A discrete time repairable multi-state series-parallel system with fuzzy data is considered. The lifetimes and the repair times of the system components are assumed to be geometrically distributed random variables with fuzzy parameters, and the performance levels of the system components are presented as fuzzy values. Fuzzy set theory is employed to deal with the uncertain problem of the system. A discrete time fuzzy Markov model is proposed to assess the fuzzy state probability of binary component. The fuzzy universal generating function technique is adopted to evaluate fuzzy state probability and fuzzy performance level of the entire system. An availability assessment approach is proposed to compute the dynamic fuzzy availability of the system with the fuzzy consumer demand. Based on the extension principle, the α - cuts of some indices for the system are obtained by using parametric programming technique. Finally, a numerical example is presented.
BalakrishnanN., LimniosN. and PapadopoulosC., Basic probabilistic models in reliability, in: Handbook of Statistics Advanced in Reliability, BalakrishnanN. and RaoC.R., eds., Elsevier, Nord-Holland, 2001, pp. 1–42.
2.
BamrungsetthapongW. and PongpullponsakA., Parameter interval estimation of system reliability for repairable multistate series-parallel system with fuzzy data, The Scientific World Journal2014, Article ID 275374, 10 pages, 2014. doi: 10.1155/2014/275374.
3.
BarbuV. and LimniosN., Empirical estimation for discrete time semi-Markov processes with applications in reliability, Journal of Nonparametric Statistics18(7-8) (2006), 483–498.
4.
BarbuV. and LimniosN., Reliability of semi-Markov systems in discrete time: Modeling and estimation, Handbook of Performability Engineering, 2008, pp. 369–380.
5.
BracquemondC. and GaudoinO., A survey on discrete lifetime distributions, International Journal on Reliability, Quality, and Safety Engineering10(1) (2003), 69–98.
6.
CaiK., WenC. and ZhangM., Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets and Systems42(2) (1991), 145–172.
7.
CaiK., Introduction to Fuzzy Reliability, Kluwer Academic Publishers, Dordrecht, 1996.
8.
ChryssaphinouO. and LimniosN., Multi-state reliability systems under discrete time semi-Markovian hypothesis, IEEE Transactions on Reliability60(1) (2011), 80–87.
9.
CsenkiA., Transition analysis of semi-Markov reliability models - a tutorial review with emphasis on discrete-parameter approaches, in: Stochastic Models in Reliability and Maintenance, OsakiS., ed., Springer, Berlin, 2002, pp. 219–251.
10.
DaQ.L. and LiuX.W., Interval number linear programming and its satisfactory solution, Systems Engineering Theory and Practice19(4) (1999), 3–7.
11.
DingY. and LisnianskiA., Fuzzy universal generating functions for multi-state system reliability assessment, Fuzzy Sets and Systems159(3) (2008), 307–324.
12.
DingY., ZuoM.J., LisnianskiA. and TianZ., Fuzzy multi-state systems: General definitions, and performance assessment, IEEE Transactions on Reliability57(4) (2008), 589–594.
13.
DümmlerM.A. and SchömigA.K., Using discrete-time analysis in the performance evaluation of manufacturing systems, International Conference on Semiconductor Manufacturing Operational Modeling and Simulation, San Francisco, California, 1999.
14.
EryilmazS., On mean residual life of discrete time multi-state systems, Quality Technology and Quantitative Management10(2) (2013), 241–249.
15.
Faghih-RoohiS., XieM., NgK.M. and YambR.C., Dynamic availability assessment and optimal component design of multistate weighted k-out-of-n systems, Reliability Engineering and System Safety123 (2014), 57–62.
16.
GiriB.C. and DohiT., Determining economic manufacturing quantity for an unreliable manufacturing system in discrete time setting, International Journal of Operational Research3(5) (2008), 557–574.
17.
HuL., YueD. and TianR., Fuzzy availability assessment for a repairable multistate series-parallel system, Discrete Dynamics in Nature and Society2015 , Article ID 156059, 11 pages, 2015. doi: 10.1155/2015/156059.
18.
