Multicriteria Methodology for Decision Aiding

We analyze the concept of decision and show that it cannot be completely separated from that of decision process. We then propose that the set of critical points in the course of a decision process determines the comprehensive or final decision. This comprehensive decision results from the interactions among the stakeholders (individuals, entities, communities) and the conflicts among the preferences of the actors (stakeholders, third parties,...). Finally, we introduce an example concerning the purchase of a family car to illustrate the chaotic and often unforeseen course that the process may take.

Before describing how this book approaches the idea of decision aiding, we must examine the idea of a model. We define a model in Section 2.1 and discuss the types of models treated in this book — namely, conscious models possessing some explicit form.

We define decision aiding in Section 2.2 as a model-based activity designed to answer questions posed by some stakeholders in the decision process. The answers sought should clarify the decision and help identify behavior that will increase the compatibility of the process with the stakeholders’ objectives and value systems. We analyze the roles of three major actors — the decision maker, the analyst, and the client — and discuss the issues of neutrality and objectivity in the modeling effort.

Consider a decision process concerning investment or production, marketing or finance, distribution or procurement, machine or personnel management. The stakeholder — who may come from a firm, from a community group, from a government administration,... — will automatically be faced with a certain family of questions putting him or her in contact with a particular class of phenomena. In some cases, simple observation is all that he or she needs to get a good handle on these phenomena and work effectively with their causes and effects. In other cases, however, more formal models can be extremely useful in understanding and working with these phenomena or in developing appropriate answers and convincing others to accept them. As with all scientific approaches, decision aiding relies heavily on relatively explicit and formal models.

We introduce 12 reference examples to illustrate the notions discussed in the first two chapters. We shall continue to develop the examples throughout the remaining chapters of the book in an effort to clarify and highlight concepts as they are presented.

In Section 4.1, we introduce the basic ideas of value system, informational system, and relational network; define the term actor; and present the interrelated concepts of study phase and decision process development state (PDS).

In Section 4.2, we sketch out the major options of an approach designed to assist the analyst in recommending or simply participating in the process, while acknowledging the decision maker’s ultimate freedom. These options serve as a guide for the proposed methodology and the organization of the remainder of the book.

In Section 5.1.1, we define the term action. We distinguish between actual and dummy actions and realistic and unrealistic actions and illustrate these concepts by continuing the Industrial Development example. In Section 5.1.2, we distinguish between a comprehensive action, the execution of which excludes the execution of any other action introduced in the model, and a fragmented action, which can be combined with other actions for joint execution in the framework of the final decision. We illustrate the distinction in the continuations of the Agricultural Development and Research Project examples.

In Section 5.2, we define a potential action as an action that is temporarily assumed to be possible for the decision aid. We also present the idea of the set of potential actions on which the decision aiding effort is based during a phase of the study, which we denote by A. We present the conditions of internal and external stability and define A as stable (fixed and permanent) when these conditions are fulfilled, and A as evolving (revisable or transitory) when they are not. The concepts of Section 5.2 are illustrated through continuations of the the Commuter Rail Line, Product Composition, and Engine Assignment examples.

In Section 6.1, we define and illustrate four reference problematics — P.α, P.β, P.γ, P.δ. The objective of P.α is to aid the decision maker by the choice of a subset that is as small as possible so that a single action can eventually be chosen. This subset contains “best” actions (optima) or, perhaps, “satisfactory” actions (satisficing solutions). The result of P.α is a choice or a selection procedure.

The objective of P.β is to aid the decision maker by a sorting that leads to an assignment of each action to a category, where the categories are defined beforehand as a function of certain norms that deal with the ultimate fate of the actions that will be assigned to them. The result of P.β is a sorting or an assignment procedure.

The objective of P.γ is to aid the decision maker through a ranking that is obtained by placing all, or simply the “most attractive,” actions into equivalence classes that are completely or partially ordered according to preferences. P.γ results in a ranking or an ordering procedure.

The objective of P.δ is to aid the decision maker by developing a description of the actions and their consequences in appropriate terms. It results in a description or a cognitive procedure.

In Section 6.2 we discuss how the problematic chosen for a given phase of the analysis can correspond to one of the four reference problematics, to a sequence of two or more of the problematics, or to a mixed problematic. We illustrate the last two cases with new examples.

In the first section we present concepts that describe an actor Z’s stated preference judgments when comparing two actions of A. In Section 7.1.1.1 we use examples to introduce the basic preference situations with which Z can be faced. We then state the axiom of limited comparability (Axiom 7.1.1), which serves as a point of departure from classic decision theory.

Based on the four binary relations of indifference (I), strict preference (P), weak preference (Q), and incomparability (R), we show in Section 7.1.2.2 that all of Z’s preferences on A can be modeled by a basic system of preference relations, denoted BSPR, or, if necessary, by a consolidated system of preference relations, denoted CSPR. We discuss the two nontraditional relations Q and R in Section 7.1.2.3 and discuss transitivity of the relations I, P, Q, and R in Section 7.1.2.4.

We then investigate the situations and consolidated relations defined in Table 7.1.5 and the most noteworthy CSPR’s that they generate. The one on which classical decision theory is based consists of two consolidated relations: ~ (nonpreference) and ≻ (preference). In Section 7.1.3.2 we discuss how the axiom of limited comparability is replaced by the axiom of complete transitive comparability in this theory. After discussing the three consolidated relations J, K, and S, in Section 7.1.3.3, we introduce a final CSPR, one in which the outranking relation S plays a fundamental role. The section ends by examining relationships among the various situations and motivating certain choices of SPR models.

We use the first part of the second section to introduce graphical conventions used in the rest of the book. We also present a new example concerning a mayor’s preferences. In the two subsections that follow, we present the principal structures associated with the most interesting SPR’s and their functional representations. In Section 7.2.2 we look at those that exclude incomparability, and in Section 7.2.3 at those that acknowledge it. The following table synthesizes the principal structures studied in these two subsections in terms of the relations that constitute the SPR.

We touch upon the fundamental problem of comparing preference differences or exchanges (elements of A × A) in Section 7.2.4.1 and examine the relationships between SPR’s on A and on A × A in Section 7.2.4.2.

Readers put off by the terse nature of Section 2 should only skim its first two subsections.

Constructing any of the systems of preference relations on A requires a model of the information that affects the formation, justification, and evolution of an actor’s preferences. This information is rarely available in a well-structured, quantified, or organized form, and what the analyst can use is often subject to imprecision, uncertainty, and inaccurate determination. In this chapