Research ArticleA MOLP Method for Solving Fully Fuzzy LinearProgramming with LR Fuzzy Parameters

Xiao-Peng Yang12 Xue-Gang Zhou13 Bing-Yuan Cao1 and S H Nasseri4

1 School of Mathematics and Information Science Key Laboratory of Mathematics and Interdisciplinary Sciences of GuangdongHigher Education Institutes Guangzhou University Guangzhou 510006 China

2Department of Mathematics and Statistics Hanshan Normal University Chaozhou 521041 China3Department of Applied Mathematics Guangdong University of Finance Guangzhou 510521 China4Department of Mathematics Mazandaran University Babolsar 47416-95447 Iran

Correspondence should be addressed to Bing-Yuan Cao caobingy163com

Received 22 March 2014 Accepted 15 September 2014 Published 29 September 2014

Academic Editor Yang Xu

Copyright copy 2014 Xiao-Peng Yang et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Kaur and Kumar 2013 use Meharrsquos method to solve a kind of fully fuzzy linear programming (FFLP) problems with LR fuzzyparameters In this paper a new kind of FFLP problems is introduced with a solutionmethod proposedThe FFLP is converted intoamultiobjective linear programming (MOLP) according to the order relation for comparing the LR flat fuzzy numbers Besides theclassical fuzzy programming method is modified and then used to solve the MOLP problem Based on the compromised optimalsolution to the MOLP problem the compromised optimal solution to the FFLP problem is obtained At last a numerical exampleis given to illustrate the feasibility of the proposed method

1 Introduction

The research on fuzzy linear programming (FLP) has risenhighly since Bellman and Zadeh [1] proposed the conceptof decision making in fuzzy environment The FLP problemis said to be a fully fuzzy linear programming (FFLP)problem if all the parameters and variables are consideredas fuzzy numbers In recent years some researchers suchas Lofti and Kumar were interested in the FFLP problemsand some solution methods have been obtained to thefully fuzzy systems [2ndash4] and the FFLP problems [5ndash13]FFLP problems can be divided in two categories (1) FFLPproblemswith inequality constraints (2) FFLP problemswithequality constraints If the FFLP problems are classified bythe types of the fuzzy numbers they will include the nextthree classes (1) FFLP problems with all the parameters andvariables represented by triangular fuzzy numbers (2) FFLPproblems with all the parameters and variables representedby trapezoidal fuzzy numbers (3) FFLP problems with all

the parameters and variables expressed by 119871119877 fuzzy numbers(or 119871119877 flat fuzzy numbers)

Fuzzy programmingmethod is a classicalmethod to solvemultiobjective linear programming (MOLP) [14 15] In thispaper the fuzzy programming method is modified and thenused to obtain a compromised optimal solution of theMOLPThe modified fuzzy programming method is shown in Steps4ndash10 of the proposed method in Section 3

Dehghan et al [2ndash4] employed several methods to findsolutions of the fully fuzzy linear systems Hosseinzadeh Lotfiet al [6] used the lexicography method to obtain the fuzzyapproximate solutions of the FFLP problems Allahviranlooet al [7] and Kumar et al [5 8] solved the FFLP problem byuse of a ranking function

Fan et al [12] adopted the 120572-cut level to deal with ageneralized fuzzy linear programming (GFLP) probelm Thefeasibility of fuzzy solutions to theGFLPwas investigated anda stepwise interactive algorithm based on the idea of designof experiment was advanced to solve the GFLP problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 782376 10 pageshttpdxdoiorg1011552014782376

2 Mathematical Problems in Engineering

Kaur and Kumar [9] introduced Meharrsquos method to theFFLP problems with 119871119877 fuzzy parameters They consider thefollowing model

Maximize (orMinimize)119899

sum

119895=1

((119901119895 119902119895 1205721015840

119895 1205731015840

119895)119871119877

⊙ (119909119895 119910119895 12057210158401015840

119895 12057310158401015840

119895)119871119877

)

subject to119899

sum

119895=1

((119886119894119895 119887119894119895 120572119894119895 120573119894119895)119871119877

⊙(119909119895 119910119895 12057210158401015840

119895 12057310158401015840

119895)119871119877

) ⪯ asymp ⪰ (119887119894 119892119894 120574119894 120575119894)119871119877

119894 = 1 2 119898

(1)

where the parameters and variables are 119871119877 flat fuzzy numbersand the order relation for comparing the numbers is definedas follows

(i) ⪯ V if and only ifR() le R(V)

(ii) ⪰ V if and only ifR() ge R(V)

(iii) asymp V if and only ifR() = R(V)

Here and V are two arbitrary 119871119877 flat fuzzy numbersIn our study we consider a new kind of FFLP problems

with 119871119877 flat fuzzy parameters as follows

min (or max)

119911 (119909) = 1198881otimes 1199091oplus 1198882otimes 1199092oplus sdot sdot sdot oplus 119888

119899otimes 119909119899

subject to

1198861198941otimes 1199091oplus 1198861198942otimes 1199092oplus sdot sdot sdot oplus 119886

119894119899otimes 119909119899le 119894

119894 = 1 2 119898

119909119895ge 0 119895 = 1 2 119899

(2)

where the parameters and variables are 119871119877 flat fuzzy numbersand the order relation shown in Definition 4 is different fromthe one above

In this paper we modify the classical fuzzy programmingmethod The FFLP is changed into a MOLP problem solvedby the modified fuzzy programming method We get thecompromised optimal solution to theMOLP and generate thecorresponding compromised optimal solution to the FFLP

The rest of the paper is organized as follows In Section 2the basic definitions and the FFLP model are introduced InSection 3 we propose a MOLP method to solve the FFLPproblems Some results are discussed from the solutionsobtained by the proposed method In Section 4 a numericalexample is given to illustrate the feasibility of the proposedmethod In Section 5 we show some short concludingremarks

2 Preliminaries

21 Basic Notations

Definition 1 (119871119877 fuzzy number see [2]) A fuzzy number issaid to be an 119871119877 fuzzy number if

(119909) =

119871(119898 minus 119909

120572) 119909 le 119898 120572 gt 0

119877(119909 minus 119898

120573) 119909 ge 119898 120573 gt 0

(3)

where119898 is the mean value of and 120572 and 120573 are left and rightspreads respectively and function 119871(sdot) means the left shapefunction satisfying

(1) 119871(119909) = 119871(minus119909)

(2) 119871(0) = 1 and 119871(1) = 0

(3) 119871(119909) is nonincreasing on [0infin)

Naturally a right shape function 119877(sdot) is similarly definedas 119871(sdot)

Definition 2 (119871119877 flat fuzzy number see [9 16]) A fuzzynumber denoted as (119898 119899 120572 120573)

119871119877 is said to be an 119871119877

flat fuzzy number if its membership function (119909) is givenby

(119909) =

119871(119898 minus 119909

120572) 119909 le 119898 120572 gt 0

119877(119909 minus 119899

120573) 119909 ge 119899 120573 gt 0

1 119898 le 119909 le 119899

(4)

Definition 3 (see [5 9]) An 119871119877 flat fuzzy number =

(119898 119899 120572 120573)119871119877

is said to be nonnegative 119871119877 flat fuzzy numberif 119898 minus 120572 ge 0 and is said to be nonpositive 119871119877 flat number if119899 + 120573 le 0

We define = (119898 119899 0 0)119871119877

as an 119871119877 fuzzy number withmembership function

(119909) = 1 119898 le 119909 le 119899

0 otherwise(5)

and denote (0 0 0 0)119871119877

as 0

Mathematical Problems in Engineering 3

22 Arithmetic Operations Let = (1198981 1198991 1205721 1205731)119871119877

and V =(1198982 1198992 1205722 1205732)119871119877

be two 119871119877 flat fuzzy numbers 119896 isin 119877 Thenthe arithmetic operations are given as follows [9 16]

oplus V = (1198981+ 1198982 1198991+ 1198992 1205721+ 1205722 1205731+ 1205732)119871119877

⊖ V = (1198981minus 1198992 1198991minus 1198982 1205721+ 1205732 1205722+ 1205731)119871119877

119896 = (1198961198981 1198961198991 1198961205721 1198961205731)119871119877 119896 ge 0

(1198961198991 1198961198981 minus1198961205731 minus1198961205721)119877119871 119896 lt 0

otimes V =

(11989811198982 11989911198992 11989811205722+ 12057211198982 11989911205732+ 12057311198992)119871119877

ge 0 V ge 0(11989811198992 11989911198982 12057211198992minus 11989811205732 12057311198982minus 11989911205722)119871119877

le 0 V ge 0(11989911198982 11989811198992 11989911205722minus 12057311198982 11989811205732minus 12057211198992)119871119877

ge 0 V le 0(11989911198992 11989811198982 minus11989911205732minus 12057311198992 minus11989811205722minus 12057211198982)119871119877

le 0 V le 0(6)

It is easy to verify that the operator oplus satisfies associativelaw Hence the formula sum119899

119895=1119895= 1oplus 2oplus sdot sdot sdot oplus

119899is

reasonable where 1 2

119899are 119871119877 flat fuzzy numbers

23 Order Relation for Comparing the LR Flat Fuzzy NumbersFor comparing the 119871119877 flat fuzzy numbers we introduce theorder relation as follows

Definition 4 Let = (1198981 1198991 1205721 1205731)119871119877

and V = (1198982 1198992 1205722

1205732)119871119877

be any 119871119877 flat fuzzy numbers Then

(i) = V if and only if 1198981= 1198982 1198991= 1198992 1205721= 1205722

1205731= 1205732

(ii) le V if and only if 1198981le 1198982 1198991le 1198992 1198981minus 1205721le

1198982minus 1205722 1198991+ 1205731le 1198992+ 1205732

(iii) ge V if and only if 1198981ge 1198982 1198991ge 1198992 1198981minus 1205721ge

1198982minus 1205722 1198991+ 1205731ge 1198992+ 1205732

Based on the definition of order le we may obtain that(i) is nonnegative if and only if ge 0 (ii) is nonpositiveif and only if le 0

The following propositions are given to show the proper-ties of the order relation defined above

Proposition 5 Let V 119908 be four arbitrary 119871119877 flat fuzzynumbers and 119896 an arbitrary real number Then

(1) le V 119908 le 997904rArr oplus 119908 le V oplus

(2) le V 997904rArr

119896 le 119896V 119896 ge 0

119896 ge 119896V 119896 le 0

(7)

Proof Suppose = (1198981 1198991 1205721 1205731)119871119877 V = (119898

2 1198992 1205722 1205732)119871119877

119908 = (1198983 1198993 1205723 1205733)119871119877 and = (119898

4 1198994 1205724 1205734)119871119877

(1) It is obvious that oplus119908 = (1198981+1198983 1198991+1198993 1205721+1205723 1205731+

1205733)119871119877 V oplus = (119898

2+ 1198984 1198992+ 1198994 1205722+ 1205724 1205732+ 1205734)119871119877 Since

le V 119908 le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198983le 1198984 119899

3le 1198994 119898

3minus 1205723le 1198984minus 1205724

1198993+ 1205733le 1198994+ 1205734

(8)

So

1198981+ 1198983le 1198982+ 1198984

1198991+ 1198993le 1198992+ 1198994

(1198981+ 1198983) minus (120572

1+ 1205723) le (119898

2+ 1198984) minus (120572

2+ 1205724)

(1198991+ 1198993) + (120573

1+ 1205733) le (119899

2+ 1198994) minus (120573

2+ 1205734)

(9)

This indicates that oplus 119908 le V oplus (2) It is clear that 119896 = (119896119898

1 1198961198991 1198961205721 1198961205731)119871119877 119896V =

(1198961198982 1198961198992 1198961205722 1198961205732)119871119877 From le V we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

(10)

