Mathematical and Computer Modelling 49 (2009) 1030–1043

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

Some concepts of the fuzzy multicommodity flow problem and theirapplication in fuzzy network designMehdi Ghatee, S. Mehdi Hashemi ∗Department of Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran

a r t i c l e i n f o

Article history:Received 3 February 2008Received in revised form 25 June 2008Accepted 7 August 2008

Keywords:Uncertain costImprecise demandPath enumerationNetwork planning

a b s t r a c t

We treat with the Minimal Cost Multicommodity Flow Problem (MCMFP) in the setting offuzzy sets, by forming a coherent algorithmic framework referred to as a fuzzy MCMFP.Given the character of granular information captured by fuzzy sets, the objective is tofind multiple flows satisfying the demands of commodities, by using available suppliesconsuming the least possible cost. With this regard, the supply and demand of nodesmay be presented linguistically; the travel cost and capacity of links can be definedunder uncertainty as well. To solve this problem, two efficient algorithms are motivated.In the first, we utilize fuzzy shortest paths and K -shortest paths to generate preferredand absorbing paths, and then we find the flow on them by solving a classic MCMFP.The second algorithm exhibits with fuzzy supply-demand, and employs a total order ontrapezoidal fuzzy numbers to reduce the fuzzy MCMFP into four classic MCMFPs. Someexamples are solved to demonstrate the performance of the presented methods. Amongthe various applications of this scheme in providing a suitable interface between themodeland physicalworld, we focus on network design under fuzziness. The granular nature of thedescription of the future travel demand contributes to the generality of the planningmodel,and determines a certain perspective from which we will looking at the network.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In many physical world problems we face, granular information includes inexact and unsure data, but optimizationtechniques need to be well-defined and to be given precise data. The use of fuzzy sets as a technology coping with granularinformation is sound [27,9]. Granular computing, an emerging computing paradigm of information processing, concerns theprocessing of complex information entities called information granules, which arise in the process of data abstraction andderivation of knowledge from information. Generally speaking, information granules are collections of entities that usuallyoriginate at the numeric level, and are arranged together due to their similarity, functional adjacency, indistinguishability,coherency, or the like [4]. As a theoretical perspective, it encourages an approach to data that recognizes and exploits theknowledge present in data at various levels of resolution or scales. In this sense, it encompasses all methods which provideflexibility and adaptability in the resolution at which knowledge or information is extracted and represented, see Bargielaand Pedrycz [2] for details.Following the principle of information granulation, it is important to know the solution of an optimization problem [27].

The concept of the solution of these programs, usually, is constructed on a rank for comparing fuzzy data, which arevarious in literature [15]. However, in contrast to the classical fuzzy optimization programming problems, the concept of

∗ Corresponding author. Tel.: +98 21 64542522; fax: +98 21 66497930.E-mail addresses: ghatee@aut.ac.ir, ghatee@cic.aut.ac.ir (M. Ghatee), hashemi@aut.ac.ir (S.M. Hashemi).URL: http://math-cs.aut.ac.ir/cs/∼ghatee/ (M. Ghatee).

0895-7177/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2008.08.009

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1031

Fig. 1. An LR type flat fuzzy number a = (a1, a2, a3, a4)LR .

solution of fuzzy networks is limited, see e.g., fuzzy shortest path problem [25], or fuzzy minimal cost flow problem [30,15,16]. These applied problems can be studied through a general model entitled: the Minimal Cost Multicommodity FlowProblem (MCMFP). The target of this problem is to find the least cost of the shipment of some commodities througha capacitated network with respect to the available supplies at certain nodes, which should be transmitted to fulfilldemands at other nodes. The modeling, optimality conditions, solution algorithms and duality concepts of MCMFP has beenintroduced [1, sec. 17], however, based on the best our knowledge, MCMFP has not been studied in a fuzzy way, whilethis program is influenced by a manifestation of the complex human, social, economic and political interactions associatedwith uncertainties, vagueness and ambiguities, and so it is desirable to model and to solve this problem with flexible fuzzyframework.In introducing fuzzyMCMFP, we consider two relevant concepts in this article; fuzzy travel cost and fuzzy travel demand.

In both of the cases we present associated embedded programswhich can be solved efficiently. In the first scheme, the fuzzyshortest paths [25] and K -shortest paths [1,17] are pursued to generate the reasonable paths, and then a classic MCMFP isused to determine the flows on these preferred and absorbing paths. In the second scheme, we utilize a Hukuhara difference[22]. The authors illustrated that a Hukuhara difference has a better interpretation in comparison with Zadeh’s difference infuzzy transportation [15]. Thenwe seek a total order on LR type flat fuzzy numbers to find the solutions of this combinatorialoptimization problem. Ourwork is similar to that ofWu [33]which employs an embedded function. Since thesemethods areembedded in classic MCMFP in the final part, each improvement in MCMFP can improve the performance of the presentedschemes. Furthermore, since the MCMFP has an essential role in network design problems [5], the results of this paper canbe implemented in network design under fuzziness.We interpret the application of fuzzy network design in urban planning.Given the above stated objectives, we structure the material in the following manner. Some basic definitions and results

on fuzzy numbers are given in the next section. In Section 3 a total order on LR type flat fuzzy numbers is introduced,with some instructive results on its performance. In Section 4, the MCMFP is modeled and the effect of fuzziness in suchtransportation problems is investigated. Section 5 includes two variants of fuzzy MCMFP to handle uncertain travel costsand demands. Also some numerical examples are given, to show the reasonability of contribution. Section 6 consists of theapplication of fuzzy MCMFP in a real network design. The final section provides a brief conclusion and future directions.

2. Fuzzy numbers and arithmetic

A fuzzy number is a convex normalized fuzzy set of the real line R, whose membership function is piecewise continuous[7]. The set of fuzzy numbers on R is denoted with F (R). An LR type flat fuzzy number [25], is denoted as a =(a1, a2, a3, a4)LR, if

µa(x) =

L(a2 − xa2 − a1

), a1 ≤ x ≤ a2,

1, a2 ≤ x ≤ a3,

R(x− a3a4 − a3

), a3 ≤ x ≤ a4,

(1)

where the symmetric non-increasing function L : [0,∞] 7→ [0, 1] is the left shape function, that L(0) = 1. Naturally, aright shape function R(.) is similarly defined as L(.), (see Fig. 1). We denote the LR type flat fuzzy numbers on real line withLR(R).According to the extension principle, the binary operation ? ∈ {+,−,×,÷} onR can be extended to the binary operation

? ∈ {+, −, ×, ÷} on fuzzy numbers a and bwith the membership functions µa(x) and µb(x), with the following:

µa?b(z) = supz=x?y

min(µa(x), µb(y)). (2)

Also the fuzzy scalar product of two fuzzy vectors x = (x1, . . . , xn) and y = (y1, . . . , yn) inF (Rn) is defined [32] as follows:

x×y =+∑

i=1,...,n

xi×yi. (3)

