A new algorithm to solve fully fuzzy linear ...

Applied Mathematical Modelling xxx (2013) xxx–xxx

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A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem R. Ezzati a,⇑, E. Khorram b,1, R. Enayati a a b

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Avenue, 15914 Tehran, Iran

a r t i c l e

i n f o

Article history: Received 13 June 2011 Received in revised form 21 January 2013 Accepted 2 March 2013 Available online xxxx Keywords: Fully fuzzy linear programming problem Multi-objective linear programming problem Lexicographic ordering on triangular fuzzy numbers

a b s t r a c t Recently, two new algorithms have been proposed to solve a fully fuzzy linear programming (FFLP) problem by Lotfi et al. [F.H. Lotfi, T. Allahviranloo, M.A. Jondabeha, L. Alizadeh, Solving a fully fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Model. 33 (2009) 3151–3156] and Kumar et al. [A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Model. 33 (2011) 817–823]. In this paper, based on a new lexicographic ordering on triangular fuzzy numbers, a novel algorithm is proposed to solve the FFLP problem by converting it to its equivalent a multi-objective linear programming (MOLP) problem and then it is solved by the lexicographic method. By a theorem, it is shown that the lexicographic optimal solution of MOLP problem can be considered as an optimal solution of the FFLP problem. Then, a simple example and two real problems, as two case studies, will be used to illustrate our algorithm and compare it with the existing methods. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Nowadays, the decision of human is increasingly depend on information more than ever. But, most of information is not deterministic and in this situation human has a capability to make a rational decision based on this uncertainty. This is hard challenge for decision maker to design an intelligent system which make a decision the same as the human. So, it was led to propose a new concept of decision making in fuzzy environment by Bellman and Zadeh [1]. One of interesting concepts in fuzzy optimization problems is to deal with fuzzy linear programming (FLP) problems. Heretofore, a number of methods have been proposed to solve the FLP problems [2–11]. The FLP problem is said to be a fully fuzzy linear programming (FFLP) problem if all parameters and variables are considered as fuzzy numbers. Recently, two methods have been introduced to solve the FFLP problems by Lotfi et al. [12] and Kumar et al. [13]. In the first method [12], the parameters of FFLP problem have been approximated to the nearest symmetric triangular fuzzy numbers. After that, a fuzzy optimal approximation solution has been achieved by solving a multi-objective linear programming (MOLP) problem. The shortcoming exists of it is that the optimal solution of FFLP is not exact. So, it is not reliable solution for decision maker. In the second method [13], an exact optimal solution is achieved using a linear ranking function. In this method, the linear ranking function has been used to convert the fuzzy objective function to the crisp objective function. The shortcoming exists of it is that the fuzziness of objective function has been neglected by the linear ranking function. ⇑ Corresponding author. Tel.: +98 912 3618518. 1

E-mail addresses: [email protected] (R. Ezzati), [email protected] (E. Khorram), [email protected] (R. Enayati). Tel.: +98 21 64542549.

0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.03.014

Please cite this article in press as: R. Ezzati et al., A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.03.014

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R. Ezzati et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

In this paper our motivation is to modify these two methods. First, the FFLP problem will be converted to a MOLP problem with three objective functions by use of a new lexicographic ordering on triangular fuzzy numbers. Then, the lexicographic method will be used to find a lexicographic optimal solution of MOLP problem. We prove that this lexicographic optimal solution can be considered as an exact optimal solution of FFLP problem. Finally, to show our advantages algorithm over existing methods [12,13], a simple example and two real world problems, namely the fuzzy transportation problem and the fuzzy investment problem, are used and compered our results with them. This paper is organized as follows: In Section 2, some definitions and notations of the fuzzy numbers and also a new definition related to lexicographic ordering on triangular fuzzy numbers are presented. In Section 3, a new algorithm is proposed to solve the FFLP problem by converting to its equivalent MOLP problem. In Section 4, a simple example and two two case studies are used to illustrate the proposed algorithm. In Section 5, Lotfi’s method, Kumar’s method and the proposed algorithm will be compared to each other. Finally, conclusions are derived in Section 6. 2. Preliminaries In this section, some definitions related to the fuzzy set theory, which will be used in the rest of paper, are given. Definition 2.1. A fuzzy number is a fuzzy set like u : R ! I ¼ ½0; 1 which satisfies, 1. u is upper semi-continuous, 2. uðxÞ ¼ 0 outside some interval ½c; d, 3. There are real numbers a; b such that c 6 a 6 b 6 d and 3.1 uðxÞ is monotonic increasing on ½c; a, 3.2 uðxÞ is monotonic decreasing on ½b; d, 3.3 uðxÞ ¼ 1; a 6 x 6 b. The set of all these fuzzy numbers is denoted by FðRÞ. An equivalent parametric form of that is presented as follows: e in parametric form is a pair ðu; uÞ of functions uðrÞ , uðrÞ , 0 6 r 6 1, which these Definition 2.2. [14] A fuzzy number u functions satisfy the following requirements: 1. uðrÞ is a bounded monotonic increasing left continuous function, 2. uðrÞ is a bounded monotonic decreasing left continuous function, 3. uðrÞ 6 uðrÞ , 0 6 r 6 1. e ¼ ða; b; c; dÞ with interval defuzzifier ½b; c and left fuzziness A popular fuzzy number is the trapezoidal fuzzy number u ðb aÞ and right fuzziness ðd cÞ, where its membership function is given as follows:

8 ba ; > xa > > < 1; leu ðxÞ ¼ dc > ; > > : dx 0;

if a 6 x 6 b; if x 2 ½b; c; if c 6 x 6 d; Otherwise:

e ¼ ða; b; c; dÞ can also signify a Its parametric form is uðrÞ ¼ a þ ðb aÞr; uðrÞ ¼ d þ ðc dÞr where 0 6 r 6 1. In particular, u e ¼ ða; b; dÞ ¼ ððuÞl ; ðuÞc ; ðuÞu Þ if b ¼ c. The set of all these triangular fuzzy numbers is denoted by triangular fuzzy number u TFðRÞ. e ¼ ðx1 ; y1 ; z1 Þ is said to be a non-negative triangular fuzzy number if and Definition 2.3. [14] A triangular fuzzy number u only if x1 P 0. The set of all these triangular fuzzy numbers is denoted by TFðRÞþ . e ¼ ðx2 ; y2 ; z2 Þ are said to be equal, u e¼v e , if and only if e ¼ ðx1 ; y1 ; z1 Þ and v Definition 2.4. [14] Two triangular fuzzy numbers u x1 ¼ x2 ; y1 ¼ y2 and z1 ¼ z2 . Definition 2.5. [14] The arithmetic operations between two triangular fuzzy numbers are defined by the extension principle and can be equivalently represented as follows: e ¼ ðx2 ; y2 ; z2 Þ be two triangular fuzzy numbers and k 2 R. Define. e ¼ ðx1 ; y1 ; z1 Þ and v Let u (i) (ii) (iii) (iv)

e ¼ ðkx1 ; ky1 ; kz1 Þ, k P 0; k u e ¼ ðkz1 ; ky1 ; kx1 Þ, k 6 0; k u ev e ¼ ðx1 þ x2 ; y1 þ y2 ; z1 þ z2 Þ, u ev e ¼ ðx1 z2 ; y1 y2 ; z1 x2 Þ, u



