Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2157937, 10 pages

https://doi.org/10.1155/2018/2157937

## Robustness Analysis of an Outranking Model Parameters’ Elicitation Method in the Presence of Noisy Examples

Correspondence should be addressed to Nelson Rangel-Valdez

Received 20 June 2017; Revised 19 November 2017; Accepted 10 December 2017; Published 3 January 2018

Academic Editor: Danielle Morais

Copyright © 2018 Nelson Rangel-Valdez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

One of the main concerns in Multicriteria Decision Aid (MCDA) is robustness analysis. Some of the most important approaches to model decision maker preferences are based on fuzzy outranking models whose parameters (e.g., weights and veto thresholds) must be elicited. The so-called preference-disaggregation analysis (PDA) has been successfully carried out by means of metaheuristics, but this kind of works lacks a robustness analysis. Based on the above, the present research studies the robustness of a PDA metaheuristic method to estimate model parameters of an outranking-based relational system of preferences. The method is considered robust if the solutions obtained in the presence of noise can maintain the same performance in predicting preference judgments in a new reference set. The research shows experimental evidence that the PDA method keeps the same performance in situations with up to 10% of noise level, making it robust.

#### 1. Introduction

In Multicriteria Decision Aid (MCDA), one of its main concerns is the robustness of methods developed in this field. The term robust refers to the capacity for withstanding “vague approximations” and/or “zones of ignorance” to prevent the degradation of the properties that must be maintained [1]. Having this idea in mind, it is important to depict how robust a new method in MCDA is.

A wide variety of problems in decision aiding often involve multiple objectives to be minimized or maximized simultaneously. Because of the conflicting nature of the criteria, it is not possible to obtain a single optimum, and consequently, the ideal solution to a multiobjective optimization problem (MOP) cannot be reached. Therefore, the analysts resort to approaches that can handle multiple criteria and at the same time can shrink the number of solutions they provide to those concerning specific interests of a decision maker.

Several approaches that solve MOPs are based on Multiobjective Evolutionary Algorithms (MOEAs) and assume a model of the decision makers’ (DM) preferences. This work focuses on the preference model proposed by Fernandez et al. in 2011. The model uses fuzzy outranking relations to incorporate preferences in MOEAs, such as the strict, weak, and -preference. Also, it allows the mapping of a many-objective problem into a surrogate problem with only three objectives. The method has been applied in a wide variety of problems, including the portfolio problems with many objectives and project partial support [2].

In order to apply the preference model of Fernandez et al. [3], the outranking model’s parameters must be elicited, for example, weights and thresholds required by the index of credibility of the outranking, a cutting level, and some additional symmetric and asymmetric parameters.

Information about the model’s parameters can be obtained either directly or indirectly. On the one hand, the direct eliciting method has been criticized by Marchant [4] and Pirlot [5] arguing that the only valid preference input information is that arising from the DM’s preference judgments about actions or pairs of actions. As stated by Covantes et al. [6] and Doumpos et al. [7], these criticisms are even more significant in the frame of outranking methods, since the DM must set parameters that are very unfamiliar to her/him (e.g., veto thresholds). On the other hand, indirect elicitation methods use regression-inspired techniques for inferring the model’s parameters from a set of decision examples [6, 7].

In the frame of outranking methods, preference-disaggregation analysis (PDA) approaches were pioneered by Mousseau and Slowinski [8]. They proposed to infer the ELECTRE TRI model’s parameters (except veto thresholds) from a set of assignment examples by using nonlinear programming. Mousseau et al. [9] proposed a method to infer the weights from assignment examples through linear programming. Ngo The and Mousseau [10] used assignment examples to elicit the boundary profiles in ELECTRE TRI. Methods dealing with indirect elicitation of weights under inconsistent sets of assignment examples have been addressed by Mousseau et al. [11, 12]. Dias et al. [13] integrate interactively the elicitation phase with a robustness analysis.