1032 M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043

Let a = (a1, a2, a3, a4)LR and b = (b1, b2, b3, b4)LR belong to LR(R). The exact formula for the extended addition andapproximated formula for the extended multiplication are as follows,

(a1, a2, a3, a4)LR+(b1, b2, b3, b4)LR = (a1 + b1, a2 + b2, a3 + b3, a4 + b4)LR (4)(a1, a2, a3, a4)LR ⊗ (b1, b2, b3, b4)LR = (a1.b1, a2.b2, a3.b3, a4.b4)LR (5)

where a ≥ 0 and b ≥ 0. For the application of these arithmetic operators in fuzzy matrix computations see e.g. [7]. It isimportant to note that multiplication of two LR fuzzy numbers does not produce LR fuzzy number. In these situations, onecan use interpolation methods to get a fuzzy number, whose some α-cuts are equal to the multiplication of two α-cuts ofcorresponding operands. This idea was followed in [8] with details.Also scalar multiplication is derived as follows,λ(a1, a2, a3, a4)LR = (λa1, λa2, λa3, λa4)LR,

where, λ ≥ 0.As mentioned by Diamond and Korner [10], b−a is not really a difference, and is rather unnatural with respect to a

linear structure. For example, b+(−1)a is not compatible with the difference in the function space. But the Hukuharadifference [22] defined as the solution for x in the equation a+x = b if it exists, does coincide with the difference in thefunction space. This property justifies the application of this difference instead of the fuzzy number b+(−1)a. We presentthis operator according to the work by Hukuhara [22], Diamond and Korner [10], Ghatee et al. [16] entitled as Hukuharadifference as follows:

Definition 2.1. For a and b belong toLR(R) if the Hukuhara difference bH a exists, it is given by

[bH a]α = {ς ∈ R | [a]α+{ς} ⊆ [b]α}, (6)

where + in the above equation for two sets X and Y means as follows:

X+Y = {x+ y|x ∈ X, y ∈ Y }.

For some cases, it can be proved that (see [10, Proposition 1]):(b1, b2, b3, b4)LRH(a1, a2, a3, a4)LR = (c1, c2, c3, c4)LR,

wherec1 = b1 − a1, c2 = b2 − a2, c3 = b3 − a3, c4 = b4 − a4.

It is important to note that with respect to this definition, aH a is zero, while by Zadeh’s difference it is not true. Thereforethis operator for solving systems including flow transshipment has more accurate and reasonable results in comparisonwith Zadeh’s difference.When Hukuhara difference is not well-defined, by utilizing L2 norm, it may be found an x that fulfills in a+x = b, see [10]

for details. But we utilize the above result in all of the situations for getting a simple and applicable expression in flowmodeling, and adopt the following definition, see [15,16].

Definition 2.2. For a and b belong toLR(R), we have:

bH a = (b1, b2, b3, b4)LRH(a1, a2, a3, a4)LR= (b1 − a1, b2 − a2, b3 − a3, b4 − a4)LR.

Through to the end of this paper, we eliminate the subscript ′LR′ of numbers belonging to LR(R), wherever there is noambiguity.

3. Order on fuzzy numbers

In order to define the inequality relation between two fuzzy numbers, many methods have been proposed in theliterature [15]. But perhaps the most convenient and directive method in this area is based on the concept of comparison offuzzy numbers by the use of ranking functions, in which a ranking function R : F (R)→ R that maps each fuzzy numberinto the real line is defined for ordering the elements ofF (R), see e.g., [15]. We pursue this approach in this article, but firstnote the following order, which was utilized by Okada and Soper [25] for the fuzzy shortest path problem:

Definition 3.1. Let a = (a1, a2, a3, a4) and b = (b1, b2, b3, b4) belong toLR(R). Then a�b if the following four inequalitieshold simultaneously:

a1 ≤ b1, a2 ≤ b2, a3 ≤ b3, a4 ≤ b4.

It is obvious that this order cannot rank all elements of LR(R). Thus, in [25], this order was only utilized to define Paretooptimal or non-dominant solution(s) for the fuzzy shortest path problem. Such non-dominant paths are utilized to solvefuzzyMCMFP in the future. But before discussingmore, we introduce the following flexible order analogues to that of Ghateeand Hashemi [15]:

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1033

Definition 3.2. Consider four real positive numbers k1, k2, k3, and k4. The less than or equal relation≤k1,k2,k3,k4 onLR(R),is defined as follows:

(a1, a2, a3, a4)≤k1,k2,k3,k4(b1, b2, b3, b4),

if and only if

k1.a1 + k2.a2 + k3.a3 + k4.a4 ≤ k1.b1 + k2.b2 + k3.b3 + k4.b4where≤means the common less than or equal relation on real numbers R.

This definition permits one to exhibit with the risk averse, risk neutral, and risk seeker decision makers [15], i.e. since thelower and upper bounds of fuzzy numbers, can be interpreted as the bounds of the pessimistic and the optimistic attitudes[15], by trading off amongweights k1, . . . , k4, it is possible to take such risks into account utilizing≤k1,k2,k3,k4 in comparisons.

Proposition 3.3. Order relation≤k1,k2,k3,k4 onLR(R) is a reflexive and transitive relation.

In order to make a total order [15], we select ki, i = 1, 2, 3, 4 more carefully, limiting our choice to non-algebraic numbers.

Definition 3.4. A complex number Z is named algebraic if and only if, it is a root of a non-zero polynomial equation byinteger coefficients, else named non-algebraic or transcendental.

Denoting the set of rational numbers and positive rational numbers with Q and Q+, respectively. We have the followinginstructive result.

Proposition 3.5. Consider non-algebraic real positive number ϑ . Let ki = qiϑni , for i = 1, 2, 3, 4, where qi ∈ Q+ are positiverational numbers and n1 6= n2 6= n3 6= n4 are nonnegative integer numbers. Then ≤k1,k2,k3,k4 on TQ = {(a1, a2, a3, a4) ∈LR(R)|ai ∈ Q, ∀ i = 1, . . . , 4} is a total order.Proof. The proof of this proposition is similar to that of Proposition (2.18) in Ghatee and Hashemi [15]. �

Remark 3.6. Working on TQ has no restriction in practice, because only the numbers belonging to TQ with finite floatingpoint can be registered, and also be saved in current computers. Furthermore, for implementing non-algebraic numbers incomputer programming, one can use double precision.

Property 3.7. Let a = (a1, a2, a3, a4) and b = (b1, b2, b3, b4) belong to TQ, then

a = b

if and only if

k1.a1 + k2.a2 + k3.a3 + k4.a4 = k1.b1 + k2.b2 + k3.b3 + k4.b4,

if and only if

a1 = b1, a2 = b2, a3 = b3, a4 = b4,

where k1, k2, k3, k4 satisfy the assumptions of Proposition 3.5.

