Optimal age replacement policy for discrete time parallel systems

Abstract

In the case of discrete age replacement policy, a system whose lifetime is measured by the number cycles is replaced preventively after a specific number of cycles or correctively at failure, whichever occurs first. Under the discrete setup, the policy has been mostly considered for single unit systems. In this paper, a discrete time age replacement policy is studied for a parallel system that consists of components having discretely distributed lifetimes. In particular, the necessary conditions for the unique and finite replacement cycle that minimize the expected cost rate are obtained. The theoretical results are illustrated with numerical examples to observe the effect of the cost values and the mean lifetime of the components on the optimal replacement cycle.

A. Appendix

Proof of Proposition 1

Define $v(N)=1-(1-p)^{N}.$ Then, the function defined by (4) can be rewritten as

$$\begin{aligned} r(N)=1-\frac{\left( 1-\left[ v(N)\right] ^{n}\right) }{\left( 1-\left[ 1- \frac{1-v(N)}{1-p}\right] ^{n}\right) }. \end{aligned}$$

Because p is known and satisfies $0<p<1$, the proof follows by applying the L’hospital rule and noting that $v(N)\longrightarrow 1$ as $N\longrightarrow \infty .$

Proof of Proposition 2

Manifestly,

$$\begin{aligned} P\left\{ S(T)=n,T\le N\right\} +P\left\{ S(T)\ne n,T\le N\right\} =P\left\{ T\le N\right\} . \end{aligned}$$

Because $P\left\{ S(T)\ne n,T\le N\right\} =0$,

$$\begin{aligned} P\left\{ S(T)=n,T\le N\right\} =P\left\{ T\le N\right\} . \end{aligned}$$

Thus, we have

$$\begin{aligned} P\left\{ S(T)=n\mid T\le N\right\} =\frac{P\left\{ T\le N\right\} }{ P\left\{ T\le N\right\} }=1 \end{aligned}$$

which implies (5). On the other hand, from Corollary 1 of Jasinski (2021), we have

$$\begin{aligned} P\left\{ S(N)=i\mid T>N\right\} =\frac{\left( {\begin{array}{c}n\\ i\end{array}}\right) F^{i}(N)(1-F(N))^{n-i}}{ 1-F^{n}(N)}, \end{aligned}$$

for $i=0,1,...,n-1$. Thus

$$\begin{aligned} E(S(N) \mid T>N)= & {} \frac{1}{1-F^{n}(N)}\sum \limits _{i=0}^{n-1}i\left( {\begin{array}{c}n\\ i\end{array}}\right) F^{i}(N)(1-F(N))^{n-i} \\= & {} \frac{nF(N)(1-F^{n-1}(N))}{1-F^{n}(N)}, \end{aligned}$$

where the last equation follows from

$$\begin{aligned} \sum \limits _{i=0}^{n-1}\left( {\begin{array}{c}n\\ i\end{array}}\right) ia^{i}(1-a)^{n-i}=na(1-a^{n-1}). \end{aligned}$$

Proof of Theorem 1

A necessary condition for the existence of finite and unique $N^{*}$ minimizing C(N) is that an $N^{*}$ satisfies

$$\begin{aligned} C(N+1)\ge C(N), \end{aligned}$$

(A.1)

and

$$\begin{aligned} C(N)<C(N-1), \end{aligned}$$

(A.2)

for $N=1,2,...$. From (8) and (A.1), we have that

$$\begin{aligned} a\sum \nolimits _{t=0}^{N-1}\left[ 1-F^{n}(t)\right] +b(1-F^{n}(N+1))\sum \nolimits _{t=0}^{N-1}\left[ 1-F^{n}(t)\right] \\ +nc_{a}F(N+1)(1-F^{n-1}(N+1))\sum \nolimits _{t=0}^{N-1}\left[ 1-F^{n}(t) \right] -a\sum \nolimits _{t=0}^{N}\left[ 1-F^{n}(t)\right] \\ -b(1-F^{n}(N))\sum \nolimits _{t=0}^{N}\left[ 1-F^{n}(t)\right] -nc_{a}F(N)(1-F^{n-1}(N))\sum \nolimits _{t=0}^{N}\left[ 1-F^{n}(t)\right] \ge 0 \end{aligned}$$

