Research Article A MOLP Method for Solving Fully …downloads.hindawi.com/journals/mpe/2014/782376.pdf · A MOLP Method for Solving Fully Fuzzy Linear Programming with LR Fuzzy Parameters
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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]
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
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
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
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)
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
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
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]
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
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
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
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)
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
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
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
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
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
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)
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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)
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
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
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
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
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
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