Therefore

1198961198981le 1198961198982 119896119899

1le 1198961198992 119896119898

1minus 1198961205721le 1198961198982minus 1198961205722

1198961198991+ 1198961205731le 1198961198992+ 1198961205732

(11)

for 119896 ge 0 and

1198961198981ge 1198961198982 119896119899

1ge 1198961198992 119896119898

1minus 1198961205721ge 1198961198982minus 1198961205722

1198961198991+ 1198961205731ge 1198961198992+ 1198961205732

(12)

for 119896 le 0 This indicates that

119896 le 119896V 119896 ge 0

119896 ge 119896V 119896 le 0

(13)

Proposition 6 Let V 119908 be three arbitrary 119871119877 flat fuzzynumbers Then

(1) le

(2) le V V le rArr = V

(3) le V V le 119908 rArr le 119908

Proof Suppose = (1198981 1198991 1205721 1205731)119871119877 V = (119898

2 1198992 1205722 1205732)119871119877

and 119908 = (1198983 1198993 1205723 1205733)119871119877

(1) Obviously = hence we have le

4 Mathematical Problems in Engineering

(2) Since le V V le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198981 119899

2le 1198991 119898

2minus 1205722le 1198981minus 1205721

1198992+ 1205732le 1198991+ 1205731

(14)

This means

1198981= 1198982 119899

1= 1198992 119898

1minus 1205721= 1198982minus 1205722

1198991+ 1205731= 1198992+ 1205732

(15)

That is

1198981= 1198982 119899

1= 1198992 120572

1= 1205722 120573

1= 1205732 (16)

Therefore we have = V(3) From le V V le 119908 we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198983 119899

2le 1198993 119898

2minus 1205722le 1198983minus 1205723

1198992+ 1205732le 1198993+ 1205733

(17)

This indicates

1198981le 1198983 119899

1le 1198993 119898

1minus 1205721le 1198983minus 1205723

1198991+ 1205731le 1198993+ 1205733

(18)

Therefore we have le 119908

From Proposition 6 we know that the order relation le isa partial order on the set of all 119871119877 fuzzy numbers

24 Fully Fuzzy Linear Programming with LR Fuzzy Parame-ters In this paper we will consider the following model thatis

min 119911 (119909) = 119888 otimes 119909

st 119860 otimes 119909 le

119909 ge 0

(19)

or

min 119911 (119909) = 1198881otimes 1199091oplus 1198882otimes 1199092oplus sdot sdot sdot oplus 119888

119899otimes 119909119899

st 1198861198941otimes 1199091oplus 1198861198942otimes 1199092oplus sdot sdot sdot oplus 119886

119894119899otimes 119909119899le 119894

119894 = 1 2 119898

119909119895ge 0 119895 = 1 2 119899

(20)

where 119888 = [119888119895]1times119899

= [119894]119898times1

119860 = [119886119894119895]119898times119899

and 119909 = [119909119895]119899times1

represent 119871119877 fuzzy matrices and vectors and 119888119895 119894 119886119894119895 and 119909

119895

are 119871119877 flat fuzzy numbers The order relations for comparing

the 119871119877 flat fuzzy numbers both in the objective function andthe constraint inequalities are as shown in Definition 4

3 Proposed Method

Steps of the proposed method are given to solve problem(20) as follows This method is applicable to minimizationof FFLP problems and the solution method of maximizationproblems is similar to that of minimization ones

Step 1 If all the parameters 119888119895 119894 119886119894119895 119909119895are represented by

119871119877 flat fuzzy numbers (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877 (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

(1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877 and (119909

1198951 1199091198952 120572119909119895 120573119909119895)119871119877 then the FFLP

(20) can be written as

min 119911 (119909) =

119899

sum

119895=1

((1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

st119899

sum

119895=1

((1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(21)

Step 2 Calculate (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877 respectively

and suppose that (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

=

(1199091015840

1198951 1199091015840

1198952 1205721015840

119909119895 1205731015840

119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895

120573119909119895)119871119877

= (119901119894119895 119902119894119895 120574119894119895 120575119894119895)119871119877 then the FFLP problem obtained

in Step 1 can be written as

min 119911 (119909) = (

119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

1205721015840

119909119895

119899

sum

119895=1

1205731015840

119909119895)

119871119877

st (

119899

sum

119895=1

119901119894119895

119899

sum

119895=1

119902119894119895

119899

sum

119895=1

120574119894119895

119899

sum

119895=1

120575119894119895)

119871119877

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(22)

Mathematical Problems in Engineering 5

Step 3 According to the order relation defined above theproblem obtained in Step 2 is equivalent to

min119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

(1199091015840

1198951minus 1205721015840

119909119895)

119899

sum

119895=1

(1199091015840

1198952+ 1205731015840

119909119895)

st119899

sum

119895=1

119901119894119895le 1198871198941 119894 = 1 2 119898

119899

sum

119895=1

119902119894119895le 1198871198942 119894 = 1 2 119898

119899

sum

119895=1

(119901119894119895minus 120574119894119895) le (119887

1198941minus 120572119887119894) 119894 = 1 2 119898

119899

sum

119895=1

(119902119894119895+ 120575119894119895) le (119887

1198942+ 120573119887119894) 119894 = 1 2 119898

1199091198951le 1199091198952 120572

119909119895ge 0 120573

119909119895ge 0

1199091198951minus 120572119909119895ge 0 119895 = 1 2 119899

(23)

We denote 119883 = (11990911 11990912 1205721199091 1205731199091 11990921 11990922 1205721199092 1205731199092

1199091198991 1199091198992 120572119909119899 120573119909119899)119879 1199111(119883) = sum

119899

119895=11199091015840

1198951 1199112(119883) = sum

119899

119895=11199091015840

1198952

1199113(119883) = sum

119899

119895=1(1199091015840

1198951minus 1205721015840

119909119895) 1199114(119883) = sum

119899

119895=1(1199091015840

1198952+ 1205731015840

119909119895) and

119863 = 119883 | 119883 satisfies the constraints of programming (23)Programming (23) may be written as the programming (24)below for short as follows

min 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(24)

Obviously programming (24) is a crisp multiobjectivelinear programming problem In fact we have 119911(119909) =

(1199111(119883) 119911

2(119883) 119911

1(119883) minus 119911

3(119883) 119911

4(119883) minus 119911

2(119883))

Step 4 Solve the subproblems

min 119911119905(119883)

st 119883 isin 119863

(25)

where 119905 = 1 2 3 4 We find optimal solutions 1198831 1198832 1198833

and 1198834 respectively And the corresponding optimal values

will be 119911min1

= 1199111(1198831) 119911min2

= 1199112(1198832) 119911min3

= 1199113(1198833) and

119911min4

= 1199114(1198834)

Step 5 Let 119911max119905

= max119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834) 119905 =

1 2 3 4 and the membership function of 119911119905(119883) is given by

120583119911119905(119911119905(119883)) =

1 119911119905(119883) lt 119911

min119905

119911max119905

minus 119911119905(119883)

119911max119905

minus 119911min119905

119911min119905

le 119911119905(119883) le 119911

max119905

0 119911119905(119883) gt 119911

max119905

(26)

where 119905 = 1 2 3 4

Step 6 Let 1198680= 1 2 3 4 the MOLP problem obtained in

Step 3 can be equivalently written as

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

0

119883 isin 119863

(27)

Suppose1198831 is one of the optimal solutions (if there exits onlyone optimal solution 1198831 is the unique one) and 1205821lowast is theoptimal objective value (in fact the optimal solution shouldbe written as (1198831 1205821lowast) Since 120582 is an auxiliary variable wedenote (1198831 1205821lowast) as 1198831 for simplicity) Then 120583

1199111199041(1199111199041(1198831

)) =

1205821lowast for at least one 119904

1in 1198680 (1199041is an arbitrary element in the

set 119869 = 119895 | 120583119911119895(119911119895(1198831

)) = 1205821lowast

)

Step 7 Let 1198681= 1198680minus 1199041 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

1

1205831199111199041(1199111199041(119883)) = 120582

1lowast

119883 isin 119863

(28)

If 1198832 is one of the optimal solutions and 1205822lowast is the optimalobjective value then 120583

1199111199042(1199111199042(1198832

)) = 1205822lowast for at least one 119904

2in

1198681

Step 8 Let 1198682= 1198680minus 1199041 1199042 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

2

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

119883 isin 119863

(29)

6 Mathematical Problems in Engineering

Suppose 1198833 is one of the optimal solutions and 1205823lowast is the

optimal objective value Then 1205831199111199043(1199111199043(1198833

)) = 1205823lowast for at least

one 1199043in 1198682

Step 9 Let 1198683= 1198680minus 1199041 1199042 1199043 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

3

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

1205831199111199043(1199111199043(119883)) = 120582

3lowast

119883 isin 119863

(30)

Suppose 1198834 is one of the optimal solutions and 1205824lowast is the

optimal objective value Then 1205831199111199044(1199111199044(1198834

)) = 1205824lowast with 119904

4in

1198683

Step 10 Take119883lowast = 1198834 as the compromised optimal solutionto programming (23) and generate the compromised optimalsolution 119909lowast to programming (21) by119883lowast Assuming

119883lowast

= (119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091 11990921 119909lowast

22 120572lowast

1199092

120573lowast

1199092 119909

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119879

(31)

we may obtain

119909lowast

= (119909lowast

1 119909lowast

2 119909

lowast

119899)119879

= ((119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091)119871119877

(119909lowast

21 119909lowast

22 120572lowast

1199092 120573lowast

1199092)119871119877

(119909lowast

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119871119877

)119879

(32)

and the corresponding objective value 119911lowast = 119911(119909lowast)

Remark 7 (1199041 1199042 1199043 1199044 = 119868 = 1 2 3 4) Some properties

of the solutions obtained in Steps 6ndash10 are shown in thefollowing proposition

Proposition 8 Suppose 120583119911119904119895(119911119904119895(119883lowast

)) 119883119895 120582119895lowast (119895 = 1 2 3 4)and 119883lowast are the notations described in Steps 1ndash10 then

(1) 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205831199111199043(1199111199043(119883lowast

)) = 1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(2) 1205821lowast

le 1205822lowast

le 1205823lowast

le 1205824lowast

(33)

Proof (1) From the results of Steps 6ndash9 it is obviously clearthat

1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(34)

and 1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast with 119883

lowast

= 1198834 Since 119883lowast = 119883

4

is an optimal solution to programming (30) we know that119883lowast satisfies the constraints of programming (30) and so

1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast 1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast and 120583

1199111199043(1199111199043(119883lowast

)) =

1205823lowast(2) In fact (1198831 1205821lowast) is an optimal solution to program-

ming (27) therefore it is a feasible solution We have

120583119911119905(119911119905(1198831

)) ge 1205821lowast

119905 isin 1198681sube 1198680

1198831

isin 119863

(35)

and it is obvious that 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast from the result of

Step 6 Hence (1198831 1205821lowast) is a feasible solution to programming(28) The objective value of (1198831 1205821lowast) is 1205821lowast and the optimalobjective value of programming (28) is 1205822lowast so we get 1205821lowast le1205822lowast It is similar to prove 1205822lowast le 1205823lowast and 1205823lowast le 1205824lowast

4 Numerical Example

In this section we present a numerical example to illustratethe feasibility of the solution method proposed in Section 3

We aim to find the compromised optimal solution andcorresponding objective value of the following fully fuzzylinear programming problem

max 119911 (119909) = 119911 (1199091 1199092)

= (6 7 1 2)119871119877otimes 1199091oplus (7 9 1 2)

119871119877otimes 1199092

st (9 10 2 1)119871119877otimes 1199091oplus (1 1 1 1)

119871119877otimes 1199092

le (50 55 4 3)119871119877

(2 3 1 1)119871119877otimes 1199091oplus (4 5 1 2)

119871119877otimes 1199092

le (66 70 3 5)119871119877

1199091ge 0 119909

2ge 0

(36)

where 1199091= (11990911 11990912 1205721 1205731)119871119877

and 1199092= (11990921 11990922 1205722 1205732)119871119877

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Kaur and Kumar [9] introduced Meharrsquos method to theFFLP problems with 119871119877 fuzzy parameters They consider thefollowing model

