Stochastic comparisons of replacement policies in coherent systems under minimal repair

Abstract

In this paper, we study stochastic comparisons of the coherent systems obtained under different repair policies. We consider minimal repairs of the components, that is, a failed component is replaced by another component which is similar (ie, it has the same distribution) to the replaced one and with the same age. Under these assumptions, we study how to determine the best replacement policies. Two general methods are proposed, one for systems with identically distributed components and another one for systems with ordered (in different stochastic senses) components. Both methods can be applied to systems with independent or dependent components. We use these methods to determine the optimal repair policies in some particular cases. We pay special attention to series and parallel systems with independent or dependent components. An algorithm to find the optimal replacement policy (when it exists) is provided.

Optimal allocation of minimal repairs in parallel and series systems

“…components. Our findings strengthen and generalize some corresponding ones in Shaked and Shanthikumar () and solve the open problems posed in Chahkandi et al () and Arriaza et al ().…”

“…We prove that both the reversed hazard function of T p ( k ) and the hazard rate function of T s ( k ) are Schur‐convex in k , which means that the optimal allocation policy can be respectively reached by putting all minimal repairs in one component for the parallel system in the sense of the reversed hazard rate ordering, and allocating the minimal repairs in balance as much as possible for the series system in the sense of the hazard rate ordering. These results not only strengthen the counterparts of Shaked and Shanthikumar () in the usual stochastic order to the reversed hazard rate order (for parallel system) and hazard rate order (for series system), respectively, but also solve the open problems proposed by Arriaza et al () and Chahkandi et al (). With the help of some numerical examples, we show that the hazard (reversed hazard) rate order does not hold for parallel (series) systems with more than three components when an allocation vector is more heterogeneous than another one.…”

“…According to Theorem , the optimal allocation strategy of minimal repairs for a series system can be obtained in terms of the hazard rate ordering if the minimal repairs are allocated in balance, which strengthens Results 2.4(s) of Shaked and Shanthikumar () and extends Theorem 3.3 of Arriaza et al (). Moreover, Theorem answers the open problems proposed in Chahkandi et al () and Arriaza et al () for series system from the perspective of the hazard rate order.…”

“…For parallel systems with stochastically ordered components, Shaked and Shanthikumar () showed that all minimal repairs should be put in the component having the largest lifetime by means of the usual stochastic ordering; see also Corollary 4.2 in Arriaza et al (). It is worth generalizing these results by means of some stronger stochastic orderings.…”

“…The aforementioned studies on allocations of minimal repairs mainly focus on reliability systems with independent components. Very recently, Arriaza, Navarro, and Suárez‐Llorens () studied stochastic comparisons of different allocation policies of minimal repairs for coherent systems with heterogeneous/homogeneous and possibly dependent components via distortion functions. Sufficient conditions on distortion functions are established for stochastic comparisons of system reliabilities under different allocation vectors.…”