Research Article Solving a Fully Fuzzy Linear Programming ...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 726296 10 pageshttpdxdoiorg1011552013726296

Research ArticleSolving a Fully Fuzzy Linear Programming Problem throughCompromise Programming

Haifang Cheng1 Weilai Huang1 and Jianhu Cai2

1 School of Management Huazhong University of Science and Technology Wuhan 430074 China2 College of Economics and Management Zhejiang University of Technology Hangzhou 310023 China

Correspondence should be addressed to Jianhu Cai hzdcjhyahoocom

Received 22 February 2013 Accepted 9 May 2013

Academic Editor Reinaldo Martinez Palhares

Copyright copy 2013 Haifang Cheng et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the current literatures there are several models of fully fuzzy linear programming (FFLP) problems where all the parametersand variables were fuzzy numbers but the constraints were crisp equality or inequality In this paper an FFLP problem withfuzzy equality constraints is discussed and a method for solving this FFLP problem is also proposed We first transformthe fuzzy equality constraints into the crisp inequality ones using the measure of the similarity which is interpreted as thefeasibility degree of constrains and then transform the fuzzy objective into two crisp objectives by considering expected valueand uncertainty of fuzzy objective Since the feasibility degree of constrains is in conflict with the optimal value of objectivefunction we finally construct an auxiliary three-objective linear programming problem which is solved through a compromiseprogramming approach to solve the initial FFLP problem To illustrate the proposed method two numerical examples aresolved

1 Introduction

Linear programming (LP) has important applications inmany areas of engineering and management In these appli-cations since the real-world problems are very complex theparameters of LP are usually represented by fuzzy numbersTherefore many researchers have shown interest in the areaof fuzzy linear programming (FLP)

Recently fuzzy set theory has been applied in manyresearch regions since fuzzy set theory is effective to solve thedecision-making problems with imprecise data [1ndash3] Severalkinds of the FLP problems have appeared in the literature[4ndash16] Delgado et al [4] have proposed a general model forthe FLP problems in which constraints are fuzzy inequalityand the parameters of constraints are fuzzy numbers but theparameters of the objective function are crisp Rommelfanger[5] has also proposed a general model for the FLP problemsand the main difference compared with [4] is that hereparameters of the objective function are also fuzzy numbersConsidering the different hypotheses researchers [6ndash13] haveproposed some particular FLP problems which can be

deduced from the general model In order to solve theseFLP problems different approaches have been proposed tooSome methods are based on the concepts of the superiorityand inferiority of fuzzy numbers [7] the degrees of feasibility[8] the satisfaction degree of the constraints [10] and thestatistical confidence interval [11] Other kinds of methodsare multiobjective optimization method [6] penalty method[12] and semi-infinite programming method [13] Similarother interesting works also can be found in the literature[14ndash16] Mahdavi-Amiri and Nasseri [14] develop a new dualalgorithm for solving the FLP problem directly Ganesan andVeeramani [15] propose a method for solving fuzzy linearprogramming problems without converting them to crisplinear programming problems Maleki et al [16] propose agood method for solving an FLP problem and an auxiliaryproblem is introduced in their model

In recent years several kinds of the fully fuzzy linearprogramming (FFLP) problems in which all the parame-ters and variables are represented by fuzzy numbers haveappeared in the literature [17ndash21] Some authors [17 18] havediscussed FFLP problems with crisp inequality constraints

2 Journal of Applied Mathematics

and obviously different methods for solving them have beenproposed In these methods the fuzzy optimal solutions ofthe FFLP problems are obtained by converting FFLP probleminto crisp linear programming (CLP) problem Other authors[19 20] have discussed FFLP problems with crisp equalityconstraints and different methods for solving them have beenproposed too In their methods the method proposed byLotfi et al [19] can only obtain the approximate solution ofthe FFLP problems but the method proposed by Kumar etal [20] can find the fuzzy optimal solution which satisfies theconstraints exactly Guo and Shang [21] propose the comput-ing model to the positive fully fuzzy linear matrix equationand the fuzzy approximate solution is obtained by usingpseudoinverse However in most of previous literatures allconstraints of FFLP problems have the crisp form

In this paper an FFLP problem with fuzzy equalityconstraints has been considered In order to solve it wefirst transform the FFLP problem into a crisp three-objectiveLP model by considering expected value and uncertainty offuzzy objective and the feasibility degree of fuzzy constrainsThen we solve it using a compromise programming (CP)approach

This paper is organized as follows In Section 2 some basicdefinitions arithmetic operations and comparison opera-tions between two triangular fuzzy numbers are reviewedIn Section 3 the processes for transforming the FFLP prob-lem with fuzzy equality constraints into crisp problem aredescribed In Section 4 a crisp three-objective LP model tofind the fuzzy optimal solution of the FFLP problem is builtand themodel is solved through CP in Section 5 In Section 6a numerical example is given Conclusions are discussed inSection 7

2 Preliminaries

21 Basic Definitions In this paper the triangular fuzzynumbers are considered because this form of fuzzy numbersis very simple and popular Moreover we can express andestimate many other types of fuzzy numbers with triangularfuzzy number [7] The triangular fuzzy numbers are definedas follows

Definition 1 (see [20 22]) A fuzzy number 119860 = (119886

(1) 119886

(2)

119886

(3)) is said to be a triangular fuzzy number if its membership

function is given by

120583119860 (119909) =

119909 minus 119886

(1)

119886

(2)minus 119886

(1) 119886

(1)le 119909 le 119886

(2)

119909 minus 119886

(3)

119886

(2)minus 119886

(3) 119886

(2)le 119909 le 119886

(3)

0 otherwise

(1)

Definition 2 (see [20 22]) A triangular fuzzy number (119886(1)119886

(2) 119886

(3)) is said to be nonnegative fuzzy number if and only

if 119886(1) ge 0

Definition 3 (see [23]) Let 119860 = (119886(1) 119886(2) 119886(3)) and 119861 = (119887(1)119887

(2) 119887

(3)) be two triangular fuzzy numbers and then the

similarity between 119860 and 119861 can be defined as

119878 (

119860

119861) = 1 minus

1

4119906

times (

1003816

1003816

1003816

1003816

1003816

119886

(1)minus 119887

(1)10038161003816

1003816

1003816

1003816

+ 2

1003816

1003816

1003816

1003816

1003816

119886

(2)minus 119887

(2)10038161003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

119886

(3)minus 119887

(3)10038161003816

1003816

1003816

1003816

)

(2)

where 119906 = max(119886(3) 119887(3)) minusmin(119886(1) 119887(1))

Definition 4 (see [20 24]) A ranking function is a functionR 119865(119877) rarr 119877 where 119865(119877) is a set of fuzzy numbers definedon set of real numbers which maps each fuzzy numberinto the real line where a natural order exists Let 119860 =

