Optimal replacement policy for a deteriorating system with increasing repair times

Zong, Shengliang; Chai, Guorong; Zhang, Zhe George; Zhao, Lei

doi:10.1016/j.apm.2013.05.019

Cited by 19 publications

(11 citation statements)

References 25 publications

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“…However, the implementation of a policy N is clearly easier than that of a policy T . This explains the consideration of a waiting time for repair and the adoption of N ‐policy in the analysis. Most of the research works on shock models assume the shock arrival process to be Poisson for mathematical convenience so that the intershock arrival times follow an exponential distribution (see the works of Lam and Zhang, Lam, Zong et al, etc) or a renewal process (eg, the work of Tang and Lam). For example, a precision instrument and meter may be placed in an environment with a high temperature and humidity, such as in naval vessels.…”

Section: Model Assumptionsmentioning

confidence: 99%

“…Case

\begin{matrix} Let G_{n} (x) & = 1 - e^{- θ {normala}^{n - 1} x}, a, θ > 0; \\ p_{n} & = \int_{0}^{\infty} λ {normale}^{- λ t} {normale}^{- θ {normala}^{n - 1} t} dt = \frac{λ}{λ + θ a^{n - 1}}, q_{n} = 1 - p_{n} = \frac{θ a^{n - 1}}{λ + θ a^{n - 1}} and \\ L_{n} & = - λ + \frac{λ θ a^{n - 1}}{λ + θ a^{n - 1}} = \frac{- λ^{2}}{λ + θ a^{n - 1}} \end{matrix}

so that

E ((), W_{n}) = \frac{λ + θ a^{n - 1}}{λ^{2}} .

Equation agrees with Equation in Zong et al In what follows, the optimal replacement policy N * is presented.…”

Section: Long‐run Average Cost Per Unit Timementioning

confidence: 99%

“…The objective is to choose the optimal replacement policies T * and N *, respectively, such that the long‐run average cost per unit time is minimized. Other works on the geometric process model in maintenance analysis include those of Lam, Stanley, Pérez‐Ocón and Torres‐Castro, Wang and Zhang, Liang et al, Zong et al, and Cheng et al…”

Section: Introductionmentioning

confidence: 99%

“…Liang et al spelt an optimal replacement policy N * in a geometric process model with a δ ‐ shock for a deteriorating and improving system. Retaining the geometric repair times modeling and assuming the shock arrival process to be a Poisson process with the threshold value to be an exponential random variable, Zong et al dealt with the determination of the optimal replacement policy N * to minimize the long‐run average cost rate. However, in realistic systems, phase‐type distribution, introduced by Neuts, plays an important role and has been applied in telecommunication, reliability, queueing, and inventory theory.…”

Section: Introductionmentioning

confidence: 99%

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On a random lead time and threshold shock model using phase‐type geometric processes

