Using evolutionary computation to infer the decision maker’s preference model in presence of imperfect knowledge: A case study in portfolio optimization

Highlights

• We present a novel method to infer the parameters of decision makers’ systems of preferences using evolutionary algorithms.

• The method’s main contribution is coping with imperfect knowledge on the decision maker’s preferences and criterion scores.

• An application in an extensive case study on optimization of stock portfolios is provided.

• The results support the convenience of the proposed method in situations of uncertainty over all the benchmarks used.

Abstract

It is usually very difficult to elicit the parameter values of models representing decision makers’ preferences. Consequently, some imprecision, ill-determination and arbitrariness are unavoidable. Moreover, such elicitation cannot be performed by traditional optimization techniques in a reasonable time. Therefore, we present here a novel elicitation method guided by a genetic algorithm whose main contribution is coping with imperfect knowledge. The latter is done by using interval numbers representing all the possible values that the parameters can attain. The assessment of the method showed its high ability to reproduce the decision maker’s preferences. Finally, as the method proposed in this paper is the complement of the authors’ previous work regarding the optimization of stock portfolios, we provide a case study in such a field. We use differential evolution to obtain the most satisfactory portfolio. The results reported here show that the best portfolio returns are obtained when the elicitation method is exploited, and we conclude that the new overall approach might be an interesting alternative to the already-existing methods.

Introduction

Multi-criteria decision aiding (MCDA) provides a wide range of appropriate methods for choosing, ranking and sorting (ordinal classification) problematics. However, the aid provided by MCDA is not effective unless the aggregation model appropriately represents the decision maker’s (DM) preferences. Generally, the MCDA models use many parameters to represent the DM’s preferences. The values of these parameters can be obtained either with direct or indirect elicitation methods. In the former, the decision maker, often aided by a decision analyst, has to directly assign the parameter values to the preference model. Whereas in indirect elicitation methods, the parameter values are deduced from a battery of easy-to-make decisions made by the DM.

Eliciting the parameter values is very important in developing a multi-criteria decision aiding approach. Some authors (for example [1]) consider the direct elicitation method as less adequate relative to indirect elicitation. Some limitations of the former are the following: i) the preference model’s parameters are meaningless as long as the multi-criteria aggregation procedure in which they are used has not been specified; ii) holistic decisions made by the DM using his/her own judgment procedure when comparing pairs of actions and/or assigning actions to categories/classes are more appropriate; iii) the DM may not be accessible (e.g., the manager of an international company) or may be an ill-defined entity (e.g., a heterogeneous group); iv) the DM usually has difficulties to explicitly specify numerical parameters and the time and cognitive effort required to do so may be inhibitory.

The indirect elicitation methods constitute the well-known preference disaggregation analysis (PDA) paradigm. PDA methods analyze decisions made by the DM in order to identify the aggregation model that underlies the outcome of the known decisions. The indirect elicitation methods infer the decision model’s parameters from holistic decisions provided by the DM and use regression-like methods to produce a decision model as consistent as possible with the set of reference (training) decisions. The PDA paradigm is of growing interest because it requires less cognitive effort from the DM. The main reason is that DMs frequently prefer making decision judgments than explaining them.

Indirect elicitation approaches have been used for decades to build functional or utility decision models (e.g. Refs. [[2], [3], [4]]). In MCDA, Jacquet-Lagreze and Siskos [5] pioneered the UTA method. Regarding the outranking approach, indirect elicitation methods are even more significant, because the DM must establish parameter values that are very unfamiliar to her/him (e.g., veto thresholds). In this frame, some important references are the works of Mousseau and Słowiński [6], Doumpos et al. [7], and Fernandez et al. [8]. Indirect elicitation approaches have been satisfactorily used by many authors in the context of financial decision making (e.g. Refs. [[9], [10], [11], [12]]) and particularly in the context of portfolio selection (e.g. Refs. [13,14]). All these proposals identify punctual values for the model’s parameters, which are supposed to be appropriate to explain or suggest new decisions.

Despite the wide use of indirect elicitation methods, they cannot avoid certain imperfect information in setting the model’s parameters; the concept of what is the appropriate value of a decision model parameter is poorly-defined due to several reasons: a) the DM’s decision policy may not match with the model’s assumptions and its mathematical structure; b) the DM’s preferences are ill-defined (e.g., a heterogeneous group); c) the DM is a mythical or inaccessible person (e.g., public opinion); d) often many parameter settings reproduce the known decision examples; and e) imprecise (even missing) information on criterion scores. Thus, there is always imprecision, uncertainty, ill-definition or arbitrariness (imperfect knowledge, according to Roy et al. [15]) to be handled by the PDA when it infers the values of the parameters.

Recently, Fernandez et al. [16] presented an extension of the outranking approach that is able to deal with imperfect knowledge on the parameters of the model and on criterion scores. Although the DM likely feels more comfortable making a direct elicitation of model parameter values as interval numbers, this approach does not avoid the convenience of indirect setting [16]. It would be more convenient if, instead of punctual values, the indirect elicitation method offers the flexibility to consider the parameters as ranges of numbers, where imperfect knowledge is contained within intervals. Such a method would combine the advantages of the indirect elicitation with the flexibility of the interval outranking approach.

The interval outranking approach was recently applied to solve a many-objective stock portfolio optimization problem in Ref. [17]. Such paper proposed an interesting approach to select the best stock portfolio considering imperfect knowledge (in the sense of [15]) that characterizes the DM’s implicit model of preferences, performing a pressure toward the DM’s most preferred portfolios; it represents the DM’s conservatism to risk, and the portfolios’ expected return and risk. However, a direct elicitation of the interval-outranking model’s parameters representing the DM’s preferences was performed there. Therefore, an interesting research question is if the application of evolutionary computation to the indirect elicitation of these parameters implies that i) the interval-outranking finds more preferred solutions, and/or ii) the portfolios found by the overall approach generate greater returns. Our main objective is thus addressing this question. We do it by proposing and assessing a novel method that indirectly elicits the interval-outranking model’s parameters using a set of judgments made by the DM.

The rest of the paper is structured as follows. In Section 2 we briefly describe some previous related work. In Section 3, we present our proposal to get an approximation to the DM’s model of preferences when the parameters are described as numerical ranges. In Section 4 we describe some experiments to validate the proposal whose results are shown in Section 5. In Section 6, a case study is presented where elicited preference parameter values are used to select the most preferred portfolios. Finally, we conclude this paper in Section 7.