An approach to the comparison of fuzzy numbers in decision-making problems

In decision-making problems, when a decision maker has information about unmanageable factors in fuzzy numbers, the problem of its comparison appears. Nowadays, a lot of different methods of comparing fuzzy numbers have been proposed. However, none of them is universal. Moreover, almost each method has pitfalls such as the difficulty of interpretation and inconsistency with human intuition. In the decision making theory the character of the applied problem is a dominant factor of choosing the method of comparing fuzzy numbers.
In this paper an approach of comparing fuzzy numbers has been proposed based on the comparison of $\alpha$-cuts. Conceptions of strong and soft preferences are proposed. According to these definitions trapezoidal and bell-shaped fuzzy numbers have been compared. This method leads to finding the solution in the lexicographic meaning of a certain multi-objective problem for some classes of fuzzy numbers. Geometrical interpretation has been given for trapezoidal and bell-shaped fuzzy numbers.