Maintenance Overtime Policy with Cumulative Damage

We propose an extended maintenance overtime policy for the cumulative damage model where an operating unit suffers some damage due to shocks. It is assumed that the total damage is additive, and the unit fails when the total damage has exceeded a prespecified damage level. It is supposed that we start to observe occurrence of shocks after time T, and the unit is replaced at Nth (N = 1, 2, ...) shock over time T or at failure, whichever occurs first. That is, we propose a new policy by extending maintenance overtime policy. One example is a rental of system such as industrial equipment with some reservations. For such systems, they should be maintained or replaced at a prespecified number of uses over a scheduled time. For such a model, we obtain the mean time to replacement and the expected costs rate. Further, we discuss about optimal number N and time T which minimizes the expected cost rate when shocks occur in a Poisson process. Finally, numerical examples are given, and suitable discussions are made. Keyword Overtime policy, Replacement policy, Shock model, Cumulative damage


Introduction
We consider a modified maintenance overtime policy in which an operating unit is replaced at shocks and a damage level. In recent years, equipment management has become more important to complete projects rapidly, safety and accurately. Furthermore, the equipment has become more complexity, and more difficult to check its state by looking the appearance, which includes information unit for software development. We consider therefore that the equipment is replaced at a completion of uses to avoid interruption of work on the way of using cycles. Such a model is called as maintenance overtime policy (Nakagawa and Zhao, 2015). Furthermore, we suppose that the equipment suffers damage at every use, and fails when the total damage has exceeded a prespecified level. Such a model is called as shock and damage model (Nakagawa, 2007). A shock model in which interval times of shocks depend on stochastically the previous shock was proposed in Markovian environment (Cha and Finkelstein, 2013;Eryilmaz, 2016).
We can see many studies of maintenance policies applied stochastic processes such as renewal theory (Barlow and Proschan, 1965;Nakagawa, 2005). The maintenance models that the unit is replaced at a random operating time are studied (Nakagawa, 2014;Chen et al., 2010a;Chen et al., 2010b). Maintenance overtime policies where the unit is replaced at the first time of a completion time of works over planned time have been discussed (Nakagawa and Zhao, 2015;Zhao and Nakagawa, 2013;Zhao et al., 2014). In the model, the unit is not replaced while it is working, because it has much loss cost to stop its work. When the unit is used for several works, we should  We propose a maintenance policy of an operating unit which extends the overtime policy to a cumulative damage model: It would be reasonable for the unit to make appropriate policies with scheduled time or number of shocks to maintain or replace it. One example is a rental of equipments with some reservations. For such equipments, they should be maintained or replaced at a prespecified number of uses over a scheduled time. On the other hand, when the possibility of failures before a scheduled time is very low, we do not need to count the number of shocks before this time and can save time. From the above viewpoints, we suppose that the unit is replaced at th ( = 1, 2, …) shock over a scheduled time or at failure, whichever occurs first. Fig. 1 shows that the preventive replacement is done at th shock over time , and Fig. 2 shows that it is replaced at failure. In the figures, horizontal axis presents the process of time, and vertical axis presents the amount of total damage due to shocks. This policy includes several maintenance policies for a cumulative damage model (Nakagawa, 2007). For the above model, we obtain the expected costs rate and discuss optimal policies which minimize them. Section 2 shows the assumptions and notations, and obtains the mean time to replacement and the expected cost rate. Section 3 discusses optimal number * and time * which minimize the expected cost rate when shocks occur in a Poisson process. Section 4 gives numerical examples of optimal * and * when each damage is exponential. We investigate several tendencies for suitable parameters in numerical examples. Section 5 takes up a new replacement policy over damage , and obtains the expected cost rates.
(iii) Let ( ) denote a random variable which is the total number of shocks up to time . Then, we define a random variable ( ) as follows: which represents the total damage at time . The unit fails when the total damage has exceeded a prespecified damage level (0 < < ∞).
(v) Cost is a replacement cost when the unit fails, and cost is a replacement cost when the unit is replaced at th shock over time , where > .
From the above assumptions, we obtain the mean time to replacement and the expected cost rate analytically as follows: The probability that the unit is replaced before failure at shock over time is (1) and the probability that it is replaced at failure is where (1) + (2) = 1. The mean time to replacement is Therefore, the expected cost rate is, from (1) and (3), When the unit is replaced only at th shock,  (Nakagawa, 2007). When the unit is replaced only at the first shock over time is which agrees with (3.50) of (Nakagawa, 2007).

