Silvia Angilella

Assessing non-additive utility for multicriteria decision aid

Abstract In the framework of Multi-Attribute Utility Theory (MAUT) several methods have been proposed to build a Decision-Makers (DM) utility function representing his/her preferences. Among such methods, the UTA method infers an additive utility function from a set of exemplary decisions using linear programming. However, the UTA method does not guarantee to find a utility function which is coherent with the available information. This drawback is due to the underlying utility model of UTA, viz. the additive one, which does not allow to include additional information such as an interaction among criteria. In this paper we present a methodology for building a non-additive utility function, in the framework of the so called fuzzy integrals, which permits to model preference structures with interaction between criteria. Like in the UTA method, we aim at searching a utility function representing the DMs preferences, but unlike UTA, the functional form is a specific fuzzy integral (Choquet integral). As a result, we obtain weights which can be interpreted as the “importance” of coalitions of criteria, exploiting the potential interaction between criteria, as already proposed by other authors. However, within the same framework, we obtain also the marginal utility functions relative to each one of the considered criteria, that are evaluated on a common scale, as a consequence of the implemented methodology. Finally, we illustrate our approach with an example.

Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem

The Choquet integral preference model is adopted in Multiple Criteria Decision Aiding (MCDA) to deal with interactions between criteria, while the Stochastic Multiobjective Acceptability Analysis (SMAA) is an MCDA methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters. In this paper, we propose to combine the Choquet integral preference model with the SMAA methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the Decision Maker (DM). In case the criteria are on a common scale, one has to elicit only a set of non-additive weights, technically a capacity, compatible with the DM’s preference information. Instead, if the criteria are on different scales, besides the capacity, one has to elicit also a common scale compatible with the preferences given by the DM. Our approach permits to explore the whole space of capacities and common scales compatible with the DM’s preference information.

The Financing of Innovative SMEs: a multicriteria credit rating model

Small and Medium-sized Enterprises (SMEs) face many obstacles when they try to access the credit market. These obstacles increase if the SMEs are innovative. In this case, financial data are insufficient or even unreliable. Thus, building a judgmental rating model, mainly based on qualitative criteria (soft information), is very important to finance SMEs’ activities. Till now, there has not been a multicriteria credit risk model based on soft information for innovative SMEs. In this paper, we try to fill this gap by presenting a multicriteria credit risk model named ELECTRE-TRI. A SMAA-TRI analysis is also implemented to obtain robust SMEs’ assignments to the risk classes. SMAA-TRI incorporates ELECTRE-TRI by considering different sets of preference parameters and uncertainty in the data via Monte Carlo simulations. Finally, we carry out a real case study with the aim of illustrating the multicriteria credit risk model.

Non Additive Robust Ordinal Regression for urban and territorial planning: an application for siting an urban waste landfill

In this paper we deal with an urban and territorial planning problem by applying the Non Additive Robust Ordinal Regression (NAROR). NAROR is a recent extension of the Robust Ordinal Regression family of Multiple Criteria Decision Aiding methods to the Choquet integral preference model which permits to represent interaction between considered criteria through the use of a set of non-additive weights called capacity or fuzzy measure. The use of NAROR permits the Decision Maker (DM) to give preference information in terms of preferences between pairs of alternatives with which she is familiar, and relative importance and interaction of considered criteria. The basic idea of NAROR is to consider the whole set of capacities that are compatible with the preference information given by the DM. In fact, the recommendation supplied by NAROR is expressed in terms of necessary preferences, in case an alternative is preferred to another for all compatible capacities, and of possible preferences, in case an alternative is preferred to another for at least one compatible capacity. In the considered case study, several sites for the location of a landfill are analyzed and compared through the use of the NAROR on the basis of different criteria, such as presence of population, hydrogeological risk, interferences on transport infrastructures and economic cost. This paper is the first application of NAROR to a real-world problem, even if not already with real DMs, but with a panel of experts simulating the decision process.

Multiple Criteria Hierarchy Process for the Choquet Integral

Interaction between criteria and hierarchical structure of criteria are nowadays two important issues in Multiple Criteria Decision Analysis (MCDA). Interaction between criteria is often dealt with fuzzy integrals, especially the Choquet integral. To handle the hierarchy of criteria in MCDA, a methodology called Multiple Criteria Hierarchy Process (MCHP) has been recently proposed. It permits consideration of preference relations with respect to a subset of criteria at any level of the hierarchy. In this paper, we propose to apply MCHP to the Choquet integral. In this way, using the Choquet integral and the MCHP, it is possible to compare two alternatives not only globally, but also partially, taking into account a particular subset of criteria and the possible interaction between them.

