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.

中文翻译：

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求解连通-$(r,s)$-out-of-$(m,n)$:F 格系统中最优排列问题的算法

本文考虑了连通域中的最优排列问题$(r,s)$-在......之外-$(m,n)$:F 晶格系统。这个问题的目的是在给定组件可靠性的假设下找到一种组件排列来最大化系统可靠性。虽然现有的研究开发了一些算法来寻找连接的最佳排列 -$(r,s)$-在......之外-$(m,n)$:F 格系统，这些现有算法只能在特殊情况下找到最佳排列。在本文中，我们开发了一种改进算法，用于有效地找到一般连接的最佳排列 -$(r,s)$-在......之外-$(m,n)$:F 晶格系统。它基于分支定界方法。首先，我们根据最优安排的现有条件推导出修剪分支的三个条件。这些条件可以减少最佳安排的候选数量。此外，我们通过合并 Omura 给出的导出条件来改进算法 等。在改进算法中，我们通过记住特定值然后利用它们来有效地计算系统可靠性。数值实验结果表明，该算法比Omura提出的算法更有效 等。在计算时间方面。结果表明，提出的算法可以找到最优排列，而中村提出的算法 等。找不到。所提出的算法提供的最优安排及其系统可靠性也可用于评估（元）启发式算法获得的解决方案的质量。