This paper is concerned with two optimization problems for a k-out-of-n system consisting of dependent components such as finding the number of elements in the system that minimize the system’s mean cost rate and the system’s optimal replacement time. In previous studies, either system consisting of independent components or parallel systems, a particular case of the present study, was examined. In particular, we numerically examine how the components’ dependence affects the optimal number of units and replacement time for the system, minimizing mean cost rates. We consider when the components are exchangeable and dependent, that is, the system consists of dependent components. For three vastly used Clayton, Gumbel, and FGM copula functions, comparative numerical results are presented.
NakagawaT. Optimization problems in k-out-of-n systems. IEEE Trans Reliab1985; R-34: 248–250.
2.
NakagawaTZhaoX. Optimization problems of a parallel system with a random number of units. IEEE Trans Reliab2012; 61: 543–548.
3.
EryilmazS. A note on optimization problems of a parallel system with a random umber of units. Int J Reliab Qual Saf Eng2017; 24: 1750022.
4.
YangLZhaoYMaX. An optimal inspection and replacement policy for a two-unit system. Proc IMechE, Part O: J Risk Reliability2018; 232(6): 766–776.
5.
LeeH. State-dependent age replacement policy for a system subject to cascading failures. Proc IMechE, Part O: J Risk Reliability2020; 234: 359–376.
6.
WangHYWangZQ. A condition-based preventive replacement policy with imperfect manual inspection for a two-stage deterioration process. Proc IMechE, Part O: J Risk Reliability2020; 227: 499–512.
7.
LevitinGFinkelsteinMDaiY. Optimal preventive replacement policy for homogeneous cold standby systems with reusable elements. Reliab Eng Syst Saf2020; 204: 107135.
8.
BairamovI. Reliability and mean residual life of complex systems with two dependent components per element. IEEE Trans Reliab2013; 62: 276–285.
9.
EryilmazS. Reliability of systems with multiple types of dependent components. IEEE Trans Reliab2016; 65: 1022–1029.
10.
NavarroJ. Distribution free comparisons of residual lifetimes of coherents systems based on copula properties. Stat Pap2018; 59: 781–800.
11.
EryilmazS. Modeling systems with two dependent components under bivariate shock models. Commun Stat Simul Comput2019; 48: 1714–1728.
12.
GebizliogluOLEryilmazS. The maximum surplus in a finite-time interval for a discrete-time risk model with exchangeable, dependent claim occurrences. Appl Stoch Models Bus Ind2019; 35: 858–870.
13.
EryilmazSOzkutM. Optimization problems for a parallel system with multiple types of dependent components. Reliab Eng Syst Saf2020; 199: 106911.
14.
SafaeiFChateletEAhmadiJ. Optimal age replacement policy for parallel and series systems with dependent components. Reliab Eng Syst Saf2020; 197: 106798.
15.
NelsenRB. An introduction to copulas. New York, NY: Springer, 2006.
16.
NavarroJSpizzichinoF. Comparisons of series and parallel system with components sharing the same copula. Appl Stoch Models Bus Ind2010; 26: 775–791.
17.
EryilmazS. Estimation in coherent reliability system through copulas. Reliab Eng Syst Saf2011; 188: 47–61.
18.
ZhangYGomesATBeerM, et al. Reliability analysis with consideration of asymmetrically dependent variables: discussion and application to geotechnical examples. Reliab Eng Syst Saf2019; 185: 261–277.
19.
StefanescuCTurnbullBW. Multivariate frailty models for exchangeable survival data with covariates. Technometrics2006; 48: 411–417.
20.
NavarroJRychlikT. Reliability and expectation bounds for coherent systems with exchangeable components. J Multivar Anal2007; 98: 102–113.
21.
JiaXCuiLYanJ. A study on the reliability of consecutive k-out-of-n:g systems based on copula. Commun Stat Theory Methods2010; 39: 2455–2472.
22.
JiaXCuiL. Reliability research of k-out-of-n: g supply chain system based on copula. Commun Stat Theory Methods2012; 41: 4023–4033.
23.
TangXSLiDQZhouCB, et al. Bivariate distribution models using copulas for reliability analysis, proceedings of the institution of mechanical engineers. Proc IMechE, Part O: J Risk Reliability2013; 227: 499–512.
24.
BayramogluIOzkutM. The reliability of coherent systems subjected to marshall-olkin type shocks. IEEE Trans Reliab2015; 64: 435–443.
25.
NakagawaT. Optimal number of units for a parallel system. J Appl Probab1984; 21: 431–436.
26.
BarlowREProschanF. Mathematical theory of reliability, vol. 17. New York, NY: Siam, 1965.
27.
ZhaoXAl-KhalifaKNHamoudaAM, et al. Age replacement models: a summary with new perspectives and methods. Reliab Eng Syst Saf2017; 161: 95–105.
28.
SheuSHKuoCM. Optimization problems in k-out-of-n systems with minimal repair. Reliab Eng Syst Saf1994; 44: 77–82.
