Optimal number of minimal repairs before replacement based on a cumulative repair-cost limit policy

In this paper, we consider a replacement model with minimal repair based on a cumulative repair-cost limit policy, where the information of all repair costs is used to decide whether the system is repaired or replaced. As a failure occurs, the system experiences one of the two types of failures: a type-I failure (repairable) with probability q, rectified by a minimal repair; or a type-II failure (non-repairable) with probability p (=1 − q) that calls for a replacement. Under such a policy, the system is replaced anticipatively at the nth type-I failure, or at the kth type-I failure (k < n) at which the accumulated repair cost exceeds the pre-determined threshold, or any type-II failure, whichever occurs first. The object of this paper is to find the optimal number of minimal repairs before replacement that minimizes the long-run expected cost per unit time of this polish. Our model is a generalization of several classical models in maintenance literature, and a numerical example is presented for illustration.     

A multi-criteria optimal replacement policy for a system subject to shocks

A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As these shocks occur, the system experiences one of two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. In this study, we consider a multi-criteria replacement policy based on system age, nature of failure, and entire repair-cost history. Under such a policy, the system is replaced at planned life time T, or at the nth type-I failure, or at the kth type-I failure (k < n) at which the accumulated repair cost exceeds the pre-determined limit, or at the first type-II failure, whichever occurs first. An optimal policy over the control parameters is studied analytically by showing its existence, uniqueness, and structural properties. This model is a generalization of several existing models in the literature. Some numerical examples are presented to show several useful insights.     

Extended optimal replacement policy for a system subject to non-homogeneous pure birth shocks

This article studies the optimal replacement policy with general repairs for an operating system subject to shocks occurring to a non-homogeneous pure birth process (NHPBP). A shock causes that the system experiences one of two types of failures: type-I failure (minor failure) is rectified by a general repair, or type-II failure (catastrophic failure) is removed by an unplanned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a bivariate replacement policy (n, T) under which the system is replaced at planned life age T, or at the nth type-I failure, or at any type-II failure, whichever occurs first. The optimal replacement schedule which minimizes the expected cost rate model is derived analytically and discussed numerically.     

Optimum preventive maintenance policies for systems subject to random working times,replacement, and minimal repair

This paper proposes, from the economical viewpoint of preventive maintenance in reliability theory, several preventive maintenance policies for an operating system that works for jobs at random times and is imperfectly maintained upon failure. As a failure occurs, the system suffers one of two types of failure based on a specific random mechanism: type-I (repairable) failure is rectified by a minimal repair, and type-II (non-repairable) failure is removed by a corrective replacement. First, a modified random and age replacement policy is considered in which the system is replaced at a planned time T, at a random working time, or at the first type-II failure, whichever occurs first. Next, as one extended model, the system may work continuously for N jobs with random working times. Finally, as another extended model, we might consider replacing an operating system at the first working time completion over a planned time T. For each policy, the optimal schedule of preventive replacement that minimizes the mean cost rate is presented analytically and discussed numerically. Because the framework and analysis are general, the proposed models extend several existing results.     

A bivariate optimal imperfect preventive maintenance policy for a used system with two-type shocks

This article studies an optimal imperfect preventive maintenance policy based on a cumulative damage model for a used system with initial variable damage. The used system is subject to shocks occurring to a non-homogeneous Poisson process, and suffers one of two types of shocks with stochastic probability: type-I shock (minor) yields a random amount of additive damage of the system, or type-II shock (catastrophic) causes the system to fail. A bivariate preventive maintenance schedule (n, T) is presented in which the system undergoes preventive maintenance at a planned time T and the nth type-I shock, or corrective maintenance at any type-II shock and the total damage exceeds a threshold level, whichever occurs first. The optimal preventive maintenance schedule which minimizes the expected cost rate is derived analytically and discussed numerically.     

Extended periodic imperfect preventive maintenance model of a system subjected to shocks

This article deals with a periodic imperfect preventive maintenance (PM) model of a system subjected to random shocks. A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur, the system experiences one of the two types of failures: type-I failure (minor) and type-II failure (catastrophic). Type-I failures are rectified by minimal repair. The system is maintained following the occurrence of a type-II failure or at age T, whichever takes place first. At the N-th PM, the system is replaced. An approach that generalises the existing works on the periodic imperfect PM policy is proposed. The imperfect PM model adopted is hybrid in the sense that it not only reduces the effective age of the system but also alters the system hazard rate. Taking random minimal repair costs into consideration, the objective consists of finding the optimal PM and replacement schedules that minimise the expected cost per unit time over an infinite time-horizon.     