Maximize (orMinimize)119899

sum

119895=1

((119901119895 119902119895 1205721015840

119895 1205731015840

119895)119871119877

⊙ (119909119895 119910119895 12057210158401015840

119895 12057310158401015840

119895)119871119877

)

subject to119899

sum

119895=1

((119886119894119895 119887119894119895 120572119894119895 120573119894119895)119871119877

⊙(119909119895 119910119895 12057210158401015840

119895 12057310158401015840

119895)119871119877

) ⪯ asymp ⪰ (119887119894 119892119894 120574119894 120575119894)119871119877

119894 = 1 2 119898

(1)

where the parameters and variables are 119871119877 flat fuzzy numbersand the order relation for comparing the numbers is definedas follows

(i) ⪯ V if and only ifR() le R(V)

(ii) ⪰ V if and only ifR() ge R(V)

(iii) asymp V if and only ifR() = R(V)

Here and V are two arbitrary 119871119877 flat fuzzy numbersIn our study we consider a new kind of FFLP problems

with 119871119877 flat fuzzy parameters as follows

min (or max)

119911 (119909) = 1198881otimes 1199091oplus 1198882otimes 1199092oplus sdot sdot sdot oplus 119888

119899otimes 119909119899

subject to

1198861198941otimes 1199091oplus 1198861198942otimes 1199092oplus sdot sdot sdot oplus 119886

119894119899otimes 119909119899le 119894

119894 = 1 2 119898

119909119895ge 0 119895 = 1 2 119899

(2)

where the parameters and variables are 119871119877 flat fuzzy numbersand the order relation shown in Definition 4 is different fromthe one above

In this paper we modify the classical fuzzy programmingmethod The FFLP is changed into a MOLP problem solvedby the modified fuzzy programming method We get thecompromised optimal solution to theMOLP and generate thecorresponding compromised optimal solution to the FFLP

The rest of the paper is organized as follows In Section 2the basic definitions and the FFLP model are introduced InSection 3 we propose a MOLP method to solve the FFLPproblems Some results are discussed from the solutionsobtained by the proposed method In Section 4 a numericalexample is given to illustrate the feasibility of the proposedmethod In Section 5 we show some short concludingremarks

2 Preliminaries

21 Basic Notations

Definition 1 (119871119877 fuzzy number see [2]) A fuzzy number issaid to be an 119871119877 fuzzy number if

(119909) =

119871(119898 minus 119909

120572) 119909 le 119898 120572 gt 0

119877(119909 minus 119898

120573) 119909 ge 119898 120573 gt 0

(3)

where119898 is the mean value of and 120572 and 120573 are left and rightspreads respectively and function 119871(sdot) means the left shapefunction satisfying

(1) 119871(119909) = 119871(minus119909)

(2) 119871(0) = 1 and 119871(1) = 0

(3) 119871(119909) is nonincreasing on [0infin)

Naturally a right shape function 119877(sdot) is similarly definedas 119871(sdot)

Definition 2 (119871119877 flat fuzzy number see [9 16]) A fuzzynumber denoted as (119898 119899 120572 120573)

119871119877 is said to be an 119871119877

flat fuzzy number if its membership function (119909) is givenby

(119909) =

119871(119898 minus 119909

120572) 119909 le 119898 120572 gt 0

119877(119909 minus 119899

120573) 119909 ge 119899 120573 gt 0

1 119898 le 119909 le 119899

(4)

Definition 3 (see [5 9]) An 119871119877 flat fuzzy number =

(119898 119899 120572 120573)119871119877

is said to be nonnegative 119871119877 flat fuzzy numberif 119898 minus 120572 ge 0 and is said to be nonpositive 119871119877 flat number if119899 + 120573 le 0

We define = (119898 119899 0 0)119871119877

as an 119871119877 fuzzy number withmembership function

(119909) = 1 119898 le 119909 le 119899

0 otherwise(5)

and denote (0 0 0 0)119871119877

as 0

Mathematical Problems in Engineering 3

22 Arithmetic Operations Let = (1198981 1198991 1205721 1205731)119871119877

and V =(1198982 1198992 1205722 1205732)119871119877

be two 119871119877 flat fuzzy numbers 119896 isin 119877 Thenthe arithmetic operations are given as follows [9 16]

oplus V = (1198981+ 1198982 1198991+ 1198992 1205721+ 1205722 1205731+ 1205732)119871119877

⊖ V = (1198981minus 1198992 1198991minus 1198982 1205721+ 1205732 1205722+ 1205731)119871119877

119896 = (1198961198981 1198961198991 1198961205721 1198961205731)119871119877 119896 ge 0

(1198961198991 1198961198981 minus1198961205731 minus1198961205721)119877119871 119896 lt 0

otimes V =

(11989811198982 11989911198992 11989811205722+ 12057211198982 11989911205732+ 12057311198992)119871119877

ge 0 V ge 0(11989811198992 11989911198982 12057211198992minus 11989811205732 12057311198982minus 11989911205722)119871119877

le 0 V ge 0(11989911198982 11989811198992 11989911205722minus 12057311198982 11989811205732minus 12057211198992)119871119877

ge 0 V le 0(11989911198992 11989811198982 minus11989911205732minus 12057311198992 minus11989811205722minus 12057211198982)119871119877

le 0 V le 0(6)

It is easy to verify that the operator oplus satisfies associativelaw Hence the formula sum119899

119895=1119895= 1oplus 2oplus sdot sdot sdot oplus

119899is

reasonable where 1 2

119899are 119871119877 flat fuzzy numbers

23 Order Relation for Comparing the LR Flat Fuzzy NumbersFor comparing the 119871119877 flat fuzzy numbers we introduce theorder relation as follows

Definition 4 Let = (1198981 1198991 1205721 1205731)119871119877

and V = (1198982 1198992 1205722

1205732)119871119877

be any 119871119877 flat fuzzy numbers Then

(i) = V if and only if 1198981= 1198982 1198991= 1198992 1205721= 1205722

1205731= 1205732

(ii) le V if and only if 1198981le 1198982 1198991le 1198992 1198981minus 1205721le

1198982minus 1205722 1198991+ 1205731le 1198992+ 1205732

(iii) ge V if and only if 1198981ge 1198982 1198991ge 1198992 1198981minus 1205721ge

1198982minus 1205722 1198991+ 1205731ge 1198992+ 1205732

Based on the definition of order le we may obtain that(i) is nonnegative if and only if ge 0 (ii) is nonpositiveif and only if le 0

The following propositions are given to show the proper-ties of the order relation defined above

Proposition 5 Let V 119908 be four arbitrary 119871119877 flat fuzzynumbers and 119896 an arbitrary real number Then

(1) le V 119908 le 997904rArr oplus 119908 le V oplus

(2) le V 997904rArr

119896 le 119896V 119896 ge 0

119896 ge 119896V 119896 le 0

(7)

Proof Suppose = (1198981 1198991 1205721 1205731)119871119877 V = (119898

2 1198992 1205722 1205732)119871119877

119908 = (1198983 1198993 1205723 1205733)119871119877 and = (119898

4 1198994 1205724 1205734)119871119877

(1) It is obvious that oplus119908 = (1198981+1198983 1198991+1198993 1205721+1205723 1205731+

1205733)119871119877 V oplus = (119898

2+ 1198984 1198992+ 1198994 1205722+ 1205724 1205732+ 1205734)119871119877 Since

le V 119908 le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198983le 1198984 119899

3le 1198994 119898

3minus 1205723le 1198984minus 1205724

1198993+ 1205733le 1198994+ 1205734

(8)

So

1198981+ 1198983le 1198982+ 1198984

1198991+ 1198993le 1198992+ 1198994

(1198981+ 1198983) minus (120572

1+ 1205723) le (119898

2+ 1198984) minus (120572

2+ 1205724)

(1198991+ 1198993) + (120573

1+ 1205733) le (119899

2+ 1198994) minus (120573

2+ 1205734)

(9)

This indicates that oplus 119908 le V oplus (2) It is clear that 119896 = (119896119898

1 1198961198991 1198961205721 1198961205731)119871119877 119896V =

(1198961198982 1198961198992 1198961205722 1198961205732)119871119877 From le V we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

(10)

Therefore

1198961198981le 1198961198982 119896119899

1le 1198961198992 119896119898

1minus 1198961205721le 1198961198982minus 1198961205722

1198961198991+ 1198961205731le 1198961198992+ 1198961205732

(11)

for 119896 ge 0 and

1198961198981ge 1198961198982 119896119899

1ge 1198961198992 119896119898

1minus 1198961205721ge 1198961198982minus 1198961205722

1198961198991+ 1198961205731ge 1198961198992+ 1198961205732

(12)

for 119896 le 0 This indicates that

119896 le 119896V 119896 ge 0

119896 ge 119896V 119896 le 0

(13)

Proposition 6 Let V 119908 be three arbitrary 119871119877 flat fuzzynumbers Then

(1) le

(2) le V V le rArr = V

(3) le V V le 119908 rArr le 119908

Proof Suppose = (1198981 1198991 1205721 1205731)119871119877 V = (119898

2 1198992 1205722 1205732)119871119877

and 119908 = (1198983 1198993 1205723 1205733)119871119877

(1) Obviously = hence we have le

4 Mathematical Problems in Engineering

(2) Since le V V le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198981 119899

2le 1198991 119898

2minus 1205722le 1198981minus 1205721

1198992+ 1205732le 1198991+ 1205731

(14)

This means

1198981= 1198982 119899

1= 1198992 119898

1minus 1205721= 1198982minus 1205722

1198991+ 1205731= 1198992+ 1205732

(15)

That is

1198981= 1198982 119899

1= 1198992 120572

1= 1205722 120573

1= 1205732 (16)

Therefore we have = V(3) From le V V le 119908 we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198983 119899

2le 1198993 119898

2minus 1205722le 1198983minus 1205723

1198992+ 1205732le 1198993+ 1205733

(17)

This indicates

1198981le 1198983 119899

1le 1198993 119898

1minus 1205721le 1198983minus 1205723

1198991+ 1205731le 1198993+ 1205733

(18)

Therefore we have le 119908

From Proposition 6 we know that the order relation le isa partial order on the set of all 119871119877 fuzzy numbers

24 Fully Fuzzy Linear Programming with LR Fuzzy Parame-ters In this paper we will consider the following model thatis

min 119911 (119909) = 119888 otimes 119909

st 119860 otimes 119909 le

119909 ge 0

(19)

or

min 119911 (119909) = 1198881otimes 1199091oplus 1198882otimes 1199092oplus sdot sdot sdot oplus 119888

119899otimes 119909119899

st 1198861198941otimes 1199091oplus 1198861198942otimes 1199092oplus sdot sdot sdot oplus 119886

119894119899otimes 119909119899le 119894

119894 = 1 2 119898

119909119895ge 0 119895 = 1 2 119899

(20)

where 119888 = [119888119895]1times119899

= [119894]119898times1

119860 = [119886119894119895]119898times119899

and 119909 = [119909119895]119899times1

represent 119871119877 fuzzy matrices and vectors and 119888119895 119894 119886119894119895 and 119909

119895

are 119871119877 flat fuzzy numbers The order relations for comparing

the 119871119877 flat fuzzy numbers both in the objective function andthe constraint inequalities are as shown in Definition 4

3 Proposed Method

Steps of the proposed method are given to solve problem(20) as follows This method is applicable to minimizationof FFLP problems and the solution method of maximizationproblems is similar to that of minimization ones

Step 1 If all the parameters 119888119895 119894 119886119894119895 119909119895are represented by

119871119877 flat fuzzy numbers (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877 (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

(1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877 and (119909

1198951 1199091198952 120572119909119895 120573119909119895)119871119877 then the FFLP

