A Single Deteriorating System’s Replacement Model with Inspections

This study considers a generalized replacement model for a deteriorating system in which failures can only be detected by inspection. The system is assumed to have two types of failures and is replaced at the occurrence of the Nth type I failure (minor failure), or the first type II failure (catastrophic failure), or at working age T, whichever occurs first. The probability of a type I or type II failure depends on the number of type I failures since the previous replacement. Such a system can be repaired after a type I failure, but is deteriorating stochastically. That is, the operating intervals are decreasing stochastically, whereas the durations of the repairs are increasing stochastically. Based on these assumptions, we determine the expected net cost rate.


Introduction
Most replacement models assume that the repaired system or machine is returned to an "as good as new" state (or called perfect repair).This assumption means that, upon repair, an item has the same life distribution as a new item.If we only consider operating time with a negligible repair time, a renewal process is obtained.If we consider the non-zero repair times which are independent and identically distributed (i.i.d.) random variables, then an alternating renewal process with up and down periods is obtained.However, for a repairable deteriorating system, it may be more realistic to assume that a failed item will return to an "as bad as old" state after being repaired.That is, if the life distribution of a new item was F , then the item after repair will have the survival function t F , where t is the item's age at failure and . This is also known as minimal repair (see Barlow and Proschan [1,2] for more detail).
However, for some practical deteriorating systems, the system can be different from that described above.Lam [3] considered a machine maintenance problem where after each repair, the machine's operating time becomes stochastically shorter.On the other hand, to model the effect of aging, repair times become stochastically larger.Therefore, we consider a repair and replacement model with such a feature for analyzing deteriorating systems.Due to the non-increasing operating times after repairs, the system will eventually die out, so the total system life is finite.One possible way of modeling this kind of deteriorating systems is to use the non-homogeneous Poisson process (NHPP).Lam [3] formulated the other way to use geometric process replacement model.
Maintenance and replacement models are usually based on the assumption that failures are detected and repaired simultaneously.However, for some systems, failures may be only identified by inspection.Nakagawa et al. [4] summarized the policies for periodic and random inspection.Chen et al. [5] applied periodic and random inspection policies to computer systems.Cheng and Li [6] studied a geometric process repair model with inspections.In this study, we develop a model with multiple failure modes with a replacement condition for the system's working age.
The outline of the article is as follows.In section 2, the model is described, then the expected net cost per unit time is found.Finally, Section 3 concludes.

General model 2.1 Model description
The system considered is "stochastically deteriorating", in the sense that successive operating intervals after repairs are stochastically decreasing and successive repair durations are stochastically increasing.We assume that the system is subject to two types of failures, which can be detected through inspection.Type I failure (minor failure) is fixed by repair, whereas type II failure (catastrophic failure) is removed by replacement.We consider a replacement policy as follows: the system will be replaced at the N th type I failure (minor failure) or first type II failure (catastrophic failure) or at the working age T , depending on whichever occurs first.The probability of type II failure depends on the number of minor failures since the last replacement.
Let M be the number of failures until the first type II failure.Let ( ).
where 1) k Q be the number of inspections during the operating period of the system from the ( 1) k  th repair to the replacement at the N th type I failure.We calculate the ( , [] k EQ as follows: be the number of inspections during the operating period of the system from the ( 1) k  th repair to the replacement at the first type II failure.We calculate be the number of inspections during the operating period of the system from the ( 1) k  th repair to the replacement at the working age T .We calculate ] [ We assume the following cost structure is imposed: the unit of repair cost rate is t , then we have (see, e.g., Ross [7] p.52).We denote the right-hand side of Eq. ( 5) by ( , , ) B N T h .
For our model, we have and Hence, the expected net cost per unit time is given by )

Conclusions
In this study, we proposed a generalized replacement model for deteriorating systems with failures which can be detected only through inspections.The system is subject to two types of failures and is replaced at the N th type I failure (minor failure) or first type II failure (catastrophic failure) or at the working age T , whichever occurs first.The expected net cost rate was derived.

1 C, the unit of reward rate for the operating period is 2 C, the replacement cost is 3 C , the unit of penalty cost rate for the idle period is 4 C
be the repair duration after the detection of the i th failure.Further, we k P for a sequence of probabilities.The sequence   k P the ( 1) i  th repair.The cumulative distribution function (c.d.f.) of i .We assume that { i Y