Random fuzzy delayed renewal processes

Abstract

In renewal processes, fuzziness and randomness often coexist intrinsically. Based on the random fuzzy theory, a delayed renewal process with random fuzzy interarrival times is proposed in this paper. Relations between the renewal number and interarrival times in such a process are investigated, and useful theorems such as the elementary renewal theorem, Blackwell renewal theorem and Smith key renewal theorem in a conventional delayed renewal process are extended to their counterparts for random fuzzy delayed renewal processes.

Notes on random fuzzy (delayed) elementary renewal theorem

“…It follows from Lemma 1 [2] that for any α ∈ (0, 1], there exist θ i , θ i ∈ Θ i such that {ξ i (θ i )} and {ξ i (θ i )} are two iid randon variable sequences for i ≥ 2, and…”

“…To show (12) and (13), Li et al [2] claimed the following fact, which is pointed out from the proof of Theorem 3.1 [2, Theorem 5] to use the dominate convergence theorem.…”

“…We can easily check that this fact is also incorrect by taking = 1 E[ξ 2 (θ)] α and θ 1 = θ 2 as in Counterexample 1. Therefore we cannot use the dominate convergence theorem by the used fact 2 to prove Theorem 5 of Li et al [2]. But, for a slight restricted class of F, we can offer a corrected version of a random fuzzy delayed elementary renewal theorem using assumption* instead of assumption (d).…”

“…Recently, Zhao et al [1] and Li et al [2] defined a random fuzzy renewal process on the possibility space on (Θ, P(Θ), P os), where (Θ, P(Θ), P os) is the product possibility space of (Θ i , P(Θ i ), P os i ), i = 1, 2, • • •. That is, Θ = ∞ i=1 Θ i and P os on P(Θ) is define as a possibility measure such that for any A ⊂ Θ, P os{A} = sup Zhao et al [1] established a random fuzzy elementary renewal theorem and Li et al [2] established a random fuzzy delayed elementary renewal theorem in random fuzzy sense. Unfortunately, these proofs are incorrect.…”

Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty

“…In fact, some authors have applied random fuzzy variable to solve various practical problems. For example, random fuzzy renewal process was first presented by Zhao et al [42], which was further extended to random fuzzy renewal reward process [30], random fuzzy alternating renewal process [31], random fuzzy delayed renewal process [15], respectively. It was also applied to project selection [9], project scheduling [10], shock model [22], transportation planning [38], DEA model [33], location problem [35] and other optimization problems.…”