverage. Schilling et al. (1993) provide a detailed review of the covering models in facility loca-tion. Boey and Narula (1997) survey the multiobjective covering and routing problems. However, cover-ing problems and proposed techniques for solving them may be important to model the service facilitylocation problems, military logistics problems, and military targeting problems in the presence of partial

10 0.501587791 21 0.358274458 32 0.47398043611 0.527746396 22 0.343801152 33 0.512756125coverSe

Fran

xi iscentesuresporte0.616718051 12 0.569516411 23 0.3660095130.648638876 13 0.366833018 24 0.4017032240.545607304 14 0.391782644 25 0.6789176710.401226814 15 0.544899959 26 0.4605276980.726808475 16 0.426770197 27 0.4687271650.644903229 17 0.538974198 28 0.5602554520.447647134 18 0.430901064 29 0.4869036320.432739482 19 0.538031619 30 0.6041933760.356353625 20 0.502618398 31 0.463161706Table 2Alternative values obtained using TOPSIS

Alternative number Value Alternative number Value Alternative number Value

1848 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858age (Karasakal and Karasakal, 2004).t covering model is as follows (Francis et al., 1992; Drezner, 1995; Daskin, 1995; Mirchandani andcis, 1990):

min Z1 Xmi1

xi; 8

s.t. Xmi1

akixi P 1; k 1; . . . ; n; 9

xi 2 0; 1f g; i 1; . . . ; p;a binary variable that is equal to one if the feasible alternative i (a potential location for supportivers) is suitable for locating supportive centers; otherwise it is equal to 0. The objective function (8) en-that the minimum number of supportive centers are located. Constraint (9) ensures that all of the sup-d centers are covered. In this expression A = [aki] is called covering matrix; aki is equal to 1 if a

Model 0 max

s.t.

where ci has been alrea

Scena

existing facilities as much as possible. Our computational results and sensitivity

analysis show that e = 2/n is an appropriate value in this case. In summary, Phase4 can be summarized as is shown in Fig. 6.

Many optimal heuristics and meta-heuristics like GA have been proposed for solving covering problems(Aickelin, 2003; Fisher and Kedia, 1990). However, based on the selected method and the size of problemwe have used LINGO 8.00 as powerful software for solving problems (Roe, 1997). There are two objectives

in therio 2 (Model 2): There already exist some active supportive centers (existing facilities) and we try tokeep them active if possible. In this case, we consider the location of these existingfacilities as new locations. However, in Phase 4 we useModel 2 instead ofModel 0 inwhich we have used the following objective function instead of (10):

min Z1 Xn1i1

xi 1 e Xn

in11xi; 14

where n1 shows the number of existing supportive centers and (n n1) shows thenumber of new supportive centers. e is a parameter that tries to increase the coe-cient of binary variables of the new facilities; this forces the objective function to useobjective function (11) maximizes the quality of the selected facilities. The constraint (12) is similar to(9). In this problem, we have considered three scenarios as follows:

Scenario 0 (Model 0): The problem is solved for the rst time and no supportive center already exists. Inthis case, the model is the same as Model 0.

Scenario 1 (Model 1): There already exist some active supportive centers (existing facilities) and all ofthese supportive centers must carry on operating. In this case, we treat the locationof these existing facilities as new locations. But in Phase 4 we useModel 1 instead ofModel 0 in which we have added the following constraints:

xi 1 8i 2 fset of existing facilitiesg. 13model; so we facdy dened as output of Phase 3. The objective function (10) is similar to (8). TheX33i1

akixi P 1; k 1; . . . ; 233; 12

xi 2 0; 1f g; i 1; . . . ; 33;Z2 i1

cixi; 11potential supportive center located in location i (xi = 1) is able to cover the supported center located inlocation k. Generating a covering matrix A = [aki] is based on a factor called critical distance. Critical dis-tance is the maximum time or distance that a supportive center can serve. Covering models focus on theworst-case behavior of the system (Daskin, 1995). In our problem, we have used critical distances equalto 8, 12 or 24 hours. Here, there are a number of 233 supported centers. We have designed a modiedset covering model with two objective functions as follows (Model 0):

min Z1 X33i1

xi; 10

X33

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1849e a multiple objective decision making (MODM) problem. There have been many

attempts at surveying this area (Hwang and Lin, 1987; Hwang and Masud, 1979; Szidarovszky et al., 1986;Ulungu and Teghem, 1994; Zionts, 1979). In this case, we chose a simple method to treat an MODM prob-lem for each critical distance (8, 12 or 24 hours) and each scenario (0, 1 or 2) as follows.

