# Modeling Comprehensive Preferences: Three Operational Approaches for Progressing beyond the Description Problematic

• Bernard Roy
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 12)

## Summary

Comprehensive preferences consider all consequences relevant to the decision aiding study. The simplest comprehensive preference model consists of an SPR that includes only dominance and incomparability. Problematics other than P.δ require more than this very disaggregate model, however. In Section 11.1.1 we formulate the performance aggregation problem. The entire chapter is an attempt to put some structure on the numerous efforts of theoreticians and practitioners to address this problem.

Any attempt to aggregate performance levels requires the analyst to take both formal and informational positions. In his formal position, he will have to consider things such as the types of preference relations compatible with the model, the aggregation logic to be used, and the functional representations of the different criteria. In his informational position, he will have to consider the nature of the intercriterion information required, how this information will be obtained, and procedures to indicate the validity of the information obtained. In Section 11.1.2, we define the operational approach as the set of these two types of positions. For the most part, the operational approaches that we present arise directly from one of the three categories defined in Sections 11.2, 11.3, and 11.4. Others appear as ad hoc combinations of two of these categories.

Section 11.2 deals with the approach that uses a single criterion to synthesize the preference information without allowing incomparability. This first operational approach (OA1) is based on using an SPR of the form (I, P) with a complete preorder structure or possibly an SPR of the form (I, P, Q) with a pseudo-order structure. This solution to the aggregation problem allows the functional representation g(a) = V[g1(a)…, gn(a)]. We illustrate the representation V, which we call the aggregation function, through the continuation of Example 3. In Section 11.2.2, we discuss the principal types of aggregation functions — weighted sum, additive, multiplicative, lexicographic — and note that V can be defined without an explicit analytical form. The two fundamental positions that characterize OAl are: i) a position that does not allow incomparability; ii) a position that explicitly states a rule (the aggregation function) addressing the aggregation problem in a synthesizing, exhaustive, and definitive fashion.

Section 11.3 deals with an outranking approach to synthesize preference information. This second operational approach (OA2) is based on making explicit the conditions that characterize soundly established outrankings. This approach leads to an SPR of the form (S, R), with AF being contained in S. Using this SPR to answer the questions posed by the decision maker is not as straightforward as it is with OA1. It is usually necessary to adapt some procedure to the problematic at hand. Instead of an aggregation rule V, this approach leads to a set of tests T presented in (r 11.3.1) which use the conditions that must be verified for the outranking. In ELECTRE methods T uses the concepts of concordance and discordance. We illustrate the ELECTRE I method in the continuation of Example 1.

Approach OA2 is generally associated with a constructive approach and requires a robustness study of the conclusions in light of the arbitrary nature of the intercriterion information. The two fundamental positions that characterize this approach are: i) a position that accepts incomparability; ii) a position that explicitly states a rule or outranking test addressing the aggregation problem in a synthesizing, exhaustive, and definitive fashion.

Section 11.4 deals with the third operational approach (OA3) to the performance aggregation problem. Unlike the other two approaches, OA3 does not make explicit any rules to address the problem in a synthesizing, exhaustive, and definitive fashion. Rather, it is based on an interactive protocol that regulates how the different series of dialogue and processing stages are linked together to develop a solution from local judgments. The manner in which these judgments are put together to lead toward a solution is primarily based on trial and error and is similar to what would come naturally in most everyday decisions (e.g., the family car example). Still, to provide a true decision aid in more complex situations, the analyst will need some protocol that can efficiently organize the successive interactions.

In Section 11.4.2, we describe the interactive protocol phases: explanation, questioning, and processing phases. We discuss the stopping conditions of the OA3 procedures in Section 11.4.3. In a constructive approach, the procedure stops when the questioner or the questionee considers the goal to be achieved or when one of these two parties decides to stop the process. In a descriptive approach, the procedure must converge before stopping. The two fundamental positions that characterize this third approach are: i) a position that gives primary importance to local judgments dealing with a very small number of actions without considering any explicit rule attempting to aggregate, even partially or temporarily, the performance levels; ii) a position that explicitly states a protocol organizing the interaction between the questionee (the decision maker or some actor in the decision process) and the questioner (the analyst or a computer) so as to allow the recommendation to emerge for the problematic considered.

## Keywords

Operational Approach Aggregation Function Aggregation Problem Descriptive Approach Interaction Protocol
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.