Generalized ranking function for all symmetric fuzzy linear programming problems

Abstract

In this paper the generalized symmetric fuzzy numbers are brought into the knowledge and a new ranking function was proposed for it. Using this ranking function, any kind of fuzzy linear programming problem can be solved. The proposed ranking function is illustrated by the numerical example. Moreover, Comparison of optimal solution of proposed and existing methods are discussed. The proposed method is posing no difficulty to understand and execute the fuzzy optimal solution of a symmetric fuzzy linear programming problem in real world situations.

Keywords

Symmetric fuzzy valued linear programming problems Generalized fuzzy numbers Generalized membership function Generalized ranking function.

1 Introduction

The techniques of a linear programming problem are used in many economic and industrial fields. Finding the best solution is the main objective of a fuzzy valued linear programming problem in the environment of uncertain, imprecise, vague or incomplete information. Fuzzy linear programming problem varies depends up on the vagueness in the situations of the problem. When Zadeh and Bellman [1] initiated the concept of decision making in fuzzy environment, the research on fuzzy linear programming attains the tremendous growth. The first formulation of the fuzzy valued linear programming was introduced by Zimmermann [2]. Jain [19] was the one who proposed a method for ranking and Yagar [21] was the first one who used the horizontal coordinate of the centroid point method in ranking fuzzy number.

Kaufmann and Gupta [20] have also proposed the ranking of fuzzy number. In the process of solving fuzzy valued linear programming problem, Leung [16] has classified that into four categories, Luhandjula [17] has classified that into three categories and Inuiguchi et al [18] has classified into six categories. Whereas Zimmerman classified [15] fuzzy valued linear programming problem into two types, as symmetrical and non-symmetrical. In this paper, an attempt is made to propose the generalized ranking function to solve the symmetric fuzzy valued linear programming problem.

2 Preliminaries

2.1 Mathematical form of a linear programming problem

The linear programming problem or model involving n decision variables and m constraints can be written in the form of, Maximize (or) Minimize Z = c₁x₁ + c₁x₂ + … + c_nx_n

Here, x₁, x₂, …, x_n are decision variables and c₁, c₁, …, c_n are the unit contributions of the decision making variables x₁, x₂, …, x_n respectively.

Subject to the constraints,

a₁₁x₁ + a₁₂x₂ + … + a_1nx_n (≤ or = or ≥) b₁ a₂₁x₁ + a₂₂x₂ + … + a_2nx_n (≤ or = or ≥) b₂ ……… …… …… …… …… …… … ……… …… …… …… …… …… … a_m1x₁ + a_m2x₂ + … + a_mnx_n (≤ or = or ≥) b_m and, the non-negativity condition is, x₁, x₂, …, x_n ≥ 0

2.1.1 Solution

A set of values (x₁, x₂, …, x_n) is said to be a solution when it satisfies the constraints of the given linear programming problem.

2.1.2 Feasible Solution

Any solution to a linear programming problem is said to be a feasible solution when it satisfies the non negativity conditions of the same linear programming problem.

2.1.3 Optimal Solution

Any feasible solution of a linear programming problem is said to be optimal solution when it optimizes (maximizes or minimizes) the objective function of the same linear programming problem.

2.2 Fuzzy Set

If X is a collection of objects (denoted generally by x) then a fuzzy set $\tilde{A} \in X$ is a set of ordered pairs,

$\tilde{A} = {(x, μ_{\tilde{A}} (x)) | x \in X}$

Where, $μ_{\tilde{A}} (x)$ is the membership function (or grade of membership) with respect to x in $\tilde{A}$ [4, 5].

2.3 Generalized Fuzzy Number

$\tilde{A} = (a_{1}, a_{2}, a_{3}, \dots, a_{n})$ is a real fuzzy number and it is a fuzzy subset of the real number $ℝ$ with Membership function $μ_{\tilde{A}}$ satisfies the below conditions,

$μ_{\tilde{A}}$ is continuous from $ℝ$ to [0, 1]

for odd n

= {\begin{matrix} μ_{\tilde{A}} is strictly increasing and continuous on [a_{1}, a_{\frac{n + 1}{2}}] \\ μ_{\tilde{A}} is strictly decreasing and continuous on [a_{\frac{n + 1}{2}}, a_{n}] \end{matrix}

for even n

is strictly increasing and continuous on

[a_{1}, a_{\frac{n}{2}}]

