Abstract
In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric triangular interval type-2 fuzzy variables (RSTIT2-FVs), a special type of interval type-2 fuzzy variable is discovered and proved. Furthermore, for applying the two measures, we propose the operational law for the variance and semi-variances of the linear function of mutually independent RSTIT2-FVs. Some numerical examples are illustrated. The consequences of examples prove that the formulas we proposed can be effectively applied to the calculation of the variance of RSTIT2-FVs. The results indicate that they play a great role in the application of variance of type-2 fuzzy sets in various fields.
Highlights
We find that there exist some relationships between the variance and semi-variances of the RSTIT2-FV
The numerical examples showed that the formulae inferred in this paper can effectively figure out the variance and one-side semi-variances, which is of great value for studying type-2 fuzzy theory in depth, solving realistic problems such as fuzzy portfolio selection problems in finance, and evaluating healthcare equipment in medical fields
The operational law presented in this paper is limited to RSTIT2-FVs
Summary
There have been several scholars focusing their eyes on the variance of T2-FS before, the results of variance calculations they proposed are fuzzy intervals or sets rather than a specific value, and limited to a specific type of fuzzy set and cannot be widely used Few of these researchers have studied the operational laws of variance, which are often involved in the solution of practical problems. Based on the theory proposed by Li and Cai [27] the operational law of the variance and semi-variances for the linear functions of mutually independent regular symmetric triangular interval type-2 fuzzy variables are deduced and some numerical examples are introduced. The variance of T2-FS is applied in many practical fields and helps us to make variance analysis for fuzzy events, and enables us to handle fuzzy data properly
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