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Aggregation functions and their transformations have found numerous applications in various kinds of systems as well as in economics and social science. Every aggregation function is known to be bounded above and below by its super-additive and sub-additive transformations. We are interested in the 'inverse' problem of whether or not every pair consisting of a super-additive function dominating a sub-additive function comes from some aggregation function in the above sense. Our main results provide a negative answer under mild extra conditions on the super-and sub-additive pair. We also show that our results are, in a sense, best possible.

This study introduces a revised definition for fuzzy bags. It is based on the definition of bags given by Delgado et al. (Int J Intell Syst 2009;24:706–721) in which each bag has two parts: function and summary information. The new definition is given as a special case of L‐fuzzy bags, where L is a complete lattice. By some examples, the new concept is illustrated. Furthermore, the algebraic structure of L‐fuzzy bags is studied and the concept of α‐cuts, L‐fuzzy bag relations, and related theorems are given.

A multi-objective linear programming problem with a system of max-arithmetic mean relational inequalities as its constraints is considered. For each of the objective functions, the decision maker has a fuzzy goal. Treating each fuzzy goal, two kind of membership functions are considered: linear and hyperbolic as a nonlinear one. Then, using membership functions and Bellman-Zadeh decision, the multi-objective linear programming problem is converted to a conventional linear programming problem. Two examples are given to illustrate the procedure and compare the two different membership functions.