3

e ¼ ðx1 ; y1 ; z1 Þ be an arbitrary triangular fuzzy number and v e ¼ ðx2 ; y2 ; z2 Þ be a non-negative triangular fuzzy num(v) Let u ber, then

8 > < ðx1 x2 ; y1 y2 ; z1 z2 Þ if x1 P 0; e e e e u v ¼ u v ¼ ðx1 z2 ; y1 y2 ; z1 z2 Þ if x1 < 0; z1 P 0; > : ðx1 z2 ; y1 y2 ; z1 x2 Þ if z1 < 0:

In the following, we propose a new definition to compare two arbitrary triangular fuzzy numbers based on lexicographic method. e ¼ ðx1 ; y1 ; z1 Þ and v e ¼ ðx2 ; y2 ; z2 Þ be two arbitrary triangular fuzzy numbers. We say that u e is relatively Definition 2.6. Let u e , which is denoted by u ev e , if and only if: less than v (i) y1 < y2 or (ii) y1 ¼ y2 and ðz1 x1 Þ > ðz2 x2 Þ or (iii) y1 ¼ y2 ; ðz1 x1 Þ ¼ ðz2 x2 Þ and ðx1 þ z1 Þ < ðx2 þ z2 Þ.

e¼v e. Remark 2.1. It is clear that y1 ¼ y2 ; ðz1 x1 Þ ¼ ðz2 x2 Þ and ðx1 þ z1 Þ ¼ ðx2 þ z2 Þ if and only if u ev e if and only if u ev e or u e¼v e. Remark 2.2. u 3. Fully fuzzy linear programming problem One of the most important and applicable issues in optimization problems is a linear programming (LP) problem. In the LP problem, we are assume that all parameters and variables are real numbers. But in the real-life, we do not have precise information. So, the fuzzy numbers and fuzzy variables should be used in the LP problem. Therefore, we encounter with the FFLP problem. In this section, based on Definitions 2.5 and 2.6, we are going to propose a new algorithm to solve the FFLP problem such that all parameters and variables are considered as triangular fuzzy numbers. Consider the standard form of FFLP as follows:

e maxðminÞ e cT x e e e s:t: A x ¼ b;

ð1Þ

e ¼ ½e e ¼ ½e e ¼ ½e where e c T ¼ ½e c j 1 n ; x x j n 1 ; A a ij m n ; b b i m 1 ; e a ij ; e cj; e b i 2 TFðRÞ; e x j 2 TFðRÞþ ; i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n. e e e e e e It should be noted that A x b and A x b can be transformed to the standard form by introducing a vector variable e and A e respectively. ex ee e ee s ¼ ðes 1 ; es 2 ; . . . ; es m ÞT , where es j 2 TFðRÞþ j ¼ 1; 2; . . . ; m, as A s¼b xe s ¼ b, e ¼ ððx Þl ; ðx Þc ; ðx Þu Þ is said to be an exact optimal solution of problem (1) if it satisfies in the following Remark 3.1. x statements: (i) (ii) (iii)

e ¼ ½e x x j n 1 where e x j 2 TFðRÞþ j ¼ 1; 2; . . . ; n, e e e A x ¼ b, e e ee e ¼ ððxÞl ; ðxÞc ; ðxÞu Þ 2 e e (in case of minimizaxe cT x 8x S ¼ fe xjA x ¼ b; x ¼ ½e x j n 1 where e x j 2 TFðRÞþ g, we have that e cT e Te T e e e tion problem c x c x ).

e0 2 e e 0 is also x0 ¼ e x , then x S such that e cT e cT e Remark 3.2. Let e x be an exact optimal solution of problem (1) and there exist an x an exact optimal solution of problem (1) and is called an alternative exact optimal solution. In the following, we are going to introduce a new algorithm to find an exact optimal solution of FFLP problem. The steps of the proposed algorithm are given as follows: Step 1. With respect to Definitions 2.2 and 2.3, problem (1) can be shown as follows:

maxðminÞ ððcT xÞl ; ðcT xÞc ; ðcT xÞu Þ s:t: ððAxÞl ; ðAxÞc ; ðAxÞu Þ ¼ ððbÞl ; ðbÞc ; ðbÞu Þ;

ð2Þ

e ¼ ððbÞl ; ðbÞc ; ðbÞu Þ; x ee e ¼ ððcT xÞl ; ðcT xÞc ; ðcT xÞu Þ; A e ¼ ððxÞl ; ðxÞc ; ðxÞu Þ and ðxÞl P 0. Equivawhere e cT x x ¼ ððAxÞl ; ðAxÞc ; ðAxÞu Þ; b lently, with regard to Definition 2.4, problem (2) may be written as follows:

maxðminÞ ððcT xÞl ; ðcT xÞc ; ðcT xÞu Þ s:t: ðAxÞl ¼ ðbÞl ; c

l

ðAxÞc ¼ ðbÞc ;

ðxÞ ðxÞ P 0;

u

ðAxÞu ¼ ðbÞu ; c

ðxÞ ðxÞ P 0;

ð3Þ

ðxÞl P 0:


4


Step 2. Regarding to Definition 2.6, problem (3) is converted to the MOLP problem with three crisp objective functions as follows:

maxðminÞ ðcT xÞc ; minðmaxÞ ððcT xÞu ðcT xÞl Þ; maxðminÞ ððcT xÞl þ ðcT xÞu Þ; s:t: ðAxÞl ¼ ðbÞl ;

ð4Þ

ðAxÞc ¼ ðbÞc ;

ðxÞc ðxÞl P 0;

ðAxÞu ¼ ðbÞu ;

ðxÞu ðxÞc P 0;

ðxÞl P 0:

Step 3. In terms of the preference of objective functions, the lexicographic method will be used to obtain a lexicographically optimal solution of problem (4). So, we have:

maxðminÞ ðcT xÞc ; s:t: ðAxÞl ¼ ðbÞl ; c

ðAxÞc ¼ ðbÞc ;

l

ðxÞ ðxÞ P 0;

u

ðAxÞu ¼ ðbÞu ; c

ð5Þ

ðxÞl P 0:

ðxÞ ðxÞ P 0;

e ¼ ððx Þl ; ðx Þc ; ðx Þu Þ, then it is an optimal solution of problem (2) and If problem (5) has a unique optimal solution, namely x stop. Otherwise go to Step 4. Step 4. Solve the following problem over the optimal solutions that are achieved in Step 3 as follows:

minðmaxÞ ððcT xÞu ðcT xÞl Þ; s:t: ðcT xÞc ¼ m ; ðAxÞl ¼ ðbÞl ;

ðAxÞc ¼ ðbÞc ;

ðxÞc ðxÞl P 0;

ðAxÞu ¼ ðbÞu ;

