Ranking function-based solutions of fully fuzzified minimal cost flow problem

Information Sciences 177 (2007) 4271–4294

www.elsevier.com/locate/ins

Ranking function-based solutions of fully fuzzified minimalcost flow problem

Mehdi Ghatee *, S. Mehdi Hashemi

Department of Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran

Received 11 April 2006; received in revised form 12 April 2007; accepted 6 May 2007

Dedicated to Professor Mohsen Razzaghi on the occasion of his 64th birthday

Abstract

The aim of minimal cost flow problem (MCFP) is to find the least transportation cost of a single commodity through acapacitated network. This paper presents a model to deal with one particular group of such problems in which the supplyand demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. For easier reference, hereafter,we refer to this group of problems as fully fuzzified MCFP. To represent our model, Hukuhara’s difference and approx-imated multiplication are used. Thereafter, we sort fuzzy numbers by an order using a ranking function and show that it isa total order, i.e., a reflexive, anti-symmetric, transitive and complete binary relation. Utilizing the proposed ranking func-tion, we transform the fully fuzzified MCFP into three crisp problems solvable in polynomial time. From this standpoint,combinatorial algorithms are provided to solve the above-mentioned problem and find the fuzzy optimal flow. Further-more, the proposed order is related to the importance weights of the center, the left spread and the right spread of eachfuzzy number. Thus, this method is capable of handling the decision maker’s risk taking. By comparing some previousranking function-based works with our method, the efficiency of the latter is revealed. Finally, an application of our pro-posed method to petroleum industry is presented.� 2007 Elsevier Inc. All rights reserved.

Keywords: Fully fuzzified MCFP; Ranking function; Total ordering

1. Introduction

In many real decision making problems we are faced with imprecise, but optimization techniques need well-defined and precise data. Interval and fuzzy viewpoints, which are commonly used for estimating quantitieslike traffic conditions, accidents, traffic jams, and weather forecasts [27], are amongst the most attractive ideasin dealing with such problems. For example, in uncertain network theory, Dijkstra’s algorithm was general-ized by Sengupta and Pal [38] in order to deal with the interval cost shortest path problem. A similar approach

0020-0255/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2007.05.007

* Corresponding author. Fax: +98 21 6497930.E-mail addresses: [email protected], [email protected] (M. Ghatee), [email protected] (S. Mehdi Hashemi).

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4272 M. Ghatee, S. Mehdi Hashemi / Information Sciences 177 (2007) 4271–4294

was followed for minimal interval cost flow problem by Hashemi et al. [19]. The main idea of those works is todefine a special rank for sorting interval and fuzzy data, see, e.g. [37], and also Wang and Kerre [43,44] for areview of some kinds of ranks. One important class of these ranks, that has been highlighted by manyresearchers, uses the possibilistic and necessity measures, following the studies by Dubois and Prade [10].Tanaka and Asai [41] initially proposed a possibilistic linear programming formulation where the coefficientsof decision variables are crisp, and decision variables are fuzzy numbers. Afterwards, Inuiguchi and Sakawa[22] implemented optimality tests in linear programming with possibilistic and necessity measures. Also, Wu[45,46] extended a fuzzy rank based on the necessity measures by employing an embedding function whichassigns a real vector to a given fuzzy number for transforming a fuzzy optimization problem with crisp inputsand constraints into a bi-objective program. Thus, the Pareto optimal solutions of the second program areaccepted as solutions of the original fuzzy model. Furthermore, Wu [47] extended this idea to fuzzy program-ming problems with fuzzy constraints and generalized the (a,b)-optimality concept. But, in contrast to thefuzzy linear programming problems, literature on fuzzy network flow algorithms and integer programmingis not abundant, see, e.g. [30] for some label setting algorithms in shortest path problem with fuzzy costs,Lin and Wen [23] on fuzzy assignment problem and Ekel and Neto [13] for general fuzzy discrete optimizationproblems.

Although minimal cost flow problem (MCFP) is a general model in network flow theory [2, p. 294], its solu-tion in uncertain environment is almost neglected. For example, by using linear programming, Shih and Lee[40] have solved a limited fuzzy MCFP. But, as a result of non-convexity and NP-hard inherent characteristicsof multi-level programming, they were not able to create an efficient algorithm [40]. On the other hand, whensolving a discrete optimization problem, it is important that its formulation and corresponding algorithmsshould exploit those properties and peculiarities of the problem which will truncate the search space and pro-mote an effective solution [13]. Having this in mind, Hashemi et al. [19] have proposed some combinatorialalgorithms upon the well-known optimality conditions for MCFP with interval costs.

However, one can transform a fuzzy programming into a multiobjective optimization [34–36]. This inter-esting procedure has also been followed by Ekel et al. [11], Ekel [12] and Ekel and Neto [13] to transform glo-bal optimization problems and also discrete optimization programs into multiobjective decision making in afuzzy environment to reduce the decision uncertainty regions. Taking this approach into account, Noda andMortin [29] developed two network Simplex methods for the bi-objective MCFP. They found all Pareto solu-tions by the non-polynomial time algorithm.

In the present study, a generalized variant of MCFP by fuzzy costs, capacities, supplies and demands, calledfully fuzzified MCFP, is discussed. Our idea is similar to that of Perny and Spanjaard [31], which has beenapplied by Hashemi et al. [19] to MCFP with interval costs. Sengupta and Pal [38] have also used a similarapproach for a shortest path problem with interval costs. According to Perny and Spanjaard’s approach[31], in combinatorial problems we should often seek an additive cost function including a total order on solu-tions in which a total order is a reflexive, anti-symmetric, transitive and complete binary relation. They fol-lowed the preferred binary relation on the feasible set for the preferred spanning trees and the preferredpaths. Also, Wu [46] has pursued the same method by using an embedding function. Continuing these ideas,we introduce a total order on fuzzy numbers by employing non-algebraic numbers as the relative importanceweights of the center, left spread and right spread of fuzzy numbers. We show that network flows can be moreappropriately modelled with Hukuhara’s difference [21] instead of Zadeh’s difference [52,53]. Based on thisorder, we transform fully fuzzified MCFP into three crisp MCFPs solvable in polynomial time. Then, we dem-onstrate the efficiency of our method by comparing with Shih and Lee’ results [40] and some other rankingfunction-based solutions on a random network. Finally, we present an application of this study to the petro-leum industry. The rest of this paper is organized as follows:

In the following section, some basic definitions and a total order on fuzzy numbers are introduced. Alsosome important results concerning the performance of this order are provided. In Section 3, the fully fuzzifiedMCFP is presented. In Section 4, we use triangular fuzzy numbers to obtain simple formulas for some well-known ranking functions and implement some numerical tests to compare the performance of the proposedmethod with some previous established works. Section 5 consists of a case study in the petroleum industry.Finally in Section 6, we discuss some important considerations about the philosophy of such ordering andits drawbacks with a brief conclusion and suggestion of some future directions.

M. Ghatee, S. Mehdi Hashemi / Information Sciences 177 (2007) 4271–4294 4273

2. Fuzzy arithmetic operators and ordering

Fuzzy set [52,53] is defined as a subset ~a of universal set X � R by its membership function l~að�Þ, whichassigns to each element x 2 X a real number l~aðxÞ in the interval [0,1]. A fuzzy number ~a is called positive(negative), denoted by ~a > 0 ð~a < 0Þ, if its membership function l~aðxÞ satisfies:

l~aðxÞ ¼ 0 8x < 0 ð8x > 0Þ:
The a-cut or a-level of a fuzzy set ~a, which plays an essential role in fuzzy optimization, is defined as an or-dinary set ½~a�a for which the degree of its membership function exceeds the level a, [9,48]. A fuzzy number is aconvex normalized fuzzy set of the real line R; whose membership function is piecewise continuous [48]. Theset of fuzzy numbers on R is denoted by FðRÞ. It is well-defined, denoting a-level set of a fuzzy number ~a withclosed interval [aL(a),aR(a)]. A fuzzy number ~a ¼ ða; aL; aRÞLR is said to be an LR fuzzy number [9] if
l~aðxÞ ¼L a�x

aL

� �; x 6 a;

R x�aaR

� �; x P a;

(ð1Þ

where the symmetric non-increasing function L:[0,1) # [0,1] is the left shape function, with L(0) = 1. Natu-rally, a right shape function R(Æ) is similarly defined. We denote the LR fuzzy numbers on the real line withLRðRÞ.

Let ~a ¼ ða; aL; aRÞLR and ~b ¼ ðb; bL; bRÞLR be in LRðRÞ. The formula for the exact extended addition accord-ing to Zadeh’s extension principle [52] and the formula for the approximated extended multiplication [9,34,17]are as follows:

ða; aL; aRÞLR~þðb; bL; bRÞLR ¼ ðaþ b; aL þ bL; aR þ bRÞLR; ð2Þ

ða; aL; aRÞLR � ðb; bL; bRÞLR ffi ðab; a:bL þ b:aL; a:bR þ b:aRÞLR; ð3Þ

where ~a > 0 and ~b > 0. A more accurate approximation of multiplication, for positive numbers ~a and ~b inLRðRÞ is proposed by Wagenknecht et al. [42] as follows:

ða; aL; aRÞLR � ðb; bL; bRÞLR ffi ðab; a:bL þ b:aL � j1aL:bL; a:bR þ b:aR þ j2aR:bRÞLR; ð4Þ

where

j1 ¼R 1

0 ½L�1ðtÞ�3 dtR 1

0½L�1ðtÞ�2 dt

;

and

j2 ¼R 1

0 ½R�1ðtÞ�3 dtR 1

0½R�1ðtÞ�2 dt

:

Also scalar multiplication is derived as follows:

kða; aL; aRÞLR ¼ ðka; kaL; kaRÞLR;

where k P 0 and for k < 0

kða; aL; aRÞLR ¼ ðka;�kaR;�kaLÞRL:

It was pointed out by Diamond and Korner [8] that ~b ~�~a is not really a difference and it becomes rather unnat-ural when applied to a linear structure. For example, ~b ~þð�1Þ~a is not compatible with the difference in thefunction space. But the Hukuhara’s difference [21] which is defined as the solution ~x of the equation~a ~þ~x ¼ ~b (if at all existing) coincides with the difference defined in the function space. This property justifiesthe application of this difference instead of the fuzzy number ~b ~þð�1Þ~a. Therefore we opt to use this operatorfor the definition of difference as presented in the works of Hukuhara [21], Diamond and Kloeden [7],Diamond and Korner [8], entitled as Hukuhara’s difference as follows:


Definition 2.1. For ~a and ~b in LRðRÞ if Hukuhara’s difference ~bH ~a exists, it is given by

½~bH ~a�a ¼ f1 2 Rj½~a�aþf1g � ½~b�ag; ð5Þ

where þ for two sets bX and bY means
bX þbY ¼ fxþ yjx 2 bX ; y 2 bY g: Proposition 2.2 [8, Proposition 1]. Let ~a ¼ ða; aL; aRÞLR
and ~b ¼ ðb; bL; bRÞLR be in LRðRÞ. If there is a 1 2 R

and an a 2 (0,1] such that ½~a�a �þf1g � ½~b�a, then ½~bH ~a�b 6¼ ; for each b P a. Furthermore if L(a), R(a) 5 0 then

Hukuhara’s difference ~bH ~a ¼ ðc; cL; cRÞLR exists and is given by

c ¼ a� b; cL ¼ aL � bL; cR ¼ aR � bR:

Remark 2.3. The consequences of Hukuhara’s difference are similar to those of the cancellation operatordefined by Hansen [18, p. 10] and applied by Hashemi et al. [19] to MCFP with interval costs.

It should be noted that fuzzy numbers are related to interval numbers; in fact according to decompositiontheorem (see, e.g. [34]), each fuzzy set can be represented by the union of intervals. Remark (2.3) shows theanalogy between these two kinds of difference. Also, by adapting one of these differences, the other can beobtained by using the decomposition theorem.

The most important property of Hukuhara’s definition is that ~aH ~a is zero, whereas this is not true usingZadeh’s difference. Therefore, for solving linear systems, e.g., the constraint of flow transmitting, this operatoryields more accurate and interesting results than Zadeh’s definition. Also, where Hukuhara’s difference is notwell-defined, it is possible to utilize L2 norm to find an ~x which minimizes the distance between ~a ~þ~x and ~b; see[8,4] for details. In order to produce useful expressions, we utilize Proposition (2.2) in agreement with the fol-lowing definition:

Definition 2.4 (Provable in some cases). For ~a and ~b in LRðRÞ; we have:

~bH ~a ¼ ðb; bL; bRÞLRH ða; aL; aRÞLR ¼ ðb� a; bL � aL; bR � aRÞLR:

Moreover, for notation simplicity, we write H ~a in place of ~0H ~a; where ~0 ¼ ð0; 0; 0ÞLR is the null member.

The following result which can be proved easily shows the usefulness of this difference operator.

Lemma 2.5. Consider ~a; ~b 2LRðRÞ and a 2 [0,1]. We have:

½~aH~b�a ¼ ½~a�a ��½~b�a;

where �� is the cancellation operator [18, p. 10] defined for two intervals [x1,x2] and [y1,y2] as follows:

½x1; x2� ��½y1; y2� ¼ ½x1 � y1; x2 � y2�:

Proof. Let ~a ¼ ða; aL; aRÞLR and ~b ¼ ðb; bL; bRÞLR: According to the definition of LR numbers and fuzzy cuts, wecan easily verify that

½~a�a ¼ ½a� L�1ðaÞ � aL; aþ R�1ðaÞ � aR�;½~b�a ¼ ½b� L�1ðaÞ � bL; bþ R�1ðaÞ � bR�;½~aH

~b�a ¼ ½ða� bÞ � L�1ðaÞ � ðaL � bLÞ; aþ bþ R�1ðaÞ � ðaR þ bRÞ�;

which prove the assertion. h

Property 2.6. Let ~a, ~b and ~c be in LRðRÞ: By Hukuhara’s difference the following statements are true:

(i) If ~a ~þ~b ¼ ~c, then ~cH~b ¼ ~a,

(ii) ~aH ð~bH~cÞ ¼ ð~aH~bÞ ~þ~c,

(iii) H ðH ~aÞ ¼ ~a,


(iv) ~aH~b ¼ ~a ~þðH

~bÞ,(v) ~aH ~a ¼ ~0 ¼ ð0; 0; 0ÞLR;

(vi) H ð~a ~þ~bÞ ¼ ðH ~aÞ ~þðH~bÞ;

(vii) ~a� ð~bH~cÞ ¼ ~a� ~bH ~a� ~c;(viii) ~a� ð~bH~cÞ ¼ ~a� ~bH ~a� ~c:

Proof. Using Definition (2.4), Lemma (2.5) and the above original definitions, all of these assertions can bederived without difficulty. We only prove (viii).

Let ~a ¼ ða; aL; aRÞLR, ~b ¼ ðb; bL; bRÞLR and ~c ¼ ðc; cL; cRÞLR. We have

~a� ð~bH~cÞ ¼ ðaðb� cÞ;aðbL� cLÞ þ ðb� cÞaL� j1aLðbL� cLÞ;aðbR� cRÞ þ ðb� cÞaRþ j2aRðbR� cRÞÞLR

¼ ðab;abLþ baL� j1aLbL;abRþ baR þ j2aRbRÞHðac;acLþ caL� j1aLcL;acR þ caRþ j2aRcRÞLR

which is our promise. h

The above property permits us to use distribution rule for each type of multiplication.

Proposition 2.7

(i) The set LRðRÞ; with Zadeh’s addition and Hukuhara’s difference considered as inverse operations, is aBoolean group.

(ii) The set LRðRÞ; with Wagenknecht’s multiplication rule or Dubois and Prade’s definition of multiplica-tion, does not admit a group structure.

Proof. The first assertion is a simple mathematical problem. To prove the second assertion, we note that byWagenknecht’s or Dubois and Prade’ multiplication, one cannot produce the inverse element of known fuzzynumbers. Thus, the corresponding set cannot possess a group structure. h

To define the inequality relation between two fuzzy numbers, numerous methods have been proposed in theliterature [43,44]. One important category is to compare fuzzy numbers using ranking functions [26], in whicha ranking function R : FðRÞ ! R that maps each fuzzy number into the real line is defined to compare theelements of FðRÞ; i.e., using the natural order of real numbers we can compare fuzzy numbers easily as fol-lows [26,25]:

~a PR~b if and only if Rð~aÞP Rð~bÞ,

~a >R~b if and only if Rð~aÞ > Rð~bÞ,

~a ¼R~b if and only if Rð~aÞ ¼ Rð~bÞ.

We pursue this approach from the decision maker’s point of view. First, we present the following four prop-erties for an acceptable order taken from the set of conditions proposed by Wang and Kerre [43]. We shallshow that these properties which are common in crisp cases [31] can solve our network problem successfully.

Definition 2.8. A binary relation R defined on T is said to be:

• Reflexive if and only if eRe for every e in T.• Transitive if and only if for every e and f in T, if eRf and f Rg, then eRg.• Anti-symmetric if and only if for every e and f in T, if eRf and f Re, then e = f.• Complete if and only if for every e and f in T, eRf or f Re.

Definition 2.9. A partial order is a reflexive, anti-symmetric and transitive binary relation. Also, a total orderis a reflexive, anti-symmetric, transitive and complete binary relation.

Through the end of this paper, we eliminate the subscript ‘LR’ of numbers belonging to LRðRÞ whereverthere is no ambiguity.


Definition 2.10. Let ~a ¼ ða; aL; aRÞ and ~b ¼ ðb; bL; bRÞ belong to LRðRÞ. For each triple of real positivenumbers (k,l,r), the ‘‘less than or equal to’’ relation R ¼ 6k;l;r on T ¼LRðRÞ, is defined as follows:

ða; aL; aRÞ 6k;l;r ðb; bL; bRÞ;

if and only if

k � aþ l � aL þ r � aR6 k � bþ l � bL þ r � bR;

where 6 means the usual total relation on real numbers R.

Let Corð~aÞ ¼ fxjl~aðxÞ ¼ 1g denote the core of fuzzy set ~a. We can extend the above ordering to fuzzy num-bers with singleton cores.

Definition 2.11 (Extended ordering). Let ~a and ~b be two fuzzy numbers with singleton cores. Suppose thatl~aðaÞ ¼ 1 and l~bðbÞ ¼ 1. Let [~a�0 ¼ ½a� aL; aþ aR� and ½~b�0 ¼ ½b� bL; bþ bR�. For each triple of real positivenumbers (k,l,r), we say ~a6k;l;r

~b if and only if

k � aþ l � aL þ r � aR6 k � bþ l � bL þ r � bR:

Using (2.10), the decision maker can adjust the ratio of the importance of the average value to the left orright spreads. This property easily permits us to model risk averse, risk neutral, and risk seeking decision mak-ers [19]. Note that the lower and upper bounds of fuzzy numbers, related to the left and right spreads, can beinterpreted as pessimistic and optimistic status, respectively, which are taken into account in many real opti-mization problems [20,33]. The following practical results can be derived by simply integrating the (a,b)-opti-mality concept defined by Wu [47] and the optimistic and pessimistic viewpoints in [33,20,32].

Proposition 2.12. Consider the following minimization problem:

minf~c:~xj~A � ~x 6 ~b;~x P 0g: ð6Þ
Then
(i) by selecting l less (greater) than r for the constraints (objective function), we obtain a pessimistic program-

ming problem.

(ii) by selecting l greater (less) than r for the constraints (objective function), we obtain an optimistic program-ming problem.

