ORIGINAL ARTICLE
A nonlinear model for fully fuzzy linear
programming with fully unrestricted variables and
parameters
H. Saberi Najafi
a, S.A. Edalatpanah
a, H. Dutta
b,* aDepartment of Mathematics, Faculty of Sciences, University of Guilan, University Campus 2, Rasht, Iran
b
Department of Mathematics, Gauhati University, Guwahati 781014, India
Received 10 June 2015; revised 22 April 2016; accepted 28 April 2016 Available online 26 May 2016
KEYWORDS
Fully fuzzy linear program-ming;
Crisp nonlinear program-ming;
Fuzzy arithmetic; Fuzzy optimal solution; Triangular fuzzy numbers
Abstract Several methods currently exist for solving fuzzy linear programming problems under nonnegative fuzzy variables and restricted fuzzy coefficients. However, due to the limitation of these methods, they cannot be applied for solving fully fuzzy linear programming (FFLP) with unre-stricted fuzzy coefficients and fuzzy variables. In this paper a new efficient method for FFLP has been proposed, in order to obtain the fuzzy optimal solution with unrestricted variables and param-eters. This proposed method is based on crisp nonlinear programming and has a simple structure. To show the efficiency of our proposed method some numerical examples have been illustrated. Ó2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an
open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Linear Programming (LP) is one of the most important tech-niques in operational research. Many real-world problems can be transformed into a linear programming model. Hence this model is an indispensable tool for today’s applications, such as energy, transportation and manufacturing. In real world applications, certainty, reliability and precision of data are often illusory. The optimal solution of an LP only depends on a limited number of constraints; therefore, much of the col-lected information has little impact on the solution. It is useful
to consider the knowledge of experts about the parameters as fuzzy data. The concept of fuzzy set was introduced in [1]. After this, Bellman and Zadeh[2]introduced fuzzy optimiza-tion problems where they have stated that, a fuzzy decision can be viewed as the intersection of fuzzy goals and problem constraints. Many researchers such as Zimmermann [3], Tanaka and Asai [4], Campos and Verdegay [5], Rom-melfanger et al.[6], Cadenas and Verdegay[7]who were deal-ing with the concept of solvdeal-ing fuzzy optimization problems, later studied this subject. In the past few years, a growing inter-est has been shown in fuzzy optimization. Buckley and Feuring [8] introduced a general class of fuzzy linear programming, called fully fuzzified linear programming problems, where all decision parameters and variables are fuzzy numbers. Lodwick and Bachman[9]have studied large scale fuzzy and possible optimization problems. Buckley and Abdalla[10]have consid-ered Monte Carlo methods in fuzzy queuing theory. Some authors have considered fuzzy linear programming, in which * Corresponding author.
E-mail addresses:[email protected](H. Saberi Najafi),
[email protected](S.A. Edalatpanah),[email protected]
(H. Dutta).
Peer review under responsibility of Faculty of Engineering, Alexandria University.
H O S T E D BY
Alexandria University
Alexandria Engineering Journal
www.elsevier.com/locate/aej www.sciencedirect.com
http://dx.doi.org/10.1016/j.aej.2016.04.039
1110-0168Ó2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
not all parts of the problem were assumed to be fuzzy, e.g., only the right hand side and the objective function coefficients were fuzzy, or only the variables were fuzzy[11–17]. The fuzzy linear programming problems in which all the parameters and variables are represented by fuzzy numbers are known as fully fuzzy linear programming (FFLP) problems. The FFLP prob-lem with inequality constraints was studied in[8,18,19]. How-ever, the main disadvantage of the solution obtained by the existing methods is that, it does not satisfy the constraints exactly i.e. it is not possible to obtain the fuzzy number of the right hand side of the constraint by putting the obtained solution on the left hand side of the constraint.
