Decision SupportRobust ordinal regression for value functions handling interacting criteria
Introduction
Ranking, choice or sorting decision with respect to a finite set of alternatives evaluated on a finite set of criteria is a problem of uttermost importance in many real-world areas of decision-making (Ehrgott et al., 2010, Figueira et al., 2005). Among many approaches that have been designed to support the Multiple Criteria Decision Analysis (MCDA), three of them seem to prevail. The first one exploits the idea of assigning a score to each alternative, as it is the case of MAUT - Multi-Attribute Utility Theory (Keeney & Raiffa, 1976). The second relies on the principle of pairwise comparison of alternatives, as it is the case of outranking methods (Roy, 1996). The third one induces logical “if…, then…” decision rules from decision examples, as it is the case of DRSA – Dominance-based Rough Set Approach (Greco et al., 2001, Słowiński et al., 2009). The value function, the outranking relation and the set of decision rules are three preference models underlying these three main approaches. It is known that in order to build such models, the Decision Maker (DM) has to provide some preference information.
The preference information may be either direct or indirect, depending whether it specifies directly values of some parameters used in the preference model (e.g., trade-off weights, aspiration levels, discrimination thresholds, etc.) or whether it specifies some examples of holistic judgments from which compatible values of the preference model parameters are induced. Eliciting direct preference information from the DM can be counterproductive in real-world decision-making because of a high cognitive effort required. Consequently, asking directly the DM to provide values for the parameters seems to make the DM uncomfortable. Eliciting indirect preference is less demanding of the cognitive effort. Indirect preference information is mainly used in the ordinal regression paradigm. According to this paradigm, a holistic preference information on a subset of some reference or training alternatives is known first and then a preference model compatible with the information is built and applied to the whole set of alternatives in order to arrive at a ranking, choice, or sorting recommendation.
Usually, from among many sets of parameters of a preference model representing the preference information given by the DM, only one specific set is selected and used to work out a recommendation. For example, while there exist many value functions representing the holistic preference information given by the DM, only one value function is typically used to recommend the best ranking, choice, or sorting of alternatives. Since the selection of one from among many sets of parameters compatible with the preference information given by the DM is rather arbitrary, robust ordinal regression proposes taking into account all the sets of parameters compatible with the preference information, in order to give a recommendation in terms of necessary and possible consequences of applying all the compatible instances of the preference model on the considered set of alternatives.
The recently proposed MCDA methods implementing robust ordinal regression on the three above-mentioned preference models have been described in Greco, Słowiński, Figueira, and Mousseau (2010). The first method in the series, called UTAGMS (Greco, Mousseau, & Słowiński, 2008), generalizes the UTA method (Jacquet-Lagrèze & Siskos, 1982) which applies ordinal regression to assess a set of additive value functions compatible with preference information provided by the DM. UTA aims at giving a complete ranking using one compatible value function, which is the one minimizing the sum of deviation errors or minimizing the number of ranking errors in the sense of Kendall or Spearman distance. In Jacquet-Lagrèze and Siskos (1982), the authors of UTA also recommend post-optimality analysis consisting in exploration of the vertices of the polyhedron of compatible value functions, in particular, the vertices for which one or more criteria get a maximum or minimum weight. UTAGMS is considering instead the whole set of compatible additive value functions to compute necessary and possible preference relations.
Even if the additive model is among the most popular ones, some critics have been addressed to this model because it has to obey an often unrealistic hypothesis about preferential independence among criteria. In consequence, it is not able to represent interactions among criteria. For example, consider evaluation of cars using such criteria as maximum speed, acceleration and price. In this case, there may exist a negative interaction (negative synergy) between maximum speed and acceleration because a car with a high maximum speed also has a good acceleration, so, even if each of these two criteria is very important for a DM who likes sport cars, their joint impact on reinforcement of preference of a more speedy and better accelerating car over a less speedy and worse accelerating car will be smaller than a simple addition of the two impacts corresponding to each of the two criteria considered separately in validation of this preference relation. In the same decision problem, there may exist a positive interaction (positive synergy) between maximum speed and price because a car with a high maximum speed usually also has a high price, and thus a car with a high maximum speed and relatively low price is very much appreciated. Thus, the comprehensive impact of these two criteria on the strength of preference of a more speedy and cheaper car over a less speedy and more expensive car is greater than the impact of the two criteria considered separately in validation of this preference relation.