Proof. Hashemi et al. [20] have obtained the most optimistic as well as the most pessimistic satisfying solu-tions, employing fuzzy extensions of relations which can be interpreted as lenient and severe orders. Regardingtheir work and according to the ordering in Definition (2.10), when l is assumed less than r in evaluating~A:~x 6 ~b, the worst case (here the upper bound of quantities) is taken into account with greater weight. Thus,the feasible space becomes narrow. On the other hand, when l is greater than r, with the mentioned order, x ispreferred to x 0 when the lower bound of cx is better than the lower bound of cx 0. Thus, this order is severe inhandling the worst case. A severe order on a narrow feasible space coincides with pessimistic programmingproblem. Similarly, assumptions (ii) result in programming problem with an optimistic viewpoint. h

Remark 2.13. Consider the following maximization problem:

max f~c � ~x; j~A � ~x P ~b; ~x P 0g: ð7Þ
Then
(i) by selecting l less (greater) than r, for the constraints (objective function), we obtain a pessimistic pro-gramming problem.

(ii) by selecting l greater (less) than r, for the constraints (objective function), we obtain an optimistic pro-gramming problem.

In the following propositions, several important points are made about this ordering.

Proposition 2.14. Order relation R ¼ 6k;l;r on T ¼LRðRÞ is reflexive and transitive.

Corollary 2.15. The order (2.10) can be defined relative to the following linear ranking function:


R : T 7! R

Rðða; aL; aRÞÞ ¼ k � aþ l � aL þ r � aR;

and so it belongs to the class of ranking function orderings [43].

This assertion cannot be proved in general, because here the third property of ranking functions is missing.Namely, if we have Rð~aÞ ¼ Rð~bÞ, we cannot guarantee the equality of ~a and ~b (presenting a counter-example isa simple mathematical problem). Thus, we select k, l and r greater carefully, limiting our choice to non-alge-braic numbers. These are very important in number theory and polynomial rings [15]. Following that, wereview some necessary definitions and results and will show the drawbacks of such numbers as relative impor-tance weights in problems of ordering.

Definition 2.16. A complex number Z is called algebraic if and only if it is a root of a non-zero polynomialequation with integer coefficients, otherwise it is called non-algebraic or transcendental.

Proposition 2.17. The set of algebraic numbers is computable. So, the set of non-algebraic numbers is

incomputable.

Example. The ratio of the circumference of a circle to its diameter denoted by p (’3.1415) and the base of thenatural logarithm e(’2.7182) are non-algebraic. Denote the set of rational and positive rational numbers withQ and Qþ, respectively. We have the following important result.

Proposition 2.18. Let # be a non-algebraic real positive number. Put

k ¼ q1#n1 ;

l ¼ q2#n2 ;

r ¼ q3#n3 ;

8><>:
where q1; q2; q3 2 Qþ and n1 5 n2 5 n3 are non-negative integer numbers. Then R ¼ 6k;l;r on
TQ ¼ fða; aL; aRÞ 2LRðRÞja; aL; aR 2 Qg is a total order.

Proof. By Proposition (2.14), the order is reflexive and transitive. Let ~a ¼ ða; aL; aRÞ; ~b ¼ ðb; bL; bRÞ 2 TQ.

Assume ~a6k;l;r~b and ~b6k;l;r~a. We get

q1#n1 aþ q2#

n2 aL þ q3#n3 aR ¼ q1#

n1 bþ q2#n2 bL þ q3#

n3 bR;

or

q1#n1ða� bÞ þ q2#

n2ðaL � bLÞ þ q3#n3ðaR � bRÞ ¼ 0:

Note that ða� bÞ; ðaL � bLÞ; ðaR � bRÞ 2 Q; thus we can obtain an equation with integer coefficients. Since #cannot be a root of a non-zero polynomial with integer coefficients and also q1; q2; q3 2 Qþ, we have:

a ¼ b; aL ¼ bL; aR ¼ bR:

Thus ~a ¼ ~b and so 6k,l,r is anti-symmetric. Finally, since 6 is complete on real numbers, 6k,l,r is complete onTQ, too. h

Remark 2.19. Since computer programs can only take numbers with finite floating points into account, the setTQ suffices for the representation of the actual parameters of real problems. Also non-algebraic numbers canbe considered with double precision.


Lemma 2.20. If ~a 2LRðRÞ is positive for each (k,l,r) satisfying Proposition (2.18), we have ~aPk;l;r~0. But the

converse is not necessarily true.

Lemma 2.21. If ~a; ~b 2LRðRÞ are positive numbers and j1 P 1, then ~a� ~b is a positive fuzzy number.

Proof. According to Wagenknecht’s definition of multiplication (4), we should prove

ab� ðabL þ baL � j1aLbLÞP 0:

Since j1 P 1, we have:

ab� ðabL þ baL � j1aLbLÞP ða� aLÞðb� bLÞ;

which proves the assertion. h

Proposition 2.22. For two arbitrary elements ~a and ~b in TQ, we have

(i) ~aPk;l;r~b if and only if ~aH

~bPk;l;r~0 if and only if H

~bPk;l;rH ~a,(ii) If ~aPk;l;r

~b and ~cPk;l;r~d then ~aþ ~cPk;l;r

~bþ ~d,

(iii) If ~aPk;l;r~b then ~a ~�~bPk;l;r

~0,

(iv) If ~aPk;l;r~b and x is a positive scalar, then ~a:xPk;l;r

~b:x,

(v) If ~aPk;l;r~b and ~x is positive, we cannot conclude that ~a� ~xPk;l;r

~b� ~x.

Proof. Note that by Corollary (2.15), the ordering 6r,l,r on TQ can be produced by the linear ranking functionRðða; aL; aRÞÞ ¼ k � aþ l � aL þ r � aR. Thus by Lemma (3.1) in [25], (i) and (ii) are concluded. To prove (iii),from ~a ¼ ða; aL; bRÞPk;l;rðb; bL; bRÞ ¼ ~b, we have

k � ða� bÞ þ l � aL þ r � aR P l � bL þ r � bR: ð8Þ

Assume ~a ~�~b<k;l;r0, then

k � ða� bÞ þ l � aL þ r � aR < �l � bL � r � bR: ð9Þ

From (8) and (9) we get (l + r)Æ(bR + bR) < 0 which is a contradiction. To prove (v), set k = 1, l = p and r = p2.We can easily show that (2,1,7) P k,l,r(4,1,5) and (1,0,1) > 0.

But

ð2; 1; 7Þ � ð1; 0; 1Þ ¼ ð2; 1; 9Þ6k;l;rð4; 1; 9Þ ¼ ð4; 1; 5Þ � ð1; 0; 1Þ: �

Proposition 2.23. Let ~a ¼ ða; aL; aRÞ and ~x ¼ ðx; xL; xRÞ be in LRðRÞ and their cores be singletons. Suppose that ~xis positive. If ~aPk;l;r0, then ~a ~~xPk;l;r0. Also, if j1 P 1, then

~a� ~xPk;l;r0:

Proof. We can easily prove:

Corð~a ~~xÞ ¼ ax;

½~a ~~x�0 ¼ ½ax� ðaxL þ xaL � aLxLÞ; axþ ðaxR þ xaR þ aRxRÞ�:

Thus with respect to the extended definition (2.11), we should prove:

k � axþ l � ðaxL þ xaL � aLxLÞ þ r � ðaxR þ xaR þ aRxRÞP 0;

or equivalently

aðk � xþ l � xL þ r � xRÞ þ l � aLðx� xLÞ þ r:aRðxþ xRÞP 0;


which is the sum of three positive parts. Moreover, by Lemma (2.21), the second part of the proposition can beproved similarly. h

Proposition 2.24. Let ~a ¼ ða; aL; aRÞ and ~b ¼ ðb; bL; bRÞ be in LRðRÞ.

(i) If ~aPk;l;r~b and a P b, for each positive fuzzy number ~x in LRðRÞ, we have:

~a� ~xPk;l;r~b� ~x:

(ii) If ~aPk;l;r~b and a P b, aL P bL and aR P bR, then for each positive fuzzy number ~x in LRðRÞ, the following

is true:

~a� ~xPk;l;r~b� ~x:

(iii) For each positive fuzzy number ~x, if Corð~xÞ is a singleton set, then

~a ~~xPk;l;r~b ~~x:

Proof. (i) For positive ~x ¼ ðx; xL; xRÞ, we have xL,x,xR P 0, and also given that ~aPk;l;r~b, we can write

k � ða� bÞ þ l � ðaL � bLÞ þ r � ðaR � bRÞP 0;

therefore

xðkða� bÞ þ lðaL � bLÞ þ rðaR � bRÞÞ þ ða� bÞðlxL þ rxRÞP 0;

which proves the assertion. The proof of (ii) is similar. To prove (iii), taking into account Property (2.6), Prop-osition (2.22) and Lemma (2.23), the following is true and the result is clear.

~a ~~xH~b ~~x ¼ ð~aH

~bÞ ~~x P ~0: �

Remark 2.25. Result (iii) of Proposition (2.24) utilizing Wagenknecht’s multiplication (4) is true whenj1 = j2 = 1.

Property 2.26. Let ~a ¼ ða; aL; aRÞ and ~b ¼ ðb; bL; bRÞ be in TQ, then ~a ¼ ~b is equivalent to each of the

following:

(i) k Æa + l ÆaL + r ÆaR = k Æb + l ÆbL + r.bR,

(ii) a = b, aL = bL, aR = bR,

where k, l and r satisfy the assumptions of Proposition (2.18).

