A new concept for fuzzy variable based non-linear programming problem with application on system reliability via genetic algorithm approach

Abstract

Fuzziness is the primary and foremost perception of science and technology. This paper, for the first time, introduces a new concept on solution technique for fuzzy variable based non-linear programming problem with both decision variables and restriction being fuzzy in nature. First the problem is transformed in to a multi-objective non-linear programming problem, and then solving it by multiobjective genetic algorithm (MOGA) approach. The proposed procedure is applied on complex system reliability model to evaluate the system reliability in fuzzy environment, using MOGA by implementing new feature as refining operation. Numerical example is presented to illustrate proposed fuzzy system reliability model.

Appendix: Definition: symmetrical triangular fuzzy number

Parametric form: A fuzzy number is a pair $(\underline{\nu },~ \overline{\nu })$ of function $\underline{\nu }(t)$, $\overline{\nu }(t) :~ 0\le t\le ~1$ which must satisfy the following properties:

• (1) $\underline{\nu }(t)$ is bounded monotonic increasing left continuous function.

• (2) $\overline{\nu }(t)$ is bounded monotonic decreasing right continuous function.

• (3) $\underline{\nu }(t)~\le ~\overline{\nu }(t)$, $0~\le ~t~\le ~1$.

A symmetric triangular fuzzy number $\tilde{A}=[x_{0},~\omega ]$ center at $x_{0}$ with basis $2\omega$, is defined with following membership functions (Fig. 4)

$$\begin{aligned} \mu _{\tilde{A}}(x)=\left\{ \begin{array}{ll} \frac{x-x_{0}+\omega }{\omega } &{}\quad \text {for}\,x_{0}-\omega \le x\le x_{0},\\ \frac{x_{0}+\omega -x}{\omega } &{} \quad \text {for}\,x_{0}\le x\le x_{0}+\omega ,\\ \quad 0\,\, &{} \quad \text {otherwise}. \end{array} \right. \end{aligned}$$

The parametric form of the above TFN is $\underline{\nu }(t)=x_{0}-\omega +\omega t,~\overline{\nu }(t)=x_{0}+\omega -\omega t$.

Fig. 4

Membership function of symmetric TFN