Note on History of Age Replacement Policies

This paper tries to trace our research history briefly from Barlow and Proschan to attain general replacement models. We begin with a random age replacement policy that is planned at a random time Y and call it as random replacement. When the distribution of Y becomes a degenerate distribution placing unit mass at T, age replacement is formulated. We obtain the general formulas for optimum replacement times. We next suppose the unit works for a job with random works, and replacement policies with N cycles are discussed. As follows, we combine age and random replacement models and discuss replacement first, replacement last, replacement overtime, replacement overtime first and replacement overtime last. By formulating the distributions of replacement times with n variables, general replacement models with n replacement times are obtained. KeywordsAge replacement, Replacement first, Replacement last, Replacement overtime, General replacement.


Introduction
Most basic model in maintenance theory is age replacement, in which we plan to replace an operating unit before failure at an optimum time * to minimize the expected replacement cost rate. Since Barlow and Proschan (Barlow and Proschan, 1996) have introduced age replacement model in 1965, a great number of age replacement models have been discussed by researchers for a half century, which have been summarized (Barlow, and Proschan, 1996;Nakagawa, 2005;Sarkar et al., 2011). Recently, we have proposed several new notions of age replacement such as random replacement, replacement first, replacement last, replacement overtime, and replacement middle (Nakagawa, 2014;Zhao and Nakagawa, 2012;Nakagawa and Zhao, 2015;Zhao et al., 2015). General replacement models combing age and random replacement policies with replacement times have been studied (Chen et al., 2016). In Section 2, general assumptions are given for modelings. In Section 3, we introduce age and random replacement models (Barlow and Proschan, 1996). In Sections 4-6, we give models of replacement first, replacement last, replacement overtime and replacement middle, respectively, and compare their optimum policies analytically. When the costs of preventive replacement are the same, age replacement is most economical than others. However, when a unit works for a job with random works, it is not wise in practice to stop its operation and to replace it at the optimum time, e.g., planned time . In this situation, replacement overtime would be better than others. In Section 7, we propose general replacement policies and call them as redundant maintenances in general forms, as their distributions of replacement times agree with the failure distributions of a series system and a parallel system with units.
In addition, we suppose the unit operates for an infinite time horizon by means of replacement. All times for corrective replacement and preventive replacement are supposed to be neglected and their costs are constantly given respectively. Our objective is to find optimum preventive replacement times to minimize the expected cost rates.

Age and Random Replacement Models
Suppose that an operating unit is replaced at a random time (0 < ≤ ∞) or at failure, whichever occurs first. The random variable has a general distribution ( ) and is independent with the failure time . We call this policy as random replacement. The expected cost rate is (Barlow and Proschan, 1996;Nakagawa, 2014) where = replacement cost at time , = replacement cost at failure and > .
It has been shown (Barlow and Proschan, 1996;Nakagawa, 2014) that C(G) can be rewritten as Suppose that there exits a minimum value Thus, where ( ) is the degenerate distribution placing unit mass at , i.e., ( ) ≡ 1 for ≥ and ( ) ≡ 0 for < . If = ∞, then the unit is replaced at failure.
It was written in (Barlow and Proschan, 1996) that the elegant proof was due to S. Karlin, who was the author of (Karlin and Taylor, 1975). This is very easy and simple, however, it could not be occurred to anybody from (1). In other words, if preventive replacement costs are equal, the policy at time is superior to other ones, which will be shown theoretically in the following sections.
Suppose the unit is replaced at time or at failure, whichever occurs first. We call this policy as age replacement, and the expected cost rate is which is the standard formula of deriving optimum age replacement time, and the resulting cost rate is Equation (3) means physically that optimum replacement time satisfies: Failure rate × Mean replacement time − Failure probability = Replacement cost ratio. This general formula will be shown in all sections.
Suppose that the unit is replaced at cycle or at failure, whichever occurs first. We also call this policy as random replacement, and the expected cost rate is (Nakagawa, 2014).

When
= , differentiating ( , ) with respect to and setting it equal to zero, The left-hand side increases with to ∞. Thus, there exists a unique * (0 < * < ∞) which satisfies (8), and The left-hand side of (8) increases strictly with to that of (3), * decreases strictly to * in (3).
, then * < * , and replacement last is more economical. This means that if the ratio of /( − ) is large, we adopt replacement last, and if is large, replacement first is adopted.

