Available online at www.ispacs.com/jfsva Volume 2011, Year 2011 Article ID jfsva-00102, 12 pages doi:10.5899/2011/jfsva-00102 Research Article
A New Method for Solving Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers Amit Kumar 1 , Jagdeep Kaur
1 ∗
(1) School of Mathemetics and Computer Applications, Thapar University, Patiala.
c Amit Kumar and Jagdeep Kaur. This is an open access article distributed under the Copyright 2011 ⃝ Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Ganesan and Veeramani [Fuzzy linear programs with trapezoidal fuzzy numbers, Annals of Operations Research 143 (2006) 305-315.] proposed a new method for solving a special type of fuzzy linear programming problems. In this paper a new method, named as Mehar’s method, is proposed for solving the same type of fuzzy linear programming problems and it is shown that it is easy to apply the Mehar’s method as compared to the existing method for solving the same type of fuzzy linear programming problems. Keywords : Fuzzy linear programming; Trapezoidal fuzzy numbers; Ranking function. ————————————————————————————————–
1
Introduction
Any linear programming model representing real world situations involves a lot of parameters whose values are assigned by experts. However, both experts and decision maker frequently do not precisely know the value of those parameters. Therefore it is useful to consider the knowledge of experts about the parameters as fuzzy data introduced by Zadeh [30]. Bellman and Zadeh [3] proposed the concept of decision making in fuzzy environment. Tanaka et al. [28] adopted this concept for solving mathematical programming problems. Zimmerman [32] proposed the first formulation of fuzzy linear programming. Chanas [7] proposed the possibility of the identification of a complete fuzzy decision in fuzzy linear programming by use of the parametric programming technique. Werners [29] introduced an interactive system which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals. ∗
Corresponding author. Email address:
[email protected], Tel: 09463984764
1
Campos and Verdegay [6] considered linear programming problems with fuzzy constraints and fuzzy coefficients in both matrix and right hand of the constraint set. Inuiguchi et al. [13] dealt with the fuzzy linear programming problems with continuous piecewise linear membership function. Cadenas and Verdegay [5] studied a linear programming problem in which all its elements are defined as fuzzy sets. Fang et al. [10] presented a method for solving linear programming problems with fuzzy coefficients in constraints. Buckley and Feuring [4] introduced a method to find the solution for fully fuzzified linear programming problems with all the parameters and variables as fuzzy numbers by changing the objective function into a multiobjective fuzzy linear programming problem. Maleki et al. [22] solved the linear programming problems in which all decision parameters are fuzzy numbers by the comparison of fuzzy numbers. Liu [19] proposed a method for solving fuzzy linear programming problems based on the satisfaction degree of the constraints. Maleki [21] introduced a new method for solving linear programming with vagueness in constraints by using ranking function. Zhang et al. [31] proposed a method for solving fuzzy linear programming problems which involve fuzzy numbers in coefficients of objective functions. Nehi et al. [25] defined the concept of optimality for linear programming problems with fuzzy parameters by transforming fuzzy linear programming problems into multiobjective linear programming problems. Ramik [26] proposed the fuzzy linear programming problems based on fuzzy relations. Ganesan and Veeramani [11] proposed an approach to solve a fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers without converting it into crisp linear programming (CLP) problem. Hashemi et al. [12] proposed a two phase approach to find the optimal solutions of class of fuzzy linear programming problems called fully fuzzified linear programming, where all decision parameters and variables are fuzzy numbers. Rommelfanger [27] proposed a new method for solving stochastic linear programming problems with fuzzy parameters. Jimenez et al. [14] proposed a method for solving linear programming problems where all the coefficients are fuzzy numbers and used a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. Allahviranloo et al. [2] solved the fuzzy integer linear programming problem by reducing it into a crisp integer linear programming problem. Allahviranloo et al. [1] proposed a new method for solving fully fuzzy linear programming problems by the use of ranking function. Nasseri [23] proposed a method for solving fuzzy linear programming problems by solving the classical linear programming. Lotfi et al. [20] discussed fully fuzzy linear programming problems by representing all parameters and variables as triangular fuzzy numbers. Ebrahimnejad and Nasseri [8] used the complementary slackness theorem to solve fuzzy linear programming problem with fuzzy parameters without the need of a simplex tableau. Ebrahimnejad et al. [9] proposed a new primal-dual algorithm for solving linear programming problems with fuzzy variables by using duality results. Nasseri and Ebrahimnejad [24] proposed a fuzzy primal simplex algorithm for solving the flexible linear programming problem. Kumar et al. [15] proposed a new method for solving fully fuzzy linear programming problems by introducing fuzzy slack and surplus variables. Kumar et al. [16, 17] proposed a new method for solving fully fuzzy linear programming problems with inequality constraints. Kumar et al. [18] proposed a new method for finding the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints. In this paper, a new method, named as Mehar’s method, is proposed to find the fuzzy 2
optimal solution of special type of fuzzy linear programming problems and the Mehar’s method is easy to apply as compared to the existing method [11]. This paper is organized as follows: In Section 2 some basic definitions and arithmetics between two symmetric trapezoidal fuzzy numbers are reviewed. In Section 3 formulation of fuzzy linear programming problems and fuzzy optimal solution are discussed. In Section 4 a new method, named as Mehar’s method, is proposed for solving the special type of fuzzy linear programming problems. To illustrate the Mehar’s method, a numerical example is solved in Section 5. Obtained results are discussed in Section 6. Conclusions are discussed in Section 7.
2
Preliminaries In this section, some necessary backgrounds and notions of fuzzy set theory are reviewed.
2.1
Basic definitions
In this section, some basic definitions are presented [11]. Definition 2.1. A fuzzy set A˜ on real numbers R is said to be a symmetric trapezoidal fuzzy number if there exist real numbers m, n, m ≤ n and α > 0 such that x+α−m , x ∈ [m − α, m] α 1, x ∈ [m, n] µA˜ (x) = −x+n+α , x ∈ [n, n + α] α 0, otherwise It is denoted as A˜ = (m, n, α, α). Definition 2.2. A ranking function is a function ℜ : F (R) → R, where F (R) is a set of fuzzy numbers defined on set of real numbers, which maps each fuzzy number into the real line, where a natural order exists. Let A˜ = (m1 , n1 , α, α) be a trapezoidal fuzzy number ˜ = m1 +n1 . then ℜ(A) 2 ˜ = (m2 , n2 , β, β) be two symmetric trapezoidal fuzzy numbers. Let A˜ = (m1 , n1 , α, α) and B Then ˜ iff ℜ(A) ˜ ≥ ℜ(B) ˜ (i) A˜ ≽ B ˜ iff ℜ(A) ˜ > ℜ(B) ˜ (ii) A˜ ≻ B ˜ iff ℜ(A) ˜ = ℜ(B) ˜ (iii) A˜ ≈ B
2.2
Arithmetic operations
˜ = (m2 , n2 , β, β) be two symmetric trapezoidal fuzzy numbers. Let A˜ = (m1 , n1 , α, α) and B ˜ are given by [11]: Then the arithmetic operations on A˜ and B ˜ = (m1 , n1 , α, α) ⊕ (m2 , n2 , β, β) = (m1 + m2 , n1 + n2 , α + β, α + β) (i) Addition: A˜ ⊕ B ˜ = (m1 , n1 , α, α) ⊖ (m2 , n2 , β, β) = (m1 − n2 , n1 − m2 , α + β, α + β) (ii) Subtraction: A˜ ⊖ B 3
(iii) Multiplication: ˜ = (m1 , n1 , α, α) ⊗ (m2 , n2 , β, β) = (( m1 +n1 )( m2 +n2 ) − w, ( m1 +n1 )( m2 +n2 ) A˜ ⊗ B 2 2 2 2 +w, |n1 β + n2 α|, |n1 β + n2 α|) where w = ( k−h 2 ) and h = min{m1 m2 , m1 n2 , m2 n1 , n1 n2 }, k = max{m1 m2 , m1 n2 , m2 n1 , n1 n2 } (iv) Scalar multiplication: { λA˜ =
3
(λm1 , λn1 , λα, λα),
λ≥0
(λn1 , λm1 , −λα, −λα), λ ≤ 0
Formulation of fuzzy linear programming problems
A fuzzy linear programming problem with constraints and variables can be written as [11]: M aximize(orM inimize) z˜ ≈
n ∑
c˜j ⊗ x ˜j
j=1
subject to
(P1 ) n ∑
aij x ˜j ≼, ≈, ≽ ˜bi , i = 1, 2, . . . , m
j=1
x ˜j ≽ ˜0
∀j
where, x ˜j = (xj , yj , αj , αj ), ˜bi = (bi , gi , γi , γi ) and c˜j = (pj , qj , βj , βj ) are symmetric trapezoidal fuzzy numbers.
3.1
Fuzzy optimal solution of fuzzy linear programming problems
A set of symmetric trapezoidal fuzzy numbers {˜ xj } is said to be fuzzy optimal solution of (P1 ) if the following properties are satisfied: (i)
n ∑
aij x ˜j ≼, ≈, ≽ ˜bi , i = 1, 2, ...m
j=1
(ii) x ˜j ≽ ˜0 ∀ j (iii) If there exist any set of symmetric trapezoidal fuzzy numbers {˜ yj } such that
n ∑ j=1
, ≈, ≽ ˜bi , i = 1, 2, . . . , m and y˜j ≽ ˜0 ∀ j then n n ∑ ∑ c˜j ⊗ y˜j (in case of maximization problem) and c˜j ⊗ x ˜j ≽ j=1 n ∑ j=1
c˜j ⊗ x ˜j ≼
j=1 n ∑
c˜j ⊗ y˜j (in case of minimization problem)
j=1
4
aij y˜j ≼
4
Mehar’s method
In this section, a new method, named as Mehar’s method, is proposed to find the fuzzy optimal solution of fuzzy linear programming problems. The steps of the Mehar’s method are as follows: Step 1. Formulate the chosen problem into the following fuzzy linear programming problem: n ∑ M aximize(orM inimize) z˜ ≈ c˜j ⊗ x ˜j j=1
subject to n ∑
aij x ˜j ≼, ≈, ≽ ˜bi , i = 1, 2, . . . , m
j=1
x ˜j ≽ ˜0
∀j
Step 2. Substituting the values of x ˜j = (xj , yj , αj , αj ), ˜bi = (bi , gi , γi , γi ) and c˜j = (pj , qj , βj , βj ) in the fuzzy linear programming problem, obtained in Step 1, we get: n ∑
M aximize(orM inimize) z˜ ≈
(pj , qj , βj , βj ) ⊗ (xj , yj , αj , αj )
j=1
subject to n ∑
aij (xj , yj , αj , αj ) ≼, ≈, ≽ (bi , gi , γi , γi ), i = 1, 2, . . . , m
j=1
(xj , yj , αj , αj ) ≽ ˜0
∀j
Step 3. Assuming aij (xj , yj , αj , αj ) = (x′j , yj′ , αj′ , αj′ ) the fuzzy linear programming problem, obtained in Step 2, can be written as: M aximize(orM inimize) z˜ ≈
n ∑
(pj , qj , βj , βj ) ⊗ (xj , yj , αj , αj )
j=1
subject to n ∑ j=1
(x′j , yj′ , αj′ , αj′ ) ≼, ≈, ≽ (bi , gi , γi , γi ), i = 1, 2, . . . , m
(xj , yj , αj , αj ) ≽ ˜0
∀j
Step 4. Using the linearity property of ranking function [11], ℜ((pj , qj , βj , βj )⊗(xj , yj , αj , αj )) = ℜ(pj , qj , βj , βj )ℜ(xj , yj , αj , αj ), the fuzzy linear programming problem, obtained in Step
5
3, can be written as: M aximize(orM inimize) ℜ(˜ z) =
n ∑
ℜ(pj , qj , βj , βj )ℜ(xj , yj , αj , αj )
j=1
subject to ℜ(
n ∑ j=1
(x′j , yj′ , αj′ , αj′ )) ≤, =, ≥ ℜ(bi , gi , γi , γi ), i = 1, 2, . . . , m
ℜ(xj , yj , αj , αj ) ≥ 0
∀j
yj − xj ≥ 0,
αj ≥ 0
Step 5. Solve the crisp linear programming problem, obtained in Step 4, to find the optimal solution xj , yj , αj and put their values in x ˜j = (xj , yj , αj , αj ) to find the fuzzy optimal solution. Step 6. Find the fuzzy optimal value of fuzzy linear programming problem by putting n ∑ the values of x ˜j in c˜j ⊗ x ˜j . j=1
Remark 4.1. The proposed method is named as Mehar’s method for solving fuzzy linear programming problems. Mehar is a lovely daughter of Parmpreet Kaur (Research scholar under the supervision of Dr. Amit Kumar).