μ_{\tilde{A}}

is strictly decreasing and continuous on

[a_{\frac{n}{2} + 1}, a_{n}]

Membership function [13] for odd n is

μ_{\tilde{A}} (x) = {\begin{matrix} 0 & if & x \leq a_{1} \\ \frac{x - a_{1}}{(a_{2} - a_{1})} & if & a_{1} \leq x \leq a_{2} \\ \dots & \dots & \dots \\ \dots & \dots & \dots \\ 1 & if & x = a_{(\frac{n + 1}{2})} \\ \dots & \dots & \dots \\ \dots & \dots & \dots \\ \frac{a_{n} - x}{(a_{n} - a_{(n - 1)})} & if & a_{n - 1} \leq x \leq a_{n} \\ 0 & if & x \geq a_{n} \end{matrix}

Membership function for even n is

μ_{\tilde{A}} (x) = {\begin{matrix} 0 & if & x \leq a_{1} \\ \frac{x - a_{1}}{(a_{2} - a_{1})} & if & a_{1} \leq x \leq a_{2} \\ \dots & \dots & \dots \\ \dots & \dots & \dots \\ 1 & if & a_{(\frac{n}{2})} \leq x \leq a_{(\frac{n}{2}) + 1} \\ \dots & \dots & \dots \\ \dots & \dots & \dots \\ \frac{a_{n} - x}{(a_{n} - a_{(n - 1)})} & if & a_{n - 1} \leq x \leq a_{n} \\ 0 & if & x \geq a_{n} \end{matrix}

2.4 Operations on Fuzzy Number

If $\tilde{A} = (a_{1}, a_{2}, a_{3}, \dots, a_{n})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, \dots, b_{n})$ are non-negative fuzzy numbers, then the following three operations can be performed follows:

Addition:

$\tilde{A} \oplus \tilde{B} = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}, \dots, a_{n} + b_{n})$

Subtraction:

$\tilde{A} ⊖ \tilde{B} = (a_{1} - b_{1}, a_{2} - b_{2}, a_{3} - b_{3}, \dots, a_{n} - b_{n})$

Multiplication:

$\tilde{A} \otimes \tilde{B} = (a_{1} \times b_{1}, a_{2} \times b_{2}, a_{3} \times b_{3}, \dots, a_{n} \times b_{n})$

2.5 Symmetric Fuzzy Number

$\tilde{A} = (a_{1}, a_{2}, a_{3}, \dots, a_{n})$ is a fuzzy number and it is said to be symmetric fuzzy number [3] [9] if and only if $(\frac{a_{1} + a_{n}}{2}) = (\frac{a_{2} + a_{n - 1}}{2}) = \dots = (\frac{a_{(\frac{n}{2})} + a_{(\frac{n}{2}) + 1}}{2})$ for even n and $(\frac{a_{1} + a_{n}}{2}) = (\frac{a_{2} + a_{n - 1}}{2}) = \dots = (\frac{a_{(\frac{n - 1}{2})} + a_{(\frac{n + 3}{2})}}{2}) = (a_{(\frac{n + 1}{2})})$ for odd n. Where, n is the number of elements of the fuzzy number. Otherwise, $\tilde{A}$ is said to be an asymmetric fuzzy number.

(Or)

The graphical representation of Left of the fuzzy number is same as Right of the fuzzy number. That is, $(\tilde{A})_{L} ≅ (\tilde{A})_{R}$

2.6 Graphical Representation of a Symmetric Fuzzy Number

According to the symmetrical axis the graphical representation of a generalized symmetric fuzzy number (for odd n) is given below in (Fig. 1).

Fig.1

Graphical representation of a symmetric fuzzy number for odd n.

The graphical representation of a generalized symmetric fuzzy number (for even n) is given below in (Fig. 2).

Fig.2

Graphical representation of a symmetric fuzzy number for even n.

From the graphical representation, in both the cases the left of a fuzzy number is symmetric to the right of a fuzzy number.

3 Ranking function of a symmetric fuzzy number

3.1 Definition of a ranking function

Ranking [14] of a fuzzy number is a function $R : F (R) ⟶ R$ , which maps each fuzzy number into the real line. Where $F (ℝ)$ is a set of fuzzy numbers defined on the set of real numbers.