(119886

(1) 119886

(2) 119886

(3)) be a triangular fuzzy number then

R (119860) =119886

(1)+ 2119886

(2)+ 119886

(3)

4

(3)

22 Arithmetic Operations In the following arithmetic oper-ations between two triangular fuzzy numbers defined onuniversal set of real numbers 119877 are reviewed [20 22]

Let 119860 = (119886

(1) 119886

(2) 119886

(3)) and 119861 = (119887(1) 119887(2) 119887(3)) be two

triangular fuzzy numbers then

(i) 119860 oplus 119861 = (119886(1) 119886(2) 119886(3)) oplus (119887(1) 119887(2) 119887(3)) = (119886(1) + 119887(1)119886

(2)+ 119887

(2) 119886

(3)+ 119887

(3))

(ii) minus119860 = minus(119886(1) 119886(2) 119886(3)) = (minus119886(3) minus119886(2) minus119886(1))

(iii) 119860 ⊝ 119861 = (119886(1) 119886(2) 119886(3)) ⊝ (119887(1) 119887(2) 119887(3)) = (119886(1) minus 119887(3)119886

(2)minus 119887

(2) 119886

(3)minus 119887

(1))

(iv) let 119861 = (119887

(1) 119887

(2) 119887

(3)) be a nonnegative triangular

fuzzy number and then

119860 otimes

119861 =

(119886

(1)119887

(1) 119886

(2)119887

(2) 119886

(3)119887

(3)) 119886

(1)ge 0

(119886

(1)119887

(3) 119886

(2)119887

(2) 119886

(3)119887

(3)) 119886

(1)lt 0 119886

(3)ge 0

(119886

(1)119887

(3) 119886

(2)119887

(2) 119886

(3)119887

(1)) 119886

(3)lt 0

(4)

23 Comparison Operations Let 119860 = (119886

(1) 119886

(2) 119886

(3)) and

119861 = (119887

(1) 119887

(2) 119887

(3)) be two triangular fuzzy numbers and

then greater-than and less-than operations can be defined asfollows [25]

119860 ge

119861 lArrrArr 119886

(1)ge 119887

(1) 119886

(2)ge 119887

(2) 119886

(3)ge 119887

(3)

119860 le

119861 lArrrArr 119886

(1)le 119887

(1) 119886

(2)le 119887

(2) 119886

(3)le 119887

(3)

(5)

Journal of Applied Mathematics 3

3 Presentation of the Problem

The FFLP problem with fuzzy equality constraints ldquocongrdquo iswritten as follows

Max

119885 =

119862 otimes

119883

st

119860 otimes

119883 cong

119887

119883 is a nonnegative fuzzy vector

(P1)

where 119862 = (119888119895)1times119899

119883 = (119909119895)119899times1 119860 = (119886119894119895)119898times119899 119887 = (119887119894)119898times1

and 119888119895 119909119895 119886119894119895 119887119894 isin 119865(119877) The symbols ldquocongrdquo denote the fuzzifiedversions of ldquo=rdquo and can be read as ldquoapproximately equal tordquo

Substituting 119862 = (119888119895)1times119899 119883 = (119909119895)119899times1 119860 = (119886119894119895)119898times119899 119887 =(

119887119894)119898times1 (P1)may be written as follows

Max

119885 =

119899

sum

119895=1

119888119895 otimes 119909119895

st119899

sum

119895=1

119886119894119895 otimes 119909119895 cong

119887119894 119894 = 1 2 119898

119909119895 is a nonnegative fuzzy number 119895 = 1 2 119899

(P2)

Also (P2) can be expressed as follows

Max

119885 =

119899

sum

119895=1

119888119895 otimes 119909119895

st119899

sum

119895=1

119886119894119895 otimes 119909119895 ≲

119887119894 119894 = 1 2 119898

119899

sum

119895=1

119886119894119895 otimes 119909119895 ≳

119887119894 119894 = 1 2 119898

119909119895 is a nonnegative fuzzy number 119895 = 1 2 119899

(P3)

Here the symbols ldquo≲ and ≳rdquo denote the fuzzified versions ofldquo⩽ and ⩾rdquo and can be read as ldquoapproximately lessgreater thanor equal tordquo

As the decision maker (DM) knows that all the param-eters and variables in each constraint of (P3) are fuzzynumbers he may allow some violation of the right handfuzzy number in each constraint This violation can alsobe considered as a fuzzy number Let 119901119894 and 119902119894 119894 =

1 2 119898 be fuzzy numbers determined by the DM givinghis allowed maximum violation in the accomplishment ofthe 119894th constraint and the 119898 + 119894th constraint of (P3)respectively It means that the DM tolerates violations in eachconstraint of (P2) up the value 119887119894 + 119901119894 and down the value

119887119894 minus 119902119894 119894 = 1 2 119898 respectively Based on these ideas

according to the resolutionmethods proposed in [4] (P3)willbecome

Max

119885 =

119899

sum

119895=1

119888119895 otimes 119909119895

st119899

sum

119895=1

119886119894119895 otimes 119909119895 ⃝le

119887119894 +119901119894 119894 = 1 2 119898

119899

sum

119895=1

119886119894119895 otimes 119909119895 ⃝ge

119887119894 minus 119902119894 119894 = 1 2 119898

119909119895 is a nonnegative fuzzy number 119895 = 1 2 119899

(P4)

where the symbols ldquo ⃝le and ⃝gerdquo are relations between fuzzynumbers which preserve the ranking when fuzzy numbersare multiplied by positive scalars and they can be anyone theDM chooses Different kind of relation ⃝le and ⃝ge will lead todifferent models of CLP problems In this paper we assumethat the relation ⃝le and ⃝ge will be determined by using thecomparison operations defined in Section 23

Without any loss of generality we assume that 119888119895 and 119886119894119895are nonnegative Let 119885 = (119885(1) 119885(2) 119885(3)) 119888119895 = (119888

(1)

119895 119888

(2)

119895 119888

(3)

119895)

119909119895 = (119909(1)

119895 119909

(2)

119895 119909

(3)

119895) 119886119894119895 = (119886

(1)

119894119895 119886

(2)

119894119895 119886

(3)

119894119895) 119887119894 = (119887

(1)

119894 119887

(2)

119894 119887

(3)

119894)

119901119894 = (119901

(1)

119894 119901

(2)

119894 119901

(3)

119894) 119902119894 = (119902

(1)

119894 119902

(2)

119894 119902

(3)

119894) and then using the

arithmetic operations between two triangular fuzzy numbers(P4)may be written as

Max (119885

(1) 119885

(2) 119885

(3))

=

119899

sum

119895=1

(119888

(1)

119895 119909(1)

119895 119888(2)

119895 119909(2)

119895 119888(3)

119895 119909(3)

119895 )

st119899

sum

119895=1

(119886

(1)

