Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut

In this paper, a fuzzy multi-objective nonlinear programming problem is presented. All the coefficients of the nonlinear multi-objective functions and the constraints are fuzzy numbers. Here we find the solution of the nonlinear programming problem by using Interval arithmetic based on Alpha-cut. A numerical example is presented.

Introduction

The concept of fuzzy non linear programming was proposed by zimmermann [12] . Fuzzy non linear programming problem is useful in solving problems which are difficult, impossible to solve to the imprecise, subjective nature of the problem formulation or have an accurate solution. Bellman and Zadeh [1] proposed the concept of decision making in fuzzy environment. A new method for solving linear programming problem with vagueness in constraints by using ranking function was introduced by Maleki [8] . Tanaka .et.al, [11] adopted this concept for solving mathematical programming problems. Pandian and Jayalakshmi [9] proposed a new method for solving fully fuzzy linear programming problem with fuzzy variables. Compos and Vardegay [2] considered linear programming problems with fuzzy constraints and fuzzy coefficients in both left and right hand of the constraints set. Kirtiwant and Tanaji [5] introduced a new approach for finding an optimal fuzzy solution for fuzzy non linear programming problem. In this paper, we propose the basic definitions of fuzzy set and Interval arithmetic operations on fuzzy numbers based on  -cut to solve a fuzzy multi objective non linear programming problem. [4] Let A be a classical set.

Preliminaries 2.1 Definition:

Preliminaries 2.1 Definition:

Definition: [3]

A fuzzy number is a convex normalized fuzzy set of the real line R whose membership function is piecewise continuous. [12] A fuzzy number A ~ in R is said to be a triangular fuzzy number if its membership function

Definition:

4 Definition: [4] Given a fuzzy set A defined on X and any number  ɛ[0,1] , the  -cut A  is the crisp set

2.5 Definition: [9] Given a fuzzy set A defined on X and any number  ɛ[0,1] , the strong  -cut [10] Interval arithmetic, interval mathematics, interval analysis, or interval was developed by various mathematicians in between 1950 and 1960. This is an approach to putting bounds on rounding errors and measurement of errors in mathematical computation and thus developing numerical methods that yield reliable results. This represents each value as a range of possibilities. The following are the basic operations of interval arithmetic, for two variables [p, q] and [r, s] that are subsets of the real line

Interval arithmetic

min (pr, ps, qr, qs) , max(pr, ps, qr, qs)] (iv) Division:

Arithmetic operations of fuzzy numbers using  -cut method

In this section, we consider addition, subtraction, multiplication and division of fuzzy numbers using  -cut method [10].

Addition of Fuzzy Numbers

Let X = [p, q, r] and Y = [a, b, c] be two fuzzy numbers whose membership functions are defined by c -(c -b)  ] are the  -cuts of the fuzzy numbers X and Y respectively. To calculate the addition of fuzzy numbers X and Y using interval arithmetic, that is

Subtraction of Fuzzy Numbers

Let X = [p, q, r] and Y = [a, b, c] be two fuzzy numbers.

Then, c -(c -b)  ] are the  -cuts of the fuzzy numbers X and Y respectively. To calculate the subtraction of fuzzy numbers X and Y using interval arithmetic, that is

Multiplication of Fuzzy Numbers

Let X = [p, q, r] and Y = [a, b, c] be two fuzzy numbers.

Then, X and Y respectively. To calculate the division of fuzzy numbers X and Y using interval arithmetic, that is,

Multiobjective nonlinear programming (monlp) formulation

The function to be maximized is called an objective function. This is denoted by Z. c = (c1, c2, ….,cn) is called a cost vector. The matrix [aij] is called a constraint matrix and the vector bi = (b1, b2,…., bm) is called right hand side vector.

Procedure

The arithmetic operations of fuzzy numbers using  -cut operations discussed in the earlier sections are used below to solve the fuzzy nonlinear programming problem.

Step 1: Find the value of X  and Y 

Step 2: Add the  -cuts of X and Y using interval arithmetic

Step 3: The values obtained in step1 and step2 is converted into a crisp MONLPP.

Step 4: By solving the above nonlinear programming problem, we obtain the optimal solution.

7. Numerical problem [5] Using the procedure an interval arithmetic nonlinear programming problem with triangular fuzzy numbers is considered.

Maximum Z = [(5, 6, 7) + (6, 7, 8)] 1

x + [ (3, 4, 5) + (1, 2, 3)] The  -cut of the fuzzy number (5, 6, 7) is

The  -cut of the fuzzy number (6, 7, 8) is

Step 2: Adding the  -cuts of X and Y using interval arithmetic we obtain Step 4: Solving the nonlinear equation ( 4) with (6). We obtain the optimal solution. Max Z = 48, 1 x = 0, 2 x = 1.8

Solving the non linear equation ( 5) with (6), we obtain the optimal solution.

Max Z = 31.68, 1 x = 0, 2 x = 1.2

Conclusion

Fuzzy multi-objective nonlinear programming problem has been solved by using interval arithmetic based on  -cut operation without converting them into a classical NLPP. This is an easy approach for solving FNLPP by using interval arithmetic when compared to the earlier approaches .This is also can be applied in trapezoidal, hexagonal, octagonal fuzzy numbers.

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