ðxÞu ðxÞc P 0;

ð6Þ

ðxÞl P 0;

e ¼ ððx Þl ; ðx Þc ; ðx Þu Þ, where m is the optimal value of problem (5). If problem (6) has a unique optimal solution, namely x then it is also an optimal solution of problem (2) and stop. Otherwise go to Step 5. Step 5. Solve the following problem over the optimal solutions that are achieved in Step 4 as follows:

maxðminÞ ððcT xÞl þ ðcT xÞu Þ; s:t: ðcT xÞu ðcT xÞl ¼ n ; ðcxÞc ¼ m ; l

l

ðAxÞ ¼ ðbÞ ; c

l

ð7Þ c

c

u

ðAxÞ ¼ ðbÞ ;

ðxÞ ðxÞ P 0;

u

u

ðAxÞ ¼ ðbÞ ; c

ðxÞ ðxÞ P 0;

ðxÞl P 0;

e ¼ ððx Þl ; ðx Þc ; ðx Þu Þ, is obwhere n is the optimal value of problem (6). So, the optimal solution of problem (2), namely x tained by solving problem (7). Now by a theorem, it is shown that the obtained lexicographic optimal solution of problem (4) can be considered as an exact optimal solution of problem (2). e ¼ ððx Þl ; ðx Þc ; ðx Þu Þ be an optimal solution of problems (5)–(7) (and hence is a lexicographic optimal solution Theorem 3.1. If x of problem (4)), then it is also an exact optimal solution of problem (2). e ¼ ððx Þl ; ðx Þc ; ðx Þu Þ be an optimal solution of (5)–(7), but it is not the exact optimal solution Proof. By contradiction, let x e o ¼ ððxo Þl ; ðxo Þc ; ðxo Þu Þ – x e , such that of problem (2). Therefore, there exists a feasible solution of problem (2), namely x T l T c T u T o l T o c T o u T l T c ððc x Þ ; ðc x Þ ; ðc x Þ Þ ððc x Þ ; ðc x Þ ; ðc x Þ Þ (in case of minimization problem ððc x Þ ; ðc x Þ ; ðcT x Þu Þ ððcT xo Þl ; ðcT xo Þc ; ðcT xo Þu Þ). So, with respect to Definition 2.6, we have three conditions as follows: (i) Let ðcT x Þc < ðcT xo Þc (in case of minimization problem ðcT x Þc > ðcT xo Þc Þ. Also, with respect to the assumption we have e l ; ðAxo Þc ¼ ð bÞ e c ; ðAxo Þu ¼ ð bÞ e u ; ðxo Þc ðxo Þl P 0; ðxo Þu ðxo Þc P 0; ðxo Þl P 0. Therefore, ððxo Þl ; ðxo Þc ; ðxo Þu Þ is a ðAxo Þl ¼ ð bÞ feasible solution of problem (5) in which the objective value in ððxo Þl ; ðxo Þc ; ðxo Þu Þ is greater (less) than the objective value in ððx Þl ; ðx Þc ; ðx Þu Þ. But, this is a contradiction. (ii) Let ðcT x Þc ¼ ðcT xo Þc and ððcT xo Þu ðcT xo Þl < ðcT x Þu ðcT x Þl Þ (in case of minimization problem ðcT xo Þu ðcT xo Þl > ðcT x Þu ðcT x Þl Þ). Hence, ððxo Þl ; ðxo Þc ; ðxo Þu Þ is a feasible solution of problem (6) in which the objective value in ððxo Þl ; ðxo Þc ; ðxo Þu Þ is less (greater) than the objective value in ððx Þl ; ðx Þc ; ðx Þu Þ. But, this is a contradiction. (iii) Let ðcT x Þc ¼ ðcT xo Þc ; ððcT xo Þu ðcT xo Þl Þ ¼ ððcT x Þu ðcT x Þl Þ and ððcT x Þl þ ðcT x Þu Þ < ððcT xo Þl þ ðcT xo Þu Þ (in case of minimization problem ððcT x Þl þ ðcT x Þu Þ > ððcT xo Þl þ ðcT xo Þu Þ). Therefore, ððxo Þl ; ðxo Þc ; ðxo Þu Þ is a feasible solution of problem (7) in which the objective value in ððxo Þl ; ðxo Þc ; ðxo Þu Þ is greater (less) than the objective value in ððx Þl ; ðx Þc ; ðx Þu Þ. But, this is a contradiction. Please cite this article in press as: R. Ezzati et al., A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.03.014

5


e ¼ ðð x e Þl ; ð x e Þc ; ð e Therefore, x x Þu Þ is an exact optimal solution of problem (2). h 4. Examples In this section, we illustrate the proposed algorithm using a simple example and two real word problems and compare our algorithm with Kumar’s method [13]. Also, a mathematical programming solver Lingo will be used to solve the following mathematical programs. Example 4.1. [12] Consider the FFLP problem with non-negative variables as follows:

e max e cT x

ð8Þ

e ee s:t: A x ¼ b; e are given as follows: e and b where e c; A

2

ð10; 15; 17Þ

3

6 ð10; 16; 20Þ 7 6 7 e c¼6 7; 4 ð10; 14; 17Þ 5

e¼ A

ð8; 10; 13Þ

ð10; 11; 13Þ

ð9; 12; 13Þ

ð11; 15; 17Þ

ð12; 14; 16Þ ð14; 18; 19Þ ð14; 17; 20Þ ð13; 14; 18Þ

;

ð10; 12; 14Þ e ¼ ð271:75; 411:75; 573:75Þ : b ð385:5; 539:5; 759:5Þ Using Step 1, the FFLP problem may be written as follows:

max ð10ðx1 Þl þ 10ðx2 Þl þ 10ðx3 Þl þ 10ðx4 Þl ; 15ðx1 Þc þ 16ðx2 Þc þ 14ðx3 Þc þ 12ðx4 Þc ; 17ðx1 Þu ; 20ðx2 Þu ; 17ðx3 Þu þ 14ðx4 Þu Þ; s:t: ð8ðx1 Þl þ 10ðx2 Þl þ 9ðx3 Þl þ 11ðx4 Þl ; 10ðx1 Þc þ 11ðx2 Þc þ 12ðx3 Þc þ 15ðx4 Þc ; 13ðx1 Þu þ 13ðx2 Þu þ 13ðx3 Þu þ 17ðx4 Þu Þ ¼ ð271:75; 411:75; 573:75Þ; ð12ðx1 Þl þ 14ðx2 Þl þ 14ðx3 Þl þ 13ðx4 Þl ; 14ðx1 Þc þ 18ðx2 Þc þ 17ðx3 Þc þ 14ðx4 Þc ; 16ðx1 Þu þ 19ðx2 Þu þ 20ðx3 Þu þ 18ðx4 Þu Þ ¼ ð385:5; 539:5; 759:5Þ; ð9Þ c

l

u

c

l

where ðxj Þ ðxj Þ P 0; ðxj Þ ðxj Þ P 0; ðxj Þ P 0 for all j ¼ 1; 2; 3; 4. So, with respect to Definitions 2.5, 2.6 and Step 2, problem (9) is converted to the MOLP problem as follows:

15ðx1 Þc þ 16ðx2 Þc þ 14ðx3 Þc þ 12ðx4 Þc ;

max min

ð17ðx1 Þu þ 20ðx2 Þu þ 17ðx3 Þu þ 14ðx4 Þu Þ ð10ðx1 Þl þ 10ðx2 Þl þ 10ðx3 Þl þ 10ðx4 Þl Þ;

max

ðð17ðx1 Þu þ 20ðx2 Þu þ 17ðx3 Þu þ 14ðx4 Þu Þ þ ð10ðx1 Þl þ 10ðx2 Þl þ 10ðx3 Þl þ 10ðx4 Þl ÞÞ;

s:t: 8ðx1 Þl þ 10ðx2 Þl þ 9ðx3 Þl þ 11ðx4 Þl ¼ 271:75; 10ðx1 Þc þ 11ðx2 Þc þ 12ðx3 Þc þ 15ðx4 Þc ¼ 411:75; 13ðx1 Þu þ 13ðx2 Þu þ 13ðx3 Þu þ 17ðx4 Þu ¼ 573:75; 12ðx1 Þl þ 14ðx2 Þl þ 14ðx3 Þl þ 13ðx4 Þl ¼ 385:5;

ð10Þ

14ðx1 Þc þ 18ðx2 Þc þ 17ðx3 Þc þ 14ðx4 Þc ¼ 539:5; 16ðx1 Þu þ 19ðx2 Þu þ 20ðx3 Þu þ 18ðx4 Þu ¼ 759:5; ðxj Þc ðxj Þl P 0; u

c

ðxj Þ ðxj Þ P 0; l

ðxj Þ P 0;

8j ¼ 1; 2; 3; 4; 8j ¼ 1; 2; 3; 4;

8j ¼ 1; 2; 3; 4:

Using Steps 3, 4 and 5, the optimal solution of problem (10) (and hence problem (9)) is achieved as follows:

8 > e x 1 > > > > e > x 3 > > > : e x4

¼ ððx1 Þl ; ðx1 Þc ; ðx1 Þu Þ ¼ ð17:27; 17:27; 17:27Þ; ¼ ððx2 Þl ; ðx2 Þc ; ðx2 Þu Þ ¼ ð2:16; 2:16; 2:16Þ; ¼ ððx3 Þl ; ðx3 Þc ; ðx3 Þu Þ ¼ ð4:64; 9:97; 16:36Þ; ¼ ððx4 Þl ; ðx4 Þc ; ðx4 Þu Þ ¼ ð6:36; 6:36; 6:36Þ:

e in e The optimal value of objective function is obtained by putting x ce x . Therefore, the optimal value of problem (2) may be written as follows: Please cite this article in press as: R. Ezzati et al., A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.03.014

6


e ÞProposed method ¼ ðcx Þl ; ðcx Þc ; ðcx Þu ¼ ðe cx

4 4 4 X X X ðcj xj Þl ; ðcj xj Þc ; ðcj xj Þu j¼1

j¼1

! ¼ ð304:58; 509:79; 704:37Þ:

j¼1

Now, using Kumar’s method [13] the optimal solution and optimal value of objective function are given as follows:

e x ¼

8 l c u > e > > x 1 ¼ ððx4 Þ ; ðx4 Þ ; ðx4 Þ Þ ¼ ð15:28; 15:28; 15:28Þ; > > > > > > e > x 3 ¼ ððx4 Þl ; ðx4 Þc ; ðx4 Þu Þ ¼ ð6:00; 11:25; 11:25Þ; > > > > > : e x 4 ¼ ððx4 Þl ; ðx4 Þc ; ðx4 Þu Þ ¼ ð6:49; 6:49; 9:49Þ

and

e ÞKumar’ method ¼ ððcx Þl ; ðcx Þc ; ðcx Þu Þ ¼ ðe cx

4 4 4 X X X ðcj xj Þl ; ðcj xj Þc ; ðcj xj Þu j¼1

j¼1

! ¼ ð301:83; 503:23; 724:15Þ:

j¼1

By comparing the results of proposed method with Kumar’s method [13], based on Definition 2.6, we can conclude that our result is more reliable, because:

503:23 ¼ ðcx ÞcKumar’s method < ðcx ÞcProposed method ¼ 509:79; 422:32 ¼ ððcx Þu ðcx Þl ÞKumar’s method > ððcx Þu ðcx Þl ÞProposed method ¼ 399:79; e ÞProposed method ¼ ð304:58; 509:79; 704:37Þ: e ÞKumar’s method ðe cx ) ð301:83; 503:23; 724:15Þ ¼ ðe cx To show the application of our algorithm and compare it with Kumar’s method [13], the fuzzy transportation problem [15] and the fuzzy investment problem [16], chosen in Exercise 16–1 by converting it to the fuzzy problem and with some changes, are given. Example 4.2. [15] Dali Company is the leading producer of soft drinks and low-temperature foods in Taiwan. Currently, Dali plans to develop the South-East Asian market and broaden the visibility of Dali products in the Chinese market. Notably, following the entry of Taiwan to the World Trade Organization, Dali plans to seek strategic alliance with prominent international companies, and introduced international bread to lighten the embedded future impact. In the domestic soft drinks market, Dali produces tea beverages to meet demand from four distribution centers in Taichung, Chiayi, Kaohsiung, and Taipei, with production being based at three plants in Changhua, Touliu, and Hsinchu. According to the preliminary environmental information, Table 1 summarizes the potential supply available from these three plants, the forecast demand from the four distribution centers, and the unit transportation costs for each route used by Dali for the upcoming season. The environmental coefficients and related parameters generally are imprecise numbers with triangular possibility distributions over the planning horizon due to incomplete or unobtainable information. For example, the available supply of the Changhua plant is (7.2,8,8.8) thousand dozen bottles, the forecast demand of the Taichung distribution center is (6.2,7,7.8) thousand dozen bottles, and the transportation cost per dozen bottles from Changhua to Taichung is ($8,$10,$10.8). Due to transportation costs being a major expense, the management of Dali is initiating a study to reduce these costs as much as possible. This real world problem can be formulated to the following FFLP problem:

min ðð8; 10; 10:8Þe x 11 ð20:4; 22; 24Þe x 12 ð8; 10; 10:6Þe x 13 ð18:8; 20; 22Þe x 14 x 22 ð10; 12; 13Þe x 23 ð6; 8; 8:8Þe x 24 ð14; 15; 16Þe x 21 ð18:2; 20; 22Þe ð18:4; 20; 21Þe x 31 ð9:6; 12; 13Þe x 32 ð7:8; 10; 10:8Þe x 33 ð14; 15; 16Þe x 34 Þ; s:t:

e x 11 e x 12 e x 13 e x 14 ¼ ð7:2; 8; 8:8Þ; e x 22 e x 23 e x 24 ¼ ð12; 14; 16Þ; x 21 e e x 31 e x 32 e x 33 e x 34 ¼ ð10:2; 12; 13:8Þ;

