In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

For two components and one standby redundancy, we develop a characterization on the hazard rate order and the reversed hazard rate order of the redundant system lifetime in the context of mutually independent components lifetimes. Also, the likelihood ratio order is derived on the lifetime of the series system with two components lifetimes and two matched active redundancies lifetimes both following the proportional hazard model.

In this paper, we investigate inventory and order strategies of a two-echelon supply chain, which is composed of two unreliable suppliers that are subject to random disruption. We develop the gross weighted profit benchmark model and the service level constrained model of the supply chain, respectively. We derive the retailer’s optimal order quantity and analyze the retailer’ optimal order policy and also obtain the analytical closed-form solutions. In addition, some numerical examples are provided to illustrate the effect of disruption time, disruption probability and fill rate on the optimal decisions and expected profit.

This manuscript studies the stochastic comparisons of the second-order statistics from dependent or independent and heterogeneous modified proportional hazard rate observations. Some sufficient conditions on the usual stochastic order of the second-order statistics from dependent and heterogeneous observations are established under Archimedean copula. Some sufficient conditions are also provided in the hazard rate order of the second-order statistics arising from two sets of independent and heterogeneous or multiple-outlier modified proportional hazard rate observations. Some numerical examples are given to illustrate the theoretical findings.

This article discusses the stochastic behavior and reliability properties for the inactivity times of failed components in coherent systems under double monitoring. A mixture representation of reliability function is obtained for the inactivity times of failed components, and some stochastic comparison results are also established. Furthermore, some sufficient conditions are developed in terms of the aging properties of the inactivity times of failed components. Finally, some numerical examples are presented to illustrate the theoretical results.