3. Fully fuzzified minimal cost flow problem

In this section, we sketch the basic concepts of minimal cost flow problem (MCFP). This problem arisesnaturally in engineering and economics contexts; it appears in problems involving equilibrium models suchas urban transportation systems, resistive electrical networks, and production-distribution problems. Indesigning telecommunication networks, it also has an important role. The central problem in all of these isfinding the least transportation cost of a commodity through a capacitated network in order to satisfydemands at certain nodes using available supplies at other nodes. Let G = (N,A) be a directed network whereN and A denote sets of nodes and links, respectively. For example, in transportation networks, the nodes aresuppliers, demanders, or intersections and the links are roadways. In addition, for every link (i, j) in A, we con-sider a cost ci,j (e.g., the traverse time, length or toll), and also an upper and a lower capacity ui,j and li,j, respec-tively. We usually assume li,j to be zero, since we can always make it zero using a simple change of variable [2].MCFP requirements can be written in the following linear program:


minXði;jÞ2A

ci;jxi;j; ð10Þ

s:t:X

fj:ði;jÞ2Agxi;j �

Xfj:ðj;iÞ2Ag

xj;i ¼ bi; 8i 2 N ; ð11Þ

0 6 xi;j 6 ui;j: ð12Þ

In this formulation xi,j is the flow on link (i,j). Moreover, if bi < 0 (bi > 0), the corresponding node is known asdemander (supplier). Otherwise i is a transient node. We suppose that

Pi2Abi ¼ 0. The vast majority of classic

MCFP models are deterministic. In such models the traverse cost and the supply and demands of nodes areassumed to be known a priori. But in applied problems, this model is influenced by complex human, social,economic, and political interactions, and so it is dependent on user perception of information and non-deter-ministic factors of the network. Therefore, by considering different provisions of traffic information, numeroustypes of transportations might be deduced. Now, we consider a case in which the data of MCFP has beenmodelled by LRðRÞ numbers. These numbers can handle vagueness as an important category of uncertainty.The following generalized problem can be stated:

minXði;jÞ2A

ðci;j; cLi;j; c

Ri;jÞ:ðxi;j; xL

i;j; xRi;jÞ; ð13Þ

s:t:X

fj:ði;jÞ2Agðxi;j; xL

i;j; xRi;jÞH

Xfj:ðj;iÞ2Ag

ðxj;i; xLj;i; x

Rj;iÞ ¼ ðbi; b

Li ; b

Ri Þ 8i 2 N ð14Þ

~xi;j P 08ði; jÞ 2 A; ð15Þ0 6 xi;j 6 ui;j; 0 6 xL

i;j 6 uLi;j; 0 6 xR

i;j 6 uRi;j; 8ði; jÞ 2 A: ð16Þ

where ‘‘Æ’’ belongs to f ~;�;�g.Note that restrictions (16) are imposed on flows, because the average, left and right spreads of a feasible

flow have to be less than the maximal value of the corresponding quantities.

Proposition 3.1. The flow constraints (14) using Hukuhara’s difference coincide, at least for three scenariosfbc

i gi2N,fbigi2N ; fbi � bLi gi2N ; fbi þ bR

i gi2N , with crisp constraints (11).

Proof. Based on Hukuhara’s difference (2.4), we can rewrite constraints (14) in the following useful form:
Xj
xi;j �X

j

xj;i;X

j

xLi;j �

Xj

xLj;i;X

j

xRi;j �

Xj

xRj;i

!¼ ðbi; b

Li ; b

Ri Þ;

and, by Property (2.26) we have

ðaÞP

jxi;j �

Pj

xj;i ¼ bi;

ðbÞP

jxL

i;j �P

jxL

j;i ¼ bLi ;

ðcÞP

jxR

i;j �P

jxR

j;i ¼ bRi :

8>>>>><>>>>>:ð17Þ

Now by subtracting (17.a) from (17.b) and adding (17.a) to (17.c), we get

ðaÞP

jxi;j �

Pj

xLi;j

!�

Pj

xj;i �P

jxL

j;i

!¼ bi � bL

i ;

ðbÞP

jxi;j þ

Pj

xRi;j

!�

Pj

xj;i þP

jxR

j;i

!¼ bi þ bR

i :

8>>>>><>>>>>:ð18Þ

(17.a), (18.a) and (18.b) together imply that there are three flows fxi;jgði;jÞ2A; fxi;j � xLi;jgði;jÞ2A; fxi;j � xR

i;jgði;jÞ2A

which satisfy constraints (11) with fbci gi2N,fbigi2N ; fbi � bL

i gi2N ; fbi þ bRi gi2N , respectively. h


Proposition 3.2. Let the shape functions L(Æ) and R(Æ) be strictly decreasing. Then the flow constraints (14) using

Hukuhara’s difference with respect to each scenario fbci 2 ~bigi2N coincide with crisp constraints (11).

Proof. Consider the scenario fbci 2 ~bigi2N with membership level k 2 (0,1). Without loss of generality, let

bci > bi. Since R(Æ) is strictly decreasing, from l~bi

ðbci Þ ¼ k, we obtain:

bci ¼ R�1ðkÞbR

i þ bi ¼ R�1ðkÞðbi þ bRi Þ þ ð1� R�1ðkÞÞbi:

Proposition (3.1) shows that restrictions (14) for the center, lower and upper bound of f~bigi2N agree with crispconstraints (11). Thus we can write:

bci ¼ R�1ðkÞ

Xj

ðxi;j þ xRi;jÞ �

Xj

ðxj;i þ xRj;iÞ

" #þ ð1� R�1ðkÞÞ

Xj

xi;j �X

j

xj;i

" #;

or equivalently

bci ¼

Xj

ðxi;j þ R�1ðkÞxRi;jÞ �

Xj

ðxj;i þ R�1ðkÞxRj;iÞ:

Therefore, the flow fxi;j þ R�1ðkÞxRi;jgði;jÞ2A satisfies f~bigi2N and coincides with the crisp constraints (11). Simi-

larly if bci < bi; the flow fxi;j � L�1ðkÞxL

i;jgði;jÞ2A satisfies the crisp restrictions (11). h

Analogous to crisp cases, we say i is a supplier node when ðbi; bLi ; b

Ri Þ > 0 and bi; b

Li ; b

Ri ; bi � bL

i P 0. Also forthe demander node i, we have bi; b

i þ bRi 6 0; bL

i ; bRi P 0. Now, using Wagenknecht’s multiplication (4), the

objective function (13) is estimated in the following way:

minXði;jÞ2A

ðci;jxi;j; ci;j:xLi;j þ cL

i;j:xi;j � j1cLi;j:x

Li;j; ci;j:xR

i;j þ cRi;j:xi;j þ j2cR

i;j:xRi;jÞ: ð19Þ

Also, constraints (14) may be rewritten as

ðNx;NxL;NxRÞ ¼ ðb; bL; bRÞ; ð20Þ
where N is the incident matrix of the network which consists of 1 and �1 in (i,k)th and (j,k)th entries, respec-tively, where the kth link connects node i to j. Other entries are assumed to be zero. Thus, by Property (2.26)we obtain:
Nx ¼ b;

NxL ¼ bL;

NxR ¼ bR:

Therefore, the original fully fuzzified MCFP may be shown as follows:

minXði;jÞ2A

ðci;jxi;j; ci;j:xLi;j þ cL


Li;j; ci;j:xR


i;j:xRi;jÞ;

s:t: Nx ¼ b; 0 6 xi;j 6 ui;j;

NxL ¼ bL; 0 6 xLi;j 6 uL

i;j;

NxR ¼ bR; 0 6 xRi;j 6 uR

i;j:

ð21Þ

This problem has a fuzzy objective function with crisp constraints. Ekel et al. [11] have proposed an efficientmethod for a general discrete optimization model based on a multicriteria approach. Their considered formu-lation and corresponding algorithm importantly exploit good properties and peculiarities of the original dis-crete problem, thereby truncating the search space and promoting an effective solution. Later, Ekel [12]extended the same methodology for generalized decision making problems by constructing non-fuzzy equiv-alents in relation to fuzzy constraints. The same programs can be solved using Wu’s embedding idea [46] and


getting a vector-valued function that may be solved by interactive optimization techniques, see, e.g. [34]. Wecan also pursue Tanaka’s approach [41] to maximize or minimize the optimistic or pessimistic regions. In addi-tion, solving such problems with fuzzy objective functions is possible by a modification of traditional math-ematical programming methods, see [11,12]. For example, we can generalize the crisp algorithms for MCFPsuch as successive shortest path or cycle-canceling [2] to fuzzy models similar to the one pursued by Hashemiet al. [19] for MCFP with interval cost. However, in this paper, we propose a simple transformation using theproposed total order.

Theorem 3.3. Let numbers be chosen in TQ: Consider an ordering 6k,l,r satisfying the assumptions of Proposition(2.18). The optimal solution of (21) is equivalent to that of the following crisp problem:

minXði;jÞ2A

fk:ðci;j:xi;jÞ þ l:ðci;j:xLi;j þ cL


Li;jÞ þ r:ðci;j:xR


i;j:xRi;jÞg

� minXði;jÞ2A

fðk:ci;j þ l:cLi;j þ rcR

i;jÞ:xi;j þ ðl:ci;j � j1cLi;jÞ:xL

i;j þ ðr:ci;j þ j2cRi;jÞ:xR

i;jg:

s:t:

Nx ¼ b; 0 6 xi;j 6 ui;j;

NxL ¼ bL; 0 6 xLi;j 6 uL

i;j;

NxR ¼ bR; 0 6 xRi;j 6 uR

i;j:

8><>:ð22Þ

Proof. Let f~xi;jgði;jÞ2A be the optimal solution of (21). Then, for other feasible solutions f~yi;jgði;jÞ2A, we have:

Xði;jÞ2A

ci;jxi;j;Xði;jÞ2A

ci;j:xLi;j þ cL


Li;j;Xði;jÞ2A

ci;j:xRi;j þ cR

i;j:xi;j þ j2cRi;j:x

Ri;j

!

6k;l;r

Xði;jÞ2A

ci;jyi;j;Xði;jÞ2A

ci;j:yLi;j þ cL

i;j:yi;j � j1cLi;j:y

Li;j;Xði;jÞ2A

ci;j:yRi;j þ cR

i;j:yi;j þ j2cRi;j:y

Ri;j

!:

By Definition (2.10), this means that

k:Xði;jÞ2A

ci;jxi;j þ l:Xði;jÞ2A

ci;j:xLi;j þ cL


Li;j þ r:

Xði;jÞ2A

ci;j:xRi;j þ cR

i;j:xi;j þ j2cRi;j:x

Ri;j

6 k:Xði;jÞ2A

ci;jyi;j þ l:Xði;jÞ2A

ci;j:yLi;j þ cL

i;j:yi;j � j1cLi;j:y

Li;j þ r:

Xði;jÞ2A

ci;j:yRi;j þ cR

i;j:yi;j þ j2cRi;j:y

Ri;j

!:

Then fðxi;j; xLi;j; x

Ri;jÞgði;jÞ2A is an optimal solution for (22). h

Proposition 3.4

(i) The result of problem (21) is close to optimistic viewpoints [20] if we select a sufficiently large l, so as to be

able to neglect k and r.

