Journal of Systems Science and Complexity

, Volume 31, Issue 6, pp 1541–1553

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

• Qiaoqiao Gao
• Dequan Yue
• Bing Zhao
Article

## Abstract

In this paper, the optimal maintenance policy is investigated for a system with stochastic lead time and two types of failures. The system has two types of failures, one type is repairable, when the repairable failure occurs, the system will be repaired by repairman, and the system after repair is not “as good as new”. The other type of failure is unrepairable, and when the unrepairable failure occurs the system must be replaced by a new and identical one. The spare system for replacement is available only by order, and the lead time for delivering the spare system is stochastic. The successive survival times of the system form a stochastically decreasing geometric process, the consecutive repair times after failures of the system form a renewal process. By using the renewal process theory and geometric process theory, the explicit expression of the long-run average cost per unit time under ordering policy (N − 1) is derived, and the corresponding optimal can be found analytically. Finally, the numerical analyses are given.

## Keywords

Failure repair geometric process ordering policy renewal process replace

## References

1. [1]
Lam Y, Geometric process and replacement problem, Acta Mathematicae Applicatae Sinica, 1988, 4: 366–377.
2. [2]
Lam Y, A note on the optimal replacement problem, Advances in Applied Probability, 1988, 20: 479–782.Google Scholar
3. [3]
Jia J S and Wu S M, Optimizing replacement policy for a cold-standby system with waiting repair times, Applied Mathematics and Computation, 2009, 24: 133–141.
4. [4]
Zhang Y L and Wang G J, A deteriorating cold standby repairable system with priority in use, European Journal of Operational Research, 2007, 18: 278–295.
5. [5]
Wang G J and Zhang Y L, Geometric process model for a system with inspections and preventive repair, Computers & Industrial Engineering, 2014, 75: 13–19.
6. [6]
Zong S L, Chai G R, Zhang Z G, et al., Optimal replacement policy for a deteriorating system with increasing repair times, Applied Mathematical Modelling, 2013, 37: 9768–9775.
7. [7]
Zhang Y L and Wang G J, An optimal replacement policy for a multistate degenerative simple system, Applied Mathematical Modelling, 2010, 34: 4138–4150.
8. [8]
Liu H T, Meng X Y, and Wu WJ, The cold standby system with repair of non-new and repairman vacation, Journal of Computational Information Systems, 2012, 8: 1349–1357.Google Scholar
9. [9]
YuM M, Tang Y H, Liu L P, et al., A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations, European Journal of Operational Research, 2013, 25: 310–323.
10. [10]
Lin D, Zuo M J, and Yam R C M, Sequential imperfect preventive maintenance models with two categories of failure modes, Naval Research Logistics, 2001, 48: 173–183.
11. [11]
Kim M J and Makis V, Optimal maintenance policy for a multi-state deteriorating system with two types of failures under general repair, Computers and Industrial Engineering, 2009, 57: 298–303.
12. [12]
Zequeira R I and Brenguer C, Periodic imperfect preventive maintenance with two categories of competing failure modes, Reliability Engineering and System Safety, 2006, 91: 460–468.
13. [13]
Castro I T, A model of imperfect preventive maintenance with dependent failure modes, European Journal of Operational Research, 2009, 16: 217–224.
14. [14]
Cheng G Q and Li L, An optimal replacement policy for a degenerative system with two-types of failure states, Journal of Computational and Applied Mathematics, 2014, 21: 139–145.
15. [15]
Wang G J and Zhang Y L, Optimal repair-replacement policies for a system with two types of failures, European Journal of Operational Research, 2013, 26: 500–506.
16. [16]
Chien Y H, A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times, Applied Mathematical Modelling, 2009, 33: 1708–1718.
17. [17]
Chien Y H, Optimal number of minimal repairs before ordering spare for preventive replacement, Applied Mathematical Modelling, 2010, 34: 3439–3450.
18. [18]
Chien Y H and Chen J A, Optimal spare ordering policy for preventive replacement under cost effectiveness criterion, Applied Mathematical Modelling, 2010, 34: 716–724.
19. [19]
Sheu S H and Chien Y H, Optimal age-replacement policy of a system subject to shocks with random lead-time, European Journal of Operational Research, 2004, 19: 132–144.
20. [20]
Tang Y H, Yu M M, Fu Y H, et al., PH deteriorating repairable system with phase-type stochastic lead time and its optimal replacement policy, Journal of Systems Science and Mathematical Sciences, 2010, 30(9): 1222–1235 (in Chinese).
21. [21]
Yu M M, Tang Y H, Fu Y H, et al., A deteriorating repairable system with stochastic lead time and replaceable repair facility, Computer & Industrial Engineering, 2012, 62: 609–615.