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Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy

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Abstract

In this paper, we consider an age-replacement model with minimal repair based on a cumulative repair cost limit and random lead time for replacement delivery. A cumulative repair cost limit policy uses information about a system’s entire repair cost history to decide whether the system is repaired or replaced; a random lead time models delay in delivery of a replacement once it is ordered. A general cost model is developed for the average cost per unit time based on the stochastic behavior of the assumed system, reflecting the costs of both storing a spare and of system downtime. The optimal age for preventive replacement minimizing that cost rate is derived, its existence and uniqueness is shown, and structural properties are presented. Various special cases are included, and a numerical example is given for illustration. Because the framework and analysis are general, the proposed model extends several existing results.

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Abbreviations

X :

Time to failure of a new system

f(⋅):

Probability density function (pdf) of the random variable (r.v.) X

F(⋅):

Cumulative distribution function (Cdf) of the r.v. X

\(\overline{F}(\cdot)\) :

Survival function (Sf) of the r.v. X, which is 1−F(⋅)

r(t):

\(f(t)/\overline{F}(t)\) : failure (hazard) rate function of the r.v. X

Λ(t):

\(\int_{0}^{t}r(u)du\) : cumulative hazard function of the r.v. X

X L :

Lead-time of a new system for replacement

l(⋅), L(⋅):

Pdf, Cdf of the r.v. X L

\(\overline{L}(\cdot)\) :

Sf of the r.v. X L , which is 1−L(⋅)

μ L :

\(\mbox{E}(X_{L})=\int_{0}^{\infty}\overline{L}(t)dt\) : mean lead-time

T :

Replacement age of an operating system

C(T):

S-expected cost-rate for an infinite time span

T * :

T which minimizes C(T)

p(t):

P r {type II failure when the system fails at age t}

q(t)=1−p(t):

P r {type I failure when the system fails at age t}

Y :

Waiting time until the first type II failure

F p (⋅):

Cdf of the r.v. Y

\(\overline{F}_{p}(\cdot)\) :

Sf of the r.v. Y, which is 1−F p (⋅)

S j :

Waiting time until the j-th type I failure

{N(t), t≥0}:

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

c 0 :

Cost due a preventive replacement

c 1 :

Cost due a critical type I failure replacement

c 2 :

Cost due a type II failure replacement

c h :

Holding cost per spare per unit time

c s :

Downtime cost due to shortage of a spare per unit time

W :

Cost for a minimal repair

G(w):

Cdf of the r.v. W

c w :

E(W): mean cost of a minimal repair

Z j :

\(\sum_{i=1}^{j}W_{i}\) : accumulated repair cost until the j-th type I failure

G(j)(y):

Cdf of the r.v. Z j : the j-fold stieltjes convolution of the distribution G of itself

ξ :

Minimal repair cost limit

U j :

Length of successive replacement cycle j, j=1,2,…

V j :

Operational cost over U j

D(t):

S-expected cost of the operating system over \([0,\;t]\)

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Authors and Affiliations

  1. Department of Applied Statistics, National Taichung Institute of Technology, No. 129, Sec. 3, San-min Rd., Taichung City, Taichung, 404, Taiwan

    Yu-Hung Chien

  2. Department of Statistics and Informatics Science, Providence University, No. 200, Chung Chi Rd., Taichung, 43301, Taiwan

    Chin-Chih Chang & Shey-Huei Sheu

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  1. Yu-Hung Chien
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  2. Chin-Chih Chang
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Correspondence to Yu-Hung Chien.

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Chien, YH., Chang, CC. & Sheu, SH. Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy. Ann Oper Res 181, 723–744 (2010). https://doi.org/10.1007/s10479-009-0679-0

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