(ii) The result of problem (21) is close to pessimistic viewpoints [20] if we select a sufficiently large r, thereforeignoring k and l.

Proof. The proof is a simplified version of Proposition (2.12). h

This proposition shows that the corresponding transformation is capable of undertaking the risk aspects.Thus, according to the risk taking viewpoint, our scheme is more flexible than Wu’s embedding function [46]or Maleki’s ranking method [26].

In model (22), if we interpret x, xL and xR as three commodities, we acquire a multi-commodity flow prob-lem, see, e.g. [2] for a brief review. Fortunately, this program is a separable problem which may be solved withthe following three disjoint MCFP with crisp objective functions:


minXði;jÞ2A

ðk:ci;j þ l:cLi;j þ rcR

i;jÞ:xi;j; ð23Þ

s:t: Nx ¼ b; 0 6 xi;j 6 ui;j;

minXði;jÞ2A

l:ðci;j � j1cLi;jÞ:xL

i;j; ð24Þ

s:t: NxL ¼ bL; 0 6 xLi;j 6 uL

i;j:

minX

ði;jÞ2Ar:ðci;j:xR


i;j:xRi;jÞ; ð25Þ

s:t: NxR ¼ bR; 0 6 xRi;j 6 uR

i;j:

If for a link (i,j) we have xi;j � xLi;j < 0, which is equivalent to say that ~xi;j ¼ ðxi;j; xL

i;j; xRi;jÞ is not positive, then an

desired case happens. In this case, using Wagenknecht’s multiplication (4) in the process of obtaining thesemodels is not valid. Thus, we consider the following restriction together with the constraints of model (24):

xLi;j 6 minfuL

i;j; x�i;jg 8ði; jÞ 2 A; ð26Þ

where the vector x* is the optimal solution of model (23) with the objective value v*. Moreover, we denote theoptimal solution of models (24) and (25) by xL,* and xR,* with the optimal values vL,* and vR,*, respectively.Then the fuzzy optimal solution of the fully fuzzified MCFP can be represented as

ðx�; x�;L; x�;RÞ;

with

ðv�; v�;L; v�;RÞ
as the fuzzy optimal value of the objective function.
4. Computational results

In this section, we present some computational tests in order to compare the effectiveness of our proposedmethod with other schemes. These tests are carried out by considering a random network and implementingShih and Lee’s scheme [40] for the fuzzy MCFP and also solving the fully fuzzified MCFP utilizing some well-known ranking functions.

4.1. Simplifying some ranking functions

For estimating the decision factors, it is reasonable to use LR flat numbers [9] or at least trapezoidal num-bers which have been used for the same targets in [11,14,13]. Experience shows that in many cases such aspower engineering problems, the membership functions of the alternatives are flat fuzzy numbers [11]. How-ever, according to Ekel et al.’s report [11], taking these types of numbers into account may lead to some dif-ficulties in obtaining a unique solution. Thus, for convenience, we exploit triangular fuzzy numbers in our testnetworks. The average, lower and upper bounds of quantities can be expressed appropriately by these num-bers. To pursue computational concepts, it is necessary to simplify some well-known ranking functions for thiskind of fuzzy numbers, see [43] for a brief review and some important properties. Note that a triangular num-ber ~a ¼ ða; aL; aRÞ is a special number in LRðRÞ with the following membership function:

l~aðxÞ ¼

1aL ðx� aþ aLÞ; a� aL

6 x 6 a;�1aR ðx� a� aRÞ; a 6 x 6 aþ aR;

0; otherwise:

8><>: ð27Þ

In the following some ranking functions with respect to a triangular number ~a are calculated.

Definition 4.1 (Adamo’s approach [1]). Given a 2 (0,1], Adamo simply evaluates a fuzzy quantity based on theright point of the a-cut. Therefore, his ordering index is


ADað~aÞ ¼ aRðaÞ ¼ aþ ð1� aÞaR:

Definition 4.2 (Yager’s approaches [49–51]). Yager proposed four indices which may be employed for the pur-pose of ordering fuzzy quantities in [0,1]. In the following, the first and second indices are produced for arbi-trary triangular number ~a:

Y 1ð~aÞ ¼R

Suppð~aÞ xl~aðxÞdxRSuppð~aÞ l~aðxÞdx

¼ aþ 1

3ðaR � aLÞ;

Y 2ð~aÞ ¼Z Hgð~aÞ

0

aLðaÞ þ aRðaÞ2

da ¼ aþ 1

4ðaR � aLÞ:

Definition 4.3 (Chang’s approach [5]). Chang’s index is simply defined by the following integral:

Cð~aÞ ¼Z

x2Suppð~aÞxl~aðxÞdx ¼ 1

2aðaL þ aRÞ þ 1

6ððaRÞ2 � ðaLÞ2Þ

Definition 4.4 (Campos and Munoz’s approach [3]). A family of ordering indices called ‘‘average index’’ is pro-posed by Campos and Munoz to rank fuzzy numbers. In the followings, two indices of this family arerepresented:

CMk1ð~aÞ ¼

Z 1

0

ðkaRðaÞ þ ð1� kÞaLðaÞÞda ¼ aþ 1

2kðaL þ aRÞ � 1

2aL;

CMk2ð~aÞ ¼ 2

Z 1

0

aðkaRðaÞ þ ð1� kÞaLðaÞÞda ¼ aþ 1


3aL:

Definition 4.5 (Liou and Wang’s approach [24]). Let L(Æ) and R(Æ) be the left and right shape functions. Theordering index proposed by Liou and Wang is defined by the following formula:

LW kð~aÞ ¼ kZ 1

0

R�1ðaÞdaþ ð1� kÞZ 1

0

L�1ðaÞda ¼ aþ 1


2aL ¼ CMk

1ð~aÞ:

where k 2 [0,1] is the optimism index reflecting the optimism degree of a decision maker. The larger k is themore optimistic the decision maker is. The two extreme cases are: k = 0, where the decision maker is com-pletely pessimistic; and k = 1, where the decision maker is completely optimistic. The case k ¼ 1

2reflects a lin-

ear decision attitude.

Definition 4.6 (Choobineh and Li’s approach [6]). Let c and d be two numbers satisfyingc 6 inffxjx 2

S~aSuppð~aÞg and d P supfxjx 2

S~aSuppð~aÞg. Choobineh and Li evaluate ~a by the following

expression:

CLð~aÞ ¼ 1

2ðHgð~aÞ � 1

d � cðZ Hgð~aÞ

0

ðd � aRðaÞÞda�Z Hgð~aÞ

0

ðaLðaÞ � cÞdaÞÞ ¼ 4aþ aR � aL � 4c4ðd � cÞ :

Definition 4.7 (Fortemps and Roubens’ approach [16]). Let the height of fuzzy number ~a be defined asHgð~aÞ ¼ maxfl~aðxÞg. Fortemps and Roubens suggested the following evaluation of ~a:

FRð~aÞ ¼ 1

2Hgð~aÞ

Z Hgð~aÞ

0

ðaLðaÞ þ aRðaÞÞda ¼ aþ 1

4ðaR � aLÞ ¼ Y 2ð~aÞ:


Now reshape model (13) to the following matrix form considering fuzzy variable ~x ¼ ðx; xL; xRÞ:

min ~c� ~x; ð28Þs:t: N~x ¼ ~b; ð29Þ

0 6 x 6 u; 0 6 xL6 uL; 0 6 xU

6 uR; ð30Þ

where the incident matrix N is stated as (20) with Hukuhara’s difference (2.4) in place of ‘‘-’’.With ranking functions (R,ADa; Y 1; Y 2;C;CMk

1;CMk2; LW k;CL; FR), we can evaluate the objective function

for comparing solutions; i.e., we can utilize minRð~c:~xÞ instead of (28).Since these ranking functions cannot produce a total order, we cannot use RðN~xÞ ¼ Rð~bÞ in place of

N~x ¼ ~b. That is

N~x ¼ ~b;

implies that

RðN~xÞ ¼ Rð~bÞ;

but not the converse. Therefore, a feasible solution for RðN~xÞ ¼ Rð~bÞ may be infeasible for N~x ¼ ~b: Thus,with these ranking functions we cannot handle fuzzy constraints, which is an essential drawback existing in allof these ranking functions. This basic requirement is, however, accomplished by our ranking function. Tomake comparisons, we assume crisp variables and a crisp right hand side in (29) and obtain the following sim-pler problem:

min ~cx ¼ minRð~cxÞ;s:t: Nx ¼ b;

0 6 x 6 u:

ð31Þ

In the following subsection, we solve this model with respect to the presented ranking functions and comparethe results with our outcomes.

4.2. Numerical computations

Consider the network depicted in Fig. 1. Assume a fully fuzzified MCFP with the input data of Tables 1–3,where the fuzzy numbers are supposed to be triangular numbers. We have used a Pentium 4 PC with 500 MHRAM for our programming problem.

We solve our three proposed models ((23)–(25)) to find the optimal solution of the fully fuzzified MCFP(13) constrained with ((14)–(16)) and ‘‘.,�’’ as Wagenknecht’s multiplication (4). Note that we do not needto demonstrate the accuracy of the solution or convergence improvement, because according to the employedmethod, these important considerations are derived from traditional MCFP directly, see [2] among hundredsof other possible references.

To investigate the effect of the selected decision parameters on results, the proposed ordering is imple-mented with the following two triples of scalars:

Setting 1 : k ¼ p; l ¼ 1; r ¼ p5;

Setting 2 : k ¼ p; l ¼ p5; r ¼ 1:

The optimal values with respect to these settings of parameters are as follows:

v�1 ¼ ð376000; 311075; 530575Þ;v�2 ¼ ð436000; 351900; 531275Þ:

Fig. 1. A network with 16 nodes and 49 links.

Table 1The supplier nodes

b1 b3 b6 b10 b15 b16

(500,50,30) (300,40,60) (500,30,50) (700,60,30) (400,50,60) (800,50,20)

Table 2The demander nodes

b2 b4 b7 b8 b11 b13 b14

(�400,50,50) (�500,20,80) (�500,40,10) (�300,30,20) (�800,40,40) (�200,30,20) (�500,70,30)


The optimal fuzzy solution based on the first setting of parameters is presented in Table 4. Furthermore, with10,000 different triples of parameters, we tried to find other solutions, but in all of those situations, the pro-cedure terminates with one of the above values. This means that our method is not very sensitive to k, l and r

variations, and we do not need to run our program for obtaining the optimal solution of real problems withtoo many difficult settings of (k, l, r). Also, since the center, left and right spreads of the right hand side areintegers, the resulting fuzzy solution has integral center, left and right spreads, which is a general result provedfor crisp MCFP [2].

Now we need a criterion for comparing results. Note that to compare the fuzzy values of an objective func-tion, we cannot use our proposed ordering (2.10), because that ordering judges our obtained solution to be thebest, and so it is not beneficial. Thus, we propose the following criterion which compares the most possibleamount of fuzzy numbers (their centers) and also the most optimistic or pessimistic amounts of fuzzy numbers(the lower or upper bounds of the support of fuzzy numbers), respectively. In our opinion, a fuzzy number ispreferred if for this number the above quantities are better compared with the corresponding quantities forother numbers. Thus, we adopt the following componentwise comparison:

Definition 4.8. Considering two triangular fuzzy numbers ~/ ¼ ð/;/L;/RÞ and ~w ¼ ðw;wL;wRÞ, we say ~/ isbetter than ~w if /, / � /L and / + /R are better than w, w � wL and w + wR, respectively. Similarly, we say ~/is almost better than ~w if at least two components of ~/ are better than the corresponding components of ~w (seeFig. 2).

Table 3The fuzzy capacity and cost of network links

Link Index Tail node Head node Link fuzzy cost Link fuzzy capacity

1 1 2 (200,150,250) (320,300,350)2 1 3 (220,200,300) (200,200,220)3 1 5 (300,280,390) (250,200,270)4 2 3 (360,300,400) (300,300,360)5 2 4 (400,380,500) (330,300,350)6 2 8 (600,550,620) (570,500,600)7 3 2 (500,460,530) (400,400,450)8 3 6 (100,90,200) (520,500,550)9 3 9 (600,540,660) (340,300,360)10 4 5 (300,270,320) (230,200,290)11 4 7 (600,560,700) (560,500,580)12 4 11 (400,370,450) (330,300,400)13 5 1 (700,640,740) (660,600,700)14 5 4 (1500,1350,1600) (1100,1000,1140)15 5 6 (1000,900,1110) (780,700,800)16 5 12 (300,280,350) (290,200,320)17 6 5 (1300,1180,1320) (1040,1000,1100)18 6 10 (700,650,780) (630,600,660)19 6 13 (400,370,410) (540,500,600)20 7 8 (200,180,300) (120,100,170)21 8 2 (500,450,520) (480,400,500)22 8 7 (300,220,340) (220,200,240)23 8 9 (500,460,600) (350,300,380)24 8 14 (600,550,680) (340,300,400)25 9 3 (300,240,320) (250,200,300)26 9 8 (1000,930,1100) (720,700,750)27 9 10 (300,280,400) (480,400,500)28 9 15 (200,180,250) (270,200,300)29 10 9 (1200,1130,1280) (970,900,1000)30 11 4 (300,270,400) (330,300,370)31 11 7 (400,350,430) (250,250,280)32 11 12 (300,290,370) (260,200,300)33 11 14 (500,470,500) (320,300,340)34 12 5 (400,340,450) (350,300,370)35 12 11 (600,580,700) (520,500,580)36 12 13 (200,140,300) (230,200,300)37 12 16 (500,470,600) (320,300,370)38 13 6 (700,660,800) (550,500,600)39 13 10 (400,390,430) (250,200,280)40 13 12 (1000,1000,1050) (550,500,600)41 14 8 (200,170,220) (350,300,400)42 14 11 (400,300,420) (320,300,350)43 14 15 (300,290,400) (330,300,350)44 15 9 (500,420,600) (410,400,450)45 15 13 (700,650,730) (550,500,600)46 15 14 (400,370,500) (250,250,300)47 16 12 (500,440,520) (350,350,390)48 16 14 (100,90,250) (150,100,200)49 16 15 (600,550,630) (520,500,580)


In the following, we demonstrate the efficiency of our method using this criterion.Using this definition, we offer examples to compare our method with other implementable schemes.

Example 4.9. In this example, we compare our results with that of Shih and Lee [40] for the fully fuzzifiedMCFP. Shih and Lee considered an MCFP with fuzzy costs and link capacities with crisp variables. For suchproblems, they assumed an a 2 (0,1] as the decision maker’s cut-off value and solved the following crisp mixed-integer linear programming problems:

Table 4The optimal fuzzy flow by our approach with respect to k = p, l = 1 and r = p5

~x1 ~x2 ~x3 ~x4 ~x5 ~x6 ~x7

(320,300,350) (0,0,0) (180,150,180) (0,0,0) (180,180,350) (140,70,0) (400,300,450)~x8 ~x9 ~x10 ~x11 ~x12 ~x13 ~x14

(0,0,0) (0,0,0) (0,0,0) (30,30,270) (330,290,100) (0,0,0) (680,620,600)~x15 ~x16 ~x17 ~x18 ~x19 ~x20 ~x21

(0,0,0) (0,0,0) (500,470,420) (0,0,0) (0,0,130) (0,0,0) (0,0,0)~x22 ~x23 ~x24 ~x25 ~x26 ~x27 ~x28

(220,200,240) (0,0,0) (300,280,190) (100,40,90) (680,680,750) (0,0,0) (0,0,0)~x29 ~x30 ~x31 ~x32 ~x33 ~x34 ~x35

(700,640,730) (0,0,0) (250,230,0) (0,0,0) (0,0,0) (0,0,0) (520,500,580)~x36 ~x37 ~x38 ~x39 ~x40 ~x41 ~x42

(0,0,0) (0,0,0) (0,0,0) (0,0,0) (170,150,190) (0,0,0) (200,200,160)~x43 ~x44 ~x45 ~x46 ~x47 ~x48 ~x49

(0,0,0) (80,80,110) (370,320,280) (250,250,300) (350,350,390) (150,100,200) (300,300,230)

Fig. 2. Since the values for the most possible, most optimistic and most pessimistic members of ~/ are less than those for the correspondingmembers of ~w; for minimizing targets, ~/ is preferred to ~w:


minfxi;jjði;jÞ2Ag f ;

s:t:X

j:ði;jÞ2A

xi;j �X

j:ðj;iÞ2A

xj;i ¼ bi 8i 2 N ;Xði;jÞ2A

cLi;jxi;j 6 f 6

Xði;jÞ2A

ci;jxi;j;

f �Pði;jÞ2A

cLi;jxi;jP

ði;jÞ2Aci;jxi;j �

Pði;jÞ2A

cLi;jxi;j

P a;

0 6 xi;j 6 uRi;j � aðuR

i;j � uLi;jÞ 8ði; jÞ 2 A;

xi;j is integer:

ð32Þ

Shih and Lee’s strategy [40] handles only situations where the left and right hand sides are both fuzzy. Becausethey are dealing with crisp variables, the left hand side of their fuzzy MCFP (13) becomes crisp. Thus, theywere not able to consider fuzzy supply-demand. This is revealed implicitly in Example (1) of their paper[40]. Therefore, from this point of view, Shih and Lee’s method is weaker than ours. In Table 5, the optimalfuzzy transportation costs for six cut-off values a = 0.0001, 0.25, 0.5, 0.75, 0.9999, 1, are presented. Also, theoptimal solution for a = 0.5 is shown in Table 6 with f = 4952575 as the crisp optimal value of objective func-tion (32).

By comparing Shih and Lee’s results (Table 5) with ours, we arrive at the following conclusions:

• Our results in both settings are better than Shih and Lee’s results for six cut-off parameters a.• Table 5 shows that Shih and Lee’s method is very sensitive to the cut-off values. Practically, it is a difficult

job to find this parameter in the decision making processes.

Table 5The fuzzy optimal values with respect to six cut-off values according to Shih and Lee’s approach

a Optimal value a Optimal value

0.0001 (4754078.10,4358272.93,5236479.66) 0.25 (4955750.00,4545725.00,5441400.00)0.5 (5165500.00,4739650.00,5653600.00) 0.75 (5375250.00,4933575.00,5865800.00)0.9999 (5609899.60,5153904.94,6107395.65) 1 (5610000.00,5154000.00,6107500.00)

Table 6The optimal crisp flow for a = 0.5 by Shih and Lee’s approach

~x1 ~x2 ~x3 ~x4 ~x5 ~x6 ~x7 ~x8 ~x9 ~x10 ~x11 ~x12 ~x13 ~x14

325 0 175 0 325 25 425 0 0 0 280 105 0 560~x15 ~x16 ~x17 ~x18 ~x19 ~x20 ~x21 ~x22 ~x23 ~x24 ~x25 ~x26 ~x27 ~x28

0 0 385 0 115 0 0 220 0 230 125 725 0 0~x29 ~x30 ~x31 ~x32 ~x33 ~x34 ~x35 ~x36 ~x37 ~x38 ~x39 ~x40 ~x41 ~x42

700 0 0 0 0 0 540 0 0 0 0 170 0 155~x43 ~x44 ~x45 ~x46 ~x47 ~x48 ~x49

0 150 255 275 370 150 280


• Since Shih and Lee’s technique changes the structure of MCFP constraints, its linear programming variantwhich relaxes the integrality assumption does not guarantee to find an integral flow, whereas our methodfinds such a solution.

