Using NSGA-II to Solve Interactive Fuzzy Multi-objective Reliability Optimization of Complex System

Abstract

In the real-world problem, reliability enhancement is one of the primary concerns in the system design. A conflicting situation often occurs when the cost of the system is reduced and its reliability is improved simultaneously. Practically, design data included in the system are not found specific. Various types of uncertainty such as vagueness, qualitative statements, expert’s information character etc. are found in the multi-objective optimization of reliability problems. Multiple solutions (Pareto-optimal solutions) are obtained in multi-objective optimization problem (MOP) where a decision-maker (DM) plays a crucial role in decision-making process. In view of such things, a fuzzy multi-objective reliability optimization model is developed interactively. Numerical examples of complex systems are given for the illustrations. To solve the problems, an efficient multi-objective evolutionary algorithm (MOEA), namely NSGA-II is employed. Finally, we get the solutions according to the preference of the DM.

Appendix

Life-support system in a space capsule: [15]

The system reliability $R_{s}$ and system cost $C_{s}$ of system configuration given in Fig. 2a are expressed as follows.

$$ \begin{aligned} R_{s} & = 1 - R_{3} \left[ {\left( {1 - R_{1} } \right)\left( {1 - R_{4} } \right)} \right]^{2} - \left( {1 - R_{3} } \right)\left[ {1 - R_{2} \left\{ {1 - \left( {1 - R_{1} } \right)\left( {1 - R_{4} } \right)} \right\}} \right]^{2} \\ C_{s} & = 2\sum\limits_{i = 1}^{4} {K_{i} R_{i}^{{\alpha_{i} }} } , \\ \end{aligned} $$

where vectors of coefficients $K_{i}$ and $\alpha_{i}$ are $K = \left\{ {100, 100, 200, 150} \right\}$ and $\alpha = \left\{ {0.6, 0.6, 0.6, 0.6} \right\}$, respectively.

Bridge network system: [15]

The algebraic expression of $R_{s}$ and $C_{s}$ of system configuration given in Fig. 2b is given as follows.

$$ \begin{aligned} R_{s} & = R_{1} R_{4} + R_{2} R_{5} + R_{2} R_{3} R_{4} + R_{1} R_{3} R_{5} + 2R_{1} R_{2} R_{3} R_{4} R_{5} - R_{1} R_{2} R_{4} R_{5} \\ & \quad - R_{1} R_{2} R_{3} R_{4} - R_{1} R_{3} R_{4} R_{5} - R_{2} R_{3} R_{4} R_{5} - R_{1} R_{2} R_{3} R_{5} \\ C_{s} & = \sum\limits_{i = 1}^{5} {a_{i} \,{ \exp }\left[ {\frac{{b_{i} }}{{1 - R_{i} }}} \right]} , \\ \end{aligned} $$

where $a_{i} = 1$ and $b_{i} = 0.0003\,\forall i$, $i = 1, 2, \ldots ,5$.