Optimal Replacement Policies
When ( ) = 1 − − , the expected cost rate in (4) is It is assumed that increases strictly with from ̅ ( ) to 1. Noting that ( ) represents the probability that the unit fails at shock + 1, given that it has not failed until shock , this assumption would be reasonable in actual fields.

Numerical Examples
We give numerical examples when ( ) = 1 − − and ( ) = 1 − − . Then, ( ) = , increases strictly with from /( − 1) to 1, increases strictly with from ( ) to 1 and increases strictly with from ( ) to 1.      Table 1 presents optimal 0 * when = 10 for / and , and when = 0, 0 * = * . For example, when / = 20 and = 2, 0 * = 2 and we should replace the unit at 2nd shock over time before failure. We can see that 0 * decreases with / and decreases with from * to 1. This indicates that if replacement cost of failure is large, then we should replace the unit early to avoid its failure. Furthermore, if the expected number of shocks until time is large, then we should replace the unit early. Table 2 presents optimal 0 * when = 20 for / and . We can see the same tendencies with Table 1, and 0 * is large as is large. This indicates that if become large, then the risk of failure decreases and we should replace the unit late. For example, when = 20 is twice of one in Table 1, optimal 0 * in Table 2 become more than twice of ones in Table 1. Table 3 presents optimal 0 * when =5 for / and . This indicates that we should replace the unit late when is large, because means the expected number of shocks to failure. It is of interest that 0 * increase almost linearly with . Table 4 presents optimal 0 * when = 10 for / and , and when = 1, 0 * = * given (12). We can see that 0 * decreases with / and also decreases with from * to 0. This indicates that if replacement cost of failure is large, then we should replace the unit early to avoid its failure. If / and are large, optimal * = 0. This means that the unit is replaced only at shock before failure. Table 5 presents optimal 0 * when = 20 for / and . We can see the same tendencies with Table 4, and 0 * is large as is large. This indicates that if the expected number of shocks is large, then we should replace the unit late to avoid its failure. It is of interest that 0 * decreases almost linearly with from = 1 to 5, and decreases drastically from = 5 to 6, because if is large, then the risk of failures becomes large and we should replacement the unit much early. Table 6 presents optimal 0 * when = 20 for / and . We can see that 0 * increases with . This indicates that if the prespecified damage level is large, then we should set large to avoid the frequent preventive replacement. It is of interest that 0 * becomes 0 when is small. Further we can see that 0 * decreases with / . This is the same tendency with Table 4 and Table 5.

Replacement Over damage
Suppose that the unit is replaced at th ( = 1, 2, …) shock over level (0 ≤ < ) or at failure, whichever occurs first.
The probability that the unit is replaced before failure at shock over level is ), (20) and the probability that it is replaced at failure is ), where (20) + (21) = 1. The expected number of shocks until replacement is and so that, the mean time to replacement is }.
We could make similar discussions used in Section 3 and derive optimal policies which minimize the expected cost rates analytically, and give numerical examples.

Conclusions
We have proposed the extended replacement policy for the cumulative damage model over the planned replacement time in which the unit is replaced at th completion of shocks over time . We have obtained the expected cost rates, and discussed both optimal * and * which minimize them analytically and numerically. It has been shown that the replacement with shock is better than replacement overtime when both replacement costs are the same. Furthermore, we have obtained that if replacement cost of failure is large, then we should replace the unit early to avoid its failure. As another extended policy, we have proposed the replacement in which the unit is replaced at th completion of shocks over damage level and obtained the expected cost rate.
As a future work, we should modify this model more realistically for equipment management. For example, we could consider maintenance policies where equipments are replaced at a random time over planned time. Another example would be given when the number of shocks over time is random. These formulations and results would be applied to real systems such as management projects to develop information systems effectively by suitable modifications.
increases strictly with from ( ) to 1 and increases strictly with from ( ) to 1.