• Shih and Lee’s scheme does not take the level of risk into account, while our method is capable of capturingthe uncertain behavior in decision making processes.

• Shih and Lee utilized mixed-integer programming techniques in which the number of iterations in solvingsuch problems may grow exponentially with respect to the size of the problem. Thus, their method may beinefficient and computationally too expensive. On the contrary, our method consists of three crisp MCFPssolvable in polynomial time employing many already-known algorithms such as scaling cycle-canceling[2,19].

Example 4.10. In this example, we solve model (31) using R,ADa and we compare its outcomes with theresults of our fully fuzzified MCFP (13) constrained by ((14)–(16)). Since we cannot exploit the fuzzy supply-demand(s) by model (31), we consider the average of these quantities. We change a from 0.0001 to 1 in sixsteps as follows:

a ¼ 0:0001; 0:25; 0:5; 0:75; 0:9999; 1:

Concerning all of these certainty levels, the following optimal fuzzy value is provided:

v�ðADaÞ ¼ ð5319000; 4883500; 5808500Þ:
Although ADa should be capable of taking the level of certainty into consideration, these tests reveal that it
cannot successfully undertake such task. Also, our optimal values in both settings are better than its optimalsolution with these different values of a. Besides, since for each a 2 (0,1], ADa considers the greatest element offuzzy cuts, it is a pessimistic approach to a minimization problem and so produces a solution with widespreads.

Example 4.11. In this example, we solve model (31) using R,Y 1; fY 2 ¼ FRg;C;CL which are independent ofthe level of certainty.

For R ¼ CL; we use (c,d) = (0,10000). Although we use different couples for this ordering, we find the sameresults. In Table 7, the fuzzy optimal values provided by these ranking functions are represented. Note thatChang’s ordering (C) has yielded the worst optimal value in this example. This table also reveals that all ofthe orders except Chang’s approach (4.3) produce optimal values equal to v*(ADa). Therefore, these rankingfunctions are as pessimistic as ADa. Furthermore, in contrast to our ordering, these methods do not allow toidentify the level of certainty.

Table 7The fuzzy optimal values utilizing R,Y 1; fY 2 ¼ FRg;C;CL

R Optimal value R Optimal value

Y1 (5319000,4883500,5808500) {Y2 = FR} (5319000,4883500,5808500)C (5524000,5093000,6042000) CL (5319000,4883500,5808500)


Example 4.12. Now we examine our network problem using R,fCMk1 ¼ LW kg; CMk

2: Since k 2 [0,1] we usek = 0, 0.5, 1. In all of the examinations, CMk

1 and CMk2 gave the following optimal value:

TableThe fu

a

0.00010.50.9999

TableThe fu

R

Y1

C

v�ðCMk1Þ ¼ v�ðCM k

2Þ ¼ ð5319000; 4883500; 5808500Þ;
which is equal to ADa, showing that they are worse than our ranking function. Also according to the results ofExample (4.10), these ranking functions are pessimistic. In addition, since the effect of k is lost, they could notsimulate risk taking or averting.
Example 4.13. In this test, an inexact heuristic method similar to that used by Maleki [26] for solving fuzzylinear programming problem with crisp variables and fuzzy coefficients is followed. According to this heuristicmethod, the following program can be considered in place of fully fuzzified MCFP (13):

min Rð~c:xÞ; ð33Þs:t: RðNxÞ ¼ Rð~bÞ; ð34Þ

0 6 x 6 Rð~uÞ: ð35Þ

Table 8 exhibits the results of utilizing R ¼ ADa with different values of a. This table concludes that as adecreases the optimal value increases. But these values are not better than our optimal value. The fuzzy opti-mal values of model (33) when R,Y 1; fY 2 ¼ FRg;C;CL are reported in Table 9.

It is clear that Chang’s solution (C) is too far from what seems reasonable. Also for CL, we had to set(c,d) = (0,1), because for almost all of the other cases, the solutions were not acceptable. Still our optimal val-ues are preferable. In Table 10, the optimal values for k = 1,0.75,0.5,0.25,0, employing R,fCMk

1 ¼LW kg;CMk

2 are gathered.In Table 10 as k decreases, the result improves, but still our solution prevails.Thus, we believe that for solving fully fuzzified MCFP (13), our ranking function-based scheme has a more

interesting performance compared with the previous established works, because of the following reasons:

• It can exhibit uncertain supply-demand(s) as well as uncertain transportation cost.• It is solvable in polynomial time, by solving three crisp MCFPs.• It is able to consider risk taking and averting in practical problems.

8zzy optimal values of (33) by ADa with respect to six values for a

Optimal value a Optimal value

(10672464.60,9798808.42,11670613.73) 0.25 (9334500.02,8570350.01,10205525.02)(7996000.00,7341400.00,8739850.00) 0.75 (6657500.00,6112450.00,7274175.00)(5319535.40,4883991.58,5809086.27) 1 (5319000.00,4883500.00,5808500.00)

9zzy optimal values of model (33) by R,Y 1; fY 2 ¼ FRg;C;CL

Optimal value R Optimal value

(5496666.67,5048600.00,6003033.33) {Y2 = FR} (5452250.00,5007325.00,5954400.00)(5745209325.94,5238257230.15,6436565788.53) CL (5452250.00,5007325.00,5954400.00)

Table 10The fuzzy optimal values utilizing R,fCMk

1 ¼ LW kg;CMk2

k R ¼ fCMk1 ¼ LW kg R ¼ CMk

1

1 (7996000.00,7341400.00,8739850.00) (7103666.67,6522100.00,7762733.33)0.75 (6724125.00,6174362.50,7347125.00) (6255750.00,5744075.00,6834250.00)0.5 (5452250.00,5007325.00,5954400.00) (5407833.33,4966050.00,5905766.67)0.25 (4180375.00,3840287.50,4561675.00) (4559916.67,4188025.00,4977283.33)0 (2918500.00,2683850.00,3180750.00) (3712000.00,3410000.00,4048800.00)


5. A case study

The approach taken in this paper can serve as a basis for solving problems in petroleum industry distrib-uting systems, see Fig. 3 which is adopted from [28] with some modifications. Petroleum exploration is at thehighest level of the chain. Petroleum may also be supplied from other companies which wish to swap theirproduction between two points. Oil tankers or pipeline networks transport petroleum to oil export terminalsor to oil pumping stations to feed refineries. Crude oil is converted into various products at refineries, whichare sent to distribution centers via pipelines, trucks, vessels or trains, depending on consumer demand. Thuswe face with two disjoint systems, with two disjoint commodities: crude oil and products. Our model is adapt-able and can solve each of these transportation problems with respect to each of the commodities, but wefocus on crude oil transportation, taking into account the uncertain conditions. Some sources of uncertaintyexisting in are as follows:

• Daily production of each unit which is dependent on many items such as the power outage or failure of fieldfacilities.

• Daily exportation of each port (terminal) which is constantly changing with oil contracts depending onuncertain customers and season variations. Also, arrival times of tankers vary according to weatherconditions.

• The level of swapping which is affected by political decisions.• The refinery intake which is dependent on the oil type and status of the refinery.• According to age and corrosion of the pipeline network, their capacities and also the ability of pump sta-

tions cannot be considered as real numbers, and only their lower and upper bounds can be predicted withthe help of experts and technical softwares.

Fig. 3. General petroleum supply chain, including oil suppliers and demanders.

Fig. 4. A simple diagram of a pilot study in Iranian petroleum industry.

Table 11The fuzzy optimal flows in a pilot study of Iranian oil industry

From! to Fuzzy flow From! to Fuzzy flow

2! 1 (489.40,474.718,538.34) 10! 9 (0.00,0,0)3! 2 (489.40,474.718,538.34) 11! 10 (0.00,0,0)3! 5 (0.00,0,0) 11! 16 (397.80,385.866,437.58)4! 3 (349.10,338.627,384.01) 12! 11 (243.60,236.292,267.96)5! 4 (349.10,338.627,384.01) 13! 11 (45.10,43.747,49.61)5! 7 (0.00,0,0) 14! 15 (46.90,45.493,51.59)6! 5 (318.90,309.333,350.79) 15! 11 (97.80,94.866,107.58)7! 6 (318.90,309.333,350.79) 16! 17 (397.80,385.866,437.58)7! 9 (0.00,0,0) 19! 18 (158.90,154.133,174.79)8! 7 (109.40,106.118,120.34) 20! 19 (158.90,154.133,174.79)9! 8 (109.40,106.118,120.34) 20! 21 (450.60,437.082,495.66)9! 11 (11.30,10.961,12.43) 21! 1 (98.60,95.642,108.46)


Therefore, one of the principal challenges for the petroleum industries is to respond to uncertain customerrequirements by decreasing the level of risk in production and business [39]. To this end, based on the resultsof this study, we have used as a pilot instance an important area in the Iranian oil industry including oil fieldsof ‘‘Ahvaz’’ and ‘‘Maroon’’, import terminal ‘‘Neka’’ and export terminal ‘‘Kharg’’ with uncertainties in sup-plies and demands. A simplified network of this region which consists of 21 main facilities and 24 main pipe-lines is depicted in Fig. 4. In this figure, nine oil fields, one swap terminal, one export terminal and tworefineries are considered. Although Iranian crude oil consists of ‘‘Bangestan’’ and ‘‘Asmari’’ kinds, their pipe-lines are segregated. Thus, it is sufficient to solve only one fully fuzzified MCFP for designing optimal dailytransportation. The obtained optimal values according to our scheme, with (k, l, r) = (p, 1,p5), is(41331.00, 9521.06,65585.12). The fuzzy optimal flows are represented in Table 11.

Although we consider only this simplified network, this approach has a more far reaching application to thepetroleum industry. It is also possible, for instance, to deal with storage tanks to save the excess of oil in sup-plier, demander and intermediate nodes. This requires further studies.

