# Towards a New Multicriteria Decision Support Method Using Fuzzy Measures and the Choquet Integral

Emdjed Alnafie, Djamila Hamdadou, Karim Bouamrane
DOI: 10.4018/IJFSA.2016010104
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## Abstract

In literature, there is a large panoply of multicriteria analysis methods (MCAM), each one is characterized by the nature of its input data, the way to edit its outputs and the operations used to perform calculations especially the performances aggregation. Aggregation is the operation consisting in grouping several quantities in a unique value in order to facilitate the manipulation and the interpretation of the original values. MCAM are classified according to the type of aggregation that they perform, so we can distinguish total, partial and local aggregation. Each MCAM has advantages and suffers from some limits. In this paper, the authors proposed a new multicriteria analysis method (AMFI) dedicated to solve ranking decision support problems. AMFI is based on the use of fuzzy measures and Choquet integral to represent interactions between criteria and improve the coherence of the results. The authors proceeded to a series of experimentations allowing highlighting theoretical elements of the proposed method and they performed sensitivity analysis to test its robustness.
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## Multicriteria Analysis: State Of The Art

Generally, solving a multicriteria problem consists in finding the “most appropriate” solution taking into account a set of criteria. It is possible to summarize this process in 4 steps. First, potential actions (alternatives) must be enumerated then we must draw up a list of criteria to be considered. Alternatives are evaluated relative to the criteria in order to generate the performances matrix. Finally, we must proceed to a multicriteria aggregation (Ben Mena, 2000).

### Multicriteria Aggregation

The multicriteria aggregation is the operation of finding a formal representation of the actions performances in relation to criteria. The origin of the performances aggregation problem is that there is rarely a consensus between the criteria in comparing alternatives and conflicts are frequently noted. Therefore, it is important to find a compromise in the light of these conflicts by looking for a decision rule for constructing a preference relation between actions.

For any pair of alternatives , a decision rule defines a binary predicate which value is a function of the performances vectors and (Grabisch & Perny, 2002). When takes its values in the set , we talk about a “net” preference relationship and there is talk of a “fuzzy” preference relationship when the predicate values belong to the interval . Indeed, the evaluation of the preference is done in two operations, namely, the aggregation of performances and and the comparison () of alternatives and through their performances vectors. The association of these two functions logically leads to two different orders in which is an increasing function of its arguments and an increasing function of its first argument and decreasing of the second and for any real :

(1)

In the “Aggregate then Compare” (AC) approach, the aggregation function allows grouping actions performances in values representing their global performances for a unique synthetic criterion. The comparison function allows measuring and evaluating the preference degree between global performances (Grabisch & Perny, 2002). The weighted sum is a good example of the “AC” approach.

The “Compare then Aggregate” (CA) approach compares actions performances in pairs. Thus, for each pair of actions and each criterion, a partial preference index is defined. The aggregation function allows aggregating partial preference indexes even if they are associated to different criteria. The ELECTRE family is a classic example of the “CA” approach.

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