On the Extension of Pseudo-Boolean Functions for the Aggregation of Interacting Criteria

The paper presents an analysis on the use of integrals defined for non-additive measures (or capacities) as the Choquet and the \Sipos{} integral, and the multilinear model, all seen as extensions of pseudo-Boolean functions, and used as a means to model interaction between criteria in a multicriteria decision making problem. The emphasis is put on the use, besides classical comparative information, of information about difference of attractiveness between acts, and on the existence, for each point of view, of a ``neutral level'', allowing to introduce the absolute notion of attractive or repulsive act. It is shown that in this case, the Sipos integral is a suitable solution, although not unique. Properties of the Sipos integral as a new way of aggregating criteria are shown, with emphasis on the interaction among criteria.

Axiomatizations of signed discrete Choquet integrals

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory

Approximations of Lovasz extensions and their induced interaction index

The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ is defined on each simplex of the standard triangulation of $[0,1]^n$ as the unique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the $n+1$ vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses $f$. In this paper we investigate the least squares approximation problem of an arbitrary Lovasz extension $\hat f$ by Lovasz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of $\hat f$ and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page

Distribution functions of linear combinations of lattice polynomials from the uniform distribution

We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovasz extensions.Comment: 11 page

Capacities and Games on Lattices: A Survey of Result

We provide a survey of recent developments about capacities (or fuzzy measures) and ccoperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.capacity, fuzzy measure, game, lattice, Choquet integral,Shapley value

Modelling fraud detection by attack trees and Choquet integral

Modelling an attack tree is basically a matter of associating a logical ÒndÓand a logical ÒrÓ but in most of real world applications related to fraud management the Ònd/orÓlogic is not adequate to effectively represent the relationship between a parent node and its children, most of all when information about attributes is associated to the nodes and the main problem to solve is how to promulgate attribute values up the tree through recursive aggregation operations occurring at the Ònd/orÓnodes. OWA-based aggregations have been introduced to generalize ÒndÓand ÒrÓoperators starting from the observation that in between the extremes Òor allÓ(and) and Òor anyÓ(or), terms (quantifiers) like ÒeveralÓ ÒostÓ ÒewÓ ÒomeÓ etc. can be introduced to represent the different weights associated to the nodes in the aggregation. The aggregation process taking place at an OWA node depends on the ordered position of the child nodes but it doesnÕ take care of the possible interactions between the nodes. In this paper, we propose to overcome this drawback introducing the Choquet integral whose distinguished feature is to be able to take into account the interaction between nodes. At first, the attack tree is valuated recursively through a bottom-up algorithm whose complexity is linear versus the number of nodes and exponential for every node. Then, the algorithm is extended assuming that the attribute values in the leaves are unimodal LR fuzzy numbers and the calculation of Choquet integral is carried out using the alpha-cuts.Fraud detection; attack tree; ordered weighted averaging (OWA) operator; Choquet integral; fuzzy numbers.

A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package

The application of multi-attribute utility theory whose aggregation process is based on the Choquet integral requires the prior identification of a capacity. The main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed. All the reviewed methods have been implemented within the Kappalab R package. Their application is illustrated on a detailed example.Multi-criteria decision aiding; Multi-attribute utility theory; Choquet integral; Free software

Bipolarization of posets and natural interpolation

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of $[0,1]^n$. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.Interpolation; Choquet integral; Lattice; Bipolar structure

Youth A Multicriteria Approach for the Evaluation of the Sustainability of Re-use of Historic Buildings in Venice

The paper presents a multiple criteria model for the evaluation of the sustainability of projects for the economic re-use of historical buildings in Venice. The model utilises the relevant parameters for the appraisal of sustainability, aggregated into three macroindicators: intrinsic sustainability, context sustainability and economic-financial feasibility. The model has been calibrated by a panel of experts and tested on two reuse hypothesis of the Old Arsenal in Venice.multiple criteria valuation, economic reuse, historical building conservation