(20) can be written as

min 119911 (119909) =

119899

sum

119895=1

((1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

st119899

sum

119895=1

((1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(21)

Step 2 Calculate (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877 respectively

and suppose that (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

=

(1199091015840

1198951 1199091015840

1198952 1205721015840

119909119895 1205731015840

119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895

120573119909119895)119871119877

= (119901119894119895 119902119894119895 120574119894119895 120575119894119895)119871119877 then the FFLP problem obtained

in Step 1 can be written as

min 119911 (119909) = (

119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

1205721015840

119909119895

119899

sum

119895=1

1205731015840

119909119895)

119871119877

st (

119899

sum

119895=1

119901119894119895

119899

sum

119895=1

119902119894119895

119899

sum

119895=1

120574119894119895

119899

sum

119895=1

120575119894119895)

119871119877

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(22)

Mathematical Problems in Engineering 5

Step 3 According to the order relation defined above theproblem obtained in Step 2 is equivalent to

min119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

(1199091015840

1198951minus 1205721015840

119909119895)

119899

sum

119895=1

(1199091015840

1198952+ 1205731015840

119909119895)

st119899

sum

119895=1

119901119894119895le 1198871198941 119894 = 1 2 119898

119899

sum

119895=1

119902119894119895le 1198871198942 119894 = 1 2 119898

119899

sum

119895=1

(119901119894119895minus 120574119894119895) le (119887

1198941minus 120572119887119894) 119894 = 1 2 119898

119899

sum

119895=1

(119902119894119895+ 120575119894119895) le (119887

1198942+ 120573119887119894) 119894 = 1 2 119898

1199091198951le 1199091198952 120572

119909119895ge 0 120573

119909119895ge 0

1199091198951minus 120572119909119895ge 0 119895 = 1 2 119899

(23)

We denote 119883 = (11990911 11990912 1205721199091 1205731199091 11990921 11990922 1205721199092 1205731199092

1199091198991 1199091198992 120572119909119899 120573119909119899)119879 1199111(119883) = sum

119899

119895=11199091015840

1198951 1199112(119883) = sum

119899

119895=11199091015840

1198952

1199113(119883) = sum

119899

119895=1(1199091015840

1198951minus 1205721015840

119909119895) 1199114(119883) = sum

119899

119895=1(1199091015840

1198952+ 1205731015840

119909119895) and

119863 = 119883 | 119883 satisfies the constraints of programming (23)Programming (23) may be written as the programming (24)below for short as follows

min 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(24)

Obviously programming (24) is a crisp multiobjectivelinear programming problem In fact we have 119911(119909) =

(1199111(119883) 119911

2(119883) 119911

1(119883) minus 119911

3(119883) 119911

4(119883) minus 119911

2(119883))

Step 4 Solve the subproblems

min 119911119905(119883)

st 119883 isin 119863

(25)

where 119905 = 1 2 3 4 We find optimal solutions 1198831 1198832 1198833

and 1198834 respectively And the corresponding optimal values

will be 119911min1

= 1199111(1198831) 119911min2

= 1199112(1198832) 119911min3

= 1199113(1198833) and

119911min4

= 1199114(1198834)

Step 5 Let 119911max119905

= max119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834) 119905 =

1 2 3 4 and the membership function of 119911119905(119883) is given by

120583119911119905(119911119905(119883)) =

1 119911119905(119883) lt 119911

min119905

119911max119905

minus 119911119905(119883)

119911max119905

minus 119911min119905

119911min119905

le 119911119905(119883) le 119911

max119905

0 119911119905(119883) gt 119911

max119905

(26)

where 119905 = 1 2 3 4

Step 6 Let 1198680= 1 2 3 4 the MOLP problem obtained in

Step 3 can be equivalently written as

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

0

119883 isin 119863

(27)

Suppose1198831 is one of the optimal solutions (if there exits onlyone optimal solution 1198831 is the unique one) and 1205821lowast is theoptimal objective value (in fact the optimal solution shouldbe written as (1198831 1205821lowast) Since 120582 is an auxiliary variable wedenote (1198831 1205821lowast) as 1198831 for simplicity) Then 120583

1199111199041(1199111199041(1198831

)) =

1205821lowast for at least one 119904

1in 1198680 (1199041is an arbitrary element in the

set 119869 = 119895 | 120583119911119895(119911119895(1198831

)) = 1205821lowast

)

Step 7 Let 1198681= 1198680minus 1199041 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

1

1205831199111199041(1199111199041(119883)) = 120582

1lowast

119883 isin 119863

(28)

If 1198832 is one of the optimal solutions and 1205822lowast is the optimalobjective value then 120583

1199111199042(1199111199042(1198832

)) = 1205822lowast for at least one 119904

2in

1198681

Step 8 Let 1198682= 1198680minus 1199041 1199042 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

2

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

119883 isin 119863

(29)

6 Mathematical Problems in Engineering

Suppose 1198833 is one of the optimal solutions and 1205823lowast is the

optimal objective value Then 1205831199111199043(1199111199043(1198833

)) = 1205823lowast for at least

one 1199043in 1198682

Step 9 Let 1198683= 1198680minus 1199041 1199042 1199043 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

3

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

1205831199111199043(1199111199043(119883)) = 120582

3lowast

119883 isin 119863

(30)

Suppose 1198834 is one of the optimal solutions and 1205824lowast is the

optimal objective value Then 1205831199111199044(1199111199044(1198834

)) = 1205824lowast with 119904

4in

1198683

Step 10 Take119883lowast = 1198834 as the compromised optimal solutionto programming (23) and generate the compromised optimalsolution 119909lowast to programming (21) by119883lowast Assuming

119883lowast

= (119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091 11990921 119909lowast

22 120572lowast

1199092

120573lowast

1199092 119909

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119879

(31)

we may obtain

119909lowast

= (119909lowast

1 119909lowast

2 119909

lowast

119899)119879

= ((119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091)119871119877

(119909lowast

21 119909lowast

22 120572lowast

1199092 120573lowast

1199092)119871119877

(119909lowast

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119871119877

)119879

(32)

and the corresponding objective value 119911lowast = 119911(119909lowast)

Remark 7 (1199041 1199042 1199043 1199044 = 119868 = 1 2 3 4) Some properties

of the solutions obtained in Steps 6ndash10 are shown in thefollowing proposition

Proposition 8 Suppose 120583119911119904119895(119911119904119895(119883lowast

)) 119883119895 120582119895lowast (119895 = 1 2 3 4)and 119883lowast are the notations described in Steps 1ndash10 then

(1) 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205831199111199043(1199111199043(119883lowast

)) = 1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(2) 1205821lowast

le 1205822lowast

le 1205823lowast

le 1205824lowast

(33)

Proof (1) From the results of Steps 6ndash9 it is obviously clearthat

1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(34)

and 1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast with 119883

lowast

= 1198834 Since 119883lowast = 119883

4

is an optimal solution to programming (30) we know that119883lowast satisfies the constraints of programming (30) and so

1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast 1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast and 120583

1199111199043(1199111199043(119883lowast

)) =

1205823lowast(2) In fact (1198831 1205821lowast) is an optimal solution to program-

ming (27) therefore it is a feasible solution We have

120583119911119905(119911119905(1198831

)) ge 1205821lowast

119905 isin 1198681sube 1198680

1198831

isin 119863

(35)

and it is obvious that 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast from the result of

Step 6 Hence (1198831 1205821lowast) is a feasible solution to programming(28) The objective value of (1198831 1205821lowast) is 1205821lowast and the optimalobjective value of programming (28) is 1205822lowast so we get 1205821lowast le1205822lowast It is similar to prove 1205822lowast le 1205823lowast and 1205823lowast le 1205824lowast

4 Numerical Example

In this section we present a numerical example to illustratethe feasibility of the solution method proposed in Section 3

We aim to find the compromised optimal solution andcorresponding objective value of the following fully fuzzylinear programming problem

max 119911 (119909) = 119911 (1199091 1199092)

= (6 7 1 2)119871119877otimes 1199091oplus (7 9 1 2)

119871119877otimes 1199092

st (9 10 2 1)119871119877otimes 1199091oplus (1 1 1 1)

119871119877otimes 1199092

le (50 55 4 3)119871119877

(2 3 1 1)119871119877otimes 1199091oplus (4 5 1 2)

119871119877otimes 1199092

le (66 70 3 5)119871119877

1199091ge 0 119909

2ge 0

(36)

where 1199091= (11990911 11990912 1205721 1205731)119871119877

and 1199092= (11990921 11990922 1205722 1205732)119871119877

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

22 Arithmetic Operations Let = (1198981 1198991 1205721 1205731)119871119877

and V =(1198982 1198992 1205722 1205732)119871119877

be two 119871119877 flat fuzzy numbers 119896 isin 119877 Thenthe arithmetic operations are given as follows [9 16]

oplus V = (1198981+ 1198982 1198991+ 1198992 1205721+ 1205722 1205731+ 1205732)119871119877

⊖ V = (1198981minus 1198992 1198991minus 1198982 1205721+ 1205732 1205722+ 1205731)119871119877

119896 = (1198961198981 1198961198991 1198961205721 1198961205731)119871119877 119896 ge 0

(1198961198991 1198961198981 minus1198961205731 minus1198961205721)119877119871 119896 lt 0

otimes V =

(11989811198982 11989911198992 11989811205722+ 12057211198982 11989911205732+ 12057311198992)119871119877

ge 0 V ge 0(11989811198992 11989911198982 12057211198992minus 11989811205732 12057311198982minus 11989911205722)119871119877

le 0 V ge 0(11989911198982 11989811198992 11989911205722minus 12057311198982 11989811205732minus 12057211198992)119871119877

ge 0 V le 0(11989911198992 11989811198982 minus11989911205732minus 12057311198992 minus11989811205722minus 12057211198982)119871119877

le 0 V le 0(6)

It is easy to verify that the operator oplus satisfies associativelaw Hence the formula sum119899

119895=1119895= 1oplus 2oplus sdot sdot sdot oplus

119899is

reasonable where 1 2

119899are 119871119877 flat fuzzy numbers

23 Order Relation for Comparing the LR Flat Fuzzy NumbersFor comparing the 119871119877 flat fuzzy numbers we introduce theorder relation as follows

Definition 4 Let = (1198981 1198991 1205721 1205731)119871119877

and V = (1198982 1198992 1205722

1205732)119871119877

be any 119871119877 flat fuzzy numbers Then

(i) = V if and only if 1198981= 1198982 1198991= 1198992 1205721= 1205722

1205731= 1205732

(ii) le V if and only if 1198981le 1198982 1198991le 1198992 1198981minus 1205721le

1198982minus 1205722 1198991+ 1205731le 1198992+ 1205732

(iii) ge V if and only if 1198981ge 1198982 1198991ge 1198992 1198981minus 1205721ge

1198982minus 1205722 1198991+ 1205731ge 1198992+ 1205732

Based on the definition of order le we may obtain that(i) is nonnegative if and only if ge 0 (ii) is nonpositiveif and only if le 0

The following propositions are given to show the proper-ties of the order relation defined above

Proposition 5 Let V 119908 be four arbitrary 119871119877 flat fuzzynumbers and 119896 an arbitrary real number Then

(1) le V 119908 le 997904rArr oplus 119908 le V oplus

(2) le V 997904rArr

119896 le 119896V 119896 ge 0

119896 ge 119896V 119896 le 0

(7)

Proof Suppose = (1198981 1198991 1205721 1205731)119871119877 V = (119898

2 1198992 1205722 1205732)119871119877

119908 = (1198983 1198993 1205723 1205733)119871119877 and = (119898

4 1198994 1205724 1205734)119871119877

(1) It is obvious that oplus119908 = (1198981+1198983 1198991+1198993 1205721+1205723 1205731+

1205733)119871119877 V oplus = (119898

2+ 1198984 1198992+ 1198994 1205722+ 1205724 1205732+ 1205734)119871119877 Since

le V 119908 le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198983le 1198984 119899