Sarada

Shenbagam

2018

Appl Stoch Models Bus & Ind

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This study investigates two random threshold shock models for a repairable deteriorating system with nonnegligible maintenance times, with and without a spare via a phase-type geometric process. The system fails whenever the intershock arrival time is less than a random threshold. The provision of stochastic lead time is incorporated in Model II so that an ordering policy N−1 and a replacement policy N based on the number of failures of the system are also considered. An explicit expression of the average cost rate is derived for both models and the optimal replacement policy N* is obtained by minimizing the long-run average cost rate analytically. The numerical illustrations and sensitivity analysis provided therein conform to the observations made in the study. KEYWORDS phase-type geometric process, random threshold, stochastic lead time Appl Stochastic Models Bus Ind. 2018;34:407-422. wileyonlinelibrary.com/journal/asmb 407 408SARADA AND SHENBAGAM this model, Lam studied two kinds of replacement policies, ie, one based on the working age T of the system (T-policy) and the other based on the failure number N of the system (N-policy). The objective is to choose the optimal replacement policies T* and N*, respectively, such that the long-run average cost per unit time is minimized. Other works on the geometric process model in maintenance analysis include those of Lam, 12 Stanley, 13 Pérez-Ocón and Torres-Castro, 14 Wang and Zhang, 15 Liang et al, 16 Zong et al, 17 and Cheng et al 18Under the assumption of geometric repair times, Lam and Zhang 9 and Tang and Lam 19 studied the -shock model, in which shocks arrive according to a Poisson process and renewal process, respectively, whereas Lam 20 introduced a geometric process -Shock model in a repairable deteriorating system and obtained an optimal replacement policy (N*). Liang et al 16 spelt an optimal replacement policy N* in a geometric process model with a -shock for a deteriorating and improving system. Retaining the geometric repair times modeling and assuming the shock arrival process to be a Poisson process with the threshold value to be an exponential random variable, Zong et al 17 dealt with the determination of the optimal replacement policy N* to minimize the long-run average cost rate. However, in realistic systems, phase-type distribution, introduced by Neuts, 21,22 plays an important role and has been applied in telecommunication, reliability, queueing, and inventory theory. It is generally assumed that the system replacement is immediate on failure, which may not be true always, as it would be expensive to maintain a spare, consequently hiking the system operating cost (eg, the work of Yu et al 23,24 ).From a thorough review of the existing literary works done, it is observed that the -shock model with a random threshold through stochastic lead time via phase-type geometric processes has not been studied from the viewpoint of deteriorating systems. Thus, in an attempt to establish the importance of the simplified computational implementation of phas...

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Section: Model Assumptionsmentioning

confidence: 99%

“…Case

\begin{matrix} Let G_{n} (x) & = 1 - e^{- θ {normala}^{n - 1} x}, a, θ > 0; \\ p_{n} & = \int_{0}^{\infty} λ {normale}^{- λ t} {normale}^{- θ {normala}^{n - 1} t} dt = \frac{λ}{λ + θ a^{n - 1}}, q_{n} = 1 - p_{n} = \frac{θ a^{n - 1}}{λ + θ a^{n - 1}} and \\ L_{n} & = - λ + \frac{λ θ a^{n - 1}}{λ + θ a^{n - 1}} = \frac{- λ^{2}}{λ + θ a^{n - 1}} \end{matrix}

so that

E ((), W_{n}) = \frac{λ + θ a^{n - 1}}{λ^{2}} .

Equation agrees with Equation in Zong et al In what follows, the optimal replacement policy N * is presented.…”

Section: Long‐run Average Cost Per Unit Timementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a random lead time and threshold shock model using phase‐type geometric processes

Sarada

Shenbagam

2018

Appl Stoch Models Bus & Ind

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“…The threshold is usually a constant. In this paper, we adopt a general -shock model by letting be an exponentially distributed random variable with parameter varying with number of repairs [6]. The power equipment after repair can be more fragile and more prone to fail again.…”

Section: Introductionmentioning

confidence: 99%

Determining Optimal Replacement Policy with an Availability Constraint via Genetic Algorithms

Zong

Chai

Su³

2017

Mathematical Problems in Engineering

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We develop a model and a genetic algorithm for determining an optimal replacement policy for power equipment subject to Poisson shocks. If the time interval of two consecutive shocks is less than a threshold value, the failed equipment can be repaired. We assume that the operating time after repair is stochastically nonincreasing and the repair time is exponentially distributed with a geometric increasing mean. Our objective is to minimize the expected average cost under an availability requirement. Based on this average cost function, we propose the genetic algorithm to locate the optimal replacement policy N to minimize the average cost rate. The results show that the GA is effective and efficient in finding the optimal solutions. The availability of equipment has significance effect on the optimal replacement policy. Many practical systems fit the model developed in this paper.

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The Ordering and Replacement Policy for the System with Two Types of Failures

2018

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Optimal replacement policy for a deteriorating system with increasing repair times

Cited by 19 publications

References 25 publications

On a random lead time and threshold shock model using phase‐type geometric processes

On a random lead time and threshold shock model using phase‐type geometric processes

Determining Optimal Replacement Policy with an Availability Constraint via Genetic Algorithms

The Ordering and Replacement Policy for the System with Two Types of Failures

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