The lambda selections of parametric interval-valued fuzzy variables and their numerical characteristics

Abstract

To model the uncertainty in the secondary possibility distributions, this paper develops a new method for handling interval-valued fuzzy variables with variable lower and upper possibility distributions. For a parametric interval-valued fuzzy variable, we define its lower selection variable, upper selection variable and lambda selection variable. The three selection variables are characterized by variable possibility distributions, and their numerical characteristics like expected values and n-th moments are important indices in practical optimization and decision-making problems. Under this consideration, we establish some useful analytical expressions of the expected values and n-th moments for the lambda selections of parametric intervalvalued trapezoidal, normal and Erlang fuzzy variables. Furthermore, we focus on the arithmetic about the sums of common parametric interval-valued fuzzy variables. Finally, we apply the proposed optimization indices to a quantitative finance problem, where the second moment is used to measure the risk of a portfolio.

A Parametric Sharpe Ratio Optimization Approach for Fuzzy Portfolio Selection Problem

“…As a special case of T2 fuzzy variable, if for any ∈ R, ∈ ⊆ [0, 1], the T2 possibility distribution functioñ( , ) = 1, theñis an interval T2 fuzzy variable [32]. with parameters , ∈ [0, 1], theñis called a parametric interval-valued fuzzy variable [32].…”

“…As a special case of T2 fuzzy variable, if for any ∈ R, ∈ ⊆ [0, 1], the T2 possibility distribution functioñ( , ) = 1, theñis an interval T2 fuzzy variable [32]. with parameters , ∈ [0, 1], theñis called a parametric interval-valued fuzzy variable [32]. If = = 0, then the reduced secondary possibility distribution is called the principle possibility distribution of̃, and the fuzzy variable associated with the principle possibility distribution is denoted by [34].…”

“…For this reason, it is reasonable to consider a more effective and flexible way to describe the uncertainty on return. Based on fuzzy possibility theory [31], Y. Liu and Y.-K. Liu [32] proposed the concept of parametric interval-valued fuzzy variable, which can handle the case of uncertainty contained in fuzzy possibility distribution. Specifically, the distribution of parametric interval-valued fuzzy variable is a variable possibility distribution.…”

A Multiproduct Single‐Period Inventory Management Problem under Variable Possibility Distributions

“…First, in this section, some basic concepts in fuzzy possibility theory are recalled [33][34][35][36].…”

“…For a PIV fuzzy variable, its lambda selection is defined in [34]. Assume that is a PIV fuzzy variable with the secondary possibility distributioñ( ) = [ ( ; ), ( ; )].…”