Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process

We propose to extend the aggregation scheme of Saaty’s AHP, from the stan- dard weighted averaging to the more general Choquet integration. In our model, a measure of inconsistency between criteria is derived from the main pairwise comparison matrix and it is used to construct a non-additive capacity, whose associated Choquet integral reduces to the standard weighted mean in the con- sistency case. In the general inconsistency case, however, the new aggregation scheme based on Choquet integration tends to attenuate (resp. emphasize) the priority values of the criteria with higher (resp. lower) average inconsistency with the remaining criteria.Aggregation Functions, Multiple Criteria Analysis, AHP, Inconsintency, non-additive measures, Choquet integral, and Shapley values.

On the priority vector associated with a fuzzy preference relation and a multiplicative preference relation.

We propose two straightforward methods for deriving the priority vector associated with a fuzzy preference relation. Then, using transformations between multiplicative preference relations and fuzzy preference relations, we study the relationships between the priority vectors associated with these two types of preference relations.pairwise comparison matrix; fuzzy preference relation; priority vector

On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper

Extended Fuzzy Analytic Hierarchy Process (E-FAHP): A General Approach

[EN] Fuzzy analytic hierarchy process (FAHP) methodologies have witnessed a growing development from the late 1980s until now, and countless FAHP based applications have been published in many fields including economics, finance, environment or engineering. In this context, the FAHP methodologies have been generally restricted to fuzzy numbers with linear type of membership functions (triangular numbers-TN-and trapezoidal numbers-TrN). This paper proposes an extended FAHP model (E-FAHP) where pairwise fuzzy comparison matrices are represented by a special type of fuzzy numbers referred to as (m,n)-trapezoidal numbers (TrN (m,n)) with nonlinear membership functions. It is then demonstrated that there are a significant number of FAHP approaches that can be reduced to the proposed E-FAHP structure. 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A Representation of Preferences by the Choquet Integral with Respect to a 2-Additive Capacity

In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions allowing to represent an ordinal preferential information provided by the decision maker by a Choquet integral w.r.t a 2-additive capacity. We provide also a characterization of this type of preferential information by a belief function which can be viewed as a capacity. These characterizations are based on three axioms, namely strict cycle-free preferences and some monotonicity conditions called MOPI and 2-MOPI.multicriteria decision making; Choquet integral; 2-additive capacity; MACBETH