ð11Þ

e x 21 e x 31 ¼ ð6:2; 7; 7:8Þ; x 11 e e x 12 e x 22 e x 32 ¼ ð8:9; 10; 11:1Þ; e x 13 e x 23 e x 33 ¼ ð6:5; 8; 9:5Þ; e x 24 e x 34 ¼ ð7:8; 9; 10:2Þ; x 14 e where e x ij 2 TFðRÞþ ; i ¼ 1; 2; 3 and j ¼ 1; 2; 3; 4. So, with respects to Definitions 2.5, 2.6 and Step 2 of the proposed algorithm, problem (11) is changed to the MOLP problem as follows: Please cite this article in press as: R. Ezzati et al., A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.03.014

7

R. Ezzati et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx Table 1 Data of Example 4.2 (in U.S. dollar). Source

Destination

Changhua Touliu Hsinchu Demand (000 dozen bottles)

Supply (000 dozen bottles)

Taichung

Chiayi

Kaohsiung

Taipei

($8,$10,$10.8) ($14,$15,$16) ($18.4,$20,$21) (6.2,7,7.8)

($20.4,$22,$24) ($18.2,$20,$22) ($9.6,$12,$13) (8.9,10,11.1)

($8,$10,$10.6) ($10,$12,$13) ($7.8,$10,$10.8) (6.5,8,9.5)

($18.8,$20,$22) ($6,$8,$8.8) ($14,$15,$16) (7.8,9,10.2)

(7.2,8,8.8) (12,14,16) (10.2,12,13.8)

min ð10ðx11 Þc þ 22ðx12 Þc þ 10ðx13 Þc þ 20ðx14 Þc þ 15ðx21 Þc þ 20ðx22 Þc þ 12ðx23 Þc þ 8ðx24 Þc þ 20ðx31 Þc þ 12ðx32 Þc þ 10ðx33 Þc þ 15ðx34 Þc Þ; max ð10:8ðx11 Þu þ 24ðx12 Þu þ 10:6ðx13 Þu þ 22ðx14 Þu þ 16ðx21 Þu þ 22ðx22 Þu þ 13ðx23 Þu þ 8:8ðx24 Þu þ 21ðx31 Þu þ 13ðx32 Þu þ 10:8ðx33 Þu þ 16ðx34 Þu Þ ð8ðx11 Þl þ 20:4ðx12 Þl þ 8ðx13 Þl þ 18:8ðx14 Þl þ 14ðx21 Þl þ 18:2ðx22 Þl þ 10ðx23 Þl þ 6ðx24 Þl þ 18:4ðx31 Þl þ 9:6ðx32 Þl þ 7:8ðx33 Þl þ 14ðx34 Þl Þ; min ð10:8ðx11 Þu þ 24ðx12 Þu þ 10:6ðx13 Þu þ 22ðx14 Þu þ 16ðx21 Þu þ 22ðx22 Þu þ 13ðx23 Þu þ 8:8ðx24 Þu þ 21ðx31 Þu þ 13ðx32 Þu þ 10:8ðx33 Þu þ 16ðx34 Þu Þ þ ð8ðx11 Þl þ 20:4ðx12 Þl þ 8ðx13 Þl þ 18:8ðx14 Þl þ 14ðx21 Þl þ 18:2ðx22 Þl þ 10ðx23 Þl þ 6ðx24 Þl þ 18:4ðx31 Þl þ 9:6ðx32 Þl þ 7:8ðx33 Þl þ 14ðx34 Þl Þ; s:t: ðx11 Þl þ ðx12 Þl þ ðx13 Þl þ ðx14 Þl ¼ 7:2; ðx11 Þc þ ðx12 Þc þ ðx13 Þc þ ðx14 Þc ¼ 8; ðx11 Þu þ ðx12 Þu þ ðx13 Þu þ ðx14 Þu ¼ 8:8; ðx21 Þl þ ðx22 Þl þ ðx23 Þl þ ðx24 Þl ¼ 12; ðx21 Þc þ ðx22 Þc þ ðx23 Þc þ ðx24 Þc ¼ 14; ðx21 Þu þ ðx22 Þu þ ðx23 Þu þ ðx24 Þu ¼ 16; ðx31 Þl þ ðx32 Þl þ ðx33 Þl þ ðx34 Þl ¼ 10:2; ðx31 Þc þ ðx32 Þc þ ðx33 Þc þ ðx34 Þc ¼ 12; ðx31 Þu þ ðx32 Þu þ ðx33 Þu þ ðx34 Þu ¼ 13:8; ðx11 Þl þ ðx21 Þl þ ðx31 Þl ¼ 6:2; ðx11 Þc þ ðx21 Þc þ ðx31 Þc ¼ 7; ðx11 Þu þ ðx21 Þu þ ðx31 Þu ¼ 7:8; ðx12 Þl þ ðx22 Þl þ ðx32 Þl ¼ 8:9; ðx12 Þc þ ðx22 Þc þ ðx32 Þc ¼ 10; ðx12 Þu þ ðx22 Þu þ ðx32 Þu ¼ 11:1; ðx13 Þl þ ðx23 Þl þ ðx33 Þl ¼ 6:5; ðx13 Þc þ ðx23 Þc þ ðx33 Þc ¼ 8; ðx13 Þu þ ðx23 Þu þ ðx33 Þu ¼ 9:5; ðx14 Þl þ ðx24 Þl þ ðx34 Þl ¼ 7:8; ðx14 Þc þ ðx24 Þc þ ðx34 Þc ¼ 9; ðx14 Þu þ ðx24 Þu þ ðx34 Þu ¼ 10:2; ðxij Þc ðxij Þl P 0; u

c

ðxij Þ ðxij Þ P 0; l

ðxij Þ P 0;

8i ¼ 1; 2; 3; 8j ¼ 1; 2; 3; 4; 8i ¼ 1; 2; 3; 8j ¼ 1; 2; 3; 4;

8i ¼ 1; 2; 3; 8j ¼ 1; 2; 3; 4: ð12Þ

Using Steps 3, 4 and 5, the optimal solution of problem (13) is obtained as follows: Please cite this article in press as: R. Ezzati et al., A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.03.014

8


8 > e x 11 > > > > > e > x 12 > > > > > e > x 13 > > > > e > x 14 > > > > > > e x > > > 21 > e > x 23 > > > > e > x 24 > > > > > > e x > > > 31 > > > e > > x 32 > > > e > x 33 > > > : e x 34