4. Minimal cost multicommodity flow problem

In this section, we sketch the basic concepts of the Minimal Cost Multicommodity Flow Problem (MCMFP) whichtransships the flows of the commodities by minimal cost, that satisfies the demand for each commodity at each vertexwithout violating the constraints imposed by the supply-demand and capacity. For a given network G = (N, A)with N andA as the sets of nodes and links, the following model represents a general MCMFP:

minT∑t=1

∑(i,j)∈A

cti,j.xti,j, (7)

s.t.

T∑t=1

xti,j ≤ ui,j, ∀(i, j) ∈ A∑j:(i,j)∈A

xti,j −∑j:(j,i)∈A

xtj,i = bti , ∀i ∈ N,∀t = 1, . . . , T .

xti,j is a positive integral regarding the amount of flow from tth commodity which streams through link (i, j), ui,j is thecapacity of link (i, j) and T is the number of commodities. Moreover for commodity t , cti,j and b

ti are the unit-cost of flow

through link (i, j) and the supply or demand of node i.Without loss of generality, assume each commodity has one resource and one customer, if not change the variables [1].

Assume r t , st and dt denote the resource node, the customer node and the travel demand for a given commodity t ,respectively. Then, we meet the following model regarding the path-flow variables:

1034 M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043

minT∑t=1

∑p∈P tctp.f

tp , (8)

s.t.

∑p∈P tf tp = d

t , ∀t = 1, . . . , T ,

xi,j =T∑t=1

∑p∈P t

δi,jp ftp ≤ ui,j, ∀(i, j) ∈ A,

where the flow of commodity t along the path p is denoted by f tp . Also δp = (δi,jp )(i,j)∈A is link-path incident vector and assigns

1 when the link (i, j) shares in path p and 0 otherwise. P t includes all of the directed paths joining r t and st .The first constraint ensures that the demand for each commodity is supplied by the path flows activated for that

commodity. Since xi,j denotes the sum of flow of individual paths including link (i, j), the second constraint implies thatthe total flow on link (i, j) is less than its capacity.The vast majority of this classic model is deterministic, as one assumes the travel cost and demand to be known a priori

for the entire links and nodes [1], while this problem is influenced by social and economic interactions and is dependenton users’ perceive information, non-deterministic nature of the problem and granular information. Granular Computingis an emerging conceptual and computing paradigm of information processing. It has been motivated by the urgent needfor intelligent processing of empirical data that is now commonly available in vast quantities, into a humanly manageableabstract knowledge. In this sense, granular computing offers a landmark change from the currentmachine-centric to human-centric approach to information and knowledge. The theoretical foundations of granular computing are exceptionallysound, and involve set theory (interval mathematics), fuzzy sets, rough sets, and random sets linked together in a highlycomprehensive treatment of this emerging paradigm [2]. In transportation concepts, because of granular information andthe different provision of traffic information, the different types of transportation might be deduced. Now, we consider acase that the data of MCMFP have been exhibited withLR(R) numbers, which handle vagueness as an important categoryof uncertainty. First, we elaborate the origin of fuzziness in real problems. In the next section, we introduce two extendedvariants to take uncertain travel cost, and uncertain travel demand into account.

4.1. Fuzziness in real transportation problems

Transportation models adopted with fuzzy relations and quantities can contribute stochastic versions in uncertaintymodeling. More specifically, since the classical random utility-based approach is not sufficient for uncertainty handling,some alternate solution to probability-based models may be obtained with possibility-based methods, see e.g., Ghatee andHashemi [17]. But as historical background, fuzzy logic tools were initially been applied to transportation problems by Daset al. [6]. Then, great attention has been focused on this problem. From a high level categorization, in real networks, such asurban networks, we are faced with the following three uncertainty sources:• Unsure network topology, (a network with fuzzy nodes and fuzzy links),• Inexact travel cost, (a network with fuzzy link cost),• Imprecise travel demand, (a network whose nodes include fuzzy excess or fuzzy deficit).

The aim of fuzzy transportation is to find the lowest transportation cost of some commodities through a capacitatednetwork when the supply and demand of nodes and the capacity and cost of links are represented as fuzzy numbers. Giventhe character of granular information captured by fuzzy sets (where one could capitalize on the nonbinary character of theirmembership functions), suchmethods are capable of handling the decisionmaker’s risk taking. This problem is a newbranchin combinatorial optimization and network flow problems [15–17]. Some application of such standpoint were presented inindustries [12,15].It is interesting to checkwhichmethods in traditional optimization problems can be extended to fuzzy ones. For example,

one can introduce the transformationwhichmaintains the nice structure of problem.When such a transformation is in hand,the valuable algorithms of classic problems can be extended to fuzzy variants. [18].In what follows, we pursue the traditional methods which can be utilized to solve fuzzy MCMFP.

5. MCMFP in fuzzy environment

MCMFP arises naturally in engineering and economics contexts; it appears in problems involving equilibrium models,such as urban transportation systems, resistive electrical networks, and production-distribution problems. In designingtelecommunication networks, it also has an important role. The central problem in all of these is finding the lowesttransportation cost of some commodities through a capacitated network in order to satisfy demands at certain nodes, usingavailable supplies at other nodes. In applied problems, this model is influenced by complex human, social, economic, andpolitical interactions, and so it is dependent on user perception of information, nondeterministic factors of the networkand granular information. Considering fuzzy numbers as the amount of supplies, demands, capacities and costs provides areasonable infrastructure to handle vagueness as an important category of uncertainty. Such an implementation of MCMFPcan be applied to recognize the appropriate models of traffic assignment with intelligent agents, see e.g., [17] for details.

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1035

5.1. MCMFP with uncertain travel cost

Due to the fuzziness of a driver’s perception over link travel time, fuzzy sets of perceived link travel time are developed toreflect a driver’s perception of link travel time under different traffic conditions. Liu et al. [24] used linguistic descriptions torepresent these fuzzy sets. Since there are various traffic conditions, it is necessary to construct fuzzy sets for most frequenttraffic conditions, such as normal, congestion, incident and construction. Obviously, more fuzzy sets can be constructed torepresent other traffic conditions (such as special event) if necessary. Henn [21] also considered the meaning of fuzzy costsin fuzzy traffic assignment models. In this paper, assume the cost, the length or the travel time of a link (i, j) for commodityt are given by

cti,j = (ct(i,j),1, c

t(i,j),2, c

t(i,j),3, c

t(i,j),4) ∈ LR(R).

Then for shipping T commodities through the network, we can write:

minT∑t=1

∑p∈P tctpf

tp , (9)

s.t.

∑p∈P tf tp = d

t , ∀t = 1, . . . , T ,

xi,j =T∑t=1

∑p∈P t

δi,jp ftp ≤ ui,j, ∀(i, j) ∈ A,

(10)

where ctp =∑+

(i,j)∈p cti,j for each p ∈ P

t .