6. Conclusion–discussion

Fuzzy programming is important in uncertainty modelling in the field of decision making. In this paper, weproposed a total ordering of fuzzy numbers that uses non-algebraic numbers instead of importance weights


relating to the center, left spread and right spread of fuzzy numbers. This order belongs to the ranking func-tion methods and can tackle risk taking aspects. Hukuhara’s difference has also been employed for the math-ematical modelling of flows. By joining these approaches, existing combinatorial algorithms can be utilized forfully fuzzified minimal cost flow problems (MCFPs), as well. We have compared our results with the previousworks on fuzzy MCFP. In addition, instead of our ordering, we derived some ranking function formulas fortriangular fuzzy numbers in order to solve fully fuzzified MCFP. Our experiments, according to the most pos-sible, pessimistic and optimistic viewpoints, show that our ordering produces better results than other meth-ods. Moreover, it is shown that some of the ranking functions exhibit weak performance when handleuncertainty. We finally illustrate the performance of our scheme for obtaining optimal transportation inthe petroleum industry. These ideas could probably be extended to fuzzy multi-commodity problems.

As a final point, we would like to mention some important considerations regarding the ranking of fuzzynumbers. Wang and Kerre [43,44] have counted more than 35 existing fuzzy number ranking indices. How-ever, unlike the case of real numbers, fuzzy quantities have no natural order. In addition Ekel and Neto[13] have indicated that any approach based on the conversion of fuzzy quantities into real numbers suffersfrom some defects because comparison of fuzzy quantities using real numbers can only focus on one specialaspect of fuzzy quantities. They have also gathered some references which indicate that many of these indicesproduce different rankings for the same problem and occasionally in some choices, they yield results inconsis-tent with intuition.

Moreover, it seems unnatural that ranking of fuzzy quantities attempts obligatorily to distinguish the alter-natives, while uncertainty of information creates the decision uncertainty regions [12]. Thus, the authors of[11–13] have utilized multicriteria choosing alternatives in a fuzzy environment because the application ofadditional criteria can serve as a convincing means to contracting the decision uncertainty regions. Themethod studied in this paper can produce these alternatives in large scale fuzzy optimization problems bychanging the triple parameters (k, l, r). Also, we have attempted to obtain a set of solutions instead of justone for fuzzy goal programming problems [20]. In [20], the optimistic and pessimistic solutions have beendetermined by employing two kinds of orders: severe order and lenient order. A convex combination of thesesolutions have then been generated as the fuzzy solution of fuzzy goal programming. Our next goals includecombining these concepts with network algorithms as well as examinations on large scale networks.

Acknowledgement

The authors express particular thanks to the three anonymous referees and Professor Witold Pedrycz fortheir valuable comments, which led us to improvements in this paper. We are also very grateful to Mr. KarimRezaei for the linguistic editing of this paper.

References

[1] J.M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems 4 (1980) 207–219.[2] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows, Prentice-Hall, Englewood cliffs, 1993.[3] L. Campos, A. Munoz, A subjective approach for ranking fuzzy numbers, Fuzzy Sets and Systems 29 (1989) 145–153.[4] P.T. Chang, E.S. Lee, Fuzzy linear regression with spreads unrestricted in sign, Computers & Mathematics with Applications 28

(1994) 61–70.[5] W. Chang, Ranking of fuzzy utilities with triangular membership functions, Proceedings of International Conference on Policy

Analysis and Systems (1981) 263–272.[6] F. Choobineh, H. Li, An index for ordering fuzzy numbers, Fuzzy Sets and Systems 54 (1993) 287–294.[7] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1995.[8] P. Diamond, R. Korner, Extended fuzzy linear models and least squares estimates, Computers & Mathematics with Applications 33

(1997) 15–32.[9] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.

[10] D. Dubois, H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Information Sciences 30 (1983) 183–224.[11] P. Ekel, W. Pedrycz, R. Schinzinger, A general approach to solving a wide class of fuzzy optimization problems, Fuzzy Sets and

Systems 97 (1998) 49–66.[12] P.Y. Ekel, Fuzzy sets and models of decision making, Computers & Mathematics with Applications 44 (2002) 863–875.[13] P.Y. Ekel, F.H.S. Neto, Algorithms of discrete optimization and their application to problems with fuzzy coefficients, Information

Sciences 176 (2006) 2846–2868.


[14] E.A. Galperin, P.Y. Ekel, Synthetic realization approach to fuzzy global optimization via Gamma algorithm, Mathematical andComputer Modelling 41 (2005) 1457–1468.

[15] M. Filaseta, Algebric Number Theory Instructor Notes, University of South Carolina, 1999.[16] P. Fortemps, M. Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems 82 (1996) 319–

330.[17] R.E. Giachetti, R.E. Young, A parametric representation of fuzzy numbers and their arithmetic operators, Fuzzy Sets and Systems 91

(1997) 185–202.[18] E. Hansen, Global Optimization Using Interval Analysis, Marcel Dekker Inc., USA, 1992.[19] S.M. Hashemi, M. Ghatee, E. Nasrabadi, Combinatorial algorithms for minimal interval cost flow problem, Applied Mathematics

and Computation 175 (2006) 1200–1216.[20] S.M. Hashemi, M. Ghatee, B. Hashemi, Fuzzy goal programming: complementary slackness conditions and computational schemes,

Applied Mathematics and Computation 179 (2006) 506–522.[21] M. Hukuhara, Integration des applications measurable dont la valeur est un compact convexe, Funkc. Ekvacioj. 10 (1967) 205–223.[22] M. Inuiguchi, M. Sakawa, Possible and necessary optimality tests in possibilistic linear programming problems, Fuzzy Sets and

Systems 67 (1994) 29–46.[23] C.J. Lin, U.P. Wen, A labeling algorithm for the fuzzy assignment problem, Fuzzy Sets and Systems 142 (2004) 373–391.[24] T. Liou, J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems 50 (1992) 247–255.[25] N. Mahdavi-Amiri, S.H. Nasseri, Duality in fuzzy number linear programming by use of a certain linear ranking function, Applied

Mathematics and Computation 180 (2006) 206–216.[26] H.R. Maleki, Ranking functions and their applications to fuzzy linear programming, Far East, Journal of Mathematical Sciences 4

(2002) 283–301.[27] R. Montemanni, L.M. Gambardella, A.V. Donati, A branch and bound algorithm for the robust shortest path problem with interval

data, Operation Research Letters 32 (2004) 225–232.[28] S.M.S. Neiro, J.M. Pinto, A general modeling framework for the operational planning of petroleum supply chains, Computers and

Chemical Engineering 28 (2004) 871–896.[29] A.S. Noda, C.G. Mortin, An algorithm for the biobjective integer minimum cost flow problem, Computers & Operation Research 28

(2001) 139–156.[30] S. Okada, T. Soper, A shortest path problem on a network with fuzzy arc lengths, Fuzzy Sets and Systems 109 (2000) 129–140.[31] P. Perny, O. Spanjaard, A preference-based approach to spanning trees and shortest paths problems, European Journal of

Operational Research 162 (2005) 584–601.[32] J. Ramik, Duality in fuzzy linear programming: some new concepts and results, Fuzzy Optimization and Decision Making 4 (2005)

25–39.[33] J. Ramik, Duality in fuzzy linear programming with possibility and necessity relations, Fuzzy Sets and Systems 157 (2006) 1283–1302.[34] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York and London, 1993.[35] M. Sakawa, I. Nishizaki, Y. Uemura, Interactive fuzzy programming for multilevel linear programming problems, Computers &

Mathematics with Applications 36 (1998) 71–86.[36] M. Sakawa, I. Nishizaki, Y. Uemura, Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy

parameters, Fuzzy Sets and Systems 115 (2000) 93–103.[37] A. Sengupta, T.K. Pal, Theory and methodology on comparing interval numbers, European Journal of Operational Research 127

(2000) 28–43.[38] A. Sengupta, T.K. Pal, Solving the shortest path problem with interval arcs, Fuzzy Optimization and Decision Making 5 (2006) 79–

99.[39] D.Y. Sha, Z.H. Che, Supply chain network design: partner selection and production/distribution planning using a systematic model,

Journal of the Operational Research Society 57 (2006) 52–62.[40] H.S. Shih, E.S. Lee, Fuzzy multi-level minimum cost flow problems, Fuzzy Sets and Systems 107 (1999) 159–176.[41] H. Tanaka, K. Asai, Fuzzy solution in fuzzy linear programming problems, IEEE Transaction on System Man Cybernet 14 (1984)

325–328.[42] M. Wagenknecht, R. Hampel, V. Schneider, Computational aspects of fuzzy arithmetics based on Archimedean t-norms, Fuzzy Sets

and Systems 123 (2001) 49–62.[43] X. Wang, E. Kerre, Reasonable properties for the ordering of fuzzy quantities(I), Fuzzy Sets and Systems 118 (2001) 375–385.[44] X. Wang, E. Kerre, Reasonable properties for the ordering of fuzzy quantities(II), Fuzzy Sets and Systems 118 (2001) 387–405.[45] H.-C. Wu, Duality theorems in fuzzy mathematical programming problems based on the concept of necessity, Fuzzy Sets and Systems

139 (2003) 363–377.[46] H.-C. Wu, Fuzzy optimization problems based on the embedding theorem and possibility and necessity measures, Mathematical and

Computer Modelling 40 (2004) 329–336.[47] H.-C. Wu, An (ab)-optimal solution concept in fuzzy optimization problems, Optimization 53 (2004) 203–221.[48] H.-C. Wu, Statistical hypotheses testing for fuzzy data, Information Sciences 175 (2005) 30–56.[49] R.R. Yager, Ranking fuzzy subsets over the unit interval, Proceedings of CDC (1978) 1435–1437.[50] R.R. Yager, On choosing between fuzzy subsets, Kybernetes 9 (1980) 151–154.[51] R.R. Yager, A procedure for ordering fuzzy sets of the unit interval, Information Sciences 24 (1981) 143–161.[52] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.[53] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) an outline, Information Sciences 172 (2005) 1–40.

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Ranking function-based solutions of fully fuzzified minimal cost flow problem