119894119895 119909(1)

119895 119886(2)

119894119895 119909(2)

119895 119886(3)

119894119895 119909(3)

119895 )

⃝le (119887

(1)

119894 + 119901(1)

119894 119887(2)

119894 + 119901(2)

119894 119887(3)

119894 + 119901(3)

119894 )

119894 = 1 2 119898

119899

sum

119895=1

(119886

(1)

119894119895 119909(1)

119895 119886(2)

119894119895 119909(2)

119895 119886(3)

119894119895 119909(3)

119895 )

⃝ge (119887

(1)

119894 minus 119902(3)

119894 119887(2)

119894 minus 119902(2)

119894 119887(3)

119894 minus 119902(1)

119894 )

119894 = 1 2 119898

(119909

(1)

119895 119909(2)

119895 119909(3)

119895 ) is a nonnegative triangular

fuzzy number 119895 = 1 2 119899

(P5)

Using the comparison operations between two triangularfuzzy numbers defined in Section 23 to deal with the

4 Journal of Applied Mathematics

inequality relation on the constraints (P5) is converted intothe following problem

Max (119885

(1) 119885

(2) 119885

(3)) =

119899

sum

119895=1

(119888

(1)

119895 119909(1)

119895 119888(2)

119895 119909(2)

119895 119888(3)

119895 119909(3)

119895 )

st119899

sum

119895=1

119886

(ℎ)

119894119895 119909(ℎ)

119895 le 119887(ℎ)

119894 + 119901(ℎ)

119894 ℎ = 1 2 3 119894 = 1 2 119898

119899

sum

119895=1

119886

(ℎ)

119894119895 119909(ℎ)

119895 ge119887(ℎ)

119894 minus119902(4minusℎ)

119894 ℎ = 1 2 3 119894 = 1 2 119898

119909

(1)

119895 ge 0 119909

(2)

119895 minus 119909(1)

119895 ge 0 119909

(3)

119895 minus 119909(2)

119895 ge 0

119895 = 1 2 119899

(P6)

Generally the DM usually knows little about the prob-lem moreover constraints of (P3) have different toleratedviolations for different DM so it is difficult for the DM todetermine reasonable values of 119901(ℎ)

119894and 119902(ℎ)

119894 ℎ = 1 2 3

119894 = 1 2 119898 In this paper the concept of similarity betweentwo triangular fuzzy numbers is introduced to solve thisproblem The key of this method is that the DM determinesan allowed similarity level instead of allowed maximumtolerated violations in each constraint of (P3) by using thefollowing inequalities

119878 (

119887119894

119887119894 +119901119894) ge 119904 119894 = 1 2 119898 (6)

119878 (

119887119894

119887119894 minus 119902119894) ge 119904 119894 = 1 2 119898 (7)

119901

(1)

119894 ge 0 119901

(2)

119894 minus 119901(1)

119894 ge 0 119901

(3)

119894 minus 119901(2)

119894 ge 0

119894 = 1 2 119898

(8)

119902

(1)

119894 ge 0 119902

(2)

119894 minus 119902(1)

119894 ge 0 119902

(3)

119894 minus 119902(2)

119894 ge 0

119894 = 1 2 119898

(9)

Here 119878(119887119894 119887119894 + 119901119894) is the similarity between two triangularfuzzy numbers 119887119894 and 119887119894 + 119901119894 119894 = 1 2 119898 119878(119887119894 119887119894 minus119902119894) is thesimilarity between two triangular fuzzy numbers119887119894 and119887119894minus119902119894119894 = 1 2 119898 and 119904 is the allowed similarity level given bythe DM

According to the definition of the similarity between twotriangular fuzzy numbers we have

119878 (

119887119894

119887119894 +119901119894) = 1 minus

119901

(1)

119894+ 2119901

(2)

119894+ 119901

(3)

119894

4119906119894

119894 = 1 2 119898

(10)

119878 (

119887119894

119887119894 minus 119902119894) = 1 minus119902

(1)

119894+ 2119902

(2)

119894+ 119902

(3)

119894

4119906

1015840119894

119894 = 1 2 119898

(11)

where 119906119894 = 119887(3)

119894+ 119901

(3)

119894minus 119887

(1)

119894 1199061015840119894 = 119887

(3)

119894+ 119902

(3)

119894minus 119887

(1)

119894 In order to

decrease the influence of variables 119901(3)119894

and 119902(3)119894

on 119878(119887119894 119887119894+ 119901119894)

and 119878(119887119894 119887119894 minus 119902119894) 119894 = 1 2 119898 respectively we assume that119906119894 = 119906

1015840119894 = 119887(3)

119894minus 119887

(1)

119894 Hence from (6) and (10) as well as (7)

and (11) we obtain the following inequalities respectively

119901

(1)

119894 + 2119901(2)

119894 + 119901(3)

119894 le 4 (1 minus 119904) (119887(3)

119894 minus 119887(1)

119894 )

119894 = 1 2 119898

119902

(1)

119894 + 2119902(2)

119894 + 119902(3)

119894 le 4 (1 minus 119904) (119887(3)

119894 minus 119887(1)

119894 )

119894 = 1 2 119898

(12)

Therefore giving the DM allowed similarity level 119904 119901119894and 119902119894 can be determined by (8) (9) and (12) Adding thoseinequalities into constrains of (P6) we obtain the followingproblem

Max (119885

(1) 119885

(2) 119885

(3)) =

119899

sum

119895=1

(119888

(1)

119895 119909(1)

119895 119888(2)

119895 119909(2)

119895 119888(3)

119895 119909(3)

119895 )

st119899

sum

119895=1

119886

(ℎ)

119894119895 119909(ℎ)

119895

le 119887

(ℎ)

119894 + 119901(ℎ)

119894 ℎ = 1 2 3 119894 = 1 2 119898

119899

sum

119895=1

119886

(ℎ)

119894119895 119909(ℎ)

119895

ge 119887

(ℎ)

119894 minus 119902(4minusℎ)

119894 ℎ = 1 2 3 119894 = 1 2 119898

119901

(1)

119894 + 2119901(2)

119894 + 119901(3)

119894

le 4 (1 minus 119904) (119887

(3)

119894 minus 119887(1)

119894 ) 119894 = 1 2 119898

119902

(1)

119894 + 2119902(2)

119894 + 119902(3)

119894

le 4 (1 minus 119904) (119887

(3)

119894 minus 119887(1)

119894 ) 119894 = 1 2 119898

119909

(1)

119895 ge 0 119909

(2)

119895 minus 119909(1)

119895 ge 0 119909

(3)

119895 minus 119909(2)

119895 ge 0

119895 = 1 2 119899

119901

(1)

119894 ge 0 119901

(2)

119894 minus 119901(1)

119894 ge 0 119901

(3)

