# Inconsistency and non-additive capacities: The Analytic Hierarchy Process in the framework of Choquet integration

Published on Feb 1, 2013in Fuzzy Sets and Systems2.907
· DOI :10.1016/j.fss.2012.08.006
Silvia Bortot3
Estimated H-index: 3
(University of Trento),
Ricardo Alberto Marques Pereira9
Estimated H-index: 9
(University of Trento)
Abstract
We examine the AHP in the framework of Choquet integration and we propose an extension of the standard AHP aggregation scheme on the basis of the Shapley values associated with the criteria. In our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the AHP. The measure of dominance inconsistency is then used to construct a non-additive capacity whose associated Shapley values reduce to the standard AHP priority weights in the consistency case. In the general inconsistency case, however, the extended aggregation scheme based on the Shapley weighted mean tends to attenuate (resp. emphasize) the priority weights of the criteria with higher (resp. lower) average dominance inconsistency with respect to the other criteria.
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• Citations (13)
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References58
Newest
#1Alessio IshizakaH-Index: 26
#2Philippe NemeryH-Index: 13
#1Brice Mayag (University of Paris)H-Index: 5
#2Michel Grabisch (University of Paris)H-Index: 41
Last. Christophe LabreucheH-Index: 16
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In the context of multiple criteria decision analysis, we present the necessary and sufficient conditions to represent a cardinal preferential information by the Choquet integral w.r.t. a 2-additive capacity. These conditions are based on some complex cycles called cyclones.
#1Brice Mayag (University of Paris)H-Index: 5
#2Michel Grabisch (University of Paris)H-Index: 41
Last. Christophe LabreucheH-Index: 16
view all 3 authors...
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 c...
#1Bice Cavallo (University of Naples Federico II)H-Index: 10
#2Livia D’Apuzzo (University of Naples Federico II)H-Index: 11
We consider the framework of pairwise comparison matrices over abelian linearly ordered groups. We introduce the notion of o-proportionality that allows us to provide new characterizations of the consistency, efficient algorithms for checking the consistency and for building a consistent matrix. Moreover, we provide a new consistency index. © 2010 Wiley Periodicals, Inc.
#1Michel Grabisch (University of Paris)H-Index: 41
The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and...
#1Michel GrabischH-Index: 41
#2Jean-Luc MarichalH-Index: 28
Last. Endre PapH-Index: 3
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Aggregation is the process of combining several numerical values into a single representative value, and an aggregation function performs this operation. These functions arise wherever aggregating information is important: applied and pure mathematics (probability, statistics, decision theory, functional equations), operations research, computer science, and many applied fields (economics and finance, pattern recognition and image processing, data fusion, etc.). This is a comprehensive, rigorous...
#1Bice CavalloH-Index: 10
#2Livia D’ApuzzoH-Index: 11
In a multicriteria decision making context, a pairwise comparison matrix A = (aij) is a helpful tool to determine the weighted ranking on a set X of alternatives or criteria. The entry aij of the matrix can assume different meanings: aij can be a preference ratio (multiplicative case) or a preference difference (additive case) or aij belongs to [0, 1] and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the ma...
#1Jiann Liang Yang (NTUST: National Taiwan University of Science and Technology)H-Index: 2
#2Huan Neng Chiu (Asia University (Japan))H-Index: 1
Last. Ruey Huei Yeh (NTUST: National Taiwan University of Science and Technology)H-Index: 1
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Vendor selection is an evaluation process that is based on many criteria that uses inaccurate or uncertain data. But while the criteria are often numerous and the relationships between higher-level criteria and lower-level sub-criteria are complex, most conventional decision models cannot help us clarify the interrelationships among the sub-criteria. Our proposed integrated fuzzy multiple criteria decision making (MCDM) method addresses this issue within the context of the vendor selection probl...
#1R. A. Marques Pereira (University of Trento)H-Index: 8
#2Rita A. Ribeiro (NOVA: Universidade Nova de Lisboa)H-Index: 22
Last. Paulo Serra (NOVA: Universidade Nova de Lisboa)H-Index: 6
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We propose an extension of the Takagi-Sugeno-Kang (TSK) fuzzy inference system, using Choquet integration for aggregating the single rule outputs. In the new Choquet-TSK fuzzy inference system, the pairwise synergies between rules are encoded in a rule correlation matrix computed from the activation pattern of the rule base. The rule correlation matrix is then used to modulate the parameters of the Choquet integration scheme in order to compensate for the effect of rule synergies, which are pres...