If there are any two fuzzy numbers then the relation between those two are, given by

$\tilde{A} ≻ \tilde{B}$ if $R (\tilde{A}) > R (\tilde{B})$

$\tilde{A} ≺ \tilde{B}$ if $R (\tilde{A}) < R (\tilde{B})$

$\tilde{A} \approx \tilde{B}$ if $R (\tilde{A}) = R (\tilde{B})$

3.2 Generalized ranking function of a Symmetric Fuzzy Number

The proposed ranking function is the generalized ranking function for the all symmetric fuzzy numbers. It is divided into two fuzzy number (odd and even) which depends on the number of elements (n) in the symmetric fuzzy number.

Let $\tilde{A} = (a_{1}, a_{2}, a_{3}, \dots, a_{n})$ be a generalized Symmetric fuzzy number. Then, the ranking function is defined as follows: $R (\tilde{A}) = {\begin{matrix} (a_{(\frac{n + 1}{2})}) & for odd n \\ \frac{1}{2} (a_{(\frac{n}{2})} + a_{(\frac{n + 2}{2})}) & for even n \end{matrix}$

4 Symmetric Fuzzy valued Linear Programming Problem

The formulation can also be expressed in a compact form using summation sign. Maximize (or) Minimize $Z = \sum_{j = 1}^{n} {\tilde{a}}_{j} {\tilde{x}}_{j}$ is the generalized fuzzy objective function of a linear programming problem.

Subject to the linear constraints

$\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j} (⪯ or \approx or) {\tilde{b}}_{i};$ i = 1, 2, 3, …, m ${\tilde{x}}_{j} 0;$ j = 1, 2, 3, …, n (Non-negativity condition) where, ${\tilde{a}}_{j}, {\tilde{a}}_{ij}, {\tilde{b}}_{i}$ (i = 1, 2, 3, …, m and j = 1, 2, 3, …, n) are symmetric fuzzy numbers (depends up on the vagueness in the problem).

5 Proposed method of solving symmetric fuzzy valued linear programming problem

A new method is proposed to get the fuzzy optimal solution of symmetric fuzzy valued linear programming problems. The steps [7] of the new proposed method are,

Step 1: Formulate the symmetric fuzzy valued linear pro-gramming problem.

Step 2: Using the proposed ranking function,

$R (\tilde{A}) = {\begin{matrix} (a_{(\frac{n + 1}{2})}) & for odd n \\ \frac{1}{2} (a_{(\frac{n}{2})} + a_{(\frac{n + 2}{2})}) & for even n \end{matrix}$

the symmetric fuzzy valued linear programming problem transform into crisp valued linear programming problem.

Step 3: Solve the crisp valued linear programming problem using simplex method (or) Big-M method.

6 Numeral Example

Consider the following numerical example [3].

Maximize $\tilde{Z}$ ≈ (13, 15, 2, 2) ⊗ ${\tilde{x}}_{1}$ ⊕ (12, 14, 3, 3) ⊗ ${\tilde{x}}_{2}$ ⊕ (15, 17, 2, 2) ⊗ ${\tilde{x}}_{3}$

Subject to

$12 {\tilde{x}}_{1} \oplus 13 {\tilde{x}}_{2} \oplus 12 {\tilde{x}}_{3} ⪯ (475, 505, 6, 6)$ $14 {\tilde{x}}_{1} \oplus 13 {\tilde{x}}_{3} ⪯ (460, 480, 8, 8)$ $12 {\tilde{x}}_{1} \oplus 15 {\tilde{x}}_{2} ⪯ (465, 495, 5, 5)$ ${\tilde{x}}_{1}, {\tilde{x}}_{2}, {\tilde{x}}_{3} 0$

Here, decision variables, unit contributions of the decision making variables, the co-efficient of the decision variable and right hand side in the constrains of the problem are all are LR-Symmetric fuzzy numbers. The above problem can be rewritten as symmetric fuzzy linear programming problem.

