Decision Support
Using multiple reference levels in Multi-Criteria Decision aid: The Generalized-Additive Independence model and the Choquet integral approaches

https://doi.org/10.1016/j.ejor.2017.11.052Get rights and content

Highlights

  • A motivation of decision strategies depending on several reference levels.

  • Representation of the reference levels-dependent Multi-Criteria Decision problems.

  • A specialization of the Generalized Additive Independence model to the presence of reference levels.

  • Description of a drawback of the Choquet integral called contamination effect.

  • A simple elicitation procedure for k-ary capacities.

Abstract

In many Multi-Criteria Decision problems, one can construct with the decision maker several reference levels on the attributes such that some decision strategies are conditional on the comparison with these reference levels. The classical models (such as the Choquet integral) cannot represent these preferences. We are then interested in two models. The first one is the Choquet with respect to a p-ary capacity combined with utility functions, where the p-ary capacity is obtained from the reference levels. The second one is a specialization of the Generalized-Additive Independence (GAI) model, which is discretized to fit with the presence of reference levels. These two models share common properties (monotonicity, continuity, properly weighted,), but differ on the interpolation means (Lovász extension for the Choquet integral, and multi-linear extension for the GAI model). A drawback of the use of the Choquet integral with respect to a p-ary capacity is that it cannot satisfy decision strategies in each domain bounded by two successive reference levels that are completely independent of one another. We show that this is not the case with the GAI model.

Introduction

Many models have been defined and studied in Multi-Criteria Decision Analysis (MCDA) with various levels of preference representation power and of elicitation complexity. While many of them are sufficient when the Decision Maker (DM) has only standard preferences, there are however situations in which the DM wishes to express much richer and specific decision strategies. We consider in this paper the case where there is an inherent complexity in the DM preferences coming from the presence of reference levels in the decision problem.

Reference elements on the attributes play an important role in MCDA. In the MACBETH approach, two reference elements are defined on each attribute, and they serve to give a semantics to the concept of weight of criteria1 (Bana e Costa, De Corte, & Vansnick, 2012). In MR-Sort, an alternative is assigned to the upper class if the set of criteria above a given threshold forms a winning coalition (Bouyssou, Marchant, 2007a, Bouyssou, Marchant, 2007b). A similar idea operates in reference point methods (Rolland, 2013). In psychology, reference levels such as the neutral level – demarcating between positive and negative stimuli (Kahneman, Slovic, Tversky, 2001, Slovic, Finucane, Peters, MacGregor, 2002) – and the satisficing level that is considered as good and completely satisfactory if the DM could obtain it, even if more attractive elements could exist (Simon, 1956), are well-developed. The existence of a neutral level characterizes bipolar scales.

Decision strategies are often affected by reference elements. Let us cite a few examples. Conditional Relative Importance depicts a decision strategy that is dependent on the value of a bipolar attribute being above or below the neutral level: the relative importance between two criteria is conditional on a third (bipolar) criterion being good (above the neutral level) or bad (below the neutral level). This strategy arises in many situations – see Labreuche and Grabisch (2007) for an example. The following example also illustrates decision strategies that are dependent on the values of some attributes.

Example 1

The chief technical officer (CTO) of a company has to decide on which products to develop and then sell to customers. Five attributes are considered to evaluate potential products:

  • Expected Revenue (ER), which is a number in M Euro;

  • Risk Level (RL), taking values in {low, medium, high};

  • Reputation of the Company (RC) on the market targeted by the product, taking qualitative values between 1 (very low) to 5 (very high);

  • Position with respect to Competitors (PC), taking values among DT(disruptive technology), SOTA(at the state of the art), BSOTA(below state of the art);

  • Risk Mitigation (RM), taking values among T(collaborate with another partner to develop the technology – called “Team” strategy, to support the development of the technology), II(internal investment), N(do nothing).

Usually, the risk and revenue attributes can be split into two domains – low/high risk and low/high revenue, respectively. The decision strategies depend on which domain the potential product belongs to. The ideal situation is clearly for low risk and high revenue products, whereas the worst situation is for high risk and low revenue products. Most products are represented by the other two situations. Low risk and low revenue products can correspond to niche products. Here the reputation (RC) is a very important criterion. The product does not necessarily need to be sophisticated, because of the “niche” situation. The last two criteria PC and RM are thus less important. Lastly, high risk and high revenue products can secure the future of the company. But the competition is high on these markets, so that criteria PC and RM are now very important.

The previous example implicitly exhibits reference levels on attributes ER and RL, and the decision strategies depend on these levels.

The question that naturally arises is how to represent such decision strategies with MCDA models. As we will see more precisely, it is not possible to represent these preferences with standard MCDA models. The additive utility model is not suitable as the relative importance between some criteria depends on the values on some other criteria, and more generally, criteria interact each other. Reference-based models cannot help to compare alternatives in the same domain (e.g. high risk and high revenue products in Example 1) (Rolland, 2013). Despite its ability to model interacting criteria, the Choquet integral fails to represent these preferences as it cannot model different decision strategies depending on whether attributes are judged good or bad (Labreuche & Grabisch, 2007). Actually the Choquet integral is based on only two levels and it cannot model various changes of preferences depending on the ranges of the attributes: The decision strategy at intermediate values is simply an interpolation of those for extreme values.