5
Numerical example
In this section, the fuzzy linear programming problem, solved by Ganesan and Veeramani [11], is solved by using the proposed method. Example 5.1. Find the fuzzy optimal solution of the following fuzzy linear programming problem [11]: M aximize ((13, 15, 2, 2) ⊗ x ˜1 ⊕ (12, 14, 3, 3) ⊗ x ˜2 ⊕ (15, 17, 2, 2) ⊗ x ˜3 subject to 12˜ x1 ⊕ 13˜ x2 ⊕ 12˜ x3 ≼ (475, 505, 6, 6) 14˜ x1 ⊕ 13˜ x3 ≼ (460, 480, 8, 8) 12˜ x1 ⊕ 15˜ x2 ≼ (465, 495, 5, 5) x ˜1 ≽ ˜0,
x ˜2 ≽ ˜0,
x ˜3 ≽ ˜0
Solution: The fuzzy optimal solution of the chosen fuzzy linear programming problem can be obtained by using the following steps
6
Step 1. The chosen fuzzy linear programming problem is: M aximize ((13, 15, 2, 2) ⊗ x ˜1 ⊕ (12, 14, 3, 3) ⊗ x ˜2 ⊕ (15, 17, 2, 2) ⊗ x ˜3 ) subject to 12˜ x1 ⊕ 13˜ x2 ⊕ 12˜ x3 ≼ (475, 505, 6, 6) 14˜ x1 ⊕ 13˜ x3 ≼ (460, 480, 8, 8) 12˜ x1 ⊕ 15˜ x2 ≼ (465, 495, 5, 5) x ˜1 ≽ ˜0,
x ˜2 ≽ ˜0,
x ˜3 ≽ ˜0
Step 2. Assuming x ˜1 = (x1 , y1 , α1 , α1 ), x ˜2 = (x2 , y2 , α2 , α2 ) and x ˜3 = (x3 , y3 , α3 , α3 ) the fuzzy linear programming problem, obtained in Step 1, can be written as: M aximize ((13, 15, 2, 2) ⊗ (x1 , y1 , α1 , α1 ) ⊕ (12, 14, 3, 3) ⊗ (x2 , y2 , α2 , α2 ) ⊕ (15, 17, 2, 2) ⊗ (x3 , y3 , α3 , α3 )) subject to 12(x1 , y1 , α1 , α1 ) ⊕ 13(x2 , y2 , α2 , α2 ) ⊕ 12(x3 , y3 , α3 , α3 ) ≼ (475, 505, 6, 6) 14(x1 , y1 , α1 , α1 ) ⊕ 13(x3 , y3 , α3 , α3 ) ≼ (460, 480, 8, 8) 12(x1 , y1 , α1 , α1 ) ⊕ 15(x2 , y2 , α2 , α2 ) ≼ (465, 495, 5, 5) (x1 , y1 , α1 , α1 ) ≽ ˜0,
(x2 , y2 , α2 , α2 ) ≽ ˜0,
(x3 , y3 , α3 , α3 ) ≽ ˜0
Step 3. Using Step 3 of the Mehar’s method the fuzzy linear programming problem, obtained in Step 2, can be written as: M aximize ((13, 15, 2, 2) ⊗ (x1 , y1 , α1 , α1 ) ⊕ (12, 14, 3, 3) ⊗ (x2 , y2 , α2 , α2 ) ⊕ (15, 17, 2, 2) ⊗ (x3 , y3 , α3 , α3 )) subject to (12x1 , 12y1 , 12α1 , 12α1 ) ⊕ (13x2 , 13y2 , 13α2 , 13α2 ) ⊕ (12x3 , 12y3 , 12α3 , 12α3 ) ≼ (475, 505, 6, 6) (14x1 , 14y1 , 14α1 , 14α1 ) ⊕ (13x3 , 13y3 , 13α3 , 13α3 ) ≼ (460, 480, 8, 8) (12x1 , 12y1 , 12α1 , 12α1 ) ⊕ (15x2 , 15y2 , 15α2 , 15α2 ) ≼ (465, 495, 5, 5) (x1 , y1 , α1 , α1 ) ≽ ˜0,
(x2 , y2 , α2 , α2 ) ≽ ˜0,
(x3 , y3 , α3 , α3 ) ≽ ˜0
Step 4. Using Step 4 of the Mehar’s method the fuzzy linear programming problem,
