In industrial engineering applications, randomly weighted $k$-out-of-$n$: G systems can model many reliability systems whose components may contribute unequally and randomly to the systems’ performance. This paper investigates optimal allocations of hot standbys for $k$-out-of-$n$: G systems with random weights. First, optimal allocation policies are presented by maximizing the total capacity according to the usual stochastic ordering and the expectation ordering when the system is constituted by independent and heterogeneous components accompanied with independent random weights. Second, we investigate hot standbys allocation for randomly weighted $k$-out-of-$n$: G systems with right [left] tail weakly stochastic arrangement increasing random weights in the sense of the usual stochastic ordering [increasing concave ordering]. Simulation studies are provided to illustrate our theoretical findings as well. These established results can provide useful guidance for system designers on how to introduce hot standbys in randomly weighted $k$-out-of-$n$: G systems in order to enhance their total capacities.

在工业工程应用中，随机加权 $克$-在......之外-$n$：G 系统可以对许多可靠性系统建模，这些系统的组件可能对系统性能的贡献不均等且随机。本文研究了热备的优化分配$克$-在......之外-$n$：具有随机权重的 G 系统。首先，当系统由具有独立随机权重的独立和异构组件构成时，根据通常的随机排序和期望排序，通过最大化总容量来提出最优分配策略。其次，我们调查随机加权的热备分配$克$-在......之外-$n$：在通常的随机排序[增加凹排序]的意义上，具有右[左]尾弱随机排列的G系统增加随机权重。还提供了模拟研究来说明我们的理论发现。这些已建立的结果可以为系统设计人员如何在随机加权中引入热备用提供有用的指导。$克$-在......之外-$n$: G 系统以提高其总容量。