Fuzzy-MACBETH Hybrid Method: Mathematical Treatment of a Qualitative Scale Using the Fuzzy Theory

Abstract

Abstract This paper describes the research procedures adopted in developing a triangular fuzzy number scale based on the semantic scale of MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique). The objective was to mathematically treat the uncertainty and subjectivity of linguistic variables used to assess a decision problem. A matrix was initially obtained based on a decision maker’s assessment of a given context analysis. This decision matrix was then fuzzified based on a triangular Fuzzy numbers scale. Next, the inference process was performed using F-LP-MACBETH linear programming problem proposed here, resulting in a Fuzzy scale. This scale was then defuzzified using the centroid method, from which a crisp basic scale emerged, which was then cardinalized. The results show that the MACBETH Fuzzy method proposed here can overcome the classical method’s cardinal inconsistency problem, which facilitates its application in complex contexts. Hence, the MACBETH Fuzzy Hybrid method generated numerical values based on the decision makers’ semantically consistent assessments in a decision matrix, which by the classical method presents cardinal inconsistency. Therefore, the advantage of the proposed method consists in the possibility of obtaining a cardinal scale aligned to the decision makers’ preferences without the need to reassess the context.

Keywords

• MACBETH method
• Semantic scale
• Cardinal scale
• Fuzzy logic
• Fuzzy-MACBETH