3.2.1. Conicting objectives

Input Output Process Value of 33 feasible

alternatives (ci) Critical distance (8, 16 or

24 hours) Covering matrix A=[aij]

The locations of all of

supportive centers

A covering model with two objective functions

Suitable MODM method and its formulas for evaluating the feasible alternatives

Fig. 6. Phase 4 of the methodology from a systematic point of view.

1850 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858It is in the nature of MODM problems to have conicting objectives; as an example, Fig. 7 depicts thevalues of the conicting objectives (normalized) in one of our models. Therefore, in this phase we face aMODM problem. Note that the rst objective function must be minimized and the second must bemaximized.

In Table 3, we have xed the value of Z1 in a constraint to nd the optimal value of Z2 (say Z2; then to

ensure the obtained values are correct, we double-checked the model by xing Z2 in a constraint to ndoptimal value of Z1 (say Z

1). Then by calculating minfZ1; Z1g, corresponding values in the table are

selected. In the table (of the scenario 0 with 8-hour critical distance) we stop at Z1 = 11 because withrespect to the constraints to cover all the supported centers at least 11 facilities must be opened.

3.2.2. Ecient solution

An ideal solution to a MODM problem is one that results in the optimum value of each of the objectivefunctions simultaneously. For instance, (Z1,Z2) = (6, 16.33952) is corresponding to an ideal solution (of thescenario 0 with 12-hour critical distance). The ideal solution is not feasible.

An ecient solution (Ulungu and Teghem, 1994) (also known as non-inferior solution or non-domi-nated solution) is one in which no one objective function can be improved without a simultaneous detri-ment to the other objective. For instance, the ecient solution of the scenario 1 with 12-hour criticaldistance is shown in Fig. 7.

1

1.20

0.2

0.4

0.6

0.8

0 1 2 3 4 5 6Z1/Z1

* (Min)

Z 2/Z

2* (M

ax)

Fig. 7. Objective function space representation of feasible area (of the scenario 0 with 12-hour critical distance).

globapossibIn our case, we have used both values for p. All of the objective functions and constraints are linear func-tions and putting p = 1 (15) becomes a linear programming problem. All LP solvers can solve such prob-lem. For instance, U1 = 0.921123 and x5 = x28 = 1 (of the scenario 0 with 24-hour critical distance); i.e.two new facilities should be located at the locations i = 5 and 28. (Note that in the scenario 0 with 24-hourcritical distance, with respect to the constraints to cover all the supported centers, at least two facilities mustbe opened).

When all the objectives and all the constraints are linear functions and p = 2, then (15) becomes a convexprogramming problem (Szidarovszky et al., 1986; Hwang and Masud, 1979); more precisely a quadraticprogramming problem. Therefore, this problem can be solved by using any solver of nding a localextreme; this local is absolute or global. For example, U2 = 0.8486647 and x5 = x28 = 1 (of the scenario0 with 24-hour critical distance); i.e. two new facilities should be located at the locations i = 5 and 28.

3.2.3.2. Utility function method. This method requires a utility function to be known prior to solving the

MODfunctrectlyassum

DM Aindhal criterion; of course these are only two of the many possible expressions that can be used. It is alsole to choose any other value for p.3.2.3. A preferred solution

A preferred solution (also known as the best solution) is an ecient solution, which is chosen by the deci-sion maker (DM) as the nal decision. We have used some of simple MODM methods and some sensibilityanalysis to choose a preferred solution. Note that our information about the importance of the objectives isnot ordinal; so, using some of the ordinal methods like Lexicographic method that requires the objectives tobe ranked in order of importance by DM is not feasible. Our case is in a military system, thus cost (corre-sponding the rst objective) is not as important as it is in a business system. However, the objective is oftype cardinal. We have used Utility function method. Forming a suitable utility function is not simple,however, we have formed a simple utility function; the idea arose from global criterion method as follows:

