Stochastics and StatisticsAn extended optimal replacement model of systems subject to shocks
Introduction
It is of great importance to avoid the failure of a system during actual operation when such an event is costly and/or dangerous. In such situations, one important area of interest in reliability theory is the study of various maintenance policies in order to reduce the operating cost and the risk of a catastrophic breakdown. Consider a system that is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement policy for the system.
Barlow and Proschan [5] discussed the traditional approach of replacing a system at failure or at age T, whichever comes first. Boland and Proschan [10] considered periodic replacement of the system and give sufficient conditions for existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [7] established similar results assuming that the cost structure is time-dependent. Abdel-Hameed [1] shows via a sample path argument that the results of Boland and Proschan [10] and Block et al. [7] hold for any counting process whose jump size is of one unit of magnitude. For previous work of a similar shock model, we refer readers to Abdel-Hammed and Proschan [2], Esary [12], Taylor [33], Feldman [13], Abdel-Hammed and Shimi [3], Zuckerman [35], [36], Brown and Proschan [8], Block et al. [6], Puri and Singh [24], Savits [26], Sheu and Griffith [30], [31] and Sheu [29].
Makabe and Morimura [14], [15], [16] proposed a new replacement model where the system is replaced at the nth failure, and discussed the optimum policy. This model has been generalized by Morimura [17], Park [22], [23], Nakagawa [18], [19], and Nakagawa and Kowada [20]. Park [22] proved that for a Weibull distribution which is widely used for failures, the optimum solution is more cost effective as compared to Barlow and Hunter’s policy. Sheu [27], [28], [29], Sheu and Griffith [31] and Sheu et al. [32] extended the model considered by Nakagawa [18] by introducing general random repair costs and age-dependent minimal repairs.
In this article a generalized replacement model where a system is replaced at age T or at nth type I failure (minor) or first type II failure (major), whichever occurs first, is presented which incorporates minimal repair, replacement, and general random repair costs. The models with two types of failures and threshold replacement policy with operating and non-zero repair phase type distributions was considered in Neuts et al. [21] This model in this paper is a generalization of the model “Policy IV” called by Morimura [17] and it can be considered as a generalization on several previously known policies for the class of systems in which the minor failure results in a minimal repair and the non-homogeneous Poisson process is used in modeling the failure process. We also assume that repairs and replacements can be done instantly. Precisely speaking, this assumption is not true in real situations. However, when these repair or replacement times are relatively small compared to the operational times, these non-operational times can be negligible. Our model is also motivated by the maintenance problem of some practical systems. For example, in one of the top 100 most wired universities in the USA ranked by Yahoo, every lecture hall has an expensive multi-media networked computer system. It is critical to maintain this system highly reliable as a system failure during a lecture can create a huge frustration to both professors and a large number of students. The system is quite expensive to replace so that a replacement occurs only when either a major non-repairable failure due to several reasons including vandalism occurs or when the system has been used for a maximum period of time, T. The system is also subject to some repairable non-major failures. If a certain number of repairable failures reaches a threshold value, n, before T, the system is also replaced because the high frequency of minor failures indicates a low reliability and may cause too many interruptions in classes. Note that in this situation, the repair times or the replacement times are negligible compared to the operational times. The model presented in this paper can help instructional media service people find an optimal maintenance policy for these expensive systems installed in all large lecture halls. To achieve this objective, we derive the expected cost rate for this model. The optimal number n∗ and optimal age T∗ which would minimize the cost rate under certain conditions are determined. Various results are obtained as special cases of our model such as Barlow and Hunter [4], Makabe and Morimura [14], [15], [16], Morimura [17], Tilquin and Cléroux [34], Cléroux [11], Park [22], [23], Nakagawa [18], and Boland [9].
The rest of this paper is organized as follows: The replacement model proposed in this paper is described in Section 2 and its formulation and analysis are given in Section 3. The total expected long-run cost per unit time is found and Theorem 1 gives general optimization results. Various special cases and some concluding remarks are given in Section 4.
Section snippets
General model
We consider a replacement model of a system subject to shocks in which minimal repair or replacement take place according to the following scheme:
- 1.
A system is subject to shocks that arrive according to a non-homogeneous Poisson process {N(t); t ⩾ 0} with intensity function r(t) and mean value function where t is the age of the system. We assume that r(t) is a continuous and positive increasing function for t ⩾ 0.
- 2.
As shocks occur the system has two types of failures: type I (minor)
Formulation
If T = ∞, the survival function of the time between successive replacements is given bywith density given by
Let random variable Y represent the time until nth type I failure or first type II failure for our model with T = ∞. That is, Y be the random variable with survival function , and let be the hazard rate function of G. Then
Special cases and concluding remarks
In this section, we demonstrate that several previous models are special cases of our model.
Case 1. T = ∞, for k = 0, 1, 2, … , and g(C(t), c(t)) = c1.
Makabe and Morimura [14], [15], [16] considered this case in which only minimal repairs with fixed cost c1 are performed on failure before the nth failure, whereas a system is replaced at the nth failure. In this case, if we put T = ∞, for k = 0, 1, 2, …), and h(t) = EC(t)[g(C(t), c(t))] = c1 in (10), then
Acknowledgements
We are very grateful to the referees for their insightful comments and suggestions which improved the paper significantly.
References (36)
- et al.
Analysis of a system with minimal repair and its application to replacement policy
European Journal of Operational Research
(1983) A generalized model for determining optimal number of minimal repairs before replacement
European Journal of Operational Research
(1993)A generalized age and block replacement of a system subject to shocks
European Journal of Operational Research
(1998)Optimal replacement of a system subject to shocks
Journal of Applied Probability
(1986)- et al.
Nonstationary shock models
Stochastic Processes and their Application
(1973) - et al.
Optimal replacement of damaged devices
Journal of Applied Probability
(1978) - et al.
Optimum preventive maintenance policies
Operations Research
(1960) - et al.
Mathematical Theory of Reliability
(1965) - et al.
Age-dependent minimal repair
Journal of Applied Probability
(1985) - et al.
A general age replacement model with minimal repair
Naval Research Logistics
(1988)
Imperfect repair
Journal of Applied Probability
Periodic replacement when minimal repair costs vary with time
Naval Research Logistics Quarterly
Optimum replacement of a system subject to shocks
Operations Research
The age replacement problem with minimal repair and random repair cost
Operations Research
Shock models and wear process
The Annals of Probability
Optimal replacement with semi-Markov shock models
Journal of Applied Probability
A new policy for preventive maintenance
Journal of Operations Research Society of Japan
On some preventive maintenance policies
Journal of Operations Research Society of Japan
Cited by (56)
Optimal replacement policy for a two-unit system subject to shocks and cumulative damage
2023, Reliability Engineering and System SafetyOptimal Warranty Policy for Consumer Electronics with Dependent Competing Failure Processes
2022, Reliability Engineering and System SafetyReliability modeling for competing failure processes with shifting failure thresholds under severe product working conditions
2021, Applied Mathematical ModellingModeling the interdependency between natural degradation process and random shocks
2020, Computers and Industrial EngineeringCitation Excerpt :A complex system can fail due to multiple competing failure modes induced either by internal degradation such as aging and fatigue (Wang and Li, 2018), or external shocks (Cha, Finkelstein, & Levitin, 2018; Chien, Sheu, Zhang, & Love, 2006; Lai & Xie, 2008; Li & Zhao, 2007).
Optimal replacement policies for a system based on a one-cycle criterion
2019, Reliability Engineering and System SafetyReliability analysis for devices subject to competing failure processes based on chance theory
2019, Applied Mathematical Modelling