3le 1198994 119898

3minus 1205723le 1198984minus 1205724

1198993+ 1205733le 1198994+ 1205734

(8)

So

1198981+ 1198983le 1198982+ 1198984

1198991+ 1198993le 1198992+ 1198994

(1198981+ 1198983) minus (120572

1+ 1205723) le (119898

2+ 1198984) minus (120572

2+ 1205724)

(1198991+ 1198993) + (120573

1+ 1205733) le (119899

2+ 1198994) minus (120573

2+ 1205734)

(9)

This indicates that oplus 119908 le V oplus (2) It is clear that 119896 = (119896119898

1 1198961198991 1198961205721 1198961205731)119871119877 119896V =

(1198961198982 1198961198992 1198961205722 1198961205732)119871119877 From le V we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

(10)

Therefore

1198961198981le 1198961198982 119896119899

1le 1198961198992 119896119898

1minus 1198961205721le 1198961198982minus 1198961205722

1198961198991+ 1198961205731le 1198961198992+ 1198961205732

(11)

for 119896 ge 0 and

1198961198981ge 1198961198982 119896119899

1ge 1198961198992 119896119898

1minus 1198961205721ge 1198961198982minus 1198961205722

1198961198991+ 1198961205731ge 1198961198992+ 1198961205732

(12)

for 119896 le 0 This indicates that

119896 le 119896V 119896 ge 0

119896 ge 119896V 119896 le 0

(13)

Proposition 6 Let V 119908 be three arbitrary 119871119877 flat fuzzynumbers Then

(1) le

(2) le V V le rArr = V

(3) le V V le 119908 rArr le 119908

Proof Suppose = (1198981 1198991 1205721 1205731)119871119877 V = (119898

2 1198992 1205722 1205732)119871119877

and 119908 = (1198983 1198993 1205723 1205733)119871119877

(1) Obviously = hence we have le

4 Mathematical Problems in Engineering

(2) Since le V V le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198981 119899

2le 1198991 119898

2minus 1205722le 1198981minus 1205721

1198992+ 1205732le 1198991+ 1205731

(14)

This means

1198981= 1198982 119899

1= 1198992 119898

1minus 1205721= 1198982minus 1205722

1198991+ 1205731= 1198992+ 1205732

(15)

That is

1198981= 1198982 119899

1= 1198992 120572

1= 1205722 120573

1= 1205732 (16)

Therefore we have = V(3) From le V V le 119908 we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198983 119899

2le 1198993 119898

2minus 1205722le 1198983minus 1205723

1198992+ 1205732le 1198993+ 1205733

(17)

This indicates

1198981le 1198983 119899

1le 1198993 119898

1minus 1205721le 1198983minus 1205723

1198991+ 1205731le 1198993+ 1205733

(18)

Therefore we have le 119908

From Proposition 6 we know that the order relation le isa partial order on the set of all 119871119877 fuzzy numbers

24 Fully Fuzzy Linear Programming with LR Fuzzy Parame-ters In this paper we will consider the following model thatis

min 119911 (119909) = 119888 otimes 119909

st 119860 otimes 119909 le

119909 ge 0

(19)

or

min 119911 (119909) = 1198881otimes 1199091oplus 1198882otimes 1199092oplus sdot sdot sdot oplus 119888

119899otimes 119909119899

st 1198861198941otimes 1199091oplus 1198861198942otimes 1199092oplus sdot sdot sdot oplus 119886

119894119899otimes 119909119899le 119894

119894 = 1 2 119898

119909119895ge 0 119895 = 1 2 119899

(20)

where 119888 = [119888119895]1times119899

= [119894]119898times1

119860 = [119886119894119895]119898times119899

and 119909 = [119909119895]119899times1

represent 119871119877 fuzzy matrices and vectors and 119888119895 119894 119886119894119895 and 119909

119895

are 119871119877 flat fuzzy numbers The order relations for comparing

the 119871119877 flat fuzzy numbers both in the objective function andthe constraint inequalities are as shown in Definition 4

3 Proposed Method

Steps of the proposed method are given to solve problem(20) as follows This method is applicable to minimizationof FFLP problems and the solution method of maximizationproblems is similar to that of minimization ones

Step 1 If all the parameters 119888119895 119894 119886119894119895 119909119895are represented by

119871119877 flat fuzzy numbers (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877 (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

(1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877 and (119909

1198951 1199091198952 120572119909119895 120573119909119895)119871119877 then the FFLP

(20) can be written as

min 119911 (119909) =

119899

sum

119895=1

((1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

st119899

sum

119895=1

((1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(21)

Step 2 Calculate (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877 respectively

and suppose that (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

=

(1199091015840

1198951 1199091015840

1198952 1205721015840

119909119895 1205731015840

119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895

120573119909119895)119871119877

= (119901119894119895 119902119894119895 120574119894119895 120575119894119895)119871119877 then the FFLP problem obtained

in Step 1 can be written as

min 119911 (119909) = (

119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

1205721015840

119909119895

119899

sum

119895=1

1205731015840

119909119895)

119871119877

st (

119899

sum

119895=1

119901119894119895

119899

sum

119895=1

119902119894119895

119899

sum

119895=1

120574119894119895

119899

sum

119895=1

120575119894119895)

119871119877

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(22)

Mathematical Problems in Engineering 5

Step 3 According to the order relation defined above theproblem obtained in Step 2 is equivalent to

min119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

(1199091015840

1198951minus 1205721015840

119909119895)

119899

sum

119895=1

(1199091015840

1198952+ 1205731015840

119909119895)

st119899

sum

119895=1

119901119894119895le 1198871198941 119894 = 1 2 119898

119899

sum

119895=1

119902119894119895le 1198871198942 119894 = 1 2 119898

119899

sum

119895=1

(119901119894119895minus 120574119894119895) le (119887

1198941minus 120572119887119894) 119894 = 1 2 119898

119899

sum

119895=1

(119902119894119895+ 120575119894119895) le (119887

1198942+ 120573119887119894) 119894 = 1 2 119898

1199091198951le 1199091198952 120572

119909119895ge 0 120573

119909119895ge 0

1199091198951minus 120572119909119895ge 0 119895 = 1 2 119899

(23)

We denote 119883 = (11990911 11990912 1205721199091 1205731199091 11990921 11990922 1205721199092 1205731199092

1199091198991 1199091198992 120572119909119899 120573119909119899)119879 1199111(119883) = sum

119899

119895=11199091015840

1198951 1199112(119883) = sum

119899

119895=11199091015840

1198952

1199113(119883) = sum

119899

119895=1(1199091015840

1198951minus 1205721015840

119909119895) 1199114(119883) = sum

119899

119895=1(1199091015840

1198952+ 1205731015840

119909119895) and

119863 = 119883 | 119883 satisfies the constraints of programming (23)Programming (23) may be written as the programming (24)below for short as follows

min 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(24)

Obviously programming (24) is a crisp multiobjectivelinear programming problem In fact we have 119911(119909) =

(1199111(119883) 119911

2(119883) 119911

1(119883) minus 119911

3(119883) 119911

4(119883) minus 119911

2(119883))

Step 4 Solve the subproblems

min 119911119905(119883)

st 119883 isin 119863

(25)

where 119905 = 1 2 3 4 We find optimal solutions 1198831 1198832 1198833

and 1198834 respectively And the corresponding optimal values

will be 119911min1

= 1199111(1198831) 119911min2

= 1199112(1198832) 119911min3

= 1199113(1198833) and

119911min4

= 1199114(1198834)

Step 5 Let 119911max119905

= max119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834) 119905 =

1 2 3 4 and the membership function of 119911119905(119883) is given by

120583119911119905(119911119905(119883)) =

1 119911119905(119883) lt 119911

min119905

119911max119905

minus 119911119905(119883)

119911max119905

minus 119911min119905

119911min119905

le 119911119905(119883) le 119911

max119905

0 119911119905(119883) gt 119911

max119905

(26)

where 119905 = 1 2 3 4

Step 6 Let 1198680= 1 2 3 4 the MOLP problem obtained in

Step 3 can be equivalently written as

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

0

119883 isin 119863

(27)

Suppose1198831 is one of the optimal solutions (if there exits onlyone optimal solution 1198831 is the unique one) and 1205821lowast is theoptimal objective value (in fact the optimal solution shouldbe written as (1198831 1205821lowast) Since 120582 is an auxiliary variable wedenote (1198831 1205821lowast) as 1198831 for simplicity) Then 120583

1199111199041(1199111199041(1198831

)) =

1205821lowast for at least one 119904

1in 1198680 (1199041is an arbitrary element in the

set 119869 = 119895 | 120583119911119895(119911119895(1198831

)) = 1205821lowast

)

Step 7 Let 1198681= 1198680minus 1199041 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

1

1205831199111199041(1199111199041(119883)) = 120582

1lowast

119883 isin 119863

(28)

If 1198832 is one of the optimal solutions and 1205822lowast is the optimalobjective value then 120583

1199111199042(1199111199042(1198832

)) = 1205822lowast for at least one 119904

2in

1198681

Step 8 Let 1198682= 1198680minus 1199041 1199042 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

2

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

119883 isin 119863

(29)

6 Mathematical Problems in Engineering

Suppose 1198833 is one of the optimal solutions and 1205823lowast is the

optimal objective value Then 1205831199111199043(1199111199043(1198833

)) = 1205823lowast for at least

one 1199043in 1198682

Step 9 Let 1198683= 1198680minus 1199041 1199042 1199043 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

3

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

1205831199111199043(1199111199043(119883)) = 120582

3lowast

119883 isin 119863

(30)

Suppose 1198834 is one of the optimal solutions and 1205824lowast is the

optimal objective value Then 1205831199111199044(1199111199044(1198834

)) = 1205824lowast with 119904

4in

1198683

Step 10 Take119883lowast = 1198834 as the compromised optimal solutionto programming (23) and generate the compromised optimalsolution 119909lowast to programming (21) by119883lowast Assuming

119883lowast

= (119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091 11990921 119909lowast

22 120572lowast

1199092

120573lowast

1199092 119909

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119879

(31)

we may obtain

119909lowast

= (119909lowast

1 119909lowast

2 119909

lowast

119899)119879

= ((119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091)119871119877

(119909lowast

21 119909lowast

22 120572lowast

1199092 120573lowast

1199092)119871119877

(119909lowast

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119871119877

)119879

(32)

and the corresponding objective value 119911lowast = 119911(119909lowast)

Remark 7 (1199041 1199042 1199043 1199044 = 119868 = 1 2 3 4) Some properties

of the solutions obtained in Steps 6ndash10 are shown in thefollowing proposition

Proposition 8 Suppose 120583119911119904119895(119911119904119895(119883lowast

)) 119883119895 120582119895lowast (119895 = 1 2 3 4)and 119883lowast are the notations described in Steps 1ndash10 then

(1) 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205831199111199043(1199111199043(119883lowast

)) = 1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(2) 1205821lowast

le 1205822lowast

le 1205823lowast

le 1205824lowast

(33)

Proof (1) From the results of Steps 6ndash9 it is obviously clearthat

1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(34)

and 1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast with 119883

lowast

= 1198834 Since 119883lowast = 119883

4

is an optimal solution to programming (30) we know that119883lowast satisfies the constraints of programming (30) and so

1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast 1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast and 120583

1199111199043(1199111199043(119883lowast

)) =

1205823lowast(2) In fact (1198831 1205821lowast) is an optimal solution to program-

ming (27) therefore it is a feasible solution We have

120583119911119905(119911119905(1198831

)) ge 1205821lowast

119905 isin 1198681sube 1198680

1198831

isin 119863

(35)

and it is obvious that 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast from the result of

