Modeling renewal processes in fuzzy decision system

Abstract

a b s t r a c tUnder expected value of fuzzy variable and continuous Archimedean triangular norms, this paper discusses a renewal process and a renewal reward process for T-independent L-R fuzzy variables in fuzzy decision systems. First, a renewal process with T-independent L-R fuzzy interarrival times is discussed, some limit theorems on renewal variable, average renewal time, and long-term renewal rate in (fuzzy) measure are obtained, and a fuzzy elementary renewal theorem is proved for the limit of the long-term expected renewal rate. Second, a renewal reward process with T-independent L-R fuzzy interarrival times and rewards is discussed, a limit theorem on reward rate in (fuzzy) measure is derived, and a fuzzy renewal reward theorem is proved for the limit value of expected reward rate. Finally, the comparison with stochastic counterparts shows an interesting and reasonable homology in convergence mode and limit value between the results obtained in fuzzy renewal processes and the corresponding results in stochastic renewal processes, though they build on two essentially different mathematical cornerstones, possibility theory and probability theory, respectively.

Renewal and renewal reward theorems for T-independent L-R fuzzy variables

“…The assumption of uniform boundedness is that it may be strong. All of the results of Wang and Liu [15] are direct consequences of our results. An illustrative example is shown.…”

“…which completes the proof. As a special case of above result, we have the following Theorem 3 of Wang and Liu [15].…”

“…Let {a n } be a sequence of uniformly bounded nonnegative real numbers, then limsup ( n i=1 α n /n) is finite and lim(max 1≤i≤n α i /n) = 0. As a special case of Theorem 1, we have the following Theorem 1 of Wang and Liu [15] from Lemma 1.…”

“…In this paper, we obtained R 1 -version of a renewal theorems and a renewal reward theorems for T -independent L-R fuzzy variables with not uniformly bounded spreads using the results of Hong [6,7]. Recently, Wang and Liu [15] proved a renewal theorems and a renewal reward theorems for T -independent L-R fuzzy variables in fuzzy decision systems under the assumption of uniformly bounded spreads of L-R fuzzy variables. The assumption of uniform boundedness is that it may be strong.…”

A Parametric Sharpe Ratio Optimization Approach for Fuzzy Portfolio Selection Problem

“…It is worth noting that, in real life, the future asset return sometimes needs to be given by the expert's judgement and estimation for no sufficient historical data or the noisy original data. Therefore, many scholars tried to model the uncertain parameter with fuzzy uncertainty [19][20][21][22], and a number of researchers have developed some downside risk measures in fuzzy decision problem [23,24]. Since the ability to measure the risk-adjusted performance of a portfolio, the Sharpe ratio was introduced to assess the portfolio in fuzzy circumstance.…”