3.2.3.1. Global criterion method. The method of global criterion for a MODM problem solves the problemgiven by following objective function subject to the model constraints (Hwang and Masud, 1979). Note thatthe rst objective is of type minimization and second is of the type maximization:

min Up Z1 Z1

=Z1 p Z2 Z2 =Z2 p 15

Boychuk and Ovchinnikov (1973) have suggested p = 1 and Salukvadze (1971) has suggested p = 2 for the

Table 3Comparison between values of objectives (of the scenario 0 with 8-hour critical distance)

Z2 16.3395 15.9957 15.6374 15.2714 14.9046 14.5128 14.1116 13.7099 13.2831 12.8522 12.4195Z1 33 32 31 30 29 28 27 26 25 24 23

Z2 11.9718 11.5113 11.0426 10.5557 10.0541 9.55146 9.02372 8.47811 7.91785 7.31366 6.66876 5.27667Z1 22 21 20 19 18 17 16 15 14 13 12 11

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1851M. The utility function would take into consideration the DMs preferences. Determination of utilityion is not simple. The major advantage of utility function method is that if the function has been cor-assessed and used, it will ensure the most satisfactory solution to the DM. The common formes that:

s utility function is additively separable with respect to the objectives;special form of utility function that has been widely used in MODM problems uses the weights wr toicate the importance (as cardinal) of each objective. It is better to get wrs from the DM. However, weve used the two following utility functions with p = 1 and 2:

min U 1 w1 X33i1

xi Z1! ,

Z1

! w2 Z2

X33i1

cixi

! ,Z2

!; 16

w1 w2 1;

min U 2 w1 X33i1

xi Z1! ,

Z1

!2 w2 Z2

X33i1

cixi

! ,Z2

!2; 17

w1 w2 1.

The advantage of using this utility function is its simplicity. The disadvantage of this approach is:

There are very few cases where utility functions are additively separable. The wrs depend upon both the achievement level of Zr itself, and the relative achievement of Zr com-pared to the achievement levels of the other objective functions.

0.4ti

Fig. 8.

1852 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 183918580

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1 1.2The weight of the first objective function

The v

alue

of

two

u The second objectivefunction value3.2.4. Sensibility analysis

We used two simple types of sensibility analysis. The rst shows the weight of the objective function ofutility function method against the weight of the rst objective function (w1) (of the all scenarios and all thecritical distances). Fig. 8 shows one of these instances. The second shows the values of normalized objectivefunctions against the value of the rst objective function. Fig. 9 shows one of these instances.

3.3. Districting model

Phase 5. After determining the locations of all the supportive centers in Phase 4, we should solve a dis-tricting or partitioning problem to assign each of the supported centers to only one of the located support-ive centers. Many algorithms have been proposed for partitioning and districting problem (Bozkaya et al.,2003; Fisher and Kedia, 1990). Here, we made a model as follows:

For the supported center i which potentially can be covered by only one of the located supportive centersj (aij = 1), there is not any problem in allocating; i.e. we have only one element 1 in column j provided

0.5

0.6

0.7

lity

func

tions

The first objectivefunction valueThe value of utility functions against the weight of the rst objective function (of the scenario 0 with 12-hour critical distance).

Fig. 9.xj = 1. The problem arises when a supported center rests in the critical distance of more than one of thelocated supportive centers.

Suppose that after solving the MODM model in Phase 4 we have found p (p 6 n) locations for locatingall of the supportive centers. In this phase, we made a new matrix Bmp = [bik] out of Amn = [aij] by delet-ing some of the columns in Amn corresponding to the variables xj = 0.

The objective functions and the constraints of the problem are as follows:

Objective function type 1: The number of assigned supported centers to each of the located supportive cen-ters must be as balanced as possible.

Objective function type 2: The dierence between the demands of dierent supportive centers must be aslittle as possible.