Dehghan et al. [20] proposed some practical methods to solve the fully fuzzy linear system (FFLS) that is comparable to the well known methods. Then they extended a new method employing Linear Programming (LP) for solving square and non-square fuzzy systems. Lotfi et al.[21]applied the concept of the symmetric triangular fuzzy number, and obtained a new method for solving FFLP by converting a FFLP into two cor-responding LPs. Kumar et al.[22]pointed out the shortcom-ings of the above methods. To overcome these shortcomshortcom-ings, they proposed a new method for finding the fuzzy optimal solution of FFLP problems with equality constraints. Saberi Najafi and Edalatpanah[23]pointed out the method of [22] needs some corrections to make the model well in general. Kaur and Kumar[24]proposed a new method for FFLP prob-lems with equality constraints having non-negative fuzzy coef-ficients and unrestricted fuzzy variables. Jayalakshmi and Pandian[25]presented a new method where, the given FFLP problem is decomposed into three crisp linear programming (CLP) problems with bounded variable constraints. Fan et al.[26] adopted thea-cut level to deal with a generalized fuzzy linear programming problem. Shamooshaki et al. [27] using L-R fuzzy numbers and ranking function established a new scheme for FFLP. Ezzati et al.[28]using a new ordering on triangular fuzzy numbers and converting FFLP to a multi-objective linear programming (MOLP) problem, presented a new method to solve FFLP. Recently, a number of researchers have exhibited their interest to solve the fuzzy linear program-ming problems[29–39]. In general, the above existing methods can be applied to the following types of FFLP problems:
(i) FFLP problem with nonnegative fuzzy coefficients and nonnegative fuzzy variables.
(ii) FFLP problem with unrestricted fuzzy coefficients and nonnegative fuzzy variables.
(iii) FFLP problem with nonnegative fuzzy coefficients and unrestricted fuzzy variables.
In crisp LP topic and application of it, we can see some LP problem in which some or all the variables are unrestricted. So these situations can be occurring in fuzzy linear programming problem, i.e. a fuzzy linear programming problem in which some or all the parameters are represented by unrestricted fuzzy numbers. As far as we are aware, there are only a few papers dealing with solution of fuzzy LP problems with fully unrestricted variables and parameters. In this paper, we focus on this problem and propose a new efficient method for solving FFLP with unrestricted variables and parameters. The proposed method is based on crisp nonlinear program-ming and has a simple and clear structure.
2. Preliminaries
In this section, we give some basic notations and preliminary results which are essential tools for describing our main results. For details, we refer to[22–24].
Definition 2.1. Let X denote a universal set. Then a fuzzy subset Ae of X is defined by its membership function
lAe: X! ½0;1; which assigns a real number le
AðxÞ in the
interval [0, 1], to each elementx2X, where the value ofle
AðxÞ
atxshows the grade of membership ofxinAe.
A fuzzy subset Aecan be characterized as a set of ordered pairs of element x and grade le
AðxÞ and is often written e
A¼ fðx;le
AðxÞÞ; x2Xg.
The class of fuzzy sets onXis denoted withCðXÞ:
Definition 2.2. A fuzzy number Ae¼ ða;b;cÞ is said to be a triangular fuzzy number if its membership function is given
leAðxÞ ¼ ðxaÞ ðbaÞ; a6x6b; ðcxÞ ðbcÞ; b6x6c; 0; else: 8 > > < > > :
Definition 2.3. A triangular fuzzy numberAe¼ ða;b;cÞis said to be nonnegative fuzzy number if and only ifaP0. The set of non-negative fuzzy numbers may be represented byF(R+).
Definition 2.4. A triangular fuzzy numberAe¼ ða;b;cÞis said to be unrestricted fuzzy number ifa;b;c2R. The set of unre-stricted fuzzy numbers can be represented byF(R).
Definition 2.5. LetAe¼ ða;b;cÞ,Be¼ ðd;e;fÞbe two triangular fuzzy numbers then;
(i) eAeB¼ ða;b;cÞ ðd;e;fÞ ¼ ðaþd;bþe;cþfÞ; (ii) eAeB¼ ða;b;cÞ ðd;e;fÞ ¼ ðaf;be;cdÞ; (iii) eABe¼ ðminðcÞ;be;maxðcÞÞwhere,c¼ fad;af;cd;cfg.
Definition 2.6. Two triangular fuzzy numbers Ae¼ ða;b;cÞ,
e
B¼ ðd;e;fÞ are said to be equal if and only if a¼d; b¼e
andc¼f.