¼ ððx11 Þl ; ðx11 Þc ; ðx11 Þu Þ ¼ ð6:2; 7; 7Þ; ¼ ððx12 Þl ; ðx12 Þc ; ðx12 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx13 Þl ; ðx13 Þc ; ðx13 Þu Þ ¼ ð1; 1; 1Þ; ¼ ððx14 Þl ; ðx14 Þc ; ðx14 Þu Þ ¼ ð0; 0; 0:8Þ: ¼ ððx21 Þl ; ðx21 Þc ; ðx21 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx22 Þl ; ðx22 Þc ; ðx22 Þu Þ ¼ ð0; 0; 1:1Þ; ¼ ððx23 Þl ; ðx23 Þc ; ðx23 Þu Þ ¼ ð4:2; 5; 5:9Þ; ¼ ððx24 Þl ; ðx24 Þc ; ðx24 Þu Þ ¼ ð7:8; 9; 9Þ: ¼ ððx31 Þl ; ðx31 Þc ; ðx31 Þu Þ ¼ ð0; 0; 0:8Þ; ¼ ððx32 Þl ; ðx32 Þc ; ðx32 Þu Þ ¼ ð8:9; 10; 10Þ; ¼ ððx33 Þl ; ðx33 Þc ; ðx33 Þu Þ ¼ ð1:3; 2; 2:6Þ; ¼ ððx34 Þl ; ðx34 Þc ; ðx34 Þu Þ ¼ ð0; 0; 0:4Þ:

e in e e as follows: The optimal value of problem (11) is achieved by putting x cx

e e

l

c

u

ð c x ÞProposed method ¼ ðcx Þ ; ðcx Þ ; ðcx Þ

¼

3 X 4 3 X 4 3 X 4 X X X ðcij xij Þl ; ðcij xij Þc ; ðcij xij Þu i¼1 j¼1

i¼1 j¼1

! ¼ ð241:98; 352; 465:18Þ:

i¼1 j¼1

Now, using Kumar’s method [13] we obtain the optimal solution and optimal value of problem (11) as follows:

8 > e > > x 11 > > > e > x 12 > > > > > > x 13 >e > > > > x 14 >e > > > > >e x 21 > > > > e > x 23 > > > > e > x 24 > > > > > > e x > > > 31 > > > e > > x 32 > > > > x 33 >e > > : e x 34

¼ ððx11 Þl ; ðx11 Þc ; ðx11 Þu Þ ¼ ð6:2; 7; 7:8Þ; ¼ ððx12 Þl ; ðx12 Þc ; ðx12 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx13 Þl ; ðx13 Þc ; ðx13 Þu Þ ¼ ð1; 1; 1Þ; ¼ ððx14 Þl ; ðx14 Þc ; ðx14 Þu Þ ¼ ð0; 0; 0Þ: ¼ ððx21 Þl ; ðx21 Þc ; ðx21 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx22 Þl ; ðx22 Þc ; ðx22 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx23 Þl ; ðx23 Þc ; ðx23 Þu Þ ¼ ð4:2; 5; 5:8Þ; ¼ ððx24 Þl ; ðx24 Þc ; ðx24 Þu Þ ¼ ð7:8; 9; 10:2Þ; ¼ ððx31 Þl ; ðx31 Þc ; ðx31 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx32 Þl ; ðx32 Þc ; ðx32 Þu Þ ¼ ð8:9; 10; 11:1Þ; ¼ ððx33 Þl ; ðx33 Þc ; ðx33 Þu Þ ¼ ð1:3; 2; 2:7Þ; ¼ ððx34 Þl ; ðx34 Þc ; ðx34 Þu Þ ¼ ð0; 0; 0Þ;

and

e e

ð c x ÞKumar’s method

¼ ðcx Þl ; ðcx Þc ; ðcx Þu ¼

3 X 4 X

ðcij xij Þl ;

3 X 4 3 X 4 X X ðcij xij Þc ; ðcij xij Þu

i¼1 j¼1

i¼1 j¼1

! ¼ ð241:98; 352; 433:46Þ:

i¼1 j¼1

Based on Definition 2.6, by comparing the results of proposed method with Kumar’s method [13], we can conclude that our result is more effective, because: Table 2 Data of Example 4.3 (in U.S. dollar). Subsidiary

Project

Rate of return

Upper limit of investment

1

1 2 3

(5%,7%,8%) (3%,5%,6%) (4%,8%,9%)

($4,$6,$7) million ($3,$5,$6) million ($8,$9,$10) million

2

4 5 6

(3%,5%,7%) (4%,7%,8%) (8%,9%,10%)


3

7 8 9

(7%,10%,11%) (6%,8%,10%) (4%,7%,8%)


4

10 11 12

(4%,6%,8%) (3%,5%,7%) (7%,9%,11%)



9


ðcx ÞcKumar’s method ¼ ðcx ÞcProposed method ¼ 352; 191:48 ¼ ððcx Þu ðcx Þl ÞKumar’s method < ððcx Þu ðcx Þl ÞProposed method ¼ 223:2; ) ð241:98; 352; 433:46Þ ¼ ðe ce x ÞKumar’s method ðe ce xÞ

Proposed method

¼ ð241:98; 352; 465:18Þ:

Example 4.3. A corporation has ($25,$30,$40) million available for the coming year to allocate to its four subsidiaries. Because of commitments to stability of personnel employment and for other reasons, the corporation has established a minimal level of funding for each subsidiary. These funding levels are ($2,$3,$5) million, ($4,$5,$6) million, ($5,$8,$9) million and ($7,$8,$14) million respectively. Each subsidiary has the opportunity to conduct various projects with the funds it receives. A rate of return (as a percent of investment) has been established for each project. In addition, certain projects permit only limited investment. The data of each project are given in Table 2. What is the best allocation to the four subsidiaries such that the maximum return is achieved for the corporation. This real world problem can be formulated to the following FFLP problem:

max ðð5; 7; 8Þe x 11 ð3; 5; 6Þe x 12 ð4; 8; 9Þe x 13 ð3; 5; 7Þe x 21 ð4; 7; 8Þe x 22 ð8; 9; 10Þe x 23 ð7; 10; 11Þe x 31 ð6; 8; 10Þe x 32 ð4; 7; 8Þe x 33 ð4; 6; 8Þe x 41 ð3; 5; 7Þe x 42 ð7; 9; 11Þe x 43 Þ; s:t:

4 X 3 X

e x ij ¼ ð25; 30; 40Þ;

3 X

i¼1 j¼1 3 X

e x 1j es 1 ¼ ð2; 3; 5Þ;

j¼1

e x 3j es 3 ¼ ð5; 8; 9Þ;

j¼1

3 X

3 X

e x 2j es 2 ¼ ð4; 5; 6Þ;

j¼1

e x 4j es 4 ¼ ð7; 8; 14Þ;