Constraints: Assign each of the existing xed supported centers to only one of the located sup-portive centers.

scenario 0 with 12-hour critical distance).Th

ykl isl; othfunctlocateportelinearbrief,0

1

2

0 0.2 0.4 0.6 0.8 1 1.2The weight of the first objective function against the other

The

norm

aliz

ed o

bjw

ith re

spec

t to

the

The normalized objective function values (using the rst utility function) against the weight of the rst objective function (of the3

4

5

6

ective

func

tion v

alues

firs

t util

ity fu

nctio

n

The first objectivefunction value

The second objectivefunction value

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1853erefore, we have a model as follows:

min U maxql1

X233k1

wkbklykl

!; 18

s.t. Xql1

bklykl 1; k 1; . . . ; 233; 19

ykl 2 0; 1f g; k 1; . . . ; 233; l 1; . . . ; q;a binary variable that is equal to 1 if the supported center k is assigned to the located supportive centererwise is equal to 0. wk is the amount of demand (or size) of the supported center k. The objectiveion (18) ensures that the amount of demand imposed by assigned supported centers to each of thed supportive centers is equaled, as much as possible. The constraint (19) ensures that each of the sup-d centers is assigned to only one of the supportive centers. Note that the objective function (18) is non-(because of max operator) and we can easily linearize it by adding new variables and constraints. InPhase 5 can be summarized as shown in Fig. 10.

the above-mentioned mathematical models and performing sensibility analysis, any commercial ILP solvercan be used. Our models were run on a PC with a Intel 1.4, Pentium 4 CPU and 256 Mb of RAM. The

summary of the decisions made are as follows.

4.1. Critical distances

Military problems are critical and sensitive. In business problems, cost generally plays very importantrole in making decisions, while in military problems cost can be only one of the factors aecting makingdecisions. As seen before, we used many factors in designing models; but because of high investment neededto build supportive centers, cost limited our decisions.

In this case, we prefer critical distance to be as small as possible. Considering the objective function (10),at least two supported centers for the 24-hour critical distance, 6 supported centers for the 12-hour criticalOur model becomes more ecient if we reduce the size of the problem, by applying the following twomodications:

Let ykl = 1 for the supported centers which can be covered only by one of the supportive centers. Dene ykl only for bkls which are equal to 1 (most of the elements in Bmq = [bkl] are zero). Therefore,sparsity of the matrix aects the number of binary variables. For instance, the sparsity of the matrix inthis case (for the selected critical distance) is 16.07%.

We solved the above-mentioned mathematical model in order to partition supported centers and assignthem to the supportive centers (whose locations were found in Phase 4).

4. Experimental results and implementation

In this case, there was a military decision maker group including some of high-level military commanderswho made some strategic decisions after investigating the proposed critical distance, the scenarios, and theoutputs of computational results. This group made their decisions in brain storming sessions. For solving

Output ProcessInput

Type of allocation orcovering domain of each

supportive center

Suitable partitioningmodel with respect toselected criteria for

appropriate allocation

Supportive centers locations Criteria for appropriate allocation

Modified covering matrixBm q=[bkl]

Fig. 10. Phase 5 of the methodology from a systematic point of view.

1854 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858distance and 11 supported centers for the 8-hour critical distance are needed. Regarding the required initialinvestment and annual operating costs, the decision maker group decided to use the 12-hour critical dis-tance (presently, critical distance is 24 hours).

4.2. Scenarios

After investigating computational results of the three scenarios using objective function (10) we saw that:

Scenario 0: At least six new facilities are needed; these facilities are all new and none of them currentlyexists;

Scenario 1: At least eight facilities are needed; 5 of them are new facilities; and 3 of them are existing facil-ities which all must continue operating;

Scenario 2: At least seven facilities are needed; ve of them are new facilities; and two are among the threeexisting facilities and must keep on operating.

Again, regarding the required initial investment and annual operating costs, the decision maker groupdecided to use the 12-hour critical distance (presently, critical distance is 24 hours).

4.3. Utility functions

The decision maker group, based on their experiences and after investigating computational results,decided to use the utility function. The members of the group believe that normally in making military,emergency and critical decisions, utility function (17) results better match with real world requirements than(16). Thus, we have used the quadratic utility function.