Definition 2.7. A ranking function is a function
R: FðRÞ !R; which maps each fuzzy number into the real line, where a natural order exists. Let Ae¼ ða;b;cÞbe any tri-angular fuzzy number, thenRðAeÞ ¼1
4ðaþ2bþcÞ:
Definition 2.8. LetAe¼ ða;b;cÞ,Be¼ ðd;e;fÞbe two triangular fuzzy numbers, then
(i) eA6Be iffRðeAÞ6RðBeÞ, (ii) eA<Be iffRðeAÞ<RðeBÞ.
3. Proposed method
In this section, first we define arithmetic operations, based on Definition 2.5.
Definition 3.1. LetAe¼ ða;b;cÞ,Be¼ ðx;y;zÞbe two triangular fuzzy numbers. Then we have,
e ABe¼ ðminðax;cxÞ;by;maxðaz;czÞÞ; if aP0; ðminðcx;azÞ;by;maxðcz;axÞÞ; if a<0; c>0; ðminðaz;czÞ;by;maxðax;cxÞÞ; if c60: 8 > < > : ð3:1Þ Then, e ABe¼ ðf0:5ðaþcÞx0:5ðcaÞjxjg;by; f0:5ðaþcÞzþ0:5ðcaÞjzjgÞ; if aP0; ðf0:5ðazþcxÞ 0:5jcxazjg;by; f0:5ðaxþczÞ þ0:5jczaxjgÞ; if a<0;c>0; ðf0:5ðaþcÞz0:5ðcaÞjzjg;by; f0:5ðaþcÞxþ0:5ðcaÞjxjgÞ; if c60: 8 > > > > > > > > < > > > > > > > > : ð3:2Þ Now, we introduce splitting technique for fuzzy coefficients matrix. LetAebemntriangular fuzzy matrix, then we splitAe
into following: e A¼AeþAeAe; where ðAeþÞmn¼ ððai;j;bi;j;ci;jÞÞmn¼ ðai;j;bi;j;ci;jÞ; if ai;jP0; ð0;0;0Þ; else: ðAeÞmn¼ ððai;j;bi;j;ci;jÞÞmn¼ ðai;j;bi;j;ci;jÞ; if ai;j60;ci;jP0; ð0;0;0Þ; else: ðAeÞmn¼ ððai;j;bi;j;ci;jÞÞmn¼ ðai;j;bi;j;ci;jÞ; if ci;j<0; ð0;0;0Þ; else:
For example, for fuzzy matrix
e A¼ ðð3;2;0Þ ð6;7;8Þ 2;4;6Þ ð2;1;2Þ , we get, e A¼AeþAeAe¼ ð0;0;0Þ ð6;7;8Þ ð2;4;6Þ ð0;0;0Þ ð0;0;0Þ ð0;0;0Þ ð0;0;0Þ ð2;1;2Þ ð3;2;0Þ ð0;0;0Þ ð0;0;0Þ ð0;0;0Þ :
Next, consider the following fully fuzzy linear program-ming (FFLP) withmconstraints andnvariables:
Min ~z¼Ce~x
s:t: Aex~¼~b; ~xis unrestricted fuzzy number: whereCeT¼ ½~c
j1nand rankðAe;b~Þ ¼rankðAeÞ ¼m. Due to the
limitation of the existing methods, these methods cannot be applied for solving above FFLP problem. Here, to overcome
the limitations of the existing methods, a new method is proposed.