One important concept of this problem is theway of generating the absorbing and preferred paths P t . Usually, the columngeneration approach [1] is employed for solving the classic MCMFP. However, to understand P t , we take path enumerationtechniques including a great number of algorithmswith the capability of finding the reasonable and preferred paths througha network, where the preference can be defined in terms of time, cost, distance, or some combination of these items. Thesereasonable paths are the base of problems consisting of path-flow variables. Following that, two applicable schemes in thiscategory are presented, the others can be meet in Ghatee and Hashemi [17].

5.1.1. Non-dominant fuzzy shortest paths schemeOkada and Soper [25] have defined the concept of the non-dominant fuzzy shortest path applying the ordermentioned in

Definition 3.1. After determining the cost of path p as fuzzy number cp, they utilize an algorithm for selecting non-dominantfuzzy numbers between such cps. Also they were developed a label-setting algorithm, analogous to that of Dijkstra [1]for this target. In virtue of the fuzziness in model (9)–(10), non-dominant fuzzy shortest path may provide an pertinentinfrastructure for solution process.

5.1.2. K-shortest paths schemeAs previous, let P t denotes the set of paths from r t to st inG = (N, A). Let ctp assigns a cost to path p ∈ P

t . In the K -shortestpaths problem, for a given positive integer K , it is intended to determine a set PK = {p1, . . . , pK }, such that:• pi is determined before pi+1 for any i ∈ {1, . . . , K − 1}• cpi ≤ cpi+1 for any i ∈ {1, . . . , K − 1}• cpK ≤ cq for any q ∈ P

t− PK .

In [1], an O(n3) algorithm for finding K -shortest paths has been studied. Also, in [17] a modification on this scheme isreported. Due to knowledge of travelers about travel times, such an idea can contribute to achieving the equilibrium state ina network, whose agents are intelligent, e.g., the role of K -shortest path is mentioned in [17] for traffic assignment modelswhich is an extension of a MCMFP considering equilibrium concept.

5.1.3. Computational method for MCMFP with uncertain costLetMt paths pt1, . . . , p

tMt with the relative costs c

t1, . . . , c

tMt are activated for transmitting the commodity t . Thus, we can

rewrite the objective function (9) as follows:T∑t=1

∑p∈P tctpf

tp =

T∑t=1

Mt∑p=1

(ctp,1, ctp,2, c

tp,3, c

tp,4)f

tp

=

T∑t=1

Mt∑p=1

(ctp,1ftp , c

tp,2f

tp , c

tp,3f

tp , c

tp,4f

tp )

=

(T∑t=1

Mt∑p=1

ctp,1ftp ,

T∑t=1

Mt∑p=1

ctp,2ftp ,

T∑t=1

Mt∑p=1

ctp,3ftp ,

T∑t=1

Mt∑p=1

ctp,4ftp

). (11)

1036 M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043

It is possible to embed such a fuzzy number into a vector in R4. Generally speaking, MCMFP with fuzzy travel cost isregarded to a vector-optimization problemwhose coefficients may be optimized simultaneously. This perspective, providesthe following 4-objectives programming problem:

minT∑t=1

Mt∑p=1

ctp,1ftp ,

minT∑t=1

Mt∑p=1

ctp,2ftp ,

minT∑t=1

Mt∑p=1

ctp,3ftp ,

minT∑t=1

Mt∑p=1

ctp,4ftp ,

(12)

subjected to the constraints (10).At the same time, the uncertainty of information generates the uncertainty decision regions. The corresponding examples

are given, for instance, in [14] for continuous optimization problems and in [12] for discrete optimization problems. Apossible way of overcoming of these situations is to reduce the problem to multicriteria decision making, because theapplication of additional criteria, including criteria of qualitative character, is a convincing means to decrease the decisionuncertainty regions, see e.g. [3] for a practical example. Taking these concepts into account, it is possible to reduce thedecision uncertainty regions when other methods for comparing fuzzy quantities are used. The total order that wasmentioned previously, can produce some non-dominant solutions in large scale fuzzy optimization problems, by changingtheweighting parameters. Also, a set of solutions, instead of just one for fuzzy goal programming problemswas investigatedin [20]. In their work, the optimistic and pessimistic solutions have been determined by employing two kinds of orders:severe order and lenient order. A convex combination of these solutions have been generated as the fuzzy solution of fuzzygoal programming. When the priorities of objective functions are well-known, preemptive ordering can be pursued, seee.g., Ghatee et al. [19]. Another approach which can be pursued, is to define a function to defuzzify the problem and thenfuzzify the solution by an inverse function. Such idea was utilized in [8] for matrix computations, which can be generalizedto find the optimal solution of such fuzzy optimization problems. Furthermore, one can deal with an optimization problemsuch as transportation problem by applying an interactive fuzzy programming method, see Sakawa et al. [29] for details.Pursuing such ideas on objective function (12) and nothing the shape of LR type flat fuzzy numbers implies that

ctp,1ftp ≤ c

tp,2f

tp ≤ c

tp,3f

tp ≤ c

tp,4f

tp ,

andwe canminimize each of these functionswithout perturbing the others. Moreover, it is worthwhile to select the strategyof aggregation of the different fuzzy criteria. Dubois and Prade [11] provide an extensive survey on fuzzy set-theoreticoperations. Several different types of aggregation operators, such as t-norms, t-conorms or mean operators have beendefined and used in the last decade, see e.g. [20]. Another important aspect, is the relative importance of each of the criteriaand how this can be incorporated [31]. This is the subject of fuzzy weighted aggregation. Inclusion is normally achieved byretaining the aggregation operators as they are defined, and then associating the weights with the membership functionsas product, power, max-min, t-conorms or t-norms. Kaymak and Sousa [23] presented an extensive analysis on differentaggregation operators and weighting factors. The authors of [31] also described the general framework of fuzzy weightedaggregation and its application to the logistic scheduling problem. Inwhat follows,weuse aweighting approachwith respectto the decision maker viewpoint. It is possible to use some advanced techniques such as fuzzy AHP [7] when the relatedimportance of weights are not to be known. We take the following objective function:

minT∑t=1

∑p∈P t

(w1.ctp,1 + w2.ctp,2 + w3.c

tp,3 + w4.c

tp,4)f

tp , (13)

which associatewith (10), is a classicmulticommodity problemand canbe solved for examplewith interior point algorithms,see e.g. [28] or traditional combinatorial algorithms, see e.g. [1].As a final point, note that by different settings of (w1, w2, w3, w4), it is possible to find all of the Pareto-optimal solutions,

but because of infeasibility of considering all of the settings, we cannot find all of the Pareto-optimal solutions. However,in real problems, the abstract knowledge of weights and an approximation of optimality are usually in hand, which may beapplied to recognize the optimal settings. Introducing an intelligent scheme for this aim is one important goal for a DecisionSupport System (DSS), and may be pursued in future researches.

