Remarks on possibilistic variances of fuzzy numbers

Carlsson and Fullér (Fuzzy Sets Syst. 122:315–326, 2001) introduced the definitions of two crisp possibilistic variances of a fuzzy number A, Var(A) and Var′(A). They showed that the subsethood does entail smaller variance in the sense of Var(?). Thus it is natural to ask whether it holds in the sense of Var′(?). Yet we are able to prove that it actually does not hold. Zhang and Wang (Appl. Math. Lett. 20:1167–1173, 2007) had introduced some conditions for which the subsethood does entail smaller variance in the sense of Var′(?). In this paper we give more generalized conditions for which it holds.