Most of the related papers elude the inference of veto thresholds because eliciting all the parameters simultaneously requires solving a very complex nonlinear programming problem. Two papers proposed the use of evolutionary algorithms (EAs) to infer the entire set of ELECTRE model’s parameters from a set of assignment examples (cf. [7, 14]). EAs are powerful tools for the treatment of nonlinearity and global optimization in polynomial time [15], as a more recent example, Álvarez et al. [16] used it to infer parameters that aid in the decision process at the collective level.

To the best of our knowledge, Fernandez et al. [17] was the first paper in which the reference information did not come from assignment examples, but from preference statements as* “** is at least as good as **”* and* “** is not at least as good as y.”* This paper infers the entire set of the ELECTRE III model’s parameters and the generalized outranking model with reinforced preferences proposed by Roy and Słowiński [18].

Cruz-Reyes et al. [19] proposed a PDA method to infer the entire parameter set of the relational system of preferences from Fernandez et al. [3]. This approach allows the introduction of the DM’s preferential judgments through pairwise comparisons of different actions. However, this work lacks a robustness analysis which would allow measuring its capacity for withstanding vague approximations and/or zones of ignorance derived from its formal representation.

Based on the above, this research proposes a method for robustness analysis of the solutions offered by PDA methods based on metaheuristics. The study case is the Genetic Algorithm from the work of Cruz-Reyes et al. [19], which is used as a PDA method for the relational system of preferences proposed by Fernandez et al. [3]. The method is considered robust if it maintains the same performance with or without noise in the reference set; otherwise, it can be concluded that the method provides sensitive solutions. As a result, the experimental design proves the method robustness by identifying that it estimates parameter value sets with a statistically nonsignificant difference when the noise levels are equal to or smaller than 10%.

Hence, the main contributions of this work are the proposed method for robustness analysis, and the noise model developed to introduce noise in a reference set. It is important to emphasize that both the method and model are the first in considering the full set of parameters in Fernandez et al. [3]. Also, an important part of the contributions of this work is the identification of the zone in the parameter space and the level of noise where the response sets are most compatible with the DM’s preferences. It needs to be noted that the noise concept is related to the inconsistencies, or errors, between the preference model and the DM.

Aside from this introduction, the paper is organized as follows. Section 2 presents the optimization approach to estimate parameter values which are subject to the robustness analysis, the associated surrogate model, and the elements required for its definition. Section 3 shows the method followed in this work to perform the robustness analysis. Sections 4 and 5 present the experimental design conducted to evaluate the robustness of the optimization approach and the results obtained from it. Finally, Section 6 brings some concluding remarks derived from the research.

#### 2. Optimization Approach for Inferring the Model’s Parameter Values

This section is organized as follows. Firstly, it gives the definition of the optimization problem used as inference approach for the estimation of parameter values. This is followed by the optimization approach used to solve the studied problem and the description of the metaheuristic used. Finally, it presents the method that served as a basis to perform the analysis of robustness of the optimization approach for inference of the parameter values.

##### 2.1. The Inference Approach

The best compromise is a solution of a problem associated with the DM’s preferences. As was stated by Branke et al. [21], there has been an increasing interest in incorporating the DM’s preference information in the search process. This situation is due to its influence on the reduction of the cognitive effort to identify a solution that best matches those preferences and to reinforce the selective pressure toward the Pareto frontier, for example, Cruz et al. [22].

A survey of strategies to incorporate preferences into multiobjective approaches can be found in [23, 24]. Particularly, this research deals with preference models based on outranking relations, such as the one developed by the works of Roy [20] and Fernandez et al. [3]. In these works, the preference model approaches situations concerning the behavior of real DMs using a relational system of preferences. The six binary relations that lie on that system are indifference, strict preference, weak preference, incomparability, -preference, and nonpreference. These relations are associated with the predicate “the DM considers that option is at least as good as ” through a degree of truth in .

Table 1 shows each outranking relation and its notation in columns one and two, and the necessary conditions that need to be satisfied for each relation in column three. The parameters* η*used in the computation of the function (cf. [19]), in combination with the credibility (), symmetry (), and asymmetry () thresholds, determine the preference relations.