5.1.4. Numerical test of uncertain costsThe performance of the proposed scheme is evaluated by using this scheme on a random produced network with 13

nodes and 31 links, which is represented in Fig. 2. Consider the data represented in Table 1. We should carry commodities

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1037

Fig. 2. Random network with 13 nodes and 31 links.

Table 1The capacity and fuzzy cost of network links are given

Link label Source Customer Capacity Fuzzy cost opt(c1) opt(c2) pes(c1) pes(c2)

1 1 3 75 (20, 30, 37, 45) 13 32 13 322 1 4 45 (30, 32, 36, 42) 45 453 1 5 93 (66, 72, 81, 89) 13 17 284 1 6 47 (28, 33, 38, 45) 19 11 425 2 3 42 (47, 56, 60, 63)6 2 4 85 (6, 14, 22, 25)7 2 5 53 (99, 104, 109, 112)8 2 6 20 (58, 58, 60, 65) 309 3 7 67 (42, 50, 57, 64) 3010 3 8 84 (52, 56, 64, 67) 13 1311 3 9 2 (33, 39, 39, 47) 2 212 4 7 68 (43, 51, 58, 64)13 4 8 38 (23, 32, 36, 40) 38 3814 4 9 83 (58, 65, 73, 80)15 4 10 50 (76, 78, 83, 88)16 4 11 71 (53, 57, 64, 68) 7 717 5 7 43 (64, 73, 77, 84)18 5 9 30 (21, 30, 33, 39) 13 17 2819 6 8 19 (38, 42, 44, 52) 19 1920 6 10 19 (78, 87, 89, 99) 1121 6 11 68 (68, 69, 76, 81) 2322 7 12 30 (46, 50, 53, 62) 30 3023 7 13 54 (57, 65, 70, 72)24 8 12 15 (79, 79, 81, 91)25 8 13 70 (6, 7, 14, 17) 70 7026 9 12 38 (60, 62, 66, 69) 19 3027 9 13 86 (5, 7, 16, 25) 1328 10 12 85 (42, 48, 57, 64) 1129 10 13 59 (30, 33, 39, 40)30 11 12 50 (87, 89, 94, 94)31 11 13 90 (2, 2, 11, 20) 7 30

Also, the optimal flows with the optimistic and the pessimistic attitudes are presented for commodities 1 and 2 denoted by (C1, C2).

1 and 2 between (1, 13) and (1, 12). Let d1 = 90 and d2 = 60. The optimal flows of these commodities according tothe optimistic and pessimistic cost functions which are the lower and upper bounds of the support of fuzzy cost function,are represented in the columns 5–8 in Table 1, link by link for each commodity. The corresponding optimal values of theoptimistic and pessimistic cases are crisp_opt = 14 008, and crisp_pes = 22 841, respectively.

1038 M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043

Table 2Ten fuzzy shortest paths and their flows

(r, s) Path Fuzzy cost Status Com. 1 Com. 2

(1, 13) 1→ 3→ 9→ 13 (58, 76, 92, 117) N-D 2(1, 13) 1→ 4→ 8→ 13 (59, 71, 86, 99) N-D 38(1, 13) 1→ 4→ 11→ 13 (85, 91, 111, 130) D 7(1, 13) 1→ 5→ 9→ 13 (92, 109, 130, 153) D(1, 13) 1→ 5→ 7→ 13 (187, 210, 228, 245) D 43

(1, 12) 1→ 3→ 7→ 12 (108, 130, 147, 171) N-D 30(1, 12) 1→ 3→ 9→ 12 (113, 131, 142, 161) N-D(1, 12) 1→ 4→ 7→ 12 (119, 133, 147, 168) N-D(1, 12) 1→ 5→ 9→ 12 (147, 164, 180, 197) D 30(1, 12) 1→ 5→ 7→ 12 (176, 195, 211, 235) D

It is dictated which of the paths is dominant (D) or non-dominant (N-D).

Table 35-shortest paths considering the optimistic attitude and their optimal flows for each commodity (C1, C2) when T1 : w1 = w2 = w3 = w4 = 0.25,T2 : w1 = 1, w2 = w3 = w4 = 0 and T3 : w4 = 1, w1 = w2 = w3 = 0

(r, s) Path Fuzzy cost T1-C1 T1-C2 T2-C1 T2-C2 T3-C1 T3-C2

(1, 13) 1→ 3→ 9→ 13 (58, 76, 92, 117) 13 2(1, 13) 1→ 4→ 8→ 13 (59, 71, 86, 99) 29 27 29(1, 13) 1→ 3→ 8→ 13 (78, 93, 115, 129) 41 43 41(1, 13) 1→ 4→ 11→ 13 (85, 91, 111, 130) 7 7 7(1, 13) 1→ 6→ 11→ 13 (98, 104, 125, 146) 11 13

(1, 12) 1→ 3→ 7→ 12 (108, 130, 147, 171) 30 30 30(1, 12) 1→ 3→ 9→ 12 (113, 131, 142, 161) 2 2(1, 12) 1→ 4→ 8→ 12 (132, 143, 153, 173) 9 11 9(1, 12) 1→ 4→ 10→ 12 (148, 159, 176, 194)(1, 12) 1→ 6→ 10→ 12 (148, 168, 184, 208) 19 19 19

To recognize the effect of fuzzy shortest paths scheme in the solution process of MCMFP with fuzzy costs, the fuzzyshortest paths are derived from Okada and Soper’s algorithm [25] and are represented in Table 2. In this table, which of thepaths is dominant or non-dominant is also presented.Considering the paths of Table 2 and solving model (13) when w1 = w2 = w3 = w4 = 0.25, we have (18644, 21337,

23843, 26481) as the fuzzy objective value. The flow paths applying these settings are shown in Table 2 for each commodity.Please note that, this scheme permits the use of both dominant and non-dominant paths. The number of non-dominantalternative paths becomes very small and it seems that they cannot cover the diversity of paths in a real environment,and so the path selection of intelligent agents is not reflected by considering only non-dominant paths in this example. Toimprove the cost value, we can also use some alternative paths from the K -shortest paths [17] which extends the domainof selection of paths. Consider, for example, the optimistic case, and apply the first component of the fuzzy link cost as anobjective function. Let us find two couples of 5-shortest paths joining (1, 13) and (1, 12) applying this objective function(Table 3). We solve model (13) regarding the following three settings of weighting parameters:

(T1) : w1 = w2 = w3 = w4 = 0.25,(T2) : w1 = 1, w2 = w3 = w4 = 0,(T3) : w1 = w2 = w3 = 0, w4 = 1.

The fuzzy values of the objective function considering these settings are as follows, respectively:

fuzzy_value_1 = (13 724, 16 138, 18 749, 21 552),fuzzy_value_2 = (14 240, 16 514, 19 192, 21 955),fuzzy_value_3 = (14 244, 16 502, 19 178, 21 929).