After some simple manipulations, we obtain

$$\begin{aligned} \left[ nc_{a}\frac{F(N+1)(1-F^{n-1}(N+1))-F(N)(1-F^{n-1}(N))}{1-F^{n}(N)}-b \frac{F^{n}(N+1)-F^{n}(N)}{1-F^{n}(N)}\right] \\ \times \sum \nolimits _{t=0}^{N-1}\left[ 1-F^{n}(t)\right] -b\left[ 1-F^{n}(N) \right] -nc_{a}F(N)(1-F^{n-1}(N))\ge a \end{aligned}$$

which yields

$$\begin{aligned}&\left[ nc_{a}\frac{H(N+1)-H(N)}{1-F^{n}(N)}-b\cdot r(N+1)\right] \sum \nolimits _{t=0}^{N-1}\left[ 1-F^{n}(t)\right] \nonumber \\&-b\left[ 1-F^{n}(N)\right] -nc_{a}H(N)\ge a, \end{aligned}$$

(A.3)

where $H(N)=F(N)(1-F^{n-1}(N)).$ Similarly, from (A.2)

$$\begin{aligned}&\left[ nc_{a}\frac{H(N)-H(N-1)}{1-F^{n}(N-1)}-b\cdot r(N)\right] \sum \nolimits _{t=0}^{N-2}\left[ 1-F^{n}(t)\right] \nonumber \\&-b\left[ 1-F^{n}(N-1)\right] -nc_{a}H(N-1)<a. \end{aligned}$$

(A.4)

Thus, if

$$\begin{aligned}&\lim _{N\longrightarrow \infty }\left[ nc_{a}\frac{H(N+1)-H(N)}{1-F^{n}(N)} -b\cdot r(N+1)\right] \sum \nolimits _{t=0}^{N-1}\left[ 1-F^{n}(t)\right] \nonumber \\&-b\left[ 1-F^{n}(N)\right] -nc_{a}H(N)\ge a, \end{aligned}$$

(A.5)

then there exists a finite and unique $N^{*}$ satisfying (A.3) and (A.4). Manifestly,

$$\begin{aligned} \lim _{N\longrightarrow \infty }\frac{H(N+1)-H(N)}{1-F^{n}(N)} =\lim _{N\longrightarrow \infty }\frac{F(N+1)-F(N)}{1-F^{n}(N)}-\frac{ F^{n}(N+1)-F^{n}(N)}{1-F^{n}(N)}. \end{aligned}$$

Using the L’hospital rule for the first fraction and Proposition 1 for the second fraction, we obtain

$$\begin{aligned} \lim _{N\longrightarrow \infty }\frac{H(N+1)-H(N)}{1-F^{n}(N)}=\left( \frac{1 }{n}-1\right) p. \end{aligned}$$

Thus the condition (A.5) becomes

$$\begin{aligned} \left[ nc_{a}\left( \frac{1-n}{n}\right) p-bp\right] E(T)>a \end{aligned}$$

which is equivalent to (9).

Proof of Proposition 3

Clearly,

$$\begin{aligned} P\left\{ S(N)=0\mid T>N\right\} =\frac{P\left\{ X_{1}>N,X_{2}>N\right\} }{ P\left\{ T>N\right\} }. \end{aligned}$$

Thus, we have

$$\begin{aligned} P\left\{ S(N)=1\mid T>N\right\}= & {} 1-\frac{P\left\{ X_{1}>N,X_{2}>N\right\} }{P\left\{ T>N\right\} } \\= & {} 1-\frac{P\left\{ X_{1}>N,X_{2}>N\right\} }{1-P\left\{ X_{1}\le N,X_{2}\le N\right\} } \end{aligned}$$

which gives the result.