Through this study, we are going to introduce a new aspect of cut sets for hesitant fuzzy sets. Compared to the rare researches in regard of cut sets for hesitant fuzzy sets, not only this study does not deal with extra parameters, but also the proposed concept of cut set conforms with the accepted principles, and further, it is extension of existing cut set concept of fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets and three-dimensional fuzzy sets. In the sequel, using proposed concept of cut sets for hesitant fuzzy sets, we extend the corresponding theorems of decomposition and representation for hesitant fuzzy sets. The current research can be a basis for the future work on the application of cut sets in multiple criteria decision making problems under hesitant fuzzy environment.

Let X be a real normed space, then C( X) is functionally convex (brie y, F-convex), if T(C)  R is convex for all bounded linear transformations T 2 B(X; R); and K( X) is functionally closed (brie y, F-closed), if T(K)  R is closed for all bounded linear transformations T 2 B(X; R). We improve the Krein-Milman theorem on  nite dimensional spaces. We partially prove the Chebyshev 60 years old open problem. Finally, we introduce the notion of functionally convex functions. The function f on X is functionally convex (brie y, F-convex) if epi f is a F-convex subset of X  R. We show that every function f: (a; b) 􀀀 ! R which has no vertical asymptote is F-convex.

One of the conventional methods for temporary support of tunnels is to use steel sets with shot crete. The nature of a temporary support system demands a quick installation of its structures. As a result, the spacing between steel sets is not a fixed amount and it can be considered as a random variable. Hence, in the reliability analysis of these types of structures, the selection of an appropriate probability distribution function of spacing of steel sets is essential. In the present paper, the distances between steel sets are collected from an under-construction tunnel and the collected data is used to suggest a proper Probability Distribution Function (PDF) for the spacing of steel sets. The tunnel has two different excavation sections. In this regard, different distribution functions were investigated and three common tests of goodness of fit were used for evaluation of each function for each excavation section. Results from all three methods indicate that the Wake by distribution function can be suggested as the proper PDF for spacing between the steel sets. It is also noted that, although the probability distribution function for two different tunnel sections is the same, the parameters of PDF for the individual sections are different from each other.