By comparing these values with the crisp optimistic and pessimistic values (crisp_opt and crisp_pess), it is easy to show thatthese optimal values are laid in a domain between the optimistic and pessimistic values. Thus, our method is capable ofestimating the optimal flows, applying 5-shortest paths. The optimal flows keeping these settings in mind, are presentedin Table 3. Note that, the optimal values are better than the obtained values those found by considering the fuzzy shortestpaths algorithm when the comparison is done componentwisely.

5.2. MCMFP with uncertain travel demand

In the real problems, applying the incomplete statistical data or granular information, the decision maker may estimatethe travel demands between each couple of network nodes. In some problems, experiments by experts are also in hand,

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1039

which are full of uncertainty. Some of the problems include linguistic descriptions which cannot be presented precisely.For example, the demand of a production unit may be increased or dropped or the supply or the amount of import of araw material is variable. This is the art of granular techniques to predict the behavior of system considering such abstractinformation. Fuzzy demand is the center ofmuch research in fuzzymathematical programming. According to the theoreticalview point, fuzzy demands and fuzzy supplies were introduced for example by Yao and Wu [34]. But its importance laysin its exclusive interpretation, i.e., since, in practice, the demands of customers, are not deterministic, their representationwith uncertain tools seems to be appropriate. Note that, although the stochastic demands of customers have been adoptedand tallied with the facts in widespread cases, stochastic assumption is not reasonable in the vast range of situations, andit is not sufficient to describe many states where the probability distribution of demands may be unknown or partiallyknown. It is usually very hard to present the precise demands of customers. For many cases, the probability distributionsfor demands cannot easily estimated, due to the lack of data. Instead, expert’s opinions can be used to provide appropriateestimations [35]. Some examples of fuzzy demand is given in literature. Ghatee and Hashemi [17] investigated the fuzzydemand in traffic assignment models. In what follows, we deal with fuzzy demand in MCMFP.

5.2.1. Modeling and embedded problemAssume the travel demand of commodity t for node i is exhibited with an LR flat fuzzy number bti = (b

ti,1, b

ti,2, b

ti,3, b

ti,4).

Let the flow on link (i, j) is given by same fuzzy number xti,j = (xt(i,j),1, x

t(i,j),2, x

t(i,j),3, x

t(i,j),4).We state:

min+∑t=1:T

+∑(i,j)∈A

cti,j.xti,j, (14)

s.t.

(a)

+∑{j:(i,j)∈A}

xti,jH+∑

{j:(j,i)∈A}

xtj,i = bti , ∀i ∈ N, ∀t = 1, . . . , T ,

(b)+∑t=1:T

xt(i,j),k ≤ u(i,j),k, ∀(i, j) ∈ A,∀k = 1, . . . , 4.

(15)

In this problem ′. ∈ {×,⊗}′, and we use Wu’s scalar product (3).Ignoring the definition of ordering inequality, we use the last simple order constraint (15)(b) on flows, because each

component of a feasible flow should be less than the maximal capacity of the corresponding link.

Proposition 5.1. The flow constraint (15) by Hukuhara difference is valid.Proof. Note that, it is not possible to take a node as supplier and demander simultaneously. Also, we consider a staticnetwork, not a dynamic one and so a given node cannot to be suppler sometimes, and a demander at other times. By theseassumptions, let i be a supplier node. Thenwe have (bti,1, b

ti,2, b

ti,3, b

ti,4) ≥ 0 and also b

ti,1 ≥ 0. In this case, the first component

of flowconstraintwhich coincides the outflow fromnode i should be greater than its inflow, i.e. each component of out flow isgreater than the corresponding component of inflow. Thus, the flow constraint states the following vector with nonnegativecomponents, which can be interpreted as a valid fuzzy number inLR(R).( ∑

{j:(i,j)∈A}

xt(i,j),1 −∑{j:(j,i)∈A}

xt(j,i),1,∑{j:(i,j)∈A}

xt(i,j),2 −∑{j:(j,i)∈A}

xt(j,i),2,

∑{j:(i,j)∈A}

xt(i,j),3 −∑{j:(j,i)∈A}

xt(j,i),3,∑{j:(i,j)∈A}

xt(i,j),4 −∑{j:(j,i)∈A}

xt(j,i),4

),

By equalizing this vector with (bi,1, bi,2, bi,3, bi,4), due to Property 3.7, one can get:∑{j:(i,j)∈A}

xt(i,j),k −∑{j:(j,i)∈A}

xt(j,i),k = bti,k, ∀k = 1, . . . , 4. (16)

This means for all of the possibilities, the flows satisfy the supply constraints.In the same fashion, for a demander node i the right hand side is negative and the first component of the constraint is

less than the second term. Also for a transient node both sides are zero. In both cases, the corresponding components areideal to interpret as the flows considering appropriate level of possibility. �

5.2.2. Computational method for MCMFP with uncertain demandApplying the multiplication (5), the objective function (14) may be approximated as follows,

minT∑t=1

∑(i,j)∈A

(ct(i,j),1xt(i,j),1, c

t(i,j),2x

t(i,j),2, c

t(i,j),3x

t(i,j),3, c

t(i,j),4x

t(i,j),4). (17)

1040 M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043

Nothing to the constraints (16), the constraints (15) may be also rewritten as follows,

(N xt1,N xt2,N x

t3,N x

t4) = (b

t1, b

t2, b

t3, b

t4),

where, N is incident matrix of network which assigns 1 and−1 to (i, k)th and (j, k)th elements respectively, if and only ifthe kth link connects node i to j, otherwise its elements are zero.Now we obtain

N xtk = btk, ∀t = 1, . . . , T , ∀k = 1, 2, 3, 4.

Therefore, the original problem may be expressed as follows:

min+∑t=1:T

+∑(i,j)∈A

(ct(i,j),1xt(i,j),1, c

t(i,j),2x

t(i,j),2, c

t(i,j),3x

t(i,j),3, c

t(i,j),4x

t(i,j),4), (18)

s.t.N xtk = b

tk, ∀t ∀k = 1, 2, 3, 4

0 ≤T∑t=1

xt(i,j),k ≤ u(i,j),k, ∀(i, j) ∈ A, ∀k = 1, 2, 3, 4.

This problem has fuzzy objective function with crisp constraints. It can be solved by using embedding function [33] to get avector-valued function, that may be solved by interactive optimization techniques [29]. Also, we can maximize (minimize)the area of optimistic (pessimistic) region [15]. But in this paper, we propose a simple transformation applying the proposedtotal order as similar to the one used by Ghatee and Hashemi [15] for fully a fuzzified minimal cost flow problem.

Theorem 5.2. Let the numbers belong to LR(R). Consider an order ≤k1,k2,k3,k4 satisfying the assumptions of Proposition 3.5.The optimal solution of (18) is equivalent to the optimal solution of the following crisp problem:

minT∑t=1

∑(i,j)∈A

k1.ct(i,j),1xt(i,j),1 + k2.c

t(i,j),2x

t(i,j),2 + k3.c

t(i,j),3x

t(i,j),3 + k4.c

t(i,j),4x

t(i,j),4, (19)

s.t.