Step 6 Hence (1198831 1205821lowast) is a feasible solution to programming(28) The objective value of (1198831 1205821lowast) is 1205821lowast and the optimalobjective value of programming (28) is 1205822lowast so we get 1205821lowast le1205822lowast It is similar to prove 1205822lowast le 1205823lowast and 1205823lowast le 1205824lowast

4 Numerical Example

In this section we present a numerical example to illustratethe feasibility of the solution method proposed in Section 3

We aim to find the compromised optimal solution andcorresponding objective value of the following fully fuzzylinear programming problem

max 119911 (119909) = 119911 (1199091 1199092)

= (6 7 1 2)119871119877otimes 1199091oplus (7 9 1 2)

119871119877otimes 1199092

st (9 10 2 1)119871119877otimes 1199091oplus (1 1 1 1)

119871119877otimes 1199092

le (50 55 4 3)119871119877

(2 3 1 1)119871119877otimes 1199091oplus (4 5 1 2)

119871119877otimes 1199092

le (66 70 3 5)119871119877

1199091ge 0 119909

2ge 0

(36)

where 1199091= (11990911 11990912 1205721 1205731)119871119877

and 1199092= (11990921 11990922 1205722 1205732)119871119877

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

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Mathematical Problems in Engineering

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4 Mathematical Problems in Engineering

(2) Since le V V le we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198981 119899

2le 1198991 119898

2minus 1205722le 1198981minus 1205721

1198992+ 1205732le 1198991+ 1205731

(14)

This means

1198981= 1198982 119899

1= 1198992 119898

1minus 1205721= 1198982minus 1205722

1198991+ 1205731= 1198992+ 1205732

(15)

That is

1198981= 1198982 119899

1= 1198992 120572

1= 1205722 120573

1= 1205732 (16)

Therefore we have = V(3) From le V V le 119908 we get

1198981le 1198982 119899

1le 1198992 119898

1minus 1205721le 1198982minus 1205722

1198991+ 1205731le 1198992+ 1205732

1198982le 1198983 119899

2le 1198993 119898

2minus 1205722le 1198983minus 1205723

1198992+ 1205732le 1198993+ 1205733

(17)

This indicates

1198981le 1198983 119899

1le 1198993 119898

1minus 1205721le 1198983minus 1205723

1198991+ 1205731le 1198993+ 1205733

(18)

Therefore we have le 119908

From Proposition 6 we know that the order relation le isa partial order on the set of all 119871119877 fuzzy numbers

24 Fully Fuzzy Linear Programming with LR Fuzzy Parame-ters In this paper we will consider the following model thatis

min 119911 (119909) = 119888 otimes 119909

st 119860 otimes 119909 le

119909 ge 0

(19)

or

min 119911 (119909) = 1198881otimes 1199091oplus 1198882otimes 1199092oplus sdot sdot sdot oplus 119888

119899otimes 119909119899

st 1198861198941otimes 1199091oplus 1198861198942otimes 1199092oplus sdot sdot sdot oplus 119886

119894119899otimes 119909119899le 119894

119894 = 1 2 119898

119909119895ge 0 119895 = 1 2 119899

(20)

where 119888 = [119888119895]1times119899

= [119894]119898times1

119860 = [119886119894119895]119898times119899

and 119909 = [119909119895]119899times1

represent 119871119877 fuzzy matrices and vectors and 119888119895 119894 119886119894119895 and 119909

119895

are 119871119877 flat fuzzy numbers The order relations for comparing

the 119871119877 flat fuzzy numbers both in the objective function andthe constraint inequalities are as shown in Definition 4

3 Proposed Method

Steps of the proposed method are given to solve problem(20) as follows This method is applicable to minimizationof FFLP problems and the solution method of maximizationproblems is similar to that of minimization ones

Step 1 If all the parameters 119888119895 119894 119886119894119895 119909119895are represented by

119871119877 flat fuzzy numbers (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877 (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

(1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877 and (119909

1198951 1199091198952 120572119909119895 120573119909119895)119871119877 then the FFLP

(20) can be written as

min 119911 (119909) =

119899

sum

119895=1

((1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

st119899

sum

119895=1

((1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

)

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(21)

Step 2 Calculate (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877 respectively

and suppose that (1198881198951 1198881198952 120572119888119895 120573119888119895)119871119877otimes (1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

=

(1199091015840

1198951 1199091015840

1198952 1205721015840

119909119895 1205731015840

119909119895)119871119877

and (1198861198941198951 1198861198941198952 120572119886119894119895 120573119886119894119895)119871119877

otimes (1199091198951 1199091198952 120572119909119895

120573119909119895)119871119877

= (119901119894119895 119902119894119895 120574119894119895 120575119894119895)119871119877 then the FFLP problem obtained

in Step 1 can be written as

min 119911 (119909) = (

119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

1205721015840

119909119895

119899

sum

119895=1

1205731015840

119909119895)

119871119877

st (

119899

sum

119895=1

119901119894119895

119899

sum

119895=1

119902119894119895

119899

sum

119895=1

120574119894119895

119899

sum

119895=1

120575119894119895)

119871119877

le (1198871198941 1198871198942 120572119887119894 120573119887119894)119871119877

119894 = 1 2 119898

(1199091198951 1199091198952 120572119909119895 120573119909119895)119871119877

ge (0 0 0 0 )119871119877

119895 = 1 2 119899

(22)

Mathematical Problems in Engineering 5

Step 3 According to the order relation defined above theproblem obtained in Step 2 is equivalent to

min119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

(1199091015840

1198951minus 1205721015840

119909119895)

119899

sum

119895=1

(1199091015840

1198952+ 1205731015840

119909119895)

st119899

sum

119895=1

119901119894119895le 1198871198941 119894 = 1 2 119898

119899

sum

119895=1

119902119894119895le 1198871198942 119894 = 1 2 119898

119899

sum

119895=1

(119901119894119895minus 120574119894119895) le (119887

1198941minus 120572119887119894) 119894 = 1 2 119898

119899

sum

119895=1

(119902119894119895+ 120575119894119895) le (119887

1198942+ 120573119887119894) 119894 = 1 2 119898

1199091198951le 1199091198952 120572

119909119895ge 0 120573

119909119895ge 0

1199091198951minus 120572119909119895ge 0 119895 = 1 2 119899

(23)

We denote 119883 = (11990911 11990912 1205721199091 1205731199091 11990921 11990922 1205721199092 1205731199092

1199091198991 1199091198992 120572119909119899 120573119909119899)119879 1199111(119883) = sum

119899

119895=11199091015840

1198951 1199112(119883) = sum

119899

119895=11199091015840

1198952

1199113(119883) = sum

119899

119895=1(1199091015840

1198951minus 1205721015840

119909119895) 1199114(119883) = sum

119899

119895=1(1199091015840

1198952+ 1205731015840

119909119895) and

119863 = 119883 | 119883 satisfies the constraints of programming (23)Programming (23) may be written as the programming (24)below for short as follows

min 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(24)

Obviously programming (24) is a crisp multiobjectivelinear programming problem In fact we have 119911(119909) =

(1199111(119883) 119911

2(119883) 119911

1(119883) minus 119911

3(119883) 119911

4(119883) minus 119911

2(119883))

Step 4 Solve the subproblems

min 119911119905(119883)

st 119883 isin 119863

(25)

where 119905 = 1 2 3 4 We find optimal solutions 1198831 1198832 1198833

and 1198834 respectively And the corresponding optimal values

will be 119911min1

= 1199111(1198831) 119911min2

= 1199112(1198832) 119911min3

= 1199113(1198833) and

119911min4

= 1199114(1198834)

Step 5 Let 119911max119905

= max119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834) 119905 =

1 2 3 4 and the membership function of 119911119905(119883) is given by

120583119911119905(119911119905(119883)) =

1 119911119905(119883) lt 119911

min119905

119911max119905

minus 119911119905(119883)

119911max119905

minus 119911min119905

119911min119905

le 119911119905(119883) le 119911

max119905

0 119911119905(119883) gt 119911

max119905

(26)

where 119905 = 1 2 3 4

Step 6 Let 1198680= 1 2 3 4 the MOLP problem obtained in

Step 3 can be equivalently written as

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

0

119883 isin 119863

(27)

Suppose1198831 is one of the optimal solutions (if there exits onlyone optimal solution 1198831 is the unique one) and 1205821lowast is theoptimal objective value (in fact the optimal solution shouldbe written as (1198831 1205821lowast) Since 120582 is an auxiliary variable wedenote (1198831 1205821lowast) as 1198831 for simplicity) Then 120583

1199111199041(1199111199041(1198831

)) =

1205821lowast for at least one 119904

1in 1198680 (1199041is an arbitrary element in the

set 119869 = 119895 | 120583119911119895(119911119895(1198831

)) = 1205821lowast

)

Step 7 Let 1198681= 1198680minus 1199041 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

1

1205831199111199041(1199111199041(119883)) = 120582

1lowast

119883 isin 119863

(28)

If 1198832 is one of the optimal solutions and 1205822lowast is the optimalobjective value then 120583

1199111199042(1199111199042(1198832

)) = 1205822lowast for at least one 119904

2in

1198681

Step 8 Let 1198682= 1198680minus 1199041 1199042 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

2

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

119883 isin 119863

(29)

6 Mathematical Problems in Engineering

Suppose 1198833 is one of the optimal solutions and 1205823lowast is the

optimal objective value Then 1205831199111199043(1199111199043(1198833

)) = 1205823lowast for at least

one 1199043in 1198682

Step 9 Let 1198683= 1198680minus 1199041 1199042 1199043 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

3

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

1205831199111199043(1199111199043(119883)) = 120582

3lowast

119883 isin 119863

(30)

Suppose 1198834 is one of the optimal solutions and 1205824lowast is the

optimal objective value Then 1205831199111199044(1199111199044(1198834

)) = 1205824lowast with 119904

4in

1198683

Step 10 Take119883lowast = 1198834 as the compromised optimal solutionto programming (23) and generate the compromised optimalsolution 119909lowast to programming (21) by119883lowast Assuming

119883lowast

= (119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091 11990921 119909lowast

22 120572lowast

1199092

120573lowast

1199092 119909

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119879

(31)

we may obtain

119909lowast

= (119909lowast

1 119909lowast

2 119909

lowast

119899)119879

= ((119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091)119871119877

(119909lowast

21 119909lowast

22 120572lowast

1199092 120573lowast

1199092)119871119877

(119909lowast

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119871119877

)119879

(32)

and the corresponding objective value 119911lowast = 119911(119909lowast)

Remark 7 (1199041 1199042 1199043 1199044 = 119868 = 1 2 3 4) Some properties

of the solutions obtained in Steps 6ndash10 are shown in thefollowing proposition

Proposition 8 Suppose 120583119911119904119895(119911119904119895(119883lowast

)) 119883119895 120582119895lowast (119895 = 1 2 3 4)and 119883lowast are the notations described in Steps 1ndash10 then

(1) 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205831199111199043(1199111199043(119883lowast

)) = 1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(2) 1205821lowast

le 1205822lowast

le 1205823lowast

le 1205824lowast

(33)

Proof (1) From the results of Steps 6ndash9 it is obviously clearthat

1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(34)

and 1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast with 119883

lowast

= 1198834 Since 119883lowast = 119883

4

is an optimal solution to programming (30) we know that119883lowast satisfies the constraints of programming (30) and so

1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast 1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast and 120583

1199111199043(1199111199043(119883lowast

)) =

1205823lowast(2) In fact (1198831 1205821lowast) is an optimal solution to program-

ming (27) therefore it is a feasible solution We have

120583119911119905(119911119905(1198831

)) ge 1205821lowast

119905 isin 1198681sube 1198680

1198831

isin 119863

(35)

and it is obvious that 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast from the result of

Step 6 Hence (1198831 1205821lowast) is a feasible solution to programming(28) The objective value of (1198831 1205821lowast) is 1205821lowast and the optimalobjective value of programming (28) is 1205822lowast so we get 1205821lowast le1205822lowast It is similar to prove 1205822lowast le 1205823lowast and 1205823lowast le 1205824lowast