119894 minus 119901(2)

119894 ge 0

119894 = 1 2 119898

119902

(1)

119894 ge 0 119902

(2)

119894 minus 119902(1)

119894 ge 0 119902

(3)

119894 minus 119902(2)

119894 ge 0

119894 = 1 2 119898

(P7)

Now we have transformed the FFLP problem (P1) into(P7) using similarity measures The main feature of (P7) isthat the constraints are crisp linear by introducing allowedsimilarity level 119904 The definition of 119904-feasible solution about(P7) is given as follows

Definition 5 Given an allowed similarity level 119904 for a decisionmatrix 1198830(119904) = (1199090119895

(ℎ)(119904))119899times3 we will say that it is 119904-feasible

solution of (P7) if1198830(119904) satisfies constraints of (P7)

Journal of Applied Mathematics 5

According to transforming processes of (P1) to (P7) wemay set the following proposition

Proposition 6 A decision matrix 1198830(119904) is 119904-fuzzy-feasiblesolution of (P1) if and only if1198830(119904) is 119904-feasible solution of (P7)

In the following the set of the 119904-feasible solution of (P7)willbe denoted by alefsym(119904) and it is evident that

1199041 lt 1199042 997904rArr alefsym(1199041) sup alefsym (1199042) (13)

Then (P7) can be rewritten as

Max (119885

(1) 119885

(2) 119885

(3)) =

119899

sum

119895=1

(119888

(1)

119895 119909(1)

119895 119888(2)

119895 119909(2)

119895 119888(3)

119895 119909(3)

119895 )

st (x(1)j x(2)j x(3)j ) isin alefsym (s) j = 1 2 n

(P7-1)

In order to transform the fuzzy objective into crisp onewe should consider expected value and uncertainty of fuzzyobjective We use a ranking function to define the expectedvalue of fuzzy objective Many ranking functions can befound in the literatures and we choose the same rankingfunction which is defined in Definition 4 used by Kumaret al [20] The uncertainty of fuzzy objective is measuredusing the difference between upper bound and lower boundof fuzzy objective value Therefore (P7)may be transformedinto the following crisp problem

Max R (119885) =1

4

119899

sum

119895=1

(119888

(1)

119895 119909(1)

119895 + 2119888(2)

119895 119909(2)

119895 + 119888(3)

119895 119909(3)

119895 )

Min Δ (

119885) =

119899

sum

119895=1

119888

(3)

119895 119909(3)

119895 minus

119899

sum

119895=1

119888

(1)

119895 119909(1)

119895

st (119909

(1)

119895 119909(2)

119895 119909(3)

119895 ) isin alefsym (119904) 119895 = 1 2 119899

(P8)

(P8) is an 119904-parametric crisp biobjective LPmodel Givingthe value of 119904 and we can solve the 119904-efficient solution whichis defined as follows

Definition 7 Giving the value of 119904 119883lowast(119904) isin alefsym(119904) is said to be119904-efficient solution to the problem (P8) if there does not existanother1198830(119904) isin alefsym(119904) such that

R (119885 (1198830(119904))) ge R (119885 (119883

lowast(119904)))

Δ (

119885 (119883

0(119904))) le Δ (

119885 (119883

lowast(119904)))

(14)

where at least one of these inequalities is strictFrom Definition 7 we have the following proposition

Proposition 8 All 119904-efficient solutions 119883lowast(119904) to the problem(P8) are 119904-fuzzy-optimal solutions to the problem (P1) andreciprocally

From Proposition 8 we can obtain the 119904-fuzzy-optimalsolutions to the problem (P1) by solving (P8)

4 The Auxiliary Three-Objective LP Model

From (12) a bigger value of 119904 implies that the DM allowsa smaller violation of the right hand fuzzy number in eachconstraint So 119904 can be interpreted as the feasibility degreeof constrains the bigger the value of 119904 is the higher thefeasibility degree of constrainswill beHowever from (13) and(P7) the bigger the value of 119904 is the worst the objective valuewill beTherefore we want to find a balance solution betweentwo goals to improve the objectives function values and toincrease the feasibility degree of constrains According to theprevious analysis (P8) can be transformed into the followingauxiliary three-objective LP model

Max R (119885) =1

4

119899

sum

119895=1

(119888

(1)

119895 119909(1)

119895 + 2119888(2)

119895 119909(2)

119895 + 119888(3)

119895 119909(3)

119895 )

Min Δ (

119885) =

119899

sum

119895=1

119888

(3)

119895 119909(3)

119895 minus

119899

sum

119895=1

119888

(1)

119895 119909(1)

119895

Max 119904

st 119904119898 le 119904 le 1

(119909

(1)

119895 119909(2)

119895 119909(3)

119895 ) isin alefsym (119904) 119895 = 1 2 119899

(P9)

where 119904119898 is the allowed minimum similarity level and it isspecified by the DM according to his interests The threeobjectives of (P9) represent the DMrsquos preference for thealternative with the higher expected value less uncertaintyof objective and the higher feasibility degree of constrainsrespectively

In the following the set of the feasible solutions of (P9)will be denoted by alefsym

Definition 9 (119883lowast 119904lowast) isin alefsym is said to be an efficient solution tothe problem (P9) if there does not exist another (1198830 1199040) isin alefsymsuch that

R (119885 (1198830)) ge R (119885 (119883

lowast))

Δ (

119885 (119883

0)) le Δ (

119885 (119883

lowast))

119904

0ge 119904

lowast

(15)

where at least one of these inequalities is strictFrom Definitions 7 and 9 and Proposition 8 we have the

following proposition

Proposition 10 All efficient solutions (119883lowast 119904lowast) to the problem(P9) are 119904lowast-fuzzy-optimal solutions to the problem (P1) andreciprocally

Proof Let (119883lowast 119904lowast) be an 119904lowast-fuzzy-optimal solutions to theproblem (P1) but not an efficient solution to the problem(P9) and then there exists another (1198830 1199040) isin alefsym such thatR(119885(1198830)) ge R(119885(119883lowast)) Δ(119885(1198830)) le Δ(119885(119883lowast)) 1199040 ge 119904lowastwhere at least one of these inequalities is strict It impliesfrom Definition 7 that (119883lowast 119904lowast) is not an 119904lowast-efficient solutionto the problem (P8) and then according to Proposition 8

6 Journal of Applied Mathematics

(119883

lowast 119904

lowast) is not an 119904lowast-fuzzy-optimal solutions to the problem

(P1)Reciprocally let (119883lowast 119904lowast) be an efficient solution to the

problem (P9) but not an 119904lowast-fuzzy-optimal solution to theproblem (P1) Then from Proposition 8 (119883lowast 119904lowast) is not 119904lowast-efficient solution to the problem (P8) So from Definition 7there exists another (1198830 119904lowast) isin alefsym(119904lowast) such that R(119885(1198830)) geR(119885(119883lowast)) Δ(119885(1198830)) le Δ(119885(119883lowast)) 119904lowast = 119904lowast where at leastone of these inequalities is strict Therefore (119883lowast 119904lowast) is not anefficient solution to the problem (P9)

