RENEWAL AND RENEWAL REWARD THEORIES FOR T-INDEPENDENT FUZZY RANDOM VARIABLES

- Journal title : Journal of applied mathematics & informatics
- Volume 33, Issue 5_6, 2015, pp.607-625
- Publisher : The Korean Society of Computational and Applied Mathematics
- DOI : 10.14317/jami.2015.607

Title & Authors

RENEWAL AND RENEWAL REWARD THEORIES FOR T-INDEPENDENT FUZZY RANDOM VARIABLES

KIM, JAE DUCK; HONG, DUG HUN;

KIM, JAE DUCK; HONG, DUG HUN;

Abstract

Recently, Wang et al. [Computers and Mathematics with Ap-plications 57 (2009) 1232-1248.] and Wang and Watada [Information Sci-ences 179 (2009) 4057-4069.] studied the renewal process and renewal reward process with fuzzy random inter-arrival times and rewards under the T-independence associated with any continuous Archimedean t-norm. But, their main results do not cover the classical theory of the random elementary renewal theorem and random renewal reward theorem when fuzzy random variables degenerate to random variables, and some given assumptions relate to the membership function of the fuzzy variable and the Archimedean t-norm of the results are restrictive. This paper improves the results of Wang and Watada and Wang et al. from a mathematical per-spective. We release some assumptions of the results of Wang and Watada and Wang et al. and completely generalize the classical stochastic renewal theorem and renewal rewards theorem.

Keywords

N(S)Fuzzy sets;(I)Stochastic processes;Fuzzy renewal process;Renewal reward theories;Possibility measure;

Language

English

References

1.

K.L. Chung, A course in probability theory, second edition, Academic Press, New York and London, 1974.

2.

G. De Cooman, Possibility theory III, International Journal of General Systems 25 (1997), 352-371.

4.

R. Fullér and E. Triesch, A note on the law of large numbers for fuzzy variables, Fuzzy Sets and Systems 55 (1993), 235-236.

5.

R. Fullér and T. Keresztfalvi, On generalization of Nguyen's theorem, Fuzzy Sets and Systems 41 (1991), 371-374.

6.

D.H. Hong and J. Lee, On the law of large numbers for mutually T-related fuzzy numbers, Fuzzy Sets and Systems 121 (2001), 537-543.

7.

D.H. Hong and C.H. Ahn, Equivalent conditions for laws of large numbers for T-related L-R fuzzy numbers, Fuzzy Sets and Systems 136 (2003), 387-395.

8.

D.H. Hong, Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards, Information Sciences 176 (2006), 2386-2395.

9.

D.H. Hong, Blackwell's Theorem for T-related fuzzy variables, Information Sciences 180 (2010), 1769-1777 .

10.

D.H. Hong, Uniform convergence of fuzzy random renewal process, Fuzzy Optimization and Decision Making 9 (2010), 275-288.

11.

D.H. Hong, Renewal process for fuzzy variables, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15 (2007), 493-501.

12.

C-M. Hwang, A theorem of renewal process for fuzzy random variables and its application, Fuzzy Sets and Systems 116 (2000), 237-244.

13.

E.P. Klement, R. Mesiar and E. Pap, Triangular norms, Trends in Logic, Vol. 8, Kluwer, Dordrecht, 2000.

14.

H. Kwakernaak, Fuzzy random variables I: Definitions and theorems, Information Sciences 15 (1978), 1-29.

15.

Y.-K. Liu, B. Liu, Fuzzy random variable: A scalar expected value operator, Fuzzy Opti-mization and Decision Making 2 (2003), 143-160.

16.

Y.-K. Liu, B. Liu, On minimum-risk problems in fuzzy random decision systems, Com-puters and Operations Research 32 (2005), 257-283.

17.

B. Liu, Uncertain Theory: An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004.

18.

B. Liu and Y.-K. Liu, Expected value of a fuzzy variable and fuzzy expected value models, IEEE Transaction on Fuzzy Systems 10 (2002), 445-450.

19.

E. Popova and H. C. Wu, Renewal reward processes with fuzzy rewards and their applica-tions to T-age replacement policies, European Journal of Operational Research 117 (1999), 606-617.

20.

M.L. Puri, D.A. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications 114 (1986), 409-422.

21.

S.M. Ross, Stochastic Processes, John Wiley and Sons, New York, 1996.

22.

P. Terán, Strong law of large numbers for t-normed arithmetics, Fuzzy Sets and Systems 159 (2008), 343-360.

23.

E. Triesch, Characterization of Archimedean t-norms and a law of large numbers, Fuzzy Sets and Systems 58 (1993), 339-342.

24.

S. Wang, J. Watada, Fuzzy random renewal reward process and its applications, Information Sciences 179 (2009), 4057-4069.

25.

S. Wang, Y-K. Liu, J. Watada, Fuzzy random renewal process with queueing applications, Computers and Mathematics with Applications 57 (2009), 1232-1248.

26.

S. Wang, J. Watada, Studying distribution functions of fuzzy random variables and its applications to critical value functions, International Journal of Innovative Computing, Information and Control 52 (2009), 279-292.

27.

L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978), 3-28.

29.

R. Zhao and B. Liu, Renewal process with fuzzy inter-arrival times and rewards, Interna-tional Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11 (2003), 573-586.