ð13Þ

j¼1

e x 12 es 6 ¼ ð3; 5; 6Þ; e x 13 es 7 ¼ ð8; 9; 10Þ; x 11 es 5 ¼ ð4; 6; 7Þ; e e x 21 es 8 ¼ ð5; 7; 8Þ; e x 22 es 9 ¼ ð8; 10; 11Þ; e x 23 es 10 ¼ ð3; 4; 5Þ; e x 31 es 11 ¼ ð4; 5; 7Þ; e x 32 es 12 ¼ ð2; 3; 6Þ; e x 33 es 13 ¼ ð4; 7; 9Þ; e x 41 es 14 ¼ ð4; 6; 7Þ; e x 42 es 15 ¼ ð4; 5; 9Þ; e x 43 es 16 ¼ ð2; 4; 5Þ; where e x ij ; es k 2 TFðRÞþ ; i ¼ 1; 2; 3; 4; j ¼ 1; 2; 3 and k ¼ 1; 2; . . . ; 16. So, with respects to Definitions 2.5, 2.6 and Step 2 of the proposed algorithm, problem (13) is converted to the MOLP problem as follows:

max ð7ðx11 Þc þ 5ðx12 Þc þ 8ðx13 Þc þ 5ðx21 Þc þ 7ðx22 Þc þ 9ðx23 Þc þ þ10ðx31 Þc þ 8ðx32 Þc þ 7ðx33 Þc þ 6ðx41 Þc þ 5ðx42 Þc þ 9ðx43 Þc Þ; u

u

u

u

u

u

u

u

u

u

u

u

min ð8ðx11 Þ þ 6ðx12 Þ þ 9ðx13 Þ þ 7ðx21 Þ þ 8ðx22 Þ þ 10ðx23 Þ þ 11ðx31 Þ þ 10ðx32 Þ þ 8ðx33 Þ þ 8ðx41 Þ þ 7ðx42 Þ þ 11ðx43 Þ Þ ð5ðx11 Þl þ 3ðx12 Þl þ 4ðx13 Þl þ 3ðx21 Þl þ 4ðx22 Þl þ 8ðx23 Þl þ 7ðx31 Þl þ 6ðx32 Þl þ 4ðx33 Þl þ 4ðx41 Þl þ 3ðx42 Þl þ 7ðx43 Þl Þ; u

u

u

u

u

u

u

u

u

u

u

max ð8ðx11 Þ þ 6ðx12 Þ þ 9ðx13 Þ þ 7ðx21 Þ þ 8ðx22 Þ þ 10ðx23 Þ þ 11ðx31 Þ þ 10ðx32 Þ þ 8ðx33 Þ þ 8ðx41 Þ þ 7ðx42 Þ

þ 11ðx43 Þu Þ þ ð5ðx11 Þl þ 3ðx12 Þl þ 4ðx13 Þl þ 3ðx21 Þl þ 4ðx22 Þl þ 8ðx23 Þl þ 7ðx31 Þl þ 6ðx32 Þl þ 4ðx33 Þl þ 4ðx41 Þl þ 3ðx42 Þl þ 7ðx43 Þl Þ; s:t:

4 X 3 X

l

ðxij Þ ¼ 25;

i¼1 j¼1 3 X

l

u

ðx1j Þ ðs1 Þ ¼ 2;

ðx2j Þl ðs2 Þu ¼ 4;

3 X

ðx3j Þl ðs3 Þu ¼ 5;

3 X

c

3 X

c


l

u

j¼1

3 X

3 X

ðx2j Þc ðs2 Þc ¼ 5;

l

c

l

ðx2j Þu ðs2 Þl ¼ 6;

j¼1 3 X

ðx3j Þc ðs3 Þc ¼ 8;

ðx3j Þu ðs3 Þl ¼ 9;

j¼1

c

3 X

c


j¼1 l

u


j¼1

j¼1


u

ðxij Þ ¼ 40;

i¼1 j¼1

j¼1

j¼1 3 X

3 X

4 X 3 X

j¼1

j¼1 3 X

c

ðxij Þ ¼ 30;

i¼1 j¼1

j¼1 3 X

4 X 3 X

u

l

ðx4j Þ ðs4 Þ ¼ 14;

j¼1 c

u

u

ðx11 Þ þ ðs5 Þ ¼ 4; ðx11 Þ þ ðs5 Þ ¼ 6; ðx11 Þ þ ðs5 Þ ¼ 7; ðx12 Þl þ ðs6 Þl ¼ 3; ðx12 Þc þ ðs6 Þc ¼ 5; ðx12 Þu þ ðs6 Þu ¼ 6;


10


ðx13 Þl þ ðs7 Þl ¼ 8; l

l

ðx13 Þc þ ðs7 Þc ¼ 9; c

ðx21 Þ þ ðs8 Þ ¼ 5;

ðx21 Þ þ ðs8 Þ ¼ 7;

ðx22 Þl þ ðs9 Þl ¼ 8;

ðx22 Þc þ ðs9 Þc ¼ 10;

l

l

ðx13 Þu þ ðs7 Þu ¼ 10;

c


ðx23 Þ þ ðs10 Þ ¼ 3;

ðx23 Þ þ ðs10 Þ ¼ 4;





l

l

c


c

ðx32 Þ þ ðs12 Þ ¼ 2;

ðx32 Þ þ ðs12 Þ ¼ 3;





l

l

c

c

ðx41 Þ þ ðs14 Þ ¼ 4;

ðx41 Þ þ ðs14 Þ ¼ 6;





l

l

ðx43 Þ þ ðs16 Þ ¼ 2; ðxij Þc ðxij Þl P 0; c

l

ðsk Þ ðsk Þ P 0;

c

c

c

c


ðx43 Þ þ ðs16 Þ ¼ 4; ðxij Þu ðxij Þc P 0; u

c

ðsk Þ ðsk Þ P 0;

ðxij Þl P 0;

8i ¼ 1; 2; 3; 4; 8j ¼ 1; 2; 3;

ðsk Þl P 0; 8k ¼ 1; 2; . . . ; 16:

Using Steps 3, 4 and 5, the optimal solution of problem (13) is obtained as follows:

8 > e x 11 > > > > > e > x 12 > > > > > e > x 13 > > > > > e x 21 > > > > > > e x > > > 22 > e > x 31 > > > > e > x 32 > > > > > > e x > > > 33 > > > e > > x 41 > > > e > x 42 > > > : e x 43

¼ ððx11 Þl ; ðx11 Þc ; ðx11 Þu Þ ¼ ð0; 0; 1Þ; ¼ ððx12 Þl ; ðx12 Þc ; ðx12 Þu Þ ¼ ð0; 0; 1Þ; ¼ ððx13 Þl ; ðx13 Þc ; ðx13 Þu Þ ¼ ð8; 9; 9Þ; ¼ ððx21 Þl ; ðx21 Þc ; ðx21 Þu Þ ¼ ð0; 0; 1Þ; ¼ ððx22 Þl ; ðx22 Þc ; ðx22 Þu Þ ¼ ð1; 1; 1Þ; ¼ ððx23 Þl ; ðx23 Þc ; ðx23 Þu Þ ¼ ð3; 4; 4Þ; ¼ ððx31 Þl ; ðx31 Þc ; ðx31 Þu Þ ¼ ð4; 5; 5Þ; ¼ ððx32 Þl ; ðx32 Þc ; ðx32 Þu Þ ¼ ð2; 2; 3Þ; ¼ ððx33 Þl ; ðx33 Þc ; ðx33 Þu Þ ¼ ð0; 0; 1Þ; ¼ ððx41 Þl ; ðx41 Þc ; ðx41 Þu Þ ¼ ð4; 4; 5Þ; ¼ ððx42 Þl ; ðx42 Þc ; ðx42 Þu Þ ¼ ð1; 1; 5Þ; ¼ ððx43 Þl ; ðx43 Þc ; ðx43 Þu Þ ¼ ð2; 3; 4Þ:

e in e e as follows: The optimal value of problem (13) is achieved by putting x cx

e e

l

c

u

ð c x ÞProposed method ¼ ðcx Þ ; ðcx Þ ; ðcx Þ

¼


i¼1 j¼1

! ¼ ð133; 245; 362Þ:

i¼1 j¼1

Also, we achieve the optimal solution and optimal value of problem (13) by use of Kumar’s method [13] as follows:

8 > e > > x 11 > > > > x 12 >e > > > > >e x 13 > > > > > e x 21 > > > > >e > x 22 > > > > e > x 31 > > > > > e x 32 > > > > > > e x 33 > > > > > > e > > x 41 > > > > x 42 >e > > : e x 43

¼ ððx11 Þl ; ðx11 Þc ; ðx11 Þu Þ ¼ ð0; 0; 1Þ; ¼ ððx12 Þl ; ðx12 Þc ; ðx12 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx13 Þl ; ðx13 Þc ; ðx13 Þu Þ ¼ ð8; 9; 10Þ; ¼ ððx21 Þl ; ðx21 Þc ; ðx21 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx22 Þl ; ðx22 Þc ; ðx22 Þu Þ ¼ ð1; 1; 1Þ; ¼ ððx23 Þl ; ðx23 Þc ; ðx23 Þu Þ ¼ ð3; 4; 5Þ; ¼ ððx31 Þl ; ðx31 Þc ; ðx31 Þu Þ ¼ ð4; 5; 6Þ; ¼ ððx32 Þl ; ðx32 Þc ; ðx32 Þu Þ ¼ ð2; 3; 3Þ; ¼ ððx33 Þl ; ðx33 Þc ; ðx33 Þu Þ ¼ ð0; 0; 0Þ; ¼ ððx41 Þl ; ðx41 Þc ; ðx41 Þu Þ ¼ ð4; 4; 5Þ; ¼ ððx42 Þl ; ðx42 Þc ; ðx42 Þu Þ ¼ ð1; 1; 5Þ; ¼ ððx43 Þl ; ðx43 Þc ; ðx43 Þu Þ ¼ ð2; 3; 4Þ

and


11


e ÞKumar’s method ¼ ððcx Þl ; ðcx Þc ; ðcx Þu Þ ¼ ðe cx


i¼1 j¼1

! ¼ ð133; 245; 371Þ:

i¼1 j¼1

By comparing the results of proposed method with Kumar’s method [13], based on Definition 2.6, we can conclude that our result is more effective, because:

ðcx ÞcKumar’s method ¼ ðcx ÞcProposed method ¼ 245; 238 ¼ ððcx Þu ðcx Þl ÞKumar’s method > ððcx Þu ðcx Þl ÞProposed method ¼ 229; e ÞProposed method ¼ ð133; 245; 362Þ: e ÞKumar’s method ðe cx ) ð133; 245; 371Þ ¼ ðe cx 5. Some advantages of the proposed method with respect to the existing methods (i) In Lotfi’s method [12], in order to solve the FFLP problem with arbitrary triangular fuzzy numbers, initially, all the triangular fuzzy numbers are approximated to the nearest symmetric triangular fuzzy numbers. Then, the FFLP problem is solved, in terms of the symmetric triangular fuzzy numbers, by the lexicographic method. One of the shortcomings of it is that the obtained optimal solution is not exact; in fact, it is considered as an approximated optimal solution of the main problem. Therefore, the optimal value of FFLP problem is not reliable for decision maker. The other shortcoming of it is that the optimal solution is not a feasible solution for the main FFLP problem, because the optimal solution does not satisfy in the constraints of main FFLP problem. But, in the proposed method, an exact optimal solution of FFLP problem is obtained by converting to its equivalent MOLP problem. (ii) In Kumar’s method [13], an exact optimal solution of FFLP problem is obtained using the arithmetic operations and linear ranking function which proposed by Liou and Wang [17]. The linear ranking function is used to convert the fuzzy objective function to the real objective function. The shortcoming of the method is that the fuzziness of the objective function is neglected by the linear ranking function. But in the proposed method, we do not restrict ourselves by use of linear ranking function. In the proposed method, by introducing a new lexicographic ordering on triangular fuzzy numbers, the FFLP problem is transformed to the MOLP problem with three crisp objective functions. 6. Conclusions In this paper, a new algorithm has been suggested to solve the FFLP problem. Based on a new lexicographic ordering on triangular fuzzy numbers, the FFLP problem is converted to its equivalent MOLP problem. By a simple example and two case studies, the obtained results of proposed algorithm with Kumar’s method have been compared and shown the reliability and applicability of our algorithm. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Manage. Sci. 17 (4) (1970) 141–164. H. Tanaka, T. Okuda, K. Asai, On fuzzy Mathematical Programming, J. Cybern. Syst. 3 (1973) 37–46. H.J. Zimmerman, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets Syst. 1 (1978) 45–55. L. Campos, J.L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets Syst. 32 (1989) 1–11. J. Buckley, T. Feuring, Evolutionary algorithm solution to fuzzy problems: fuzzy Linear Programming, Fuzzy Sets Syst. 109 (2000) 35–53. H.R. Maleki, M. Tata, M. Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets Syst. 109 (2000) 21–33. H.R. Maleki, Ranking functions and their applications to fuzzy linear programming, Far East J. Math. Sci. 4 (2002) 283–301. H.M. Nehi, H.R. Maleki, M. Mashinchi, Solving fuzzy number linear programming problem by lexicographic ranking function, Ital. J. Pure Appl. Math. 15 (2004) 9–20. T. Allahviranloo, F.H. Lotfi, M.K. Kiasary, N.A. Kiani, L. Alizadeh, Solving fully fuzzy linear programming problem by the ranking functhion, Appl. Mat. Sci. 2 (2008) 19–32. A. Ebrahimnejad, S.H. Nasseri, Using complementary slackness property to solve linear programming with fuzzy parameters, Fuzzy Inf. Eng. 1 (2009) 233–245. A. Ebrahimnejad, S.H. Nasseri, F.H. Lotfi, M. Soltanifar, A primal-dual method for linear programming problems with fuzzy variables, Eur. J. Ind. Eng. 4 (2010) 189–209. F.H. Lotfi, T. Allahviranloo, M.A. Jondabeha, L. Alizadeh, Solving a fully fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Model. 33 (2009) 3151–3156. A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Model. 35 (2011) 817–823. A. Kauffmann, M.M. Gupta, Introduction to fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1991. T.F. Liang, C.S. Chiu, H.W. Cheng, Using possibilistic linear programming for fuzzy transportation planning decisions, Hsiuping J. 11 (2005) 93–112. M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming Network Flows, fourth ed., John Wiley and Sons, New York, 2010. T.S. Liu, M.J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets Syst. 49 (3) (1992) 247–255.


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