A deteriorating repairable system with stochastic lead time and replaceable repair facility

In this paper, a deteriorating repairable system with stochastic lead time and replaceable repair facility is studied. We assume that the spare system for replacement is available only by an order and the lead time for delivering the spare follows exponential distribution. Moreover, we also suppose that the repair facility may be subject to failure during the repair period. Under these assumptions, by using the geometric process and the supplementary variable technique, some important reliability indices such as the system availability, rate of occurrence of failure (ROCOF) and the probability that the system is waiting for replacement are derived. An ordering policy N − 1 and a replacement policy N based on the number of failures of the system are also considered. Furthermore, employing several Lemmas, the explicit expression of the average cost rate is derived. Meanwhile, the optimum value N^∗ for minimizing the average cost rate could be determined numerically.     

Warranty servicing for discretely degrading items with non-zero repair time under renewing warranty

This paper considers a repair-replacement strategy for a special class of discretely degrading and repairable products under renewing free replacement warranty. Each product may experience N different working states with different exponential hazard functions, before the warranty contract expires. Once the item enters working state i (i = 1, 2, … , N), it can either fail or move to any of the subsequent working states with different probabilities, given that a transition has been made. In the former case, the best rectification action, replacement or minimal repair, regarding the failure status, i.e., the product degradation level at the failure time and the remaining warranty time, should be conducted to put the item into operation. It is assumed that replacement of the faulty item occurs instantly and the product warranty is renewed, but non-zero repair time has been included in the model. We derive the optimal replacement-repair policy to minimize the manufacturer’s expected warranty servicing cost per item sold. The Adomian decomposition method is used to find an analytic approximate solution for a special case with two working states. For N > 2, a simulation based optimization method has been developed to analyze the expected warranty cost. Some numerical examples in each section are given to clearly demonstrate the application of this model.     

Constructing vertex-disjoint paths in (n, k)-star graphs

This work describes a novel routing algorithm for constructing a container of width n − 1 between a pair of vertices in an (n, k)-star graph with connectivity n − 1. Since Lin et al. [T.C. Lin, D.R. Duh, H.C. Cheng, Wide diameter of (n, k)-star networks, in: Proceedings of the International Conference on Computing, Communications and Control Technologies, vol. 5, 2004, pp. 160-165] already calculated the wide diameters in (n, n − 1)-star and (n, 1)-star graphs, this study only considers an (n, k)-star with 2 ? k ? n − 2. The length of the longest container among all constructed containers serves as the upper bound of the wide diameter of an (n, k)-star graph. The lower bound of the wide diameter of an (n, k)-star graph with 2 ? k ? ⌊n/2⌋ and the lower bound of the wide diameter of a regular graph with a connectivity of 2 or above are also computed. Measurement results indicate that the wide diameter of an (n, k)-star graph is its diameter plus 2 for 2 ? k ? ⌊n/2⌋, or its diameter plus a value between 1 and 2 for ⌊n/2⌋ + 1 ? k ? n − 2.     

Some replacement policies with minimal repairs and repair cost limit

Some extended replacement policies based on the number of failures, incorporating the concept of repair cost limit are discussed. Three models are considered as follows: (a) a unit is replaced at the nth failure, or when the estimated minimal repair cost exceeds a particular limit c; (b) a unit has two types of failures and is replaced at the nth type 1 failure, or type 2 failure, or when the estimated repair cost of type 1 failures exceeds a predetermined limit c—type 1 failures are minimal; failures, type 2 failures are catastrophic failures and both occur with constant probability; (c) a unit has two types of failures and the type 1 and type 2 failures are age dependent—the unit is replaced at the nth type 1 failure, type 2 failure, or when the estimated repair cost due to type 1 failures exceeds a predetermined limit c. Introducing costs due to replacements, inspections, and minimal repairs, an optimal number of minimal repairs before replacement is obtained, which minimizes the expected cost rate. Some particular cases are also derived. Finally, the application of these models to computer science is discussed.