On Fuzzy Simulations for Expected Values of Functions of Fuzzy Numbers and Intervals

Abstract

Based on existing fuzzy simulation algorithms, this paper presents two innovative techniques for approximating the expected values of fuzzy numbers' monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li ( 2015) is time-consuming and also not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of α-optimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout the article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency.

An Extensive Operational Law for Monotone Functions of LR Fuzzy Intervals with Applications to Fuzzy Optimization

“…In order to solve model (33), Liu (2002) designed a hybrid intelligent algorithm (HIA) by combining stochastic discretization algorithm (SDA), neural network and genetic algorithm. However, Li (2015) and Liu et al (2020) pointed out that SDA has poor performance both on accuracy and computational time over simulating the EV. Liu et al (2020) subsequently proposed a numericalintegral based algorithm, but it is not applicable to monotone but not necessarily strictly monotone functions with regard to regular LR-FIs.…”

“…However, Li (2015) and Liu et al (2020) pointed out that SDA has poor performance both on accuracy and computational time over simulating the EV. Liu et al (2020) subsequently proposed a numericalintegral based algorithm, but it is not applicable to monotone but not necessarily strictly monotone functions with regard to regular LR-FIs. Thus this paper proposes a new numerical integration algorithm (NIA) to fill the gap.…”

“…It is worth noting that that the credibility distributions ψ(t) (or the ICDs ψ −1 (δ)) of LR-FIs in Examples 1-3 (or in Examples 5-7) are continuous and strictly increasing on the domain {t| 0 < ψ(t) < 0.5 or 0.5 < ψ(t) < 1}. For the sake of describing such kind of LR-FIs, we first introduce the definition of regular LR-FI proposed in Liu et al (2020) and then verify two equivalent conditions.…”

“…Definition 4 (Liu et al 2020) If the shape functions L and R of an LR-FI ζ are continuous and strictly decreasing on the domains {t| 0 < L(t) < 1} and {t| 0 < R(t) < 1}, respectively, then the LR-FI is regular. Theorem 2 An LR-FI is regular if and only if it satisfies any one of the following conditions, (i) The credibility distribution ψ(t) is continuous on the domain {t| 0 < ψ(t) < 1} and strictly increasing on the domain {t| 0 < ψ(t) < 0.5 or 0.5 < ψ(t) < 1};…”

“…Although the operational law in is quite practical and applicable to some fuzzy optimization problems, it is notable that the functions in the fuzzy programming models are restricted to strictly monotone ones in regard to fuzzy variables and the involving fuzzy variables are required to be regular LR fuzzy numbers such as triangular fuzzy numbers. In practice, many optimization problems cannot be modeled by strictly monotone functions like classical newsvendor problem and moreover as for the fuzzy variables there are too many cases where the operational law in Zhou et al (2016) cannot work, for example, when those fuzzy variables are regular LR-FIs defined by Liu et al (2020) like trapezoidal fuzzy numbers. Therefore, it is necessary and valuable to make an extensive study for the research of .…”