4 Numerical Example

In this section we present a numerical example to illustratethe feasibility of the solution method proposed in Section 3

We aim to find the compromised optimal solution andcorresponding objective value of the following fully fuzzylinear programming problem

max 119911 (119909) = 119911 (1199091 1199092)

= (6 7 1 2)119871119877otimes 1199091oplus (7 9 1 2)

119871119877otimes 1199092

st (9 10 2 1)119871119877otimes 1199091oplus (1 1 1 1)

119871119877otimes 1199092

le (50 55 4 3)119871119877

(2 3 1 1)119871119877otimes 1199091oplus (4 5 1 2)

119871119877otimes 1199092

le (66 70 3 5)119871119877

1199091ge 0 119909

2ge 0

(36)

where 1199091= (11990911 11990912 1205721 1205731)119871119877

and 1199092= (11990921 11990922 1205722 1205732)119871119877

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Step 3 According to the order relation defined above theproblem obtained in Step 2 is equivalent to

min119899

sum

119895=1

1199091015840

1198951

119899

sum

119895=1

1199091015840

1198952

119899

sum

119895=1

(1199091015840

1198951minus 1205721015840

119909119895)

119899

sum

119895=1

(1199091015840

1198952+ 1205731015840

119909119895)

st119899

sum

119895=1

119901119894119895le 1198871198941 119894 = 1 2 119898

119899

sum

119895=1

119902119894119895le 1198871198942 119894 = 1 2 119898

119899

sum

119895=1

(119901119894119895minus 120574119894119895) le (119887

1198941minus 120572119887119894) 119894 = 1 2 119898

119899

sum

119895=1

(119902119894119895+ 120575119894119895) le (119887

1198942+ 120573119887119894) 119894 = 1 2 119898

1199091198951le 1199091198952 120572

119909119895ge 0 120573

119909119895ge 0

1199091198951minus 120572119909119895ge 0 119895 = 1 2 119899

(23)

We denote 119883 = (11990911 11990912 1205721199091 1205731199091 11990921 11990922 1205721199092 1205731199092

1199091198991 1199091198992 120572119909119899 120573119909119899)119879 1199111(119883) = sum

119899

119895=11199091015840

1198951 1199112(119883) = sum

119899

119895=11199091015840

1198952

1199113(119883) = sum

119899

119895=1(1199091015840

1198951minus 1205721015840

119909119895) 1199114(119883) = sum

119899

119895=1(1199091015840

1198952+ 1205731015840

119909119895) and

119863 = 119883 | 119883 satisfies the constraints of programming (23)Programming (23) may be written as the programming (24)below for short as follows

min 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(24)

Obviously programming (24) is a crisp multiobjectivelinear programming problem In fact we have 119911(119909) =

(1199111(119883) 119911

2(119883) 119911

1(119883) minus 119911

3(119883) 119911

4(119883) minus 119911

2(119883))

Step 4 Solve the subproblems

min 119911119905(119883)

st 119883 isin 119863

(25)

where 119905 = 1 2 3 4 We find optimal solutions 1198831 1198832 1198833

and 1198834 respectively And the corresponding optimal values

will be 119911min1

= 1199111(1198831) 119911min2

= 1199112(1198832) 119911min3

= 1199113(1198833) and

119911min4

= 1199114(1198834)

Step 5 Let 119911max119905

= max119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834) 119905 =

1 2 3 4 and the membership function of 119911119905(119883) is given by

120583119911119905(119911119905(119883)) =

1 119911119905(119883) lt 119911

min119905

119911max119905

minus 119911119905(119883)

119911max119905

minus 119911min119905

119911min119905

le 119911119905(119883) le 119911

max119905

0 119911119905(119883) gt 119911

max119905

(26)

where 119905 = 1 2 3 4

Step 6 Let 1198680= 1 2 3 4 the MOLP problem obtained in

Step 3 can be equivalently written as

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

0

119883 isin 119863

(27)

Suppose1198831 is one of the optimal solutions (if there exits onlyone optimal solution 1198831 is the unique one) and 1205821lowast is theoptimal objective value (in fact the optimal solution shouldbe written as (1198831 1205821lowast) Since 120582 is an auxiliary variable wedenote (1198831 1205821lowast) as 1198831 for simplicity) Then 120583

1199111199041(1199111199041(1198831

)) =

1205821lowast for at least one 119904

1in 1198680 (1199041is an arbitrary element in the

set 119869 = 119895 | 120583119911119895(119911119895(1198831

)) = 1205821lowast

)

Step 7 Let 1198681= 1198680minus 1199041 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

1

1205831199111199041(1199111199041(119883)) = 120582

1lowast

119883 isin 119863

(28)

If 1198832 is one of the optimal solutions and 1205822lowast is the optimalobjective value then 120583

1199111199042(1199111199042(1198832

)) = 1205822lowast for at least one 119904

2in

1198681

Step 8 Let 1198682= 1198680minus 1199041 1199042 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

2

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

119883 isin 119863

(29)

6 Mathematical Problems in Engineering

Suppose 1198833 is one of the optimal solutions and 1205823lowast is the

optimal objective value Then 1205831199111199043(1199111199043(1198833

)) = 1205823lowast for at least

one 1199043in 1198682

Step 9 Let 1198683= 1198680minus 1199041 1199042 1199043 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

3

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

1205831199111199043(1199111199043(119883)) = 120582

3lowast

119883 isin 119863

(30)

Suppose 1198834 is one of the optimal solutions and 1205824lowast is the

optimal objective value Then 1205831199111199044(1199111199044(1198834

)) = 1205824lowast with 119904

4in

1198683

Step 10 Take119883lowast = 1198834 as the compromised optimal solutionto programming (23) and generate the compromised optimalsolution 119909lowast to programming (21) by119883lowast Assuming

119883lowast

= (119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091 11990921 119909lowast

22 120572lowast

1199092

120573lowast

1199092 119909

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119879

(31)

we may obtain

119909lowast

= (119909lowast

1 119909lowast

2 119909

lowast

119899)119879

= ((119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091)119871119877

(119909lowast

21 119909lowast

22 120572lowast

1199092 120573lowast

1199092)119871119877

(119909lowast

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119871119877

)119879

(32)

and the corresponding objective value 119911lowast = 119911(119909lowast)

Remark 7 (1199041 1199042 1199043 1199044 = 119868 = 1 2 3 4) Some properties

of the solutions obtained in Steps 6ndash10 are shown in thefollowing proposition

Proposition 8 Suppose 120583119911119904119895(119911119904119895(119883lowast

)) 119883119895 120582119895lowast (119895 = 1 2 3 4)and 119883lowast are the notations described in Steps 1ndash10 then

(1) 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205831199111199043(1199111199043(119883lowast

)) = 1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(2) 1205821lowast

le 1205822lowast

le 1205823lowast

le 1205824lowast

(33)

Proof (1) From the results of Steps 6ndash9 it is obviously clearthat

1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(34)

and 1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast with 119883

lowast

= 1198834 Since 119883lowast = 119883

4

is an optimal solution to programming (30) we know that119883lowast satisfies the constraints of programming (30) and so

1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast 1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast and 120583

1199111199043(1199111199043(119883lowast

)) =

1205823lowast(2) In fact (1198831 1205821lowast) is an optimal solution to program-

ming (27) therefore it is a feasible solution We have

120583119911119905(119911119905(1198831

)) ge 1205821lowast

119905 isin 1198681sube 1198680

1198831

isin 119863

(35)

and it is obvious that 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast from the result of

Step 6 Hence (1198831 1205821lowast) is a feasible solution to programming(28) The objective value of (1198831 1205821lowast) is 1205821lowast and the optimalobjective value of programming (28) is 1205822lowast so we get 1205821lowast le1205822lowast It is similar to prove 1205822lowast le 1205823lowast and 1205823lowast le 1205824lowast

4 Numerical Example

In this section we present a numerical example to illustratethe feasibility of the solution method proposed in Section 3

We aim to find the compromised optimal solution andcorresponding objective value of the following fully fuzzylinear programming problem

max 119911 (119909) = 119911 (1199091 1199092)

= (6 7 1 2)119871119877otimes 1199091oplus (7 9 1 2)

119871119877otimes 1199092

st (9 10 2 1)119871119877otimes 1199091oplus (1 1 1 1)

119871119877otimes 1199092

le (50 55 4 3)119871119877

(2 3 1 1)119871119877otimes 1199091oplus (4 5 1 2)

119871119877otimes 1199092

le (66 70 3 5)119871119877

1199091ge 0 119909

2ge 0

(36)

where 1199091= (11990911 11990912 1205721 1205731)119871119877

and 1199092= (11990921 11990922 1205722 1205732)119871119877

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Suppose 1198833 is one of the optimal solutions and 1205823lowast is the

optimal objective value Then 1205831199111199043(1199111199043(1198833

)) = 1205823lowast for at least

one 1199043in 1198682

Step 9 Let 1198683= 1198680minus 1199041 1199042 1199043 and solve the following crisp

programming

max 120582

st 120583119911119905(119911119905(119883)) ge 120582 119905 isin 119868

3

1205831199111199041(1199111199041(119883)) = 120582

1lowast

1205831199111199042(1199111199042(119883)) = 120582

2lowast

1205831199111199043(1199111199043(119883)) = 120582

3lowast

119883 isin 119863

(30)

Suppose 1198834 is one of the optimal solutions and 1205824lowast is the

optimal objective value Then 1205831199111199044(1199111199044(1198834

)) = 1205824lowast with 119904

4in

1198683

Step 10 Take119883lowast = 1198834 as the compromised optimal solutionto programming (23) and generate the compromised optimalsolution 119909lowast to programming (21) by119883lowast Assuming

119883lowast

= (119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091 11990921 119909lowast

22 120572lowast

1199092

120573lowast

1199092 119909

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119879

(31)

we may obtain

119909lowast

= (119909lowast

1 119909lowast

2 119909

lowast

119899)119879

= ((119909lowast

11 119909lowast

12 120572lowast

1199091 120573lowast

1199091)119871119877

(119909lowast

21 119909lowast

22 120572lowast

1199092 120573lowast

1199092)119871119877

(119909lowast

1198991 119909lowast

1198992 120572lowast

119909119899 120573lowast

119909119899)119871119877

)119879

(32)

and the corresponding objective value 119911lowast = 119911(119909lowast)

Remark 7 (1199041 1199042 1199043 1199044 = 119868 = 1 2 3 4) Some properties

of the solutions obtained in Steps 6ndash10 are shown in thefollowing proposition

Proposition 8 Suppose 120583119911119904119895(119911119904119895(119883lowast

)) 119883119895 120582119895lowast (119895 = 1 2 3 4)and 119883lowast are the notations described in Steps 1ndash10 then

(1) 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205831199111199043(1199111199043(119883lowast

)) = 1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(2) 1205821lowast

le 1205822lowast

le 1205823lowast

le 1205824lowast

(33)

Proof (1) From the results of Steps 6ndash9 it is obviously clearthat

1205821lowast

= 1205831199111199041(1199111199041(1198831

))

1205822lowast

= 1205831199111199042(1199111199042(1198832

))

1205823lowast

= 1205831199111199043(1199111199043(1198833

))

1205824lowast

= 1205831199111199044(1199111199044(1198834

))

(34)

and 1205831199111199044(1199111199044(119883lowast

)) = 1205824lowast with 119883

lowast

= 1198834 Since 119883lowast = 119883

4

is an optimal solution to programming (30) we know that119883lowast satisfies the constraints of programming (30) and so

1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast 1205831199111199042(1199111199042(119883lowast