Cited By13
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#1Silvia Bortot (University of Trento)H-Index: 3
#2Ricardo Alberto Marques Pereira (University of Trento)H-Index: 9
Last. Anastasia Stamatopoulou (University of Trento)
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We consider a set $$N = \{ 1,\ldots ,n \}$$ of interacting agents whose individual opinions are denoted by $$x_{i}$$, $$i \in N$$ in some domain $${\mathbb {D}}\subseteq {\mathbb {R}}$$. The interaction among the agents is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interacting network structure is thus that of a complete graph with edge values in (0, 1). In the Choquet framework, the interacting network structure ...
#1Abbas Mardani (UTM: Universiti Teknologi Malaysia)H-Index: 17
#2Mehrbakhsh Nilashi (UTM: Universiti Teknologi Malaysia)H-Index: 21
Last. Noriza Mohd Jamal (UTM: Universiti Teknologi Malaysia)H-Index: 2
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In many real-life decision making (DM) situations, the available information is vague or imprecise. To adequately solve decision problems with vague or imprecise information, fuzzy set theory and aggregation operator theory have become powerful tools. In last three decades, DM theories and methods under fuzzy aggregation operator have been proposed and developed for effectively solving the DM problems and numerous applications have been reported in the literature. While various aggregation opera...
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#2Shih-Wen HsiaoH-Index: 25
#1Silvia Bortot (University of Trento)H-Index: 3
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Abstract We introduce a general framework for the algebraic representation of the weighted mean in the case in which priorities and values refer to a common aggregation domain, for instance in hierarchical multicriteria decision models of the AHP type. The general framework proposed is based on the semifield structure of open interval domains and provides a natural algebraic description of weighted mean aggregation, including the essential mechanism of normalization which transforms priorities i...
#1Mustafa S. Abbas (PSU: Portland State University)H-Index: 1
#2Dundar F. Kocaoglu (PSU: Portland State University)H-Index: 14
The objective of this research is to establish consistency thresholds linked to alpha (α) levels for HDM's (Hierarchical Decision Model) judgment quantification method. Measuring consistency in order to control it is a crucial and inseparable part of any AHP/HDM experiment. The researchers on the subject recommend establishing thresholds that are statistically based on hypothesis testing, and are linked to the number of decision variables and a level. Such thresholds provide the means with which...
#1Bice Cavallo (University of Naples Federico II)H-Index: 10
#2Livia D’Apuzzo (University of Naples Federico II)H-Index: 11
In the context of Pairwise Comparison Matrices (PCMs) defined over abelian linearly ordered group, -consistency and -transitivity represent a full coherence of the Decision Maker (DM) and the minimal logical requirement that DM's preferences should satisfy, respectively. Moreover, the -mean vector wm is proposed as weighting vector for the decision elements related to the PCM. In this paper, we investigate the effects of -inconsistency of a -transitive PCM on wm and, in order to ensure its relia...
#1Salvatore Corrente (University of Catania)H-Index: 15
#2Salvatore Greco (University of Catania)H-Index: 53
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We consider multiple criteria decision aiding in the case of interaction between criteria. In this case the usual weighted sum cannot be used to aggregate evaluations on different criteria and other value functions with a more complex formulation have to be considered. The Choquet integral is the most used technique and also the most widespread in the literature. However, the application of the Choquet integral presents two main problems being the necessity to determine the capacity, which is th...
#1Bice Cavallo (University of Naples Federico II)H-Index: 10
#2Livia D’Apuzzo (University of Naples Federico II)H-Index: 11
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Consistency and transitivity are important and leading research topics in the study of decision-making in terms of pairwise comparison matrices. In this paper, we search for conditions that, in case of inconsistency, guarantee ordinal compatibility between ordinal ranking (actual ranking) derived from a transitive matrix and cardinal rankings provided by the most usual priority vectors proposed in the scientific literature. We provide the notion of weak consistency; it is a condition weaker than...
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#2Yuh-Wen ChenH-Index: 10