The steps of the proposed method are as follows:
Step 1: By using the splitting technique, write above FFLP as follows: Maxðor MinÞ ~z¼ X n j¼1 ~ cj~xj ! s:t: ðAeþAeAeÞ ~xj¼b~i; 8i¼1;2;...;m;; x~j2FðRÞ:
Step 2: If all the parameters~cj;~xj;~aþi;j;~aþ;~ai;jand~biare rep-resented by triangular fuzzy numbersðpj;qj;rjÞ;ðxj;yj;zjÞ; ðaþi;j;bþi;j;cþi;jÞ;ðai;j;bi;j;ci;jÞ;ðai;j;bi;j;ci;jÞ, and ðbi;gi;hiÞ respectively, then the FFLP problem obtained in Step 1 may be written as follows:
Maxðor MinÞ ~z¼X n j¼1 pjP0 ðpj;qj;rjÞ ðxj;yj;zjÞ Xn j¼1 pj<0 rjP0 ðpj;qj;rjÞ ðxj;yj;zjÞ Xn j¼1 rj<0 ðpj;qj;rjÞ ðxj;yj;zjÞ s:t: X n j¼1 ðaþi;j;bþi;j;cþi;jÞ ðxj;yj;zjÞ Xn j¼1 ðai;j;bi;j;ci;jÞ ðxj;yj;zjÞ X n j¼1 ðai;j;bi;j;ci;jÞ ðxj;yj;zjÞ ¼ ðbi;gi;hiÞ; ðxj;yj;zjÞ 2FðRÞ; 8i¼1;2;. . .;m:
Step 3: Using(3.1)andDefinition 3.1, the FFLP problem obtained in Step 2 can be written as follows:
Maxðor MinÞ ~z¼X n j¼1 pjP0 ðminðpjxj;rjxjÞ;qjyj;maxðpjzj;rjzjÞÞ þX n j¼1 pj<0 rjP0 ðminðpjzj;rjxjÞ;qjyj;maxðpjxj;rjzjÞÞ þX n j¼1 rj<0 ðminðpjzj;rjzjÞ;qjyj;maxðpjxj;rjxjÞÞ s:t: X n j¼1 ai;jP0 ðminðai;jxj;ci;jxjÞ;bi;jyj;maxðai;jzj;ci;jzjÞÞ Xn j¼1 ai;j<0 ci;jP0 ðminðai;jzj;ci;jxjÞ;bi;jyj;maxðai;jxj;ci;jzjÞÞ X n j¼1 ci;j<0 ðminðai;jzj;ci;jzjÞ;bi;jyj;maxðai;jxj;ci;jxjÞÞ ¼ ðbi;gi;hiÞ; ðxj;yj;zjÞ 2FðRÞ; 8i¼1;2;. . .;m:
Step 4: Using(3.2)andDefinition 3.1, the FFLP problem obtained in Step 3 can be written as follows:
Step 5: UsingDefinition 2.7, the fuzzy objective function of the FFLP problem, obtained in Step 4, can be converted into the crisp objective function and the obtained fuzzy lin-ear programming problem can be written as follows:
Step 6: Using Definitions 2.6–2.8, the fuzzy linear program-ming problem, obtained in Step 5, is converted into the fol-lowing crisp nonlinear programming problem:
Maxðor MinÞ ~z¼X n j¼1 pjP0 0:5ðpjþrjÞxj0:5ðrjpjÞjxjj;qjyj;0:5ðpjþrjÞzjþ0:5ðrjpjÞjzjj þX n j¼1 pj<0 rjP0 0:5ðpjzjþrjxjÞ 0:5pjzjrjxj;qjyj;0:5ðpjxjþrjzjÞ þ0:5jpjxjrjzjj þX n j¼1 rj<0 0:5ðpjþrjÞzj0:5ðrjpjÞjzjj;qjyj;0:5ðpjþrjÞxjþ0:5ðrjpjÞjxjj s:t: X n j¼1 ai;jP0 0:5ðai;jþci;jÞxj0:5ðci;jai;jÞjxjj;bi;jyj;0:5ðai;jþci;jÞzjþ0:5ðci;jai;jÞjzjj þX n j¼1 ai;j<0 ci;jP0 0:5ðai;jzjþci;jxjÞ 0:5jai;jzjci;jxjj;bi;jyj;0:5ðai;jxjþci;jzjÞ þ0:5jai;jxjci;jzjj þX n j¼1 ci;j<0 0:5ðai;jþci;jÞzj0:5ðci;jai;jÞjzjj;bi;jyj;0:5ðai;jþci;jÞxjþ0:5ðci;jai;jÞjxjj ¼ ðbi;gi;hiÞ; ðxj;yj;zjÞ 2FðRÞ; 8i¼1;2;. . .;m: Maxðor MinÞ R X n j¼1 pjP0 0:5ðpjþrjÞxj0:5ðrjpjÞjxjj;qjyj;0:5ðpjþrjÞzjþ0:5ðrjpjÞjzjj 0 B @ 1 C A þR X n j¼1 pj<0 rjP0 0:5ðpjzjþrjxjÞ 0:5jpjzjrjxjj;qjyj;0:5ðpjxjþrjzjÞ þ0:5jpjxjrjzjj 0 B B B @ 1 C C C A þR X n j¼1 rj<0 0:5ðpjþrjÞzj0:5ðrjpjÞjzjj;qjyj;0:5ðpjþrjÞxjþ0:5ðrjpjÞjxjj 0 B @ 1 C A s:t: X n j¼1 ai;jP0 0:5ðai;jþci;jÞxj0:5ðci;jai;jÞjxjj;bi;jyj;0:5ðai;jþci;jÞzjþ0:5ðci;jai;jÞjzj þXn j¼1 ai;j<0 ci;jP0 0:5ðai;jzjþci;jxjÞ 0:5jai;jzjci;jxjj;bi;jyj;0:5ðai;jxjþci;jzjÞ þ0:5jai;jxjci;jzjj þX n j¼1 ci;j<0 0:5ðai;jþci;jÞzj0:5ðci;jai;jÞjzjj;bi;jyj;0:5ðai;jþci;jÞxjþ0:5ðci;jai;jÞjxjj ¼ ðbi;gi;hiÞ; ðxj;yj;zjÞ 2FðRÞ; 8i¼1;2;. . .;m:
Step 7: Find the optimal solution xj;yj;zj by solving the crisp nonlinear programming problem obtained in Step 6. Step 8: Find the fuzzy optimal solution by putting the val-ues ofxj;yj;zjin~xj¼ ðxj;yj;zjÞ:
Step 9: Find the fuzzy optimal value by putting ~xj in
Pn
j¼1~cj~xj.
4. Numerical examples
Here, we give some examples to illustrate the results obtained in pervious section.
Example 4.1. Consider the following FFLP problem Max ~z¼ ðð1;2;3Þ ðx1;y1;z1ÞÞ ðð2;3;4Þ ðx2;y2;z2ÞÞ subject to ðð3;2;0Þ ðx1;y1;z1ÞÞ ðð6;7;8Þ ðx2;y2;z2ÞÞ
¼ ð30;17;3Þ;
ðð2;4;6Þ ðx1;y1;z1ÞÞ ðð2;1;2Þ ðx2;y2;z2ÞÞ ¼ ð4;8;18Þ:
whereðx1;y1;z1Þ;ðx2;y2;z2Þare unrestricted triangular fuzzy numbers:
Step 1: By using the splitting technique, write above FFLP as follows: Max z~¼ ðð1;2;3Þ ðx1;y1;z1ÞÞ ðð2;3;4Þ ðx2;y2;z2ÞÞ s:t: ð0;0;0Þ ð6;7;8Þ ð2;4;6Þ ð0;0;0Þ " # ð0;0;0Þ ð0;0;0Þ ð0;0;0Þ ð2;1;2Þ " # ð3;2;0Þ ð0;0;0Þ ð0;0;0Þ ð0;0;0Þ " #! ðx1;y1;z1Þ ðx2;y2;z2Þ " # ¼ ð30;17;3Þ ð4;8;18Þ " # ; ðxj;yj;zjÞ 2FðRÞ; j¼1;2:
Step 2: Using the arithmetic operations,Definition 3.1, the FFLP problem, obtained in Step 1 can be written as follows:
Max ~z¼ ððminðz1;3x1Þ;2y1;maxðx1;3z1Þ ðminð2x2;4x2Þ;3y2;maxð2z2;4z2ÞÞ s:t: ðminð6x2;8x2Þ;7y2;maxð6z2;8z2ÞÞ ðminð3z1;0Þ;2y1;maxð3x1;0ÞÞ ¼ ð30;17;3Þ; ðminð2x1;6x1Þ;4y1;maxð2z1;6z1ÞÞ ðminð2z2;2x2Þ;y2;maxð2x2;2z2ÞÞ ¼ ð4;8;18Þ; ðxj;yj;zjÞ 2FðRÞ; j¼1;2:
Step 3: By using(3.2),Definition 3.1, the FFLP problem, obtained in Step 2 can be written as follows:
Max ~z¼ð0:5ðz1þ3x1Þ 0:5jz1þ3x1j;2y1;0:5ðx1þ3z1Þ þ0:5jx1þ3z1jÞÞ ð3x2 jx2j;3y2;3z2þ jz2jÞ Maxðor MinÞ 1 4 Xn j¼1 pjP0 0:5ðpjþrjÞxj0:5ðrjpjÞjxjj þ2ðqjyjÞ þ0:5ðpjþrjÞzjþ0:5ðrjpjÞjzjj 0 B @ 1 C A þ1 4 Xn j¼1 pj<0 rjP0 0:5ðpjzjþrjxjÞ 0:5jpjzjrjxjj þ2ðqjyjÞ þ0:5ðpjxjþrjzjÞ þ0:5jpjxjrjzjj 0 B B B @ 1 C C C A þ1 4 Xn j¼1 rj<0 0:5ðpjþrjÞzj0:5ðrjpjÞjzjj þ2ðqjyjÞ þ0:5ðpjþrjÞxjþ0:5ðrjpjÞjxjj 0 B @ 1 C A s:t: X n j¼1 ai;jP0 0:5ðai;jþci;jÞxj0:5ðci;jai;jÞjxjj þX n j¼1 ai;j<0 ci;jP0 0:5ðai;jzjþci;jxjÞ 0:5ai;jzjci;jxj þX n j¼1 ci;j<0 0:5ðai;jþci;jÞzj0:5ðci;jai;jÞjzjj ¼bi; Xn j¼1 bi;jyj¼gi; Xn j¼1 ai;jP0 0:5ðai;jþci;jÞzjþ0:5ðci;jai;jÞjzjj þX n j¼1 ai;j<0 ci;jP0 0:5ðai;jxjþci;jzjÞ þ0:5jai;jxjci;jzjj þX n j¼1 ci;j<0 0:5ðai;jþci;jÞxjþ0:5ðci;jai;jÞjxjj ¼hi; yjxjP0; zjyjP0: wherexj;yj;zj2R 8i¼1;2;. . .;m; j¼1;2;. . .;n:
s:t: ð7x2 jx2j;8x2Þ;7y2;7z2þ jz2jÞ
ð1:5z11:5jz1j;2y1;1:5x1þ1:5jx1jÞ ¼ ð30;17;3Þ; ð4x12jx1j;4y1;4z1þ2jz1jÞ
ðz2þx2 jz2þx2j;y2;x2þz2þ jz2þx2jÞ ¼ ð4;8;18Þ; ðxj;yj;zjÞ 2FðRÞ;j¼1;2:
Step 4: By usingDefinition 2.7the fuzzy objective function of the FFLP problem, obtained in Step 3, can be converted into the crisp objective function and the obtained fuzzy linear programming problem can be written as follows:
Max Rð0:5ðz1þ3x1Þ 0:5jz1þ3x1j;2y1;0:5ðx1þ3z1Þ þ0:5jx1þ3z1jÞÞ Rð3x2 jx2j;3y2;3z2þ jz2jÞ s:t: ð7x2 jx2j;8x2Þ;7y2;7z2þ jz2jÞ ð1:5z11:5jz1j;2y1;1:5x1þ1:5jx1jÞ ¼ ð30;17;3Þ; ð4x12jx1j;4y1;4z1þ2jz1jÞ ðz2þx2 jz2þx2j;y2;x2þz2þ jz2þx2jÞ ¼ ð4;8;18Þ; ðxj;yj;zjÞ 2FðRÞ; j¼1;2:
Step 5: Using Definitions 2.6–2.8 the fuzzy linear program-ming problem, obtained in Step 4, is converted into the following crisp nonlinear programming problem:
Max 1 4ð0:5ðz1þ3x1Þ 0:5jz1þ3x1j þ4y1þ0:5ðx1þ3z1Þ þ0:5jx1þ3z1j þ3x2 jx2j þ6y2;3z2þ jz2jÞ s:t: 7x2 jx2j 1:5z11:5jz1j ¼ 30; 8x22y1¼ 17; 7z2þ jz2j 1:5x1þ1:5jx1j ¼ 3; 4x12jx1j z2þx2 jz2þx2j ¼ 4; 4y1y2¼8; 4z1þ2jz1j x2þz2þ jz2þx2j ¼18; yjxjP0;zjyjP0: wherexj;yj;zj2R 8i¼1;2;j¼1;2:
Step 6: The optimal solution of the crisp nonlinear pro-gramming problem, obtained in Step 5, is
x1¼1;y1¼1:5;z1¼2;x2¼ 3;y2¼ 2;z2¼ 0:5
Step 7: By putting the values ofxj;yj;zjin~xj, the fuzzy opti-mal solution is~x1¼ ð1;1:5;2Þ;~x2¼ ð3;2;0:5Þ: Step 8: By putting~xjin
Pn
j¼1~cj~xj, the fuzzy optimal value is~z¼ ð14;3;5Þ:
InTable 1, we solve some other FFLP problems with our
method. We can see that, when some or all variables are non-negative, the method also works well.