Now we have transformed the FFLP problem (P1) into(P9) which is a crisp three-objective LP problem Accordingto Proposition 10 we can obtain the 119904lowast-fuzzy-optimal solu-tion of initial problem (P1) by solving (P9) In the followingsection we will solve (P9) through CP approach

5 Compromise Solutions

CP is a Multiple Criteria Decision Making approach whichranks alternatives according to their closeness to the idealpointThe best alternative is the onewhose point is at the leastdistance from an ideal point in the set of efficient solutions[26]

In order to apply the CP approach to solve the problem(P9) we need to obtain the pay-off matrix For this we opti-mize each objective separately calculating the values reachedby the objectives on the optimal solution respectively Let119883

lowast119879 119879 = R Δ 119904 be the optimal solutions for each objective

and R119879 Δ119879 119904119879 the values reached by three objectives onthe optimal solution 119883lowast119879 119879 = R Δ 119904 respectively Then theobtained pay-off matrix can be expressed as in Table 1

From Table 1 we know that the elements of principaldiagonal (RR ΔΔ 119904119904) form the ideal point The anti-idealpoint is (minRΔR119904maxΔR Δ 119904min119904R 119904Δ)

The distance between each objective value and the corre-sponding ideal point is

1198631 = RR minusR (

119885) 1198632 = Δ (

119885) minus ΔΔ

1198633 = 119904119904 minus 119904

(16)

As the objectives are measured with different units it isnecessary to homogenize the distances as

1198891 =

RR minusR (

119885)

RR minusmin RΔR119904 1198892 =

Δ (

119885) minus ΔΔ

max ΔR Δ 119904 minus ΔΔ

1198893 =119904119904 minus 119904

119904119904 minusmin 119904R 119904Δ

(17)

The distance measure used in CP is the family of 119871119901-metrics given as

119871119901 = [

3

sum

119896=1

(119908119896119889119896)119901]

1119901

(18)

where 119908119896 is the weight or relative importance attached to the119896th objective and 119901 is the topological metric that is a realnumber belonging to the closed interval [1infin]

Table 1 The pay-off matrix

119883

lowastR 119883

lowastΔ 119883

lowast119904

R(119885) RR RΔ R119904

Δ(

119885) ΔR ΔΔ Δ 119904

119904 119904R 119904Δ 119904119904

A compromise solution is the one which minimizes 119871119901Therefore we have

Min 119871119901 = [

3

sum

119896=1

(119908119896119889119896)119901]

1119901

st ((119909

(ℎ)

119895 )119899times3 119904) isin alefsym

(P10)

Obviously the solution of (P10) depends on the chosenmetric The most commonly obtained compromise solutionsare for metrics 119901 = 1 and 119901 = infin because for other metricsthe nonlinear programming algorithms are needed [27] Alsoin the biobjective case they are the bounds of the wholecompromise set [28 29]

For 119901 = 1 the compromise solution closest to the idealsolution can be obtained by solving the following LP problem

Min 1198711 =

3

sum

119896=1

(119908119896119889119896)

st ((119909

(ℎ)

119895 )119899times3 119904) isin alefsym

(P10-1)

For 119901 = infin themaximum divergence between individualdiscrepancies is minimized Consequently the compromisesolution is obtained by solving the following problem

Min 119871infin = max119896=123

119908119896119889119896

st ((119909

(ℎ)

119895 )119899times3 119904) isin alefsym

(P10-2)

The previous problem is a min-max problem Let 119889119898 =max119896=123119908119896119889119896 and then it is reformulated as

Min 119871infin = 119889119898

st 119908119896119889119896 le 119889119898 119896 = 1 2 3

((119909

(ℎ)

119895 )119899times3 119904) isin alefsym

(P10-3)

For 119901 = 1 Yu showed that the solution of (P10-1) isalways Pareto efficient [30] For 119901 = infin if (P10-3) exists asa unique optimal solution then it is an efficient solution tothe problem (P9) [31] If the uniqueness is not satisfied thenthe efficiency is not guaranteed for all solutions [31 32] Inorder to obtain an efficient solution several approaches havebeen proposed in the literature [31ndash35]

Journal of Applied Mathematics 7

A composite form of CP for 119901 = 1 and 119901 = infin can beobtained byminimizing a linear combination between1198711 and119871infin that is

Min 119871119888 = (1 minus 120582) 119889119898 + 120582

3

sum

119896=1

(119908119896119889119896)

st 119908119896119889119896 le 119889119898 119896 = 1 2 3

((119909

(ℎ)

119895 )119899times3 119904) isin alefsym

(P10-4)

where 120582 isin [0 1] 120582 can be interpreted as a trade-off ormarginal rate of substitution between 1198711 and 119871infin When 120582 =1 problem (P10-4) gives the compromise solution for 119901 = 1and for 120582 = 0 (P10-4) gives the compromise solution for119901 = infin For any set of positive weights and any 120582 gt 0 thesolutions of problem (P10-4) are efficient [36]

6 Numerical Examples

In this section two numerical examples are given to illustratethe proposed model and method

Example 1 Let us solve the following FFLP problem withfuzzy equality constraints which is like Example 61 given in[20] with the main difference that here constraints are fuzzyequality not crisp equality Consider

Max

119885 = (1 6 9) otimes 1199091 oplus (2 3 8) otimes 1199092

st (2 3 4) otimes 1199091 oplus (1 2 3) otimes 1199092 cong (6 16 30)

(minus1 1 2) otimes 1199091 oplus (1 3 4) otimes 1199092 cong (1 17 30)

1199091 1199092 are nonnegative triangular fuzzy numbers(EP 1)

Solution Let 1199091 = (119909(1)1 119909(2)1 119909(3)1 ) and 1199092 = (119909

(1)2 119909(2)2 119909(3)2 )

then the given FFLP problem (EP 1)may be written as

Max

119885 = (119909

(1)

1 + 2119909(1)

2 6119909(2)

1 + 3119909(2)

2 9119909(3)

1 + 8119909(3)

2 )

st (2119909

(1)

1 + 119909(1)

2 3119909(2)

1 + 2119909(2)

2 4119909(3)

1 + 3119909(3)

2 )cong(6 16 30)

(minus119909

(3)

1 + 119909(1)

2 119909(2)

1 + 3119909(2)

2 2119909(3)

1 + 4119909(3)

2 ) cong (1 17 30)

(119909

(1)

1 119909(2)

1 119909(3)

1 ) (119909(1)

2 119909(2)

2 119909(3)

