ANALYSING A FULLY FUZZY LINEAR PROGRAMMING PROBLEM
USING TOPSIS
V. Thanvee
*1, Dr. A. Sahaya Sudha
*2*1
PG Scholar, Department of Mathematics, Nirmala college for women, Coimbatore, Tamil Nadu, India
*2Assistant Professor, Department of Mathematics, Nirmala college for women, Coimbatore,
Tamil Nadu, India.
ABSTRACT
In this paper a fuzzy to TOPSIS method is used to solve the Fully Fuzzy Linear Programming Problem (FFLPP). Here we convert a multi objective problem into a single objective problem. We use “Entropy Method” to find weight determination. Then we find both positive and negative ideal solutions by fuzzy TOPSIS method. Our method is based on membership functions. Finally, we conclude the optimal solution for FFLPP.
Keywords: Fuzzy linear programming, Triangular Fuzzy numbers, Membership function, Big-M method.
I.
INTRODUCTION
Zadeh in 1965 introduced fuzzy set theory by publishing the first article in this area [7]. Linear programming (LP) problem is mostly used in different fields of science and engineering for modelling real world problems [6]. In real world situation the available information in the system under consideration are not exact, therefore fuzzy linear programming (FLP) was introduced and studied by many researchers [2,4,8,9,10,12,13,14] Linear programming or linear optimization is one of the most practically used techniques in operation research, which finds the best extractable solution with respect to the constraints. Linear programming problem is in the two forms of classical linear programming (LP) and fuzzy linear programming problem (FLPP). In real-world problems, values of the parameters in LP problem should be precisely described and evaluated. How-ever in real-world applications, the parameters are often illusionary. The optimal solution of an LP only depends on a limited number of constraints, therefore, much of the collected information has a little impact on the solution. It is useful to consider the knowledge of experts about the parameters as fuzzy data. The concept of decision in fuzzy environment was first proposed by Bellman and Zadeh [1]. Real world problem invariably involve multiple objectives. Multi objective programming is a part of mathematical programming dealing with decision problems characterized by multiple and conflicting objective functions that are to be optimized over a feasible set of decisions. Goal programming (GP) is a multi objective programming technique [3]. When the decision-making process is used in a fuzzy environment,[1] proposed that the symmetry between goals and constraints is the most important feature. With this property of symmetry Bellman and Zadeh proposed that a fuzzy decision is defined as the fuzzy set of alternatives resulting from the intersection of the goals/objectives and constraints [1] to formulate goal programming model of the problem the aspiration levels of the objective functions are defined first and also to formulate the fuzzy goal programming model of the problem the fuzzy goal levels of the objective functions are defined first. In this paper we take one multi objective linear programming problem and solve this problem first by min sum goal programming technique. Further we solve the same problem under fuzzy environment, where the resources are triangular fuzzy number using different techniques [12]. The proposed approach is very useful due to its capacity of providing optimal results with incomplete significant information [5].
II.
PRELIMINARIES
2.1 Fuzzy set:
Let X be a non-empty set. A fuzzy set ̃ in X is represented by its membership function ̃: X and ̃
is interpreted as the degree of membership of element x in fuzzy set ̃ for each x
2.2 Normal fuzzy set:
2.3 Piecewise continuous:
A fuzzy number is a convex normalized fuzzy set of the real line R. Whose membership function is piecewise continuous.
2.4 Triangular fuzzy number:
A triangular fuzzy number ̃ = (a, b, c) is fuzzy number on R with the membership function ̃ defined by:
̃ {
A triangular fuzzy number (a, b, c) is said to be non-negative triangular fuzzy number iff a ≥ 0.
2.5 Arithmetic operation on fuzzy numbers:
Let ̃ = (a, b, c) ̃ = (d, e, f) be two triangular fuzzy numbers. Then the arithmetic operation on these two fuzzy number can be defined as follows:
1.Addition : ̃ ̃ 2.Symmetry : - ̃ = (-c, -b, -a)
3.Subtraction : ̃ ̃ 4.Equality : ̃ = ̃ iff a=d, b=e, c=f
5.Multiplication : Suppose ̃ be any triangular fuzzy number and ̃ be non-negative triangular fuzzy number, then we define:
̃ ̃ {
III.
METHODOLOGY
3.1 Proposed method for fully fuzzy linear programming problem
A FFLP Problem with m fuzzy constraints and n variables is formulated as follows: Max or Min ∑ ̃ ̃ Subject to, { ∑ ̃ ̃ ̃ ̃ ̃ ̃
̃ ̃ , ̃ ̃ ̃ ̃ , ̃ ̃ , and ̃ ̃ ̃ ̃ . According to this definition the
step of our solution algorithm are as follows:
Step 1:
Let ̃ ̃ ̃ and ̃ are represented by triangular numbers ( , , ( ),( , and ( ,
respectively. Then, by substituting these values the FFLP problem, obtained in (1) is written as follows: Max or Min ∑ ( ,
Subject to,
{
( ) ( )
( ) ( )
Step 2:
By arithmetic operations, the linear programming problem of step 1 is converted into the following equivalent problem: Max or Min ∑ ( , = ∑ Subject to, { ∑ ∑ ∑ where, ( ) ( ) ( ) ( ) ( )
Step 3: Entropy method for weight determination
a) Normalize the decision matrix ∑ b) Compute entropy ∑
h = where m = number of alternatives. c) Compute the weight vector
∑
, j=1,2, ..., n & ∑
Step 4: Fuzzy topsis method to obtain fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS).
a) Vector normalization ̅ √∑
b) We compute the distance from each alternative to the FPIS and to the FNIS.
