Algorithm for Solving Optimal Arrangement Problem in Connected-$(r,s)$-out-of-$(m,n)$:F Lattice System

Abstract

This paper considers the optimal arrangement problem in a connected- $(r,s)$ -out-of- $(m,n)$ :F lattice system. This problem aims to find a component arrangement to maximize system reliability under the assumption that the component reliabilities are given. Although existing studies developed a few algorithms for finding the optimal arrangement of connected- $(r,s)$ -out-of- $(m,n)$ :F lattice systems, these existing algorithms can only find the optimal arrangement in the special cases. In this paper, we develop an improved algorithm for efficiently finding the optimal arrangement of general connected- $(r,s)$ -out-of- $(m,n)$ :F lattice systems. It is based on the branch-and-bound method. First, we derive three conditions for pruning a branch based on the existing conditions for the optimal arrangement. The conditions can reduce the number of candidates for the optimal arrangement. Moreover, we improve the algorithm by incorporating the derived conditions as given by Omura et al. In the improved algorithm, we compute the system reliability efficiently by memorizing the specific values and then utilizing them. The result of the numerical experiment shows the proposed algorithm is more efficient than the algorithm proposed by Omura et al. in terms of the computation time. The result shows that the proposed algorithm can find the optimal arrangement whereas the algorithm proposed by Nakamura et al. cannot find. The optimal arrangement and its system reliability, which the proposed algorithm provides can also be used to evaluate the quality of solutions obtained by (meta) heuristic algorithms.