We show that two models can represent decision strategies that are conditional on the comparison with the reference levels. The first one is the Choquet integral w.r.t. a p-ary capacity (Grabisch, Labreuche, 2005, Grabisch, Labreuche, 2008), combined with utility functions. Here the p-ary capacity represents the overall value at the different reference levels. The second one is a specialization of the Generalized Additive Independence (GAI) model that is discretized to fit with the presence of reference levels.

The concept of p-ary capacity and the associated Choquet integral have been defined as a mathematical tool that naturally extends capacities. However, their complexity makes it very hard to use them in real applications. Up to our knowledge, the only attempt in this direction was for the representation of restaurant customer satisfaction (Rolland, Ah-Pine, & Mayag, 2015), although limited to bi-capacities. The aim of this paper is to show the interest of the GAI model and the Choquet integral w.r.t. p-ary capacities in applications, and especially in those exhibiting reference levels.

The main contributions of this paper can be summarized as follows:

  • We present a motivating example taken from a real application in the engineering domain, to illustrate situations where decision strategies are dependent on several reference levels (Section 3). In engineering, the different levels are successive levels of requirements from the simplest to satisfy to the most difficult one. There are different decision strategies associated to these levels. For instance, the smallest level is generally considered as a hard constraint, whereas the largest level is seen as a bonus (the customer is expected to have a crush in such a situation). This example is running throughout the document, and we show that it cannot be represented by the classical MCDA models.

  • Which model can represent the motivating example? (Section 4). We provide two models able to represent the reference levels-dependent multi-criteria decision problems: the Choquet integral w.r.t. p-ary capacities and the GAI models.

    Regarding the GAI model, we provide a specialization of this model to the presence of reference levels, which is new. The GAI model is then seen as a generalization of the UTA approach, where continuous attributes are discretized. This is important to do so in practice in order to reduce the computational complexity (Grabisch & Labreuche, 2017).

  • How to choose between the two aggregation models? (Section 5). These two models share common properties (monotonicity, continuity, properly weighted,), but differ on the interpolation method (Lovász extension for the Choquet integral, and multi-linear extension for the GAI model). A drawback of the use of the Choquet integral w.r.t. a p-ary capacity is that the possible decision strategies satisfied in each domain bounded by two successive reference levels are not completely independent of one another. We call this phenomenon contamination. This was already noted with the conditional relative importance strategy (Labreuche & Grabisch, 2007). We show that this is not the case with the GAI model.

    We also present the pros and cons of choosing each of these two models.

  • How to elicit the two models? (Section 6). The question of the elicitation of MCDA models is central in applications. This problem might be seen as intractable for the GAI model or the Choquet integral w.r.t. a p-ary capacity due to the large number of parameters. We propose a simple elicitation procedure that focuses on alternatives with balanced profiles (i.e., alternatives in-between two identical levels on all attributes) and fill the model on the other cases by using a cautious principle.

Section snippets

General model

We are given a set of n attributes indexed by N={1,,n}. Each attribute i ∈ N is represented by a set Xi. The alternatives are characterized by a value on each attribute, and are thus represented by an element in X=X1××Xn. We assume that we are given a preference relation ≿ over X. It is supposed to be represented by an overall utility function U:XIR,i.e., such that xy iff U(x) ≥ U(y).

For x, y ∈ X and AN, we denote by XA the set ∏i ∈ AXi, by xA the restriction of x on attributes A, and by (x

A motivating example

This section introduces and discusses a motivating example in which decision strategies are conditional on where the alternatives are located relatively to some reference levels.

Two aggregation models based on several reference levels

This section describes two models able to distinguish decision strategies depending on the position with respect to reference elements. The first one is an extension of the Choquet integral to cope with multiple levels (see Section 4.1). The second one is a specialization of the GAI model that allows, by construction, separate decision strategies in each domain (see Section 4.2). We will show that these models are able to capture the preferences expressed in the motivating example.

The set of

How to choose between the two aggregation models?

We have shown in Section 4 that the GAI model UΠ and the Choquet integral U w.r.t. p-ary capacities are in fact very similar models, despite the apparent difference between their mathematical formulation. Both are based on a p-ary capacity and only differ from the interpolation used. As the Lovász extension requires appropriate normalization of the attributes, which is not the case of the multi-linear extension, the Choquet integral model requires utility functions. This explains the

How to elicit the two models?

This section addresses the elicitation of the previous models, which is very important in practice. We have seen in Section 5.1 that the parameters of a GAI model can be put in the form of a p-ary capacity. Hence in order to address the elicitation of the two models that are considered in this paper, it is sufficient to consider the elicitation of a p-ary capacity.

Conclusion

This paper considers MCDA situations in which the decision strategies are conditional on the position of the values of the alternative on the attributes comparatively to several reference levels. Two examples of such situations have been given. Standard MCDA approaches cannot handle these cases. We have seen that the Choquet w.r.t. a p-ary capacity and a specialization of the GAI model, which is discretized to fit with the presence of reference levels, can fulfil these examples. The latter

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