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A new periodic damage model is proposed in this paper, where the damage produced by shocks is only measured at periodic times, without considering the stochastic shocks and the cumulative process of damage in discrete times. We model five replacement policies for the periodic damage models using three approaches of whichever triggering event occurs first, whichever triggering event occurs last, and replacing over a planned measure, which are named as replacement first, replacement last, over-damage replacement, over-damage replacement first, and over-damage replacement last, respectively. In analytical ways, we obtain their optimum replacement policies to minimize the expected cost rates and compare them to find which model is more economical. Finally, numerical examples are given to illustrate the validity of the presented models.

It has been proposed in recent literatures that if replacement time is planned too early prior to failure time, a waste of operation cost, i.e., excess costs, would incur because the system might run for an additional period of time to complete critical operations; and if the replacement time is too late after failure, a great failure cost, i.e., shortage cost, is incurred due to the delay in time of the carelessly scheduled replacement. The above notion of shortage and excess costs are taken into consideration for the replacement first, replacement last and replacement overtime models in this paper. We obtain the expected cost rates and their optimum replacement times. Comparisons among these optimum times are made analytically and numerically.

Preventive replacement policy conducted at a routine time $T^{\ast }$ has been surveyed extensively in literatures. Another two random replacement times $Y_1$ and $Y_2$ for time $T^{\ast }$ are jointly considered in this paper. We formulate the models of replacement first, replacement last and replacement middle, which are based on the respective assumptions of whichever occurs first, whichever occurs last and whichever occurs middle. Next, we modify replacement first and replacement last by using whichever occurs first and last jointly. Furthermore, we give the model of replacement overtime when the assumptions of whichever occurs first, whichever occurs last and replacing the unit over time $T$ are simultaneously considered. Comparisons among these replacement models are made to find which policy is more economical than the other.

The random age replacement policies discussed in literatures are helpful to complete the nonstopping works with random working cycles, however, maintenance policies are more easily performed at periodic times in real applications. For such a viewpoint, this paper proposes that age replacement policies are planned at periodic times while considering the random working cycles. Using the modeling approaches of replacement first and last policies, we discuss two models such that the unit is replaced at periodic times $KT$ and working cycles $N$, whichever occurs first and whichever occurs last. The expected cost rate models are obtained and their optimal solutions for $K$ and $N$ are discussed. The comparisons between the policies of $K$ and $N$, and replacement first and last are made from the point of cost. Numerical examples are illustrated when the failure time has a Weibull distribution.

In this paper, a random repairable $k$-out-of-$n$ system is studied when random replacement policies are performed, where $k$ is a stochastic parameter. It is replaced before failure at a planned time $T$ or at random working cycle $Y$, whichever occurs last or first. Furthermore, optimal replacement policies are obtained for the extended models with working numbers. This paper aims to find the optimal replacement policies when replacement first and last policies are performed. Expected cost rates for the first and last replacement policies are derived, and their optimal replacement policies are derived analytically. Numerical examples are given to illustrate the theoretical results.