2 ) are nonnegative

triangular fuzzy numbers(EP 2)

Suppose that the allowed minimum similarity level spec-ified by the DM is 119904119898 = 09 According to (P9) we will solve

the following three-objective LP model

Max R (119885)

=

1

4

(119909

(1)

1 + 2119909(1)

2 + 12119909(2)

1 + 6119909(2)

2 + 9119909(3)

1 + 8119909(3)

2 )

Min Δ (

119885) = 9119909

(3)

1 + 8119909(3)

2 minus 119909(1)

1 minus 2119909(1)

2

Max 119904

st 09 le 119904 le 1

(119909

(1)

119895 119909(2)

119895 119909(3)

119895 ) isin alefsym (119904) 119895 = 1 2

(EP 3)

where alefsym(119904) is the set of the 119904-feasible solution of (P7) whichis determined using the following constrain inequalities

2119909

(1)

1 + 119909(1)

2 le 6 + 119901(1)

1 3119909

(2)

1 + 2119909(2)

2 le 16 + 119901(2)

1

4119909

(3)

1 + 3119909(3)

2 le 30 + 119901(3)

1 minus119909

(3)

1 + 119909(1)

2 le 1 + 119901(1)

2

119909

(2)

1 + 3119909(2)

2 le 17 + 119901(2)

2 2119909

(3)

1 + 4119909(3)

2 le 30 + 119901(3)

2

2119909

(1)

1 + 119909(1)

2 ge 6 minus 119902(1)

1 3119909

(2)

1 + 2119909(2)

2 ge 16 minus 119902(2)

1

4119909

(3)

1 + 3119909(3)

2 ge 30 minus 119902(3)

1 minus119909

(3)

1 + 119909(1)

2 ge 1 minus 119902(1)

2

119909

(2)

1 + 3119909(2)

2 ge 17 minus 119902(2)

2 2119909

(3)

1 + 4119909(3)

2 ge 30 minus 119902(3)

2

119901

(1)

1 + 2119901(2)

1 + 119901(3)

1 le 96 (1 minus 119904)

119901

(1)

2 + 2119901(2)

2 + 119901(3)

2 le 116 (1 minus 119904)

119902

(1)

1 + 2119902(2)

1 + 119902(3)

1 le 96 (1 minus 119904)

119902

(1)

2 + 2119902(2)

2 + 119902(3)

2 le 116 (1 minus 119904)

119909

(1)

119895 ge 0 119909

(2)

119895 minus 119909(1)

119895 ge 0 119909

(3)

119895 minus 119909(2)

119895 ge 0 119895 = 1 2

119901

(1)

119894 ge 0 119901

(2)

119894 minus 119901(1)

119894 ge 0 119901

(3)

119894 minus 119901(2)

119894 ge 0 119894 = 1 2

119902

(1)

119894 ge 0 119902

(2)

119894 minus 119902(1)

119894 ge 0 119902

(3)

119894 minus 119902(2)

119894 ge 0 119894 = 1 2

(19)

In order to obtain the pay-off matrix we optimize eachobjective separately calculating the values reached by theobjectives on the optimal solution respectivelyThe obtainedpay-off matrix is shown in Table 2

From Table 2 we know that the elements of principaldiagonal (4134 5636 10) form the ideal pointThe anti-idealpoint is (3342 9120 09)

8 Journal of Applied Mathematics

Table 2 The pay-off matrix of (EP 2)

119883

lowastR 119883

lowastΔ 119883

lowast119904

R(119885) 4134 3342 3450Δ(

119885) 9120 5636 6600119904 09 09 10

From (16) and (17) the homogenized distance betweeneach objective value and the corresponding ideal point is

1198891=

4134 minus(14) (119909

(1)1 +2119909

(1)2 +12119909

(2)1 +6119909

(2)2 +9119909

(3)1 + 8119909

(3)2 )

4134 minus 3342

1198892 =9119909

(3)1 + 8119909

(3)2 minus 119909

(1)1 minus 2119909

(1)2 minus 5636

9120 minus 5636

1198893 =10 minus 119904

10 minus 09

(20)

The compromise solution can be obtained by solving(P10) Let 1199081 = 035 1199082 = 035 1199083 = 030 and then wehave the following

(i) For 119901 = 1 the compromise solution closest to theideal solution can be obtained by solving the followingLP problem

Min 1198711 = 0351198891 + 0351198892 + 0301198893

st ((119909

(ℎ)

119895 )2times3 119904) isin alefsym

(EP 4)

The optimal solution of (EP 4) is 119909(1)1 = 063 119909(2)1 = 233

119909

(3)1 = 273 119909(1)2 = 475 119909(2)2 = 475 119909(3)2 = 573 119904lowast = 0985

According to Proposition 10 119904lowast-fuzzy-optimal solutions tothe problem (EP 1) are 119909lowast1 (0985) = (063 233 332)119909

lowast2 (0985) = (475 475 573)Put 119909lowast1 (0985) 119909

lowast2 (0985) in objectives of (EP 2) and

(EP 3) and then we have 119885lowast = (1012 2820 7573)R(119885lowast) =3556 Δ(119885lowast) = 6561

(ii) For 119901 = infin the compromise solution is obtained bysolving the following LP problem

Min 119871infin = 119889119898

st 0351198891 le 119889119898

0351198892 le 119889119898

0301198893 le 119889119898

((119909

(ℎ)

119895 )2times3 119904) isin alefsym

(EP 5)

The optimal solution of (EP 5) is 119909(1)1 = 087 119909(2)1 = 266

119909

(3)1 = 300 119909(1)2 = 425 119909(2)2 = 425 119909(3)2 = 700 119904lowast = 0959

According to Proposition 10 119904lowast-fuzzy-optimal solutions tothe problem (EP 1) are 119909lowast1 (0959) = (087 266 300)119909

lowast2 (0959) = (425 425 700)

Put 119909lowast1 (0959) 119909lowast2 (0959) in objectives of (EP 2) and

(EP 3) and then we have 119885lowast = (938 2870 8293)R(119885lowast) =3743 Δ(119885lowast) = 7355

(iii) For a composite form of CP for 119901 = 1 and 119901 = infinthe compromise solution is obtained by solving thefollowing LP problem

Min 119871119888 = (1 minus 120582) 119889119898 + 120582 (0351198891 + 0351198892 + 0301198893)

st 0351198891 le 119889119898

0351198892 le 119889119898

0301198893 le 119889119898

((119909

(ℎ)

119895 )2times3 119904) isin alefsym

(EP 6)

Assuming that 120582 = 05 then optimal solution of (EP 6)is 119909(1)1 = 073 119909(2)1 = 276 119909(3)1 = 276 119909(1)2 = 454 119909(2)2 =