N xtk = b

tk ∀t ∀k = 1, 2, 3, 4

0 ≤T∑t=1

xt(i,j),k ≤ u(i,j),k, ∀(i, j) ∈ A, ∀k = 1, 2, 3, 4.

Proof. By Definition 3.2 and Property 3.7 the result is straightforward. �

Corollary 5.3. (i) The result of problem (18) is close to the optimistic attitude [15] if we equalize k1 to a sufficient great numbersuch that it is possible to ignore from k2, k3, k4 in comparison with k1.(ii) The result of problem (18) is close to the pessimistic attitude [15] if we select k4 sufficiently great such that it is possible

to ignore from k1, k2, k3 in comparison with k4.

Remark 5.4. Since, the corresponding approach for transformation can take the risk concept of decision making processes,it is more flexible than the Wu’s embedding function [33].

5.2.3. Numerical test of uncertain travel demandsAs a heuristic, we can decompose model (19) to four classic multicommodity flow problems with respect to each

component. To clarify the discussion, denote the optimal solution of problem of kth component with xt,∗k . Then, xt,∗k is the

lower bound of flow in problem of (k + 1)th component and also xt,∗k is the upper bound of flow in problem of (k − 1)thcomponent. It is easy to eliminate the lower bound capacity by a changing variable. Thus, a traditional algorithm for MCMFPcan be implemented for each component. In what follows, we solve a fuzzy MCMFP applying this idea.Again, consider the network in Fig. 2, and data that mentioned in Table 4, the link labels are the same as Table 1.

Assume two commodities between (1, 13) and (1, 12). Consider the demand as d(1,13) = (100, 120, 135, 150) and d(1,12) =(60, 61, 75, 100), where each fuzzy number assumed as trapezoidal number with linear left and right shape functions.We test our method using the following setting of parameters:

k1 = π, k2 = π2, k3 = π3, k4 = π5.

The optimal value of this setting is as follows:

fuzzy_value∗ = (14 944, 19 435, 26 098, 37 089).

This number is close to the optimistic value, crisp_opt, which illustrates the efficiency of the method and reasonability ofsolution process.

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1041

Table 4The topological information of network and the fuzzy flows of commodities 1 and 2

Link label Fuzzy cost Fuzzy capacity Fuzzy flow (commodity 1) Fuzzy flow (commodity 2)

1 (20, 30, 37, 45) (75, 85, 93, 95) (15.54, 21.54, 29.54, 31.78) (31.46, 32.46, 44.46, 53.22)2 (30, 32, 36, 42) (45, 47, 53, 56) (44.05, 46.05, 50.97, 51.34) (0.95, 0.95, 2.03, 4.66)3 (66, 72, 81, 89) (93, 99, 107, 114) (21.85, 21.85, 21.85, 33.81) (8.15, 8.15, 8.15, 13.19)4 (28, 33, 38, 45) (47, 52, 59, 62) (18.56, 30.56, 32.63, 33.07) (19.44, 19.44, 20.37, 28.93)5 (47, 56, 60, 63) (42, 51, 54, 59) (0, 0, 0, 0) (0, 0, 0, 0)6 (6, 14, 22, 25) (85, 93, 96, 97) (0, 0, 0, 0) (0, 0, 0, 0)7 (99, 104, 109, 112) (53, 58, 61, 71) (0, 0, 0, 0) (0, 0, 0, 0)8 (58, 58, 60, 65) (20, 20, 25, 31) (0, 0, 0, 0) (0, 0, 0, 0)9 (42, 50, 57, 64) (67, 75, 82, 86) (0, 0, 0, 0) (30, 31, 43, 48)10 (52, 56, 64, 67) (84, 88, 91, 96) (14.4, 14.4, 14.4, 14.59) (0.6, 0.6, 0.6, 3.41)11 (33, 39, 39, 47) (2, 8, 16, 19) (1.15, 7.15, 15.15, 17.19) (0.85, 0.85, 0.85, 1.81)12 (43, 51, 58, 64) (68, 76, 82, 86) (0, 0, 0, 0) (0, 0, 0, 0)13 (23, 32, 36, 40) (38, 47, 51, 53) (37.05, 37.05, 37.97, 38.34) (0.95, 0.95, 2.03, 4.66)14 (58, 65, 73, 80) (83, 90, 97, 103) (0, 0, 0, 0) (0, 0, 0, 0)15 (76, 78, 83, 88) (50, 52, 57, 65) (0, 0, 0, 0) (0, 0, 0, 0)16 (53, 57, 64, 68) (71, 75, 79, 84) (7, 9, 13, 13) (0, 0, 0, 0)17 (64, 73, 77, 84) (43, 52, 59, 65) (0, 0, 0, 0) (0, 0, 0, 0)18 (21, 30, 33, 39) (30, 39, 45, 47) (21.85, 21.85, 21.85, 33.81) (8.15, 8.15, 8.15, 13.19)19 (38, 42, 44, 52) (19, 23, 31, 35) (18.56, 19.56, 21.63, 22.07) (0.44, 0.44, 1.37, 9.93)20 (78, 87, 89, 99) (19, 28, 38, 46) (0, 0, 0, 0) (19, 19, 19, 19)21 (68, 69, 76, 81) (68, 69, 74, 81) (0, 11, 11, 11) (0, 0, 0, 0)22 (46, 50, 53, 62) (30, 34, 43, 48) (0, 0, 0, 0) (30, 31, 43, 48)23 (57, 65, 70, 72) (54, 62, 64, 70) (0, 0, 0, 0) (0, 0, 0, 0)24 (79, 79, 81, 91) (15, 15, 25, 33) (0, 0, 0, 0) (2, 2, 4, 18)25 (6, 7, 14, 17) (70, 71, 74, 75) (70, 71, 74, 75) (0, 0, 0, 0)26 (60, 62, 66, 69) (38, 40, 43, 49) (0, 0, 0, 0) (9, 9, 9, 15)27 (5, 7, 16, 25) (86, 88, 97, 98) (23, 29, 37, 51) (0, 0, 0, 0)28 (42, 48, 57, 64) (85, 91, 98, 102) (0, 0, 0, 0) (19, 19, 19, 19)29 (30, 33, 39, 40) (59, 62, 63, 66) (0, 0, 0, 0) (0, 0, 0, 0)30 (87, 89, 94, 94) (50, 52, 52, 61) (0, 0, 0, 0) (0, 0, 0, 0)31 (2, 2, 11, 20) (90, 90, 99, 99) (7, 20, 24, 24) (0, 0, 0, 0)

