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Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks

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Abstract

An operating system is subject to random shocks that arrive according to a non-homogeneous Poisson process and cause the system failed. System failures experience to be divided into two categories: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. An age-replacement model is studied by considering both a cumulative repair-cost limit and a system’s entire repair-cost history. Under such a policy, the system is replaced at age T, or at the k-th type-I failure at which the accumulated repair cost exceeds the pre-determined limit, or at any type-II failure, whichever occurs first. The object of this article is to study analytically the minimum-cost replacement policy for showing its existence, uniqueness, and the structural properties. The proposed model provides a general framework for analyzing the maintenance policies, and presents several numerical examples for illustration purposes.

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Abbreviations

{N(t):t≥0}:

Non-homogeneous Poisson process (NHPP) with intensity r(t)

Λ(t):

Mean value function of {N(t):t≥0}; \(\Lambda(t) = \int_{ 0}^{ t} r(u)\mathrm{d}u\)

S k :

Arrival instant of the kth shock for k=1,2,3,…

W i :

Minimal repair cost due to the ith type-I failure for i=1,2,3,…

G(w):

cdf (cumulative distribution function) of the r.v. (random variable) W i

c w :

Mean cost of W i ; c w =E[W i ]

Z j :

Accumulated repair cost until the jth type-I failure; \(Z_{j} = \sum_{i = 1}^{j} W_{i}\)

G(j)(z):

cdf of the r.v. Z j ; the j-fold Stieltjes convolution of the distribution G with itself

M :

Number of shocks proceeding the first type-II failure

\(\bar{P}_{k}, p_{k}\) :

sf (survival function), pmf (probability mass function) of M; \(\bar{P}_{k} = P(M > k)=\mathrm{Pr}\{\mbox{first }k\mbox{ shocks are type-I failures}\}\), where the domain of \(\bar{P}_{k}\) is {0,1,2,…} and \(1 = \bar{P}_{0} \ge\bar{P}_{1} \ge\bar{P}_{2} \ge\cdots \); \(p_{k} = P(M = k) = \bar{P}_{k - 1} - \bar{P}_{k} = \bar{P}_{k - 1}(1 - \bar{P}_{k} / \bar{P}_{k - 1})\)

\(\{ \bar{P}_{k}\}\) :

A sequence of \(\bar{P}_{k}\)

q k :

\(\mathrm{Pr}\{\mbox{a type-I failure when shock }k\mbox{ arrives}\}=\bar{P}_{k} / \bar{P}_{k - 1}\)

θ k :

Pr{a type-II failure when shock k arrives}=1−q k

T :

Replacement age of an operating system

L :

Total repair-cost limit

\(B(T;L,\{ \bar{P}_{k}\} )\) :

s-expected cost rate for an infinite time span

T ∗ :

T which minimizes \(B(T;L,\{ \bar{P}_{k}\} )\)

L ∗ :

L which minimizes \(B(T;L,\{ \bar{P}_{k}\} )\)

c 0 :

Cost of a planned replacement

c 1 :

Cost of an unplanned replacement

Y :

Waiting time until the first unplanned replacement (kth type-I failure at which the accumulated repair cost exceeds the pre-determined limit L or first type-II failure) when T=∞

h(t),H(t):

pdf (probability density function), cdf of the r.v. Y

\(\bar{H}(t)\) :

sf of Y, which is 1−H(t)

r H (t):

Failure (hazard) rate of Y; \(r_{H}(t) = h(t) / \bar{H}(t)\)

U i :

Length of successive replacement cycle for i=1,2,…

V i :

Operational cost over U i

D(t):

s-expected cost of the operating system over [0,t]

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Authors and Affiliations

  1. Department of Statistics and Informatics Science, Providence University, Taichung, 433, Taiwan

    Shey-Huei Sheu & Chin-Chih Chang

  2. Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, 106, Taiwan

    Shey-Huei Sheu

  3. Department of Applied Statistics, National Taichung Institute of Technology, Taichung, 404, Taiwan

    Yu-Hung Chien

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  1. Shey-Huei Sheu
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  2. Chin-Chih Chang
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  3. Yu-Hung Chien
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Correspondence to Shey-Huei Sheu.

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Sheu, SH., Chang, CC. & Chien, YH. Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks. Ann Oper Res 186, 317–329 (2011). https://doi.org/10.1007/s10479-011-0864-9

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