To handle the interactions among criteria, one can consider non-additive integrals, such as Choquet integral and Sugeno integral (for a comprehensive survey on the use of non-additive integrals in MCDA see Grabisch, 1996). The non-additive integrals suffer, however, from some limitations within MCDA (Roy, 2009); in particular, they need that the evaluations on all criteria are expressed on the same scale. This means that in order to apply a non-additive integral it is necessary, for example, to estimate if the maximum speed of 200 km/h is as valuable as the price of 35,000€.
In this paper, we propose a new aggregation model which modifies the usual additive value function model so as to handle interactions among criteria without the necessity of expressing all the evaluations on the same scale. The paper is organized as follows. In the next section, we introduce main concepts and notation. In Section 3, first we show by an example what means violation of the preferential independence hypothesis for an additive value function, and then we continue the same example to show that the well known Choquet integral is not able to represent properly the observed interaction between criteria. In Section 4, we recall basic concepts and properties of robust ordinal regression. Our main proposal, extending UTAGMS to the case of interacting criteria, is given in Section 5. It starts with a reminder of the principle of robust ordinal regression, and then continues with presentation of the UTAGMS–INT method. An illustrative example is provided in Section 6. The last section contains conclusions. All the proofs are deferred to the appendix.
Section snippets
Main concepts and notation
We are considering a multiple criteria decision problem where a finite set of alternatives () is evaluated on a family of n criteria (). To simplify notation, we will identify the family of criteria with set I of their indices. The family of criteria is supposed to satisfy consistency conditions (Roy & Bouyssou, 1993), i.e., completeness (all relevant criteria are considered), monotonicity (the better the evaluation of an alternative on considered
Violation of the preferential independence, and representation of preferences with interacting criteria
Let us explain, by an example inspired by Grabisch (1996), what means violation of the preferential independence hypothesis for an additive value function (1). Consider the case of a dean of some school that has to compare four students on the basis of their scores in Mathematics, Physics and Literature presented in Table 1. Here, .
Imagine the following reasoning of the dean: “considering students and , I prefer over because in case of good
Reminder on multiple criteria ranking with a set of additive value functions inferred by robust ordinal regression
Applying a set of additive value functions compatible with preference information provided by the DM to multiple criteria ranking problems was implemented first in UTAGMS (Greco et al., 2008) and GRIP (Figueira, Greco, & Słowiński, 2009) methods. These methods consist of three steps. They start with the preference elicitation process, lead through the statement of appropriate ordinal regression problems, i.e., definition of the set of compatible general additive value functions, and result in
Constraints defining compatible value functions with interacting criteria
The assumed preference model is a value function equal to not only the sum of marginal non-decreasing value functions , as considered in UT AGMS and in GRIP, but also adding the sum of functions , and subtracting the sum of functions . Functions are non-decreasing in both their two arguments, and such thatfor all
A didactic example
The dean of a school has to rank some students that are evaluated on Mathematics, Physics and Literature, as shown in Table 7.
The dean gives the following preference information:
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preference information with respect to overall comparisons of students:
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student is better than student ,
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student is better than student ,
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student is better than student ,
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preference information relative to intensity of preference in overall comparison of students:
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student is preferred to student more
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Conclusions
We presented a robust ordinal regression method, , able to handle positive and negative interactions between criteria. The methodology is based on an additive value function preference model which includes “bonus” and “penalty” components corresponding to positively and negatively interacting pairs of criteria, respectively.
The preference model compares favorably to the Choquet integral which is frequently used to handle interacting sets of criteria. It does not require all
Acknowledgements
The third author wishes to acknowledge financial support from the Polish National Science Centre, Grant DEC-2011/01/B/ST6/07318. The authors also wish to acknowledge Salvatore Corrente for his remarks and contribution to the computation of the illustrative example. The authors finally wish to thank the anonymous reviewers for their constructive comments.
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