Calculation formulas and correlation inequalities for variance bounds and semi-variances of fuzzy intervals

Variance in fuzzy set theory, generally applied in investment decision, risk evaluation, and so on, can be described as a measurement that gauges the deviation of a fuzzy number. In this paper, in order to extend the application range and enrich the research area of variance, the concepts of variance bounds and semi-variances are defined and discussed from a theoretical point of view. With respect to some frequently-used fuzzy intervals, four relatively simple calculation formulas for upper and lower bounds of variance, and upside and downside semi-variances are put forward respectively, with the aid of which several correlation inequalities are subsequently presented and proved. Besides, in order to depict the concepts and inequalities more distinctly, plenty of examples are introduced to make some numerical illustration.