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**Priyadarsini Rath ^{*} and Rajani B. Dash**

Department of Mathematics, Ravenshaw University, Cuttack, Odisha - 753003, India

- *Corresponding Author:
- Rath P

Department of Mathematics

Ravenshaw University

Cuttack, Odisha - 753003, India

**Tel:**07328083159

**E-mail:**[email protected].com

**Received Date:** October 24, 2017; **Accepted Date:** November 17, 2017; **Published Date:** November 24, 2017

**Citation:** Rath P, Dash RB (2017) Solution of Fuzzy Multi-objective Nonlinear Programming Problem Using Fuzzy Programming Techniques Based on Hyperbolic Membership Functions. Appl Sci Res Rev 4:13. DOI: 10.21767/2394-9988.100063

The main objective of this paper is three folds: (i) introduction of hyperbolic membership function in Zimmerman’s fuzzy programming technique to handle multi-objective nonlinear programming problems (ii) reduction of fuzzy multi objective nonlinear programming problem to crisp one using ranking function (iii) reduction of complexity of nonlinear problem occurring in course of application of Zimmerman’s technique by the process of linearization.

Hyperbolic membership function; Fuzzy multi-objective nonlinear programming problem; Ranking function

The real world problems with multiple conflicting objectives are more conveniently modeled into multi-objective programming problems. Further, when the parameters are imprecise numerical quantities, it is very much appropriate to implement fuzzy quantities for modeling these situations. In 1970, Bellmann and Zadeh introduced the concept of fuzzy quantities in decision making. Authors like H.R. Maleki, A. Ebrahimnejad et al., P. Fortemps et al., H. Zimmerman have introduced fuzzy programming approach to solve crisp multi-objective linear programming problem. In 1981, Leberling used a nonlinear membership function in form of hyperbolic function for solving linear programming problem. In 1991, Dhingra et al. introduced exponential, quadratic and logarithimic membership functions for optimal designing problems. In 1997, R. Verma et al. used the fuzzy programming technique based on some nonlinear membership function to solve fuzzy MOLPP. R. B. Dash et al. used defuzzification through ranking function and Zimmerman’s technique based on trapezoidal membership function for solving fuzzy MOLPP. P. Rath et al. used exponential and hyperbolic membership functions in Zimmerman’s technique to solve fuzzy MOLPP. Recently P. Rath et al. extended the idea of their paper and solved fuzzy multi-objective nonlinear programming problem (FMONLPP) through exponential membership function [1-17]. In this paper, after defuzzification of FMONLPP, the resulting nonlinear multi-objective programming problem is solved using Zimmerman’s technique through hyperbolic membership function. Also, the nonlinear terms occurring in the Zimmerman’s procedure are linearalised to marginalize the intricacy. A numerical example is given for better understanding of the technique.

We consider fuzzy multi objective nonlinear programming problem with trapezoidal fuzzy coefficients as follows

Max p=1, 2…q

i=1,2…m (2.1)

where

and and are in the above relation are in trapezoidal form as

Now the FMONLPP can be transformed to a MONLPP by applying the Rouben’s ranking function R as below.

p = 1, 2…q

i = 1, 2…m

p=1, 2…q

i=1, 2…m (2.2)

Where are real numbers corresponding to the fuzzy numbers with respect to the exponential function respectively and the Rouben’s ranking function is given below.

The ranking function suggested by F. Rouben is defined by

where

**Lemma 1**

The optimum solutions of (2.1) and (2.2) are equivalent.

**Proof**

Let M_{1}, M_{2} be set of all feasible solutions of (2.1) and (2.2) respectively.

Then x ∈ M_{1}

i=1, 2…m

(By applying the Rouben’s ranking function) i=1, 2…m

⇒ x ∈ M_{2}

Converse can be proved similarly.

Thus, M_{1}=M_{2}

Let x* ∈ X be the complete optimal solution of (2.1).

Then for all x ∈ X, where X is the set of feasible solutions.

(By applying the Rouben’s ranking function)

j=1, 2…q

j=1, 2…q

∀x

We modified the Zimmermann’s technique using hyperbolic membership function in order to solve multi-objective nonlinear programming problem (2.2).

**Step-1:**

The multi-objective linear programming problem is solved by considering one objective at a time and ignoring all others. The process is repeated q times for q different objective functions.

Let X_{1}, X_{2},…X_{q} be the ideal situations for the respective functions.

**Step-2:**

A pay-off matrix of size q by q is formed using all the ideal solutions of step-1. Then from pay-off matrix lower bounds (L_{p}) and upper bounds (U_{p}) of the objective functions are obtained.

Thus p=1, 2,…q

**Step-3:**

Using hyperbolic membership function, an equivalent crisp model for the fuzzy model can be formulated as follows:

Min λ

p=1, 2…q

i=1, 2...m

j=1, 2…n

After simplification, the above problem reduces to

Min *x _{mn+1}*

Subject to

p=1, 2…q

i=1, 2...m

x_{j} ≥0, j=1, 2…n

x_{mn+1} ≥0

Where

**Step-4:**

The crisp model is solved and the optimal compromise solution is obtained. The values of objective functions at the compromise solution are obtained.

(3.1)

where

Using ranking function, the problem reduces to

x_{1}, x_{1} ≥ 0

(3.2)

(3.3)

Solving (3.2) and (3.4) by Wolf’s method we get

Solving (3.3) and (3.4) by Wolf’s method, we get

The pay-off matrix of Lower Bounds (L.B.) and Upper Bounds (U.B.) of the objective functions Z′_{1} and Z′_{2} is given below

Function | LB | UB |
---|---|---|

Z′_{1} |
3.0655 | 3.00770 |

Z′_{2} |
3.8065 | 3.9653 |

The crisp model can be formulated as

Min *x _{3}*

Subject to

j =1, 2...m (3.5)

j =1,2...n

Putting the values of we get

Min *x _{3}*

s.t.

(3.6)

Due to presence of non-linear term x_{1}^{2} in the constraint (3.6), the problem becomes too complex to solve. To avoid the situation taking advantage of

0.1837 ≤ x_{1} ≤ 0.3637

We linearize x_{1}^{2} as follows.

Where

x_{10} = 0.1837

x_{11} = 0.2437

x_{12} = 0.3037

x_{13} = 0.3637

Then the problem (3.6) reduces to

Min: x_{3}

s.t.

1.1x_{1} + 3.9x_{2} ≤ 3.8 (3.7)

1.0x_{1} +1.1x_{2} ≤ 2.1

Solving (3.7) the optimal solution of the problem is obtained as:

Now the optimal value of the objective functions of FMOLPP (4.1) becomes

The membership functions corresponding to the fuzzy objective functions are as follows.

The closeness of the result of the numerical example of this paper with that of the previous paper [17] confirms that the method given in this paper is an alternative way of solving fuzzy multi-objective nonlinear programming problem.

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