SMAA-Choquet: Stochastic Multicriteria Acceptability Analysis for the Choquet Integral

In this paper, we extend the Choquet integral decision model in the same spirit of the Stochastic Multicriteria Acceptability Analysis (SMAA) method that takes into account a probability distribution over the preference parameters of multiple criteria decision methods. In order to enrich the set of parameters (the capacities) compatible with the DM’s preference information on the importance of criteria and interaction between couples of criteria, we put together Choquet integral with SMAA. The sampling of the compatible preference parameters (the capacities) is obtained by a Hit-and-Run procedure. Finally, we evaluate a set of capacities contributing to the evaluation of the rank acceptability indices and of the central preference parameters as done in the SMAA methods.

The most representative utility function for non-additive robust ordinal regression

Non-additive robust ordinal regression (NAROR) considers Choquet integral or one of its generalizations to represent preferences of a Decision Maker (DM). More precisely, NAROR takes into account all the fuzzy measures which are compatible with the preference information given by the DM and builds two preference relations: possible preference relation, when there is at least one compatible fuzzy measure for which an alternative is preferred to the other, and necessary preference relation, when an alternative is preferred to the other for all compatible fuzzy measures. Although it is interesting to take into consideration all the compatible fuzzy measures, in some decision problems we need to give a value to every alternative and it results necessary to obtain the most representative fuzzy measures among all the compatible ones. The aim of the paper is to propose an algorithm to the DM for selecting the most representative utility function expressed as Choquet integral from which a DMs representation of preferences is obtained.

Implementations of PACMAN

Passive and Active Compensability Multicriteria ANalysis (PACMAN) is a multiple criteria methodology based on a decision maker oriented notion of compensation, called compensability. An important feature of PACMAN is a possible asymmetry of the connected decision procedure, since compensability is determined for each ordered pair of criteria, distinguishing the compensating criterion from the compensated one. Here we give a notion of implementation of PACMAN, which allows a concrete modelization of a multiple criteria decision problem. We study regular implementations of PACMAN and their monotonicity properties. We also examine several regular implementations, which satisfy some additional properties. Particular emphasis is given to a regular implementation of PACMAN that produces the lexicographic ordering.

A linear implementation of PACMAN

PACMAN (Passive and Active Compensability Multicriteria ANalysis) is a multiple criteria methodology based on a decision maker oriented notion of compensation, called compensability. A basic step of PACMAN is the construction of compensatory functions, which model intercriteria relations for each pair of criteria on the basis of compensability. In this paper we examine a simplified version of PACMAN, which uses the so-called linear compensatory functions and consistently reduces the overall complexity of its implementation in practical cases. We use Mathematica® to develop a computer-aided graphical interface that eases the interaction among the actors of the decision process at each stage of PACMAN. We also propose the possibility to perform a sensitivity analysis in this simplified version of PACMAN as a nonlinear optimization problem.

Robust sustainable development assessment with composite indices aggregating interacting dimensions: The hierarchical-SMAA-Choquet integral approach

The evaluation of sustainable development - and, in particular, rural development - through composite indices requires taking into account a plurality of indicators, which are related to economic, social and environmental aspects. The points of view evaluated by these indices are naturally interacting: thus, a bonus has to be recognized to units performing well on synergic criteria, whereas a penalization has to be assigned on redundant criteria. An additional difficulty of the modelization is the elicitation of the parameters for the composite indices, since they are typically affected by some imprecision. In most approaches, all these critical points are usually neglected, which in turn yields an unpleasant degree of approximation in the computation of indices. In this paper we propose a methodology that allows one to simultaneously handle these delicate issues. Specifically, to take into account synergy and redundancy between criteria, we suitably aggregate indicators by means of the Choquet integral. Further, to obtain recommendations that take into account the space of fluctuation related to imprecision in nonadditive weights (capacity of the Choquet integral), we adopt the Robust Ordinal Regression (ROR) and the Stochastic Multicriteria Acceptability Analysis (SMAA). Finally, to study sustainability not only at a comprehensive level (taking into account all criteria) but also at a local level (separately taking into account economic, social and environmental aspects), we apply the Multiple Criteria Hierarchy Process (MCHP). We illustrate the advantages of our approach in a concrete example, in which we measure the rural sustainability of 51 municipalities in the province of Catania, the largest city of the East Coast of Sicily (Italy).