̃ ̃) = √ ∑ (FPIS)
̃ ̃) = √ ∑ (FNIS)
c) Compute the closeness coefficient for each alternative
Step 5: Define the membership function.
Step 6: Obtain the maximum of minimum satisfaction degree of all objectives. Step 7: Obtain the fuzzy optimum value.
IV.
NUMERICAL EXAMPLE
4.1 Consider the problem,
Min Z = (1, 2, 3) ̃ ̃ ̃ Subject to, { ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ 0 Solution: Step 1:
let ̃ ̃ ̃ be converted into a triangular fuzzy variables,
Min Z= (1, 2, 3) Subject to, { Step 2:
The objective function and the constraints are,
Min Z = ( , ) (1) Subject to, { (2)
Step 3: Weight determination (entropy method)
Table-1: Matrix form
(0, 1, 2) (1, 2, 3) (2, 3, 4)
(1, 2, 3) (2, 3, 4) (0, 1, 2)
(3, 4, 5) (0, 1, 2) (2, 3, 4)
∑ = (4, 7, 10) (3, 6, 9) (4,7,10)
∑
Table-2: Normalize decision matrix
(0, 0.14, 0.2) (0.33, 0.33, 0.33) (0.5, 0.43, 0.4) (0.5, 0.29, 0.3) (0.67, 0.5, 0.44) (0, 0.14, 0.2) (0.75, 0.57, 0.5) (0, 0.17, 0.22) (0.5, 0.43, 0.4) h = where m = 3 alternatives, h = 0.91 Entropy ∑ (0, 0.14*In0.14, 0.2*In0.2) = (0, -0.28, -0.32)
(0.33*In0.33, 0.33*In0.33, 0.33*In0.33) = (-0.37, -0.37, -0.37) Similarly, we get,
Table-3: Entropy Weights
(0, -0.28, -0.32) (-0.37, -0.37, -0.37) (-0.35, -0.36, -0.37) (-0.35, -0.36, -0.36) (-0.27, -0.35, -0.36) (0, -0.28, -0.32) (-0.22, -0.32, -0.35) (0, -0.30, -0.33) (-0.35, -0.36, -0.37) ∑ = (-0.57, -0.96, -1.03) (-0.64, -1.02, -1.06) (-0.7, -1, -1.06) Substitute in we get, Compute the weight vector using the formula
∑ j=1,2,3 1- = (0.48,0.13,0.07) + (0.42,0.08,0.04) + (0.36,0.09,0.04) ∑ = (1.26,0.3,0.15) ∑ Since ∑ Where, ,
Step 4: FPIS and FNIS
Vector normalization ̅
√∑
Similarly, we get,
Table-4: Vector normalization matrix
(0, 1, 2) (1, 2, 3) (2, 3, 4)
(1, 2, 3) (2, 3, 4) (0, 1, 2)
21, 38) (5, 14, 29) (8, 19, 36)
√ = (3.16, 4.58, 6.16) (2.24,3.74, 5.38) (2.82, 4.36, 6)
Substitute in ̅ we get,
Table-5: Normalized decision matrix
(0.38, 0.43, 0.46) (0.33, 0.26, 0.26) (0.28, 0.3, 0.26) (0, 0.22, 0.32) (0.45, 0.53, 0.56) (0.70, 0.69, 0.67) (0.32, 0.44, 0.49) (0.89, 0.80, 0.74) (0, 0.22, 0.33) (0.95, 0.87, 0.81) (0, 0.26, 0.37) (0.70, 0.69, 0.67) The value in each cell is known as a normalized performance value.
Next we multiply the weights of each criteria with the normalized performance value of each cell.
Table-6: Weighted normalized decision matrix
(0, 0.09, 0.15) (0.15, 0.14, 0.15) (0.19, 0.20, 0.17) (0.12 ,0.19, 0.23) (0.29, 0.20, 0.19) (0, 0.07, 0.09) (0.36, 0.37, 0.37) (0, 0.07, 0.09) (0.19, 0.20, 0.17) = (0.36, 0.37, 0.37) (0.29, 0.20, 0.19) (0.19, 0.20, 0.17) (0, 0.07, 0.09) (0, 0.07, 0.09) (maximum value) (minimum value)
We compute the distance from each alternative to the FPIS and to the FNIS. ̃ ̃) = √ ∑ j=1,2,3
Table-7: The distance from each alternative to the FPIS
0.29 0.09 0 0.17 0 0.14 0 0.19 0 ̃ ̃) = √ ∑ j=1,2,3 & n=1,2,3
Table-8: The distance from each alternative to the FNIS
0 0.10 0.14
0.10 0.19 0
0.29 0 0.14
=
Table-9: Closeness coefficient for each alternative
0.38 0.24 0.39
0.31 0.29 0.48
0.19 0.43 0.69
= Summation of FNIS
Step 5: Membership function
{ = { = { Step 6:
From (1) & (2) we get Min Subject to, Using Big-M method
Min Subject to, Using Big-M Method
( ) , 0) = 11 Min Subject to, Using Big-M method
Max = Min u Min + - = 1
Min + - = 1
Min + - = 1
Max Max and Max = 0.008.
Step 7:
The fuzzy objective value of the problem is,
Min Z = ( , ) = (2, 11, 28) The optimal solution of proposed method is Min Z= (2, 11, 28).
V.
CONCLUSION
In this paper a new method for solving, FFLPP is presented. This method is based on fuzzy TOPSIS. The aim of this paper is to find the maximum satisfaction degree of all objectives and also the optimal solution of FFLPP can be easily obtained.
VI.
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