The minimal repair replacement is a reasonable assumption in many practical systems. Under this assumption a failed component is replaced by another one whose reliability is the same as that of the component just before the failure, i.e., a used component with the same age. In this paper we study the minimal repair in coherent systems. We consider both the cases of independent and dependent components. Three replacement policies are studied. In the first one, the first failed component in the system is minimally repaired while, in the second one, we repair the component which causes the system failure. A new technique based on the relevation transform is used to compute the reliability of the systems obtained under these replacement policies. In the third case, we consider the replacement policy which assigns the minimal repair to a fixed component in the system. We compare these three options under different stochastic criteria and for different system structures. In particular, we provide the optimal strategies for all the coherent systems with 1–۴ independent and identically distributed components.

Introduction

Nowadays, people demand more and more reliable systems. Several techniques have been developed to model and improve the reliability of a system. The basic concepts used in Reliability Theory were introduced in the classic book by Barlow and Proschan (1975). Recent developments can be seen, for example, in Aven and Jensen (2013) and Natvig (2011). A good way to improve the reliability of a system is to consider some redundancy or maintenance actions. These actions can be performed in different ways as, for example, by planning some replacement strategies, minimal repairs, imperfect repairs, redundancies, etc. On the one hand, it is addressed in literature the concept of active or hot redundancy, where some additional components are included in the system by using parallel structures, see Valdés and Zequeira (2006), Zhao, Chan, Li, and Ng (2013), Zhao, Chan, and Ng (2012), and Zhao, Zhang, and Chen (2017), or Belzunce, Martínez-Puertas, and Ruiz (2013) and Zhang, Amini-Seresht, and Ding (2017) for systems having independent and dependent components, respectively. On the other hand, it is addressed the concept of standby or cold redundancy, where a component is replaced or repaired when it fails. Among the standby policies, many papers study the case of perfect repairs when the broken unit is replaced by a new and identical unit, see, e.g., Misra, Misra, and Dhariyal (2011), Singh and Misra (1994) and You and Li (2014). Nevertheless, there exist many options of replacement for a failed component. A nice summary of these cases is described in Aven (2014). In this paper we focus on minimal repairs as a particular case of cold redundancy. Under this assumption a failed component is repaired to be just as it was before its failure.