Fuzzy Nonparametric Predictive Inference for the Reliability of Series Systems


  • Soheil Shokri Department of Statistics, International Campus, Ferdowsi University of Mashhad,
  • Bahram Sadeghpour Gildeh Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad
  • Gholam-Reza Mohtashami Borzadaran Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad
  • Behrouz Fathi Vajargah Department of Statistics, Faculty of Mathematical Sciences, University of Guilan, Rasht


series systems, lower and upper probabilities, non-parametric predictive inference, Fuzzy number.


This paper presents fuzzy lower and upper probabilities for the reliability of series systems. Attention is restricted to series systems with exchangeable components. In this paper, we consider the problem of the evaluation of system reliability based on the nonparametric predictive inferential (NPI) approach, in which the defining the parameters of reliability function as crisp values is not possible and parameters of reliability function are described using a triangular fuzzy number. The formula of a fuzzy reliability function and its α-cut set are present. The fuzzy reliability of structures defined based on a fuzzy number. Furthermore, the fuzzy reliability functions of series systems discussed. Finally, some numerical examples are present to illustrate how to calculate the fuzzy reliability function and its α-cut set. In other words, the aim of this paper is present a new method titled fuzzy nonparametric predictive inference for the reliability of series systems.


Augustin, T. and Coolen, F.P.A., “Nonparametric predictive inference and interval probabilityâ€, Journal of Statistical Planning and Inference, 124, pp. 251-272, 2004.

Bayes, T., “An essay towards solving a problem in the doctrine of chancesâ€, Philos. Trans. Roy. Soc. London 53, p. 370-418; 54, pp. 296-325, 1763.

Birnbaum, Z. W., Esary, J. D. and Saunders, S., “Multi-component systems and structures and their reliabilityâ€, Technometrics, 3, pp. 55-77, 1961.

Cai, K-Y., Introduction to Fuzzy Reliability, Kluwer Academic Publishers, Boston, 1996.

Coolen, F.P.A., “Low structure imprecise predictive inference for Bayes' problemâ€, Statistics & Probability Letters 36, pp. 349-357, 1998.

Coolen, F.P.A., “On the use of imprecise probabilities in reliabilityâ€, Quality and Reliability Engineering International 20 , pp. 193–202, 2004.

Coolen, F.P.A., Nonparametric Predictive Inference. Wiley Series in Probability and Statistics, Wiley, 2015.

Coolen, F.P.A., Coolen-Schrijner, P., Yan, K.J., “Nonparametric predictive inference in reliabilityâ€, Reliability Engineering and System Safety 78, pp. 185-193, 2002.

Coolen, F.P.A. and Coolen-Schrijner, P., “Nonparametric predictive inference for k-out-of-m systemsâ€, In Advances in mathematical modeling for reliability, (Eds T. Bedford, et al.) pp. 185–192 (IOS Press, Amsterdam), 2008.

Coolen F.P.A., Utkin L.V., “Imprecise reliabilityâ€, In: International Encyclopedia of Statistical Science, M. Lovric (Ed.). Springer, pp. 649-650, 2011.

Coolen-Schrijner, P., Coolen, F.P.A. and MacPhee, I.M., “Nonparametric predictive inference for systems reliability with redundancy allocationâ€, Journal of Risk and Reliability 222, pp. 463-476, 2008.

De Finetti, B., Theory of Probability, 2 vols. Wiley, London, 1974.

Dubois, D. and Prade, H., Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988.

Geisser, S., Predictive Inference: An Introduction. Chapman & Hall, London, 1993.

Hill, B.M., “Posterior distribution of percentiles: Bayes' theorem for sampling from a populationâ€, Journal of the American Statistical Association 63, pp. 677-691, 1968.

Hill, B.M., “De Finetti's theorem, induction, and A(n) or Bayesian nonparametric predictive inference (with discussion)â€, In J.M. Bernardo, et al. (Eds.), Bayesian Statistics 3, pp. 211-241. Oxford University Press, 1988.

Tian, Z., Zuo, M. J. and Yam, R. C., “Multi-state k-out-of-n systems and their performance evaluationâ€, IIE Transactions, 41, pp. 32-44, 2009.

Walley, P., Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London, 1991.

Weichselberger, K., “Axiomatic foundations of the theory of interval-probabilityâ€, In: Mammitzsch, V., SchneeweiU, H. (Eds.), Proceedings of the Second GauU Symposion, Section B. De Gruyter, Berlin, pp. 47–64, 1995.

Weichselberger, K., “The theory of interval-probability as a unifying concept for uncertaintyâ€, Int. J. Approx. Reason. 24, pp. 149–170, 2000.

Zadeh, L.A., Fuzzy sets. Information and Control 8, pp. 338–353, 1965.

Zimmermann, H.J., Fuzzy set theory and its applications. Kluwer Academic Publishers, Dordrecht, 1991.




How to Cite

Shokri, S., Gildeh, B. S., Borzadaran, G.-R. M., & Vajargah, B. F. (2017). Fuzzy Nonparametric Predictive Inference for the Reliability of Series Systems. Asian Journal of Fuzzy and Applied Mathematics, 4(6). Retrieved from https://ajouronline.com/index.php/AJFAM/article/view/4298