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Different definitions of the concept of a fuzzy semiorder are compared. It is proved that their α-cuts are crisp binary relations that may fail to be Ferrers and semitransitive, in general. Consequently, we analyze the preservation of semiorders when coming back from the fuzzy to the crisp setting using α-cuts. In the final sections, a discussion is developed about the extension to the fuzzy setting of the concept of a threshold of utility discrimination, and its corresponding numerical representability of fuzzy semiorders by means of the representability of their α-cuts as crisp binary relations.

An approach using defuzzifying methods is proposed for the fuzzy multiple attribute decision-making (MADM) problems. The computing effectiveness of the proposed defuzzifying methods combined with the simple additive weighting (SAW) method and the technique for order preference by similarity to ideal solution (TOPSIS) method are evaluated based on a comparison to the improved fuzzy weighted average (IFWA) followed by a ranking method. Both SAW and TOPSIS methods are two of classic MADM methods. The purpose of this application is to make the method easier to program and data easier to manipulate. This results in a more practical method for fuzzy decisions. A numerical example and experiment are discussed to demonstrate the implementation of the methods in different input conditions.

Most of the real-world multicriteria decision-making (MCDM) problems contain a mixture of quantitative and qualitative criteria; therefore quantitative MCDM methods are inadequate for handling this type of decision problems. In this paper, a MCDM method based on the Fuzzy Sets Theory and on the Analytic Hierarchy Process (AHP) is proposed. This method incorporates a number of perspectives on how to approach the fuzzy MCDM problem, as follows: (1) combining quantitative and qualitative criteria; (2) expressing criteria pair-wise comparison in linguistic terms and performance of the alternative on each criterion in linguistic terms or exact values when criterion is qualitative or quantitative, respectively; (3) converting all the assessments into trapezoidal fuzzy numbers; (4) using the difference minimization method to calculate the local weight of criteria, employing the algebraic operations of fuzzy numbers based on the concept of α-cuts; (5) calculating the global weight of criteria and the global performance of each alternative using geometric mean and the weighted sum, respectively; (6) using the centroid method to rank the alternatives. Finally, an illustrative example on evaluation of several combined cooling, heat, and power production systems is used to demonstrate the effectiveness of the proposed methodology.