7
obtained in Step 3, can be written as the following crisp linear programming: M aximize (7x1 + 7y1 +
13 2 x2
+
13 2 y2
+ 8x3 + 8y3 )
subject to 12x1 + 12y1 + 13x2 + 13y2 + 12x3 + 12y3 ≤ 980 14x1 + 14y1 + 13x3 + 13y3 ≤ 940 12x1 + 12y1 + 15x2 + 15y2 ≤ 960 x1 + y1 ≥ 0, x2 + y2 ≥ 0, x3 + y 3 ≥ 0 x1 ≤ y 1 , x2 ≤ y 2 , x3 ≤ y 3 α1 ≥ 0,
α2 ≥ 0,
α3 ≥ 0
Step 5. Solving the crisp linear programming obtained in Step 4, we get x1 = 0,
y1 = 0,
α1 = 0,
x2 =
730 169 ,
y2 =
730 169 ,
α2 = 0,
x3 =
470 13 ,
y3 =
470 13 ,
α3 = 0.
Putting these values in x ˜1 = (x1 , y1 , α1 , α1 ), x ˜2 = (x2 , y2 , α2 , α2 ), x ˜3 = (x3 , y3 , α3 , α3 ) the fuzzy optimal solution is x ˜1 = (0, 0, 0, 0), 730 x ˜2 = ( 730 169 , 169 , 0, 0), 470 x ˜3 = ( 470 13 , 13 , 0, 0)
Step 6. Putting the values of x ˜1 , x ˜2 and x ˜3 in ((13, 15, 2, 2) ⊗ x ˜1 ⊕ (12, 14, 3, 3) ⊗ x ˜2 ⊕ (15, 17, 2, 2) ⊗ x ˜3 ) 114090 14410 14410 the fuzzy optimal value is ( 100410 169 , 169 , 169 , 169 ).
8
6
Results and discussions
The results of the fuzzy linear programming problem, chosen in Example (5.1), obtained by using the existing method [11] and the Mehar’s method are shown in Table 1. Table 1 Results of Existing method [11] and Mehar’s method
Example
5.1
Existing method [11] x ˜j ℜ(˜ xj ) x ˜1 = (0, 0, 0, 0) 0 415 1045 174 174 730 x ˜2 = ( 169 , 169 , 169 , 169 ) 169 480 8 8 470 x ˜3 = ( 460 , , , ) 13 13 13 13 13
Mehar’s method x ˜j ℜ(˜ xj ) x ˜1 = (0, 0, 0, 0) 0 730 730 730 x ˜2 = ( 169 , 169 , 0, 0) 169 470 470 x ˜3 = ( 470 , , 0, 0) 13 13 13
It is obvious from the results, shown in Table 1, that by using both the existing method [11] and Mehar’s method the the rank of the fuzzy optimal solution and fuzzy optimal value are same while it is much easy to apply the Mehar’s method as compared to the existing method [11].
7
Conclusions
In this paper, a new method, named as Mehar’s method, is proposed for solving the special type of fuzzy linear programming problems. The Mehar’s method is easy as compared to the existing method [11].
Acknowledgements The authors would like to thank to the Editor-in-Chief for various suggestions which have led to an improvement in both the quality and clarity of the paper.
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