It is important to know that how the results look like if we increase the spread of LR flat trapezoidal fuzzy numbers. In theliterature, the soundmotivation behind the common use of triangular and trapezoidalmembership functionswas addressedby Pedrycz [26]. Also, the evaluations on triangular fuzzy numbers sensitive to their spread have been deeply analyzed [13].Note that, the flat values of a fuzzy number almost equal to the most plausible amount of this number (say the averagevalues) and the spreads could take the place of standard deviation in measuring the vagueness. Therefore, the spreads of afuzzy number represent an opportunity to provide a mental image of the vagueness, although it is only a statistical imageor description of the shape of membership function. It is important, practically, to pay attention to the different weights ofthe spreads in the objective function, i.e., when the spreads are large, it is necessary to chose a small weight for the left orthe right spreads to decrease the effect of risk in decisionmaking process. More specifically, when the decisionmaker is riskseeker or risk averse, the corresponding weights of the left and the right spreads are adopted to control the amount of risk.Clearly, a risk neutral decision maker gets lower numbers as the weights of the left and the right spreads.As a final point, since the proposed scheme in this paper takes the extreme points of LR type flat numbers into

consideration, the approximated multiplication does not impact on results, i.e., utilizing exact formulas also produces thesame resultswhen the presented approach is followed. Furthermore, when the exact formula is used, the fuzzyMCMFP has anon-linear structure and some approximation needs to bemet to reshape the nonlinear problem into a linear dependent one,to solve by an efficient algorithm. Thus, it seems that applying an approximation in the modeling phase is more reasonablethan such an approximation after validating the model.

6. Application of fuzzy MCMFP in network design

The network design problem is one of the most important problems in combinatorial optimization, which has beenconsidered by many researchers and engineers [5]. The aim of this problem is to design a network to satisfy the demandof users who wish to travel through the network by consuming minimum construction cost. Minimum spanning tree andSteiner tree are two traditional variants of this problem, which have been applied to design communication and waternetworks [1]. In some cases, a network exists, and the management wishes to extend the network to satisfy future demand.The extension of a network can be also considered by interchanging the objectives such as cost, reliability, mobility andaccessibility. To consider such concepts, the designer need to consider a great number of concepts. One important point inthis problem is the granular information about this problem, because designing is done for a long-term period and the futurestatues cannot to be considered in a clear manner. It is worth reporting the sources of uncertainty in measurement of datain this problem:

1042 M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043

• The demand for transportation of each node, which is dependent on many items such as time, season and weatherconditions.• Linguistic definition of important travel.• Imperfect realization of the definition of important travel.• Non-representative sampling (the sample measured may not represent the defined travels).• Inadequate knowledge of the effects of environmental conditions on the measurement, or imperfect measurement ofenvironmental conditions.• Personal bias in reading analogue instruments.• Finite instrument resolution or discrimination threshold.• Inexact values of measurement standards.• Inexact values of traffic constants, and other parameters obtained from external sources and used in the data-reductionalgorithm.• Approximations and assumptions incorporated in the measurement method and procedure.• Variations in repeated observations of travel under apparently identical conditions.

To design a network after processing these granular information, one can consider fuzzy numbers as transportation andconstruction costs and with respect to these parameters, the following model may be expressed:

min+∑t=1:T

+∑(i,j)∈A

cti,j.xti,j +

∑(i,j)∈A

µi,jzi,j, ∀k = 1, . . . , K , (20)

s.t.

+∑{j:(i,j)∈A}

xti,jH+∑

{j:(j,i)∈A}

xtj,i = bti , ∀i ∈ N, ∀t = 1, . . . , T ,

+∑t=1:T

xt(i,j),k ≤ u(i,j),k, ∀(i, j) ∈ A, ∀k = 1, . . . , 4,

0 ≤+∑

t=1,...,T

xi,j ≤ u(i,j),kzi,j, ∀(i, j) ∈ A, ∀k = 1, . . . , 4

(21)

where x ti,j is the fuzzy flow of commodity t and variables zi,j ∈ {0, 1} are associated with the construction of link (i, j) ∈ A:zi,j = 1 if (i, j) belongs to the final solution, otherwise zi,j = 0. The objective function (20) is the sum of variable and fixedfuzzy costs. In this function, cti,j is the linear fuzzy cost associated with fuzzy flow of tth commodity through link (i, j) andµi,j is the fixed fuzzy cost associated with the selection of link (i, j) in the final solution. θ is a control parameter implyingthe trade-off between two objective functions. Other parameters are analogues to before. The following bi-level model canbe considered, when the priority of construction costs is essentially greater than transportation costs:

min∑(i,j)∈A

µi,jzi,j, (22)

s.t.

min+∑t=1:T

+∑(i,j)∈A

cti,j.xti,j

s.t.+∑

{j:(i,j)∈A}

xti,jH+∑

{j:(j,i)∈A}

xtj,i = bti , ∀i ∈ N

+∑t=1:T

xt(i,j),k ≤ u(i,j),k, ∀(i, j) ∈ A, ∀k = 1, . . . , 4.

0 ≤ xi,j ≤ u(i,j),kzi,j.

(23)

It is easy to follow an enumeration scheme such as branch and bound or genetic algorithm on the upper-level model, whosesub-procedures are fuzzy MCMFPs.As a final point, it is worthwhile to note that, in some cases when the network has been constructed, one can extend the

network to satisfy the new demands produced. Whether or not, with a feasible fuzzy flow in hand, a sensitivity analysis canbe utilized associate with the methodology of this paper to provide appropriate solution(s) which are referred to the nextwork.

7. Conclusion and future work

Fuzzy programming is an important area when the granular information is in hand and the uncertainty is capturedwith fuzzy quantities. Keeping this point of view in mind, two generalized variants of Minimal Cost Multicommodity FlowProblem (MCMFP) are described to handle the imprecise travel cost and inexact supply-demand. For the first variant, two

M. Ghatee, S.M. Hashemi / Mathematical and Computer Modelling 49 (2009) 1030–1043 1043

heuristicmethods, based on fuzzy shortest paths andK -shortest paths are investigatedwhen a total order is applied for fuzzynumber comparisons. For the second variant with uncertain travel demands, considered as LR type flat fuzzy numbers, weuse the Hukuhara difference in place of Zadeh’s difference. Then, we reduce the fuzzy MCMFP into a traditional MCMFPwhich can be solved efficiently. Note that, each improvement in classic MCMFP can be utilized for the related fuzzy one. Ourexperiments demonstrate the performance of this schemes. Because, the proposed scheme considers the extreme points ofLR type flat numbers, using an exact or an approximated multiplication provides the same results. The application of thisstudy in network design is also mentioned. Implementing this approach for real and large scale networks, and comparisonbetween different path enumeration techniques, such as dissimilar paths, gateway paths and shortest paths with iterativepenalties, and also other ranking functions are left to the next work. Duality concepts the same as one has introduced in [16]for fuzzy MCMFP with single commodity can be extended for cases with multiple commodities in future works.

Acknowledgement

The authors would like to express particular thanks to the anonymous referees and Editor-in-Chief for their valuablecomments, which led us to improvements in this paper.

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