)) = 1205822lowast and 120583

1199111199043(1199111199043(119883lowast

)) =

1205823lowast(2) In fact (1198831 1205821lowast) is an optimal solution to program-

ming (27) therefore it is a feasible solution We have

120583119911119905(119911119905(1198831

)) ge 1205821lowast

119905 isin 1198681sube 1198680

1198831

isin 119863

(35)

and it is obvious that 1205831199111199041(1199111199041(119883lowast

)) = 1205821lowast from the result of

Step 6 Hence (1198831 1205821lowast) is a feasible solution to programming(28) The objective value of (1198831 1205821lowast) is 1205821lowast and the optimalobjective value of programming (28) is 1205822lowast so we get 1205821lowast le1205822lowast It is similar to prove 1205822lowast le 1205823lowast and 1205823lowast le 1205824lowast

4 Numerical Example

In this section we present a numerical example to illustratethe feasibility of the solution method proposed in Section 3

We aim to find the compromised optimal solution andcorresponding objective value of the following fully fuzzylinear programming problem

max 119911 (119909) = 119911 (1199091 1199092)

= (6 7 1 2)119871119877otimes 1199091oplus (7 9 1 2)

119871119877otimes 1199092

st (9 10 2 1)119871119877otimes 1199091oplus (1 1 1 1)

119871119877otimes 1199092

le (50 55 4 3)119871119877

(2 3 1 1)119871119877otimes 1199091oplus (4 5 1 2)

119871119877otimes 1199092

le (66 70 3 5)119871119877

1199091ge 0 119909

2ge 0

(36)

where 1199091= (11990911 11990912 1205721 1205731)119871119877

and 1199092= (11990921 11990922 1205722 1205732)119871119877

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 The optimal values and solutions of the four subproblems

The optimal objective value The optimal solution(a) 119911

max1

= 70 1198831= (0 03223 0 02887 10 101163 70903 01898)

119879

(b) 119911max2

= 1033188 1198832= (16015 37101 16039 0 42500 85942 31084 0)

119879

(c) 119911max3

= 50 1198833= (0 02544 0 02973 10 100893 0 02179)

119879

(d) 119911max4

= 1425882 1198834= (14272 42157 12516 0 0 0 0 116275)

119879

According to Steps 1 and 2 in the proposed method weobtain the following programming

max 119911 = (611990911+ 711990921 711990912+ 911990922 61205721+ 71205722

+11990911+ 211990921 71205731+ 91205732+ 211990912+ 11990922)119871119877

st (911990911+ 11990921 1011990912+ 11990922 91205721+ 1205722+ 211990911

+11990921 101205731+ 1205732+ 11990912+ 11990922)119871119877

le (50 55 4 3)119871119877

(211990911+ 411990921 311990912+ 511990922 21205721+ 41205722+ 11990911

+11990921 31205731+ 51205732+ 11990912+ 211990922)119871119877

le (66 70 3 5)119871119877

(11990911 11990912 1205721 1205731)119871119877ge (0 0 0 0)

119871119877

(11990921 11990922 1205722 1205732)119871119877ge (0 0 0 0)

119871119877

(37)

By Step 3 the programming above is transformed into thefollowing programming

max 1199111= 611990911+ 711990921

1199112= 711990912+ 911990922

1199113= 511990911+ 511990921minus 61205721minus 71205722

1199114= 911990912+ 10119909

22+ 71205731+ 91205732

st 911990911+ 11990921le 50

1011990912+ 11990922le 55

711990911minus 91205721minus 1205722le 46

1111990912+ 211990922+ 10120573

1+ 1205732le 58

211990911+ 411990921le 66

311990912+ 511990922le 70

11990911+ 311990921minus 21205721minus 41205722le 63

411990912+ 711990922+ 31205731+ 51205732le 75

11990911minus 1205721ge 0 119909

21minus 1205722ge 0

11990911le 11990912 119909

21le 11990922

1205721 1205722 1205731 1205732ge 0

(38)

Programming (38) can be abbreviated to the followingprogramming

max 1199111(119883)

1199112(119883)

1199113(119883)

1199114(119883)

st 119883 isin 119863

(39)

where119883 = (11990911 11990912 1205721 1205731 11990921 11990922 1205722 1205732)119879

Solve the following subproblems

(a) max 1199111(119883)

st 119883 isin 119863

(b) max 1199112(119883)

st 119883 isin 119863

(c) max 1199113(119883)

st 119883 isin 119863

(40)

(d) max 1199114(119883)

st 119883 isin 119863

(41)

respectively and we obtain the optimal objective value andone of the optimal solutions as shown in Table 1

According to 119911min119905

= min119911119905(1198831) 119911119905(1198832) 119911119905(1198833) 119911119905(1198834)

we acquire the lower objective values 119911min1

= 85631 119911min2

=

295098 119911min3

= 03681 and 119911min4

= 1072245 withcorresponding membership functions given below Consider

1205831199111(1199111(119883)) =

1

1199111(119883) gt 70

1199111(119883) minus 85631

70 minus 85631

85631 le 1199111(119883) le 70

0

1199111(119883) lt 85631

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

1205831199112(1199112(119883)) =

1

1199112(119883) gt 1033188

1199112(119883) minus 295098

1033188 minus 295098

295098 le 1199112(119883) le 1033188

0

1199112(119883) lt 295098

1205831199113(1199113(119883)) =

1

1199113(119883) gt 50

1199113(119883) minus 03681

50 minus 03681

03681 le 1199113(119883) le 50

0

1199113(119883) lt 03681

1205831199114(1199114(119883)) =

1

1199114(119883) gt 1425882

1199114(119883) minus 1072245

1425882 minus 1072245

1072245 le 1199114(119883) le 1425882

0

1199114(119883) lt 1072245

(42)

By Steps 4ndash6 we get

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245ge 120582

119883 isin 119863

(43)

The optimal objective value is 1205821lowast = 06033 and oneof the optimal solutions is 1198831 = (25901 39107 04927 0

47997 51848 03109 46127)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198831 and we get 120583

1199111(1199111(1198831

)) =

06604 1205831199112(1199112(1198831

)) = 06033 1205831199113(1199113(1198831

)) = 06336 and1205831199114(1199114(1198831

)) = 06033

Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(44)

The optimal objective value is 1205822lowast = 06033 and one ofthe optimal solutions is 1198832 = (28 39107 03135 0 47161

51850 04310 46124)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198832 and we get 120583

1199111(1199111(1198832

)) =

06714 1205831199112(1199112(1198832

)) = 06033 1205831199113(1199113(1198832

)) = 06511 and1205831199114(1199114(1198832

)) = 06033Solve the following problem

max 120582

st 1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631ge 120582

1205831199113(1199113(119883))

=511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(45)

The optimal objective value is 1205823lowast = 07126 and one ofthe optimal solutions is1198833 = (26750 39107 01158 0000551850 51850 02608 46125)

119879Calculate the value of the membership function of

119911119905(119883) (119905 = 1 2 3 4) at 119883 = 119883

3 and we get 1205831199111(1199111(1198833

)) =

07126 1205831199112(1199112(1198833

)) = 06033 1205831199113(1199113(1198833

)) = 07274 and1205831199114(1199114(1198833

)) = 06033

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 2 Values of the four membership functions at 119883119895

1205831199111(1199111(119883)) 120583

1199112(1199112(119883)) 120583

1199113(1199113(119883)) 120583

1199114(1199114(119883))

119883 = 1198831

06604 06033 06336 06033

119883 = 1198832

06714 06033 06511 06033

119883 = 1198833

07126 06033 07274 06033

119883 = 1198834

07126 06033 07844 06033

Table 3 Values of the objective function 119911(119909) at 119909119895

119911(119909)

119909 = 1199091

(4913 7403 1732 3896)119871119877

119909 = 1199092

(4981 7403 1713 3896)119871119877

119909 = 1199093

(5234 7403 1556 3896)119871119877

119909 = 1199094

(5234 7403 1304 3896)119871119877

Solve the following problem

max 120582

st 1205831199113(1199113(119883)) =

511990911+ 511990921minus 61205721minus 71205722minus 03681

50 minus 03681ge 120582

1205831199111(1199111(119883)) =

611990911+ 711990921minus 85631

70 minus 85631= 07126

1205831199112(1199112(119883)) =

711990912+ 911990922minus 295098

1033188 minus 295098= 06033

1205831199114(1199114(119883))

=911990912+ 10119909

22+ 71205731+ 91205732minus 1072245

1425882 minus 1072245= 06033

119883 isin 119863

(46)

The optimal objective value is 1205824lowast = 07844 and one ofthe optimal solutions is 1198834 = (26754 39106 0 0 51843

51848 0 46128)119879

Calculate the value of the membership function of 119911119905(119883)

(119905 = 1 2 3 4) at 119883 = 1198834 and we get 120583

1199111(1199111(1198834

)) =

07126 1205831199112(1199112(1198834

)) = 06033 1205831199113(1199113(1198834

)) = 07844 and1205831199114(1199114(1198834

)) = 06033Generate 119909119895 by 119883119895 (119895 = 1 2 3 4) following Step 10 and

calculate the value of 119911(119909119895) As shown in Tables 2 and 3 thesolution 119909119895+1 (or119883119895+1) is better than 119909119895 (or119883119895) 119895 = 1 2 3

Following Step 10 we find

119883lowast

= 1198834

= (26754 39106 0 0 51843 51848 0 46128)119879

(47)

Therefore

119909lowast

= 1199094

= ((26754 39106 0 0)119871119877

(51843 51848 0 46128)119871119877)119879

(48)

serves as the compromised optimal solution with corre-sponding objective value

119911lowast

= 119911 (119909lowast

) = (5234 7403 1304 3896)119871119877 (49)

5 Concluding Remarks

To the end we show the following concluding remarks

(1) In this paper we proposed a new method to findthe compromised optimal solution to the fully fuzzylinear programming problemswith all the parametersand variables represented by 119871119877 flat fuzzy numbersThe solution is also an 119871119877 flat fuzzy number In thissense we get an exact solution which may give morehelp to the decision makers

(2) The order relation in the objective function is thesame as that in the constraint inequalities Based onthe definition of the order relation the FFLP can beequivalently transformed into a MOLP which is acrisp programming that is easy to be solved

(3) Considering the MOLP problem classical fuzzy pro-gramming method is modified for obtaining thecompromised optimal solution

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous reviewers for their valuable comments which havebeen very helpful in improving the paper This work is sup-ported by the PhD Start-up Fund of Natural Science Founda-tion of Guangdong Province China (S2013040012506) andthe China Postdoctoral Science Foundation Funded Project(2014M562152)

References

[1] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquoManagement Science vol 17 pp B141ndashB164 1970

[2] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[3] M Dehghan and B Hashemi ldquoSolution of the fully fuzzylinear systems using the decomposition procedurerdquo Applied

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Mathematics and Computation vol 182 no 2 pp 1568ndash15802006

[4] M Dehghan B Hashemi and M Ghatee ldquoSolution of the fullyfuzzy linear systems using iterative techniquesrdquo Chaos Solitonsamp Fractals vol 34 no 2 pp 316ndash336 2007

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

[6] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[7] 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

[8] J Kaur and A Kumar ldquoExact fuzzy optimal solution of fullyfuzzy linear programming problems with unrestricted fuzzyvariablesrdquo Applied Intelligence vol 37 no 1 pp 145ndash154 2012

[9] J Kaur and A Kumar ldquoMeharrsquos method for solving fully fuzzylinear programming problems with L-R fuzzy parametersrdquoApplied Mathematical Modelling vol 37 no 12-13 pp 7142ndash7153 2013

[10] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[11] 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

[12] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[13] S H Nasseri and F Zahmatkesh ldquoHuang method for solvingfully fuzzy linear system of equationsrdquo Journal of Mathematicsand Computer Science vol 1 pp 1ndash5 2010

[14] H-J Zimmermann ldquoFuzzy programming and linear program-ming with several objective functionsrdquo Fuzzy Sets and Systemsvol 1 no 1 pp 45ndash55 1978

[15] B-Y Cao Fuzzy Geometric Programming Kluwer AcademicPublishers Boston Mass USA 2002

[16] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of