4.4. Objective values

Using sensibility analysis on the objective function values and considering the selected critical distance,the selected scenario and the selected utility function, the decision maker group decided to use valuesw1 = 0.65 and w2 = 0.35. Our investigations show that making decision around w1 = 0.1 is too sensitive;fortunately the opinion of the group about w1 is signicantly far from 0.1.

4.5. Benets

The major benets of the proposed model in comparison with the current situation are as follows:

In the current situation, three supportive centers serve in Logistics Department system. Using the pro-posed model, this number increases; but critical distance in serving the supported centers improves byabout 105%.

With respect to the selected critical distance, currently about 9.5% of existing supported centers are outof critical distance (coverage radius); while in the proposed model, there are not any supported centersout of the critical distance of their assigned supportive center.

Currently, the workload of supportive centers are very unbalanced; for instance, the amount of work-load of the largest supportive center is 2.4 times greater than the smallest one; while in the proposed sys-tem this ratio has decreased to 0.35.

4.6. Implementation problems

Some important issues should be considered in using the proposed model:

The model works only at the time of peace; i.e. in war condition most of supported centers (corps andbrigades) move to the related war region; so they enter the region assigned to the other supportive cen-ters which is not considered in the model. Obviously, the war situation is a critical situation and its plan-ning issues (response to such events) will dier from peace condition.

There are two types of supported centers: mobile and xed. Most of the supported centers are xed inlocation. Mobile supported centers move all around the country based on their missions; so, those will

R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858 1855use the materials stored in the related supportive centers of the region where they are positioned

will be used in such conditions by mobile facilities.

building such facilities simultaneously is impossible. Based on the time needed to build these facilitiesand time constraints to prepare the facilities, designing a prioritization plan for installing the proposed

and maximum quality of locations in a military logistics system was investigated. The supportive centers

1856 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858should cover all of the predetermined supported centers; in addition, each supported center must be laid incritical distance of only one of the located supportive centers. However, in such cases due to the size, complex-ity and large number of attributes in the problem, it is solved sequentially, as we did here. In each phase, wexed some of the variables; then we solved another model in which the xed variables were the input data forthe next phase. In this paper, we presented a ve-stage procedure that in each stage regarding the situation weused dierent tools and models like MADM, covering, districting, MODM, binary programming and qua-dratic programming. Solving the problem regarding to various scenarios, studying benets of the method,investigating implementation problems and proposing future works were the last stages of this paper.

Acknowledgments

We thank Prof. Ferenc Szidarovszky and Nasim Akbarzadeh for their valuable comments on this work;their suggestions are appreciated. The authors also thank two anonymous referees for their constructivesuggestions, which improved the content substantially. The authors are responsible for all results and opin-ions expressed in this paper.

References

Aickelin, U., 2003. An indirect genetic algorithm for set covering problem. Journal of the Operations Research Society 53 (10), 1118supportive centers is an important research area. Critical distance is a dynamic parameter; for instance, based on the dierence between velocity of thevehicles moving from the supportive centers to the supported centers in winter and summer, some prob-lems will occur. Thus, solving a dynamic (time dependent) covering problem is a worthy research area.

In the proposed model, the relationship between the supportive centers (via moving needed materialsand goods) is not considered; this will be an important issue especially in the war conditions when manymobile facilities accumulate in an emergency region. Thus, considering relationships between the sup-portive centers in the covering problem is another important future research area.

5. Conclusion

In this paper, a case study about nding the locations of some supportive centerswith theminimumnumber4.7. Future works

Based on the experiences gained during this case study, following researches are proposed:

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1858 R.Z. Farahani, N. Asgari / European Journal of Operational Research 176 (2007) 18391858

Combination of MCDM and covering techniques in a hierarchical model for facility location: A case studyIntroductionOrigins (sources for supplying all needed goods and materials)Supportive centers (warehouses, repair shops and terminals for receiving, storing and shipping all needed goods and materials to supported centers)Supported centers (centers that consume the shipped goods and materials)

The dataThe methodMADMMODMConflicting objectivesEfficient solutionA preferred solutionGlobal criterion methodUtility function method

Sensibility analysis

Districting model

Experimental results and implementationCritical distancesScenariosUtility functionsObjective valuesBenefitsImplementation problemsFuture works

ConclusionAcknowledgmentsReferences