Example 4.2. Max z~¼ ðð5;2;1Þ ðx1;y1;z1ÞÞ ðð3;0;1Þ ðx2;y2;z2ÞÞ subject to 1;2;5 2 ðx1;y1;z1Þ 3 2 ; 1 2 ;0 ðx2;y2;z2Þ ¼ 15 2 ; 5 4 ;7 ; ðð1;3;5Þ ðx1;y1;z1ÞÞ 14;12;1 ðx2;y2;z2Þ ¼ 6;5 4 ;9 :
whereðx1;y1;z1Þ;ðx2;y2;z2Þare unrestricted triangular fuzzy numbers: Example 4.3. Max z~¼ ðð1;2;3Þ ðx1;y1;z1ÞÞ ðð2;3;4Þ ðx2;y2;z2ÞÞ subject to ðð0;1;2Þ ðx1;y1;z1ÞÞ ðð6;4;2Þ ðx2;y2;z2ÞÞ ¼ 18;5;11 2 ; ðð5;3;2Þ ðx1;y1;z1ÞÞ ðð1;2;3Þ ðx2;y2;z2ÞÞ ¼ 59 4;5;21 :
whereðx1;y1;z1Þ;ðx2;y2;z2Þare nonnegative and unrestricted fuzzy numbers;respectively:
Example 4.4. Min ~z¼ ðð1;3;9Þ ðx1;y1;z1ÞÞ ðð2;4;8Þ ðx2;y2;z2ÞÞ subject to ðð1;3;5Þ ðx1;y1;z1ÞÞ ðð2;3;4Þ ðx2;y2;z2ÞÞ ¼ ð1;9;22Þ; ðð1;2;3Þ ðx1;y1;z1ÞÞ ðð2;3;4Þ ðx2;y2;z2ÞÞ ¼ ð1;8;18Þ:
whereðx1;y1;z1Þ;ðx2;y2;z2Þare nonnegative fuzzy numbers:
5. Conclusions and future work
In the past few years, a growing interest has been shown in fuzzy linear programming and currently there are several methods for solving FFLP. However, no paper has discussed the solving FFLP with unrestricted fuzzy coefficients and fuzzy variables yet. In this paper a new efficient method for FFLP has been proposed, in order to obtain the fuzzy optimal solu-tion with unrestricted variables and parameters. Furthermore, the limitations of other existing methods have been pointed out. To show the efficiency of the proposed method some numerical examples have been illustrated.
In addition, although the model, arithmetic operations and results presented here have demonstrated the effectiveness of our approach, it could be further applied in other fuzzy opti-mization problem such as the fuzzy bounded problems, fuzzy assignment, fuzzy travelling salesman and fuzzy generalized assignment. As future researches, we intend to study these problems.
Table 1 Results of the chosen problems.
Example Fuzzy optimal solution Fuzzy optimal value 4.2 x~1¼ 53;21;1 ~ x2¼ ð3;12;4Þ ~z¼ ð17;1;12Þ 4.3 x~1¼ ð3;1;3Þ x~2¼ 14;1;2 ~ z¼ ð5 2;2;11Þ 4.4 x~1¼ ð0;1;2Þ x~2¼21;2;3 ~z¼ ð1;11;42Þ
Acknowledgments
The authors would like to thank all the three reviewers for their valuable suggestions which have led to an improvement in both the quality and clarity of this paper.
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