HuJ., ChenD. and DuJ., State estimation for a class of discrete nonlinear systems with randomly occurring uncertainties and distributed sensor delays, International Journal of General Systems43(3-4) (2014), 387–401.
19.
HuJ., WangZ., ShenB. and GaoH., Gain-constrained recursive filtering with stochastic nonlinearities and probabilistic sensor delays, IEEE Transactions on Signal Processing61(5) (2013), 1230–1238.
20.
HuJ., WangZ., ShenB. and GaoH., Quantized recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements, International Journal of Control86(4) (2013), 650–663.
21.
KeJ.C., HuangH.I. and LinC.H., Fuzzy analysis for steadystate availability: A mathematical programming approach, Engineering Optimization38(8) (2006), 909–921.
22.
KeJ.C., HuangH.I. and LinC.H., Analysis on a queue system with heterogeneous servers and uncertain patterns, Journal of Industrial and Management Optimization6(1) (2010), 57–71.
23.
KolowrockiK. and Kwiatuszewska-SarneckaB., Reliability and risk analysis of large systems with ageing components, Reliability Engineering and System Safety93(12) (2008), 1821–1829.
24.
ŁazarskiK. and StettnerŁ., Average cost per unit time control of discrete time unreliable manufacturing systems with Markov demand, Mathematical Methods of Operations Research49(3) (1999), 457–473.
25.
LevitinG., Universal Generating Function and Its Application, Springer, Berlin, 2005.
26.
LiC.Y., ChenX., YiX.S. and TaoJ.Y., Interval-valued reliability analysis of multi-state systems, IEEE Transactions on Reliability60(1) (2011), 323–330.
27.
LisnianskiA. and LevitinG., Multi-state System Reliability Assessment, Optimization, Applications, World Scientific, Singapore, 2003.
28.
LiuB., Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain Systems4(3) (2010), 163–170.
29.
LiuY., LiX.Z. and DuZ.P., Reliability analysis of a random fuzzy repairable parallel system with two non-identical components, Journal of Intelligent & Fuzzy Systems27(6) (2014), 2775–2784.
30.
LiuY., HuangH. and LevitinG., Reliability and performance assessment for fuzzy multi-state elements, Journal of Risk and Reliability222(4) (2008), 675–686.
31.
LiuY. and HuangH., Reliability assessment for fuzzy multistate systems, International Journal of Systems Science41(4) (2010), 365–379.
32.
OnisawaT., An approach to human reliability in man-machine systems using error possibility, Fuzzy Sets and Systems27(2) (1988), 87–103.
33.
PlatisA., LimniosN. and LeM., Du, Hitting times in a finite non-homogeneous Markov chain with applications, Applied Stochastic Models and Data Analysis14(3) (1998), 241–253.
34.
Rocha-MartinezJ.M. and ShakedM., A discrete-time model of failure and repair, Applied Stochastic Models and Data Analysis11(2) (1995), 167–180.
35.
SadekA. and LimniosN., Asymptotic properties for maximum likelihood estimators for reliability and failure rates of Markov chains, Communications in Statistics-Theory and Methods31(10) (2002), 1837–1861.
36.
UshakovI., Universal generating function, Soviet Journal of Computer and System Sciences24(5) (1986), 118–129.
37.
UtkinL.V. and GurovS.V., A general formal approach for fuzzy reliability analysis in the possibility context, Fuzzy Sets and Systems83(2) (1996), 203–213.
38.
WangJ., HuangY. and DoT.V., A single-server discrete-time queue with correlated positive and negative customer arrivals, Applied Mathematical Modelling37(9) (2013), 6212–6224.
39.
WangJ., WangN. and AlfaA.S., Discrete-time GI/G/1 retrial queues with time-controlled vacation policies, Acta Mathematica Applicatae Sinica29(4) (2013), 689–704.
40.
YueD. and ZhangF., A discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times, Journal of Systems Science and Complexity26(4) (2013), 556–571.
41.
ZadehL.A., Fuzzy sets, Information and Control8(3) (1965), 338–353.
42.
ZadehL.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems1(1) (1978), 3–28.