Maximize $\tilde{Z}$ ≈ (11, 13, 15, 17) ⊗ ${\tilde{x}}_{1}$ ⊕ (9, 12, 14, 17) ⊗ ${\tilde{x}}_{2}$ ⊕ (13, 15, 17, 19) ⊗ ${\tilde{x}}_{3}$

Subject to

$12 {\tilde{x}}_{1} \oplus 13 {\tilde{x}}_{2} \oplus 12 {\tilde{x}}_{3} ⪯ (469, 475, 505, 511)$ $14 {\tilde{x}}_{1} \oplus 13 {\tilde{x}}_{3} ⪯ (452, 460, 480, 488)$ $12 {\tilde{x}}_{1} \oplus 15 {\tilde{x}}_{2} ⪯ (460, 465, 495, 500)$ ${\tilde{x}}_{1}, {\tilde{x}}_{2}, {\tilde{x}}_{3} 0$

After using proposed ranking function for even n, the objective function and the constrains become as given below,

Maximize $\tilde{Z} \approx 14 \otimes {\tilde{x}}_{1} \oplus 13 \otimes {\tilde{x}}_{2} \oplus 16 \otimes {\tilde{x}}_{3}$

Subject to

$12 {\tilde{x}}_{1} \oplus 13 {\tilde{x}}_{2} \oplus 12 {\tilde{x}}_{3} ⪯ 490$ $14 {\tilde{x}}_{1} \oplus 13 {\tilde{x}}_{3} ⪯ 470$ $12 {\tilde{x}}_{1} \oplus 15 {\tilde{x}}_{2} ⪯ 480$ ${\tilde{x}}_{1}, {\tilde{x}}_{2}, {\tilde{x}}_{3} 0$

Using Simplex method of solving we obtain the optimal solution is,

${\tilde{x}}_{1} \approx (0, 0, 0, 0), {\tilde{x}}_{2} \approx (\frac{655}{169}, \frac{673}{169}, \frac{787}{169}, \frac{805}{169}), {\tilde{x}}_{3} \approx (\frac{452}{13}, \frac{460}{13}, \frac{480}{13}, \frac{488}{13})$ and $\tilde{Z} \approx (\frac{7905}{13}, \frac{8015}{13}, \frac{8485}{13}, \frac{8595}{13})$

7 Comparison and advantages

The optimal solution that we got from proposed ranking method is $\tilde{Z} \approx (\frac{7905}{13}, \frac{8015}{13}, \frac{8485}{13}, \frac{8595}{13})$ . At the same time optimal solution using Mehar’s method is $\tilde{Z} \approx (\frac{8500}{169}, \frac{100410}{169}, \frac{114090}{169}, \frac{128500}{169})$ . It is clear that the spread of the optimal solution by Mehar’s method is bigger than the optimal solution that we obtained by proposed ranking method. Though the spread is different, the ranking value of the both solutions is same $R (\tilde{Z}) = \frac{8250}{13}$ . One can be the alternate solution to the other.

Using proposed method, we can get optimal solution with less spread which makes most trustful and less oscillation in it. This ranking method can also be used to solve any kind of symmetric fuzzy linear programming problems without posing any difficult in the solving procedure.

Following figure will show the spread difference of the both optimal solutions in (Fig. 3).

Fig.3

Graphical representation of optimal solution.

The following (Table 1) shows there is no such difference in the optimal solution between existing and proposed method. The main advantage of this proposed method is that it can be used well instead of all the existing methods. Furthermore, this ranking function can be used to any symmetrical type fuzzy number such as symmetric triangular fuzzy numbers, symmetric trapezoidal fuzzy numbers etc.

Table 1

Difference between ranking values of optimal solutions

Existing	Proposed
method	method
A. Ebrahimnejad [8]	$\frac{28}{5}$ or (5.6)	$\frac{28}{5}$ or (5.6)
Akbar Hashemi Borzabadi and	5.6	5.6
Hossein Alemy [10]
K.Dhurai and A.Karpagam [6]	$\frac{11232}{132}$	$\frac{11232}{132}$
Amit Kumar and Jagdeep Kaur [3]	$\frac{1616}{11}$	$\frac{1616}{11}$
Rajarajeswari.P and Sahaya Sudha.A [11]	5067.86	5067.86
Sahaya Sudha.A and
Anitha.N [12]	35	35

8 Conclusion

In this paper, we have proposed a new ranking function for solving a symmetric fuzzy linear programming problem. Using this ranking function, all symmetric fuzzy valued linear programming problems can be converted into crisp valued linear programming problems and then the problem will be solved by the traditional way. The numerical example is illustrated and the advantages are pointed out between the existing method and proposed method. From this, it is guaranteed that the proposed method provides the optimal solution with less spread in it. Hence, this method is lot easier to get the optimal solution for a symmetric fuzzy valued linear programming problem.

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