454 119909(3)2 = 677 119904lowast = 0957 According to Proposition 10119904

lowast-fuzzy-optimal solutions to the problem (EP 1) are119909

lowast1 (0957) = (073 276 276) 119909

lowast2 (0957) = (454 454 677)

Put 119909lowast1 (0957) 119909lowast2 (0957) in objectives of (EP 2) and

(EP 3) and then we have 119885lowast = (981 3019 7904)R(119885lowast) =3731 Δ(119885lowast) = 6923

Given different values of 120582 we can also obtain the 119904lowast-fuzzy-optimal solutions to the problem (EP 1) see Table 3Comparing these solutions with the ideal point DM maychoose an acceptable optimal solution

Example 2 Let us solve the following FFLP problem withcrisp equality constraints [20]

Max

119885 = (1 6 9) otimes 1199091 oplus (2 3 8) otimes 1199092

st (2 3 4) otimes 1199091 oplus (1 2 3) otimes 1199092 = (6 16 30)

(minus1 1 2) otimes 1199091 oplus (1 3 4) otimes 1199092 = (1 17 30)

1199091 1199092 are nonnegative triangular fuzzy numbers(21)

Solution Let 1199091 = (119909(1)1 119909(2)1 119909(3)1 ) and 1199092 = (119909

(1)2 119909(2)2 119909(3)2 )

and then the previous FFLP problem may be written as

Max

119885 = (119909

(1)

1 + 2119909(1)

2 6119909(2)

1 + 3119909(2)

2 9119909(3)

1 + 8119909(3)

2 )

st (2119909

(1)

1 + 119909(1)

2 3119909(2)

1 + 2119909(2)

2 4119909(3)

1 + 3119909(3)

2 )=(6 16 30)

(minus119909

(3)

1 + 119909(1)

2 119909(2)

1 + 3119909(2)

2 2119909(3)

1 + 4119909(3)

2 ) = (1 17 30)

(119909

(1)

1 119909(2)

1 119909(3)

1 ) (119909(1)

2 119909(2)

2 119909(3)

2 ) are nonnegative

triangular fuzzy numbers(22)

As the constraints are crisp equality 119904 = 1 Accordingto (P9) we can solve the following two-objective LP model

Journal of Applied Mathematics 9

Table 3 The 119904lowast-fuzzy-optimal solutions with different value of 120582

120582 119904

lowast119909

lowast1 (119904lowast) 119909

lowast2 (119904lowast)

119885

lowast R(119885lowast) Δ(

119885

lowast)

0 0959 (087 266 300) (425 425 700) (938 2870 8293) 3743 735501 0959 (090 282 282) (420 420 700) (930 2949 8136) 3741 720702 0957 (073 276 276) (454 454 677) (981 3019 7904) 3731 692303 0957 (073 276 276) (454 454 677) (981 3019 7904) 3731 692304 0957 (073 276 276) (454 454 677) (981 3019 7904) 3731 692305 0957 (073 276 276) (454 454 677) (981 3019 7904) 3731 692306 0957 (073 276 276) (454 454 677) (981 3019 7904) 3731 692307 0985 (063 233 332) (475 475 573) (1012 2820 7573) 3556 656108 0985 (063 233 332) (475 475 573) (1012 2820 7573) 3556 656109 0985 (063 233 332) (475 475 573) (1012 2820 7573) 3556 65611 0985 (063 233 332) (475 475 573) (1012 2820 7573) 3556 6561

to obtain the fuzzy optimal solution of the previous FFLPproblem

Max R (119885)

=

1

4

(119909

(1)

1 + 2119909(1)

2 + 12119909(2)

1 + 6119909(2)

2 + 9119909(3)

1 + 8119909(3)

2 )

Min Δ (

119885) = 9119909

(3)

1 + 8119909(3)

2 minus 119909(1)

1 minus 2119909(1)

2

st 2119909

(1)

1 + 119909(1)

2 = 6 3119909

(2)

1 + 2119909(2)

2 = 16

4119909

(3)

1 + 3119909(3)

2 = 30 minus119909

(3)

1 + 119909(1)

2 = 1

119909

(2)

1 + 3119909(2)

2 = 17 2119909

(3)

1 + 4119909(3)

2 = 30

119909

(1)

119895 ge 0 119909

(2)

119895 minus 119909(1)

119895 ge 0 119909

(3)

119895 minus 119909(2)

119895 ge 0

119895 = 1 2

(23)

In order to obtain the pay-off matrix we optimize eachobjective separately As the optimal solution is same for twoobjective we have obtained the optimal solution of previoustwo-objective LP model 119909(1)1 = 1 119909

(2)1 = 2 119909

(3)1 = 3 119909

(1)2 = 4

119909

(2)2 = 5 119909(3)2 = 6 R(119885lowast) = 345 Δ(119885lowast) = 66 Therefore

the fuzzy optimal solution of the given FFLP problem is 1199091 =(1 2 3) 1199092 = (4 5 6) 119885

lowast= (9 27 75)

From the analysis and solution processes of the numericalexamples we have summarized the following advantages

(i) The auxiliary model (P9) to solve the FFLP problem(P1) is linear so it is very ease to solve

(ii) The FFLP problem with crisp equality constraints in[20] is a special case of the FFLP problems with fuzzyequality constraints (119904 = 1)

(iii) In order to solve the FFLP problems with fuzzyequality constraints the DM only determines oneparameter instead of 6119898 parameters Moreover theDM can get different fuzzy optimal solution givingdifferent similarity level

The main disadvantage of proposed method is that thenumber of constraints is increased when the FFLP problemswith fuzzy equality constraints are converted into crisp one

7 Conclusions

In this paper an FFLP problem with fuzzy equality con-straints has been presented Also an approach has beengiven to solve it We first transform the FFLP problem withfuzzy equality constraints into the crisp three-objective LPmodel by considering expected value anduncertainty of fuzzyobjective and the feasibility degree of fuzzy constrains Thenwe solve it using a CP approach To illustrate the proposedmethod two numerical examples are solved

For different comparison relations of fuzzy number anddifferent measure methods of similarity an FFLP problemcan be converted into different auxiliary problemsThe studyof those problems will be the aim of a forthcoming paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (no 71202140) and the Human-ities and Social Sciences Project of Ministry of Educationunder Grants 10YJC630009 and 12YJCZH065

References

[1] Y Tan and Y Long ldquoOption-game approach to analyze technol-ogy innovation investment under fuzzy environmentrdquo Journalof Applied Mathematics vol 2012 Article ID 830850 9 pages2012

[2] L Zhang X Xu and L Tao ldquoSome similarity measures for tri-angular fuzzy number and their applications inmultiple criteriagroup decision-makingrdquo Journal of Applied Mathematics vol2013 Article ID 538261 7 pages 2013

[3] H Tsai and T Chen ldquoA fuzzy nonlinear programming approachfor optimizing the performance of a four-objective fluctuationsmoothing rule in awafer fabrication factoryrdquo Journal of AppliedMathematics vol 2013 Article ID 720607 15 pages 2013

[4] M Delgado J L Verdegay andM A Vila ldquoA general model forfuzzy linear programmingrdquo Fuzzy Sets and Systems vol 29 no1 pp 21ndash29 1989

10 Journal of Applied Mathematics

[5] H Rommelfanger ldquoFuzzy linear programming and applica-tionsrdquo European Journal of Operational Research vol 92 no 3pp 512ndash527 1996

[6] G Zhang Y-H Wu M Remias and J Lu ldquoFormulationof fuzzy linear programming problems as four-objective con-strained optimization problemsrdquo Applied Mathematics andComputation vol 139 no 2-3 pp 383ndash399 2003

[7] N van Hop ldquoSolving fuzzy (stochastic) linear programmingproblems using superiority and inferiority measuresrdquo Informa-tion Sciences vol 177 no 9 pp 1977ndash1991 2007

[8] M Jimenez M Arenas A Bilbao andM V Rodrıguez ldquoLinearprogramming with fuzzy parameters an interactive methodresolutionrdquo European Journal of Operational Research vol 177no 3 pp 1599ndash1609 2007

[9] H-C Wu ldquoOptimality conditions for linear programmingproblems with fuzzy coefficientsrdquo Computers amp Mathematicswith Applications vol 55 no 12 pp 2807ndash2822 2008

[10] X Liu ldquoMeasuring the satisfaction of constraints in fuzzy linearprogrammingrdquo Fuzzy Sets and Systems vol 122 no 2 pp 263ndash275 2001

[11] J Chiang ldquoFuzzy linear programming based on statisticalconfidence interval and interval-valued fuzzy setrdquo EuropeanJournal of Operational Research vol 129 no 1 pp 65ndash86 2001

[12] K D Jamison and W A Lodwick ldquoFuzzy linear programmingusing a penalty methodrdquo Fuzzy Sets and Systems vol 119 no 1pp 97ndash110 2001

[13] T Leon and E Vercher ldquoSolving a class of fuzzy linear programsby using semi-infinite programming techniquesrdquo Fuzzy Setsand Systems vol 146 no 2 pp 235ndash252 2004

[14] N Mahdavi-Amiri and S H Nasseri ldquoDuality results and adual simplex method for linear programming problems withtrapezoidal fuzzy variablesrdquo Fuzzy Sets and Systems vol 158 no17 pp 1961ndash1978 2007

[15] K Ganesan and P Veeramani ldquoFuzzy linear programs withtrapezoidal fuzzy numbersrdquo Annals of Operations Research vol143 no 1 pp 305ndash315 2006

[16] H RMalekiM Tata andMMashinchi ldquoLinear programmingwith fuzzy variablesrdquo Fuzzy Sets and Systems vol 109 no 1 pp21ndash33 2000

[17] S M Hashemi M Modarres E Nasrabadi and M MNasrabadi ldquoFully fuzzified linear programming solution anddualityrdquo Journal of Intelligent and Fuzzy Systems vol 17 no 3pp 253ndash261 2006

[18] T Allahviranloo F H Lotfi M K Kiasary N A Kiani and LAlizadeh ldquoSolving fully fuzzy linear programming problem bythe ranking functionrdquoAppliedMathematical Sciences vol 2 no1ndash4 pp 19ndash32 2008

[19] F H Lotfi T Allahviranloo M A Jondabeh and L AlizadehldquoSolving a full fuzzy linear programming using lexicographymethod and fuzzy approximate solutionrdquoAppliedMathematicalModelling vol 33 no 7 pp 3151ndash3156 2009

[20] A Kumar J Kaur and P Singh ldquoA newmethod for solving fullyfuzzy linear programming problemsrdquo Applied MathematicalModelling vol 35 no 2 pp 817ndash823 2011

[21] X Guo and D Shang ldquoFuzzy approximate solution of positivefully fuzzy linear matrix equationsrdquo Journal of Applied Mathe-matics vol 2013 Article ID 178209 7 pages 2013

[22] A Kaufmann and M M Gupta Introduction to Fuzzy Arith-metic Theory and Applications Van Nostrand Reinhold NewYork NY USA 1985

[23] H-S Lee ldquoOptimal consensus of fuzzy opinions under groupdecisionmaking environmentrdquo Fuzzy Sets and Systems vol 132no 3 pp 303ndash315 2002

[24] T-S Liou and M J J Wang ldquoRanking fuzzy numbers withintegral valuerdquo Fuzzy Sets and Systems vol 50 no 3 pp 247ndash255 1992

[25] S-J Chen and C-L Hwang Fuzzy Multiple Attribute DecisionMaking Methods and Applications vol 375 of Lecture Notesin Economics and Mathematical Systems Springer Berlin Ger-many 1992

[26] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[27] M Arenas Parra A Bilbao Terol B Perez Gladish and M VRodrıguez Urıa ldquoSolving a multiobjective possibilistic problemthrough compromise programmingrdquo European Journal of Oper-ational Research vol 164 no 3 pp 748ndash759 2005

[28] M Freimer and P L Yu ldquoSome new results on compromisesolutions for group decision problemsrdquo Management Sciencevol 22 no 6 pp 688ndash693 1976

[29] F Blasco E Cuchillo-Ibanez M A Moron and C RomeroldquoOn the monotonicity of the compromise set in multicriteriaproblemsrdquo Journal of OptimizationTheory andApplications vol102 no 1 pp 69ndash82 1999

[30] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

[31] M Jimenez and A Bilbao ldquoPareto-optimal solutions in fuzzymulti-objective linear programmingrdquo Fuzzy Sets and Systemsvol 160 no 18 pp 2714ndash2721 2009

[32] B Werners ldquoInteractive multiple objective programming sub-ject to flexible constraintsrdquo European Journal of OperationalResearch vol 31 no 3 pp 342ndash349 1987

[33] E S Lee and R-J Li ldquoFuzzy multiple objective programmingand compromise programming with Pareto optimumrdquo FuzzySets and Systems vol 53 no 3 pp 275ndash288 1993

[34] C Hu Y Shen and S Li ldquoAn interactive satisficing methodbased on alternative tolerance for fuzzy multiple objectiveoptimizationrdquo Applied Mathematical Modelling vol 33 no 4pp 1886ndash1893 2009

[35] M A Yaghoobi and M Tamiz ldquoA method for solving fuzzygoal programming problems based on MINMAX approachrdquoEuropean Journal of Operational Research vol 177 no 3 pp1580ndash1590 2007

[36] F J Andre and C Romero ldquoComputing compromise solutionson the connections between compromise programming andcomposite programmingrdquo Applied Mathematics and Computa-tion vol 195 no 1 pp 1ndash10 2008