Replacement Overtime
We suppose the unit is replaced preventively at the first completion of working cycles ( = 1,2 ⋯ ) over . We call this policy as replacement overtime, and the expected cost rate is .

Replacement Overtime First
Suppose the unit is replaced over (0≤ ≤∞) or at cycle ( = 1,2, ⋯ ), whichever occurs first. Then, the expected cost rate is .  The left-hand side of (19) increases with to ℎ(∞). Thus, there exists a unique * (0 < * < ∞) which satisfies (19), and the resulting cost rate is Noting that the left-hand side of (19) increases strictly with to that of (16), * decreases strictly with to * given in (16).

Replacement Overtime Last
Suppose the unit is replaced over time or at cycle , whichever occurs last. Then, the expected cost rate is .

Replacement Middle
Suppose the unit is replaced at T, 1 or 2 , whichever occurs first, where each ( = 1,2) has distribution ( ) ≡ Pr{ ≤ }. Then, the expected cost rate is where = replacement cost at with < ( = 1,2) and and are given in (2).
Next, suppose the unit is replaced at times T, 1 or 2 , whichever occurs last. Then, the expected cost rate is The left-hand side of (30) increases strictly with T to ℎ(∞) = ∞. Thus, there exists a finite and unique 2 * (0 < 2 * <∞) which satisfies (30), and the resulting cost rate is Finally, suppose the unit is replaced at times T, 1 or 2 , whichever occurs middle. We call this policy as replacement middle , and the expected cost rate is The left-hand side of (33) increases with T to ∞. Thus, there exists a unique 2 * (0 < 2 * < ∞) which satisfies (33), and
, then the ranking is replacement middle, first, last.

General Age Replacement
Suppose the unit is replaced at random times , where ( = 1, 2, … , ) are independent with each other and have general distributions ( ) with finite means 1/ , respectively. Denoting ≡ min{ 1, 2 , … , }, it has a distribution and denoting ≡ max{ 1, 2 , … , }, it has a distribution Note that (35) and (36) correspond to the respective failure distributions of a series system and a parallel system with units when ( ) is the failure distribution of unit i (Nakagawa, 2008), so that we call the above policies as redundant replacement.
Next, we consider the following four replacement policies: (a) Replacement first: The unit is replaced at time T or at time , whichever occurs first. Then, the unit is replaced at time ≡ min{ , } with a general distribution (b) Modified replacement first: The unit is replaced at time T or at time , whichever occurs first. Then, the unit is replaced at time ̃≡ min{ , } with a general distribution (c) Replacement last: The unit is replaced at time T or at time , whichever occurs last. Then, the unit is replaced at time ≡ max{ , } with a general distribution Then, the unit is replaced at time ̃≡ max{ , } with a general distribution From the definitions of and ̃( = , ), ≤̃≤̃≤ , and hence, ( ) ≥̃( ) ≥ ( ) ≥ ( ).

Replacement First
Suppose the unit is replaced at time in case (a). Then, replacing ( ) with ( ) in (37), the expected cost rate is (Chen et al., 2016) where = replacement cost at time with < .

Conclusions
Age replacement model has been already challenged that the constant planned T can be a random variable Y to meet the random perforation need. In this paper, the so called replacement that is planned at time T and random replacement planned at random time Y have been introduced. Another random replacement that is planned at working cycles N are also obtained. If we need to consider both age and random replacement policies with working cycles, we next have discussed replacement first, replacement last, replacement overtime and replacement middle. The motivations of these methods can be found in our literatures, so that we give the expected cost rates directly and discuss their optimum policies to minimize them analytically.
Another contribution of the paper is to give comparisons for these replacement policies. For examples, it has been shown that when the costs of preventive replacement are equal, age replacement is more economical than the random policy, age replacement is more economical than replacement overtime, replacement overtime is more economical than the random policy. Also, we have compared replacement first and replacement last analytically and showed that both have advantages in cost rates.
In order to formulate general replacement models of replacement first and last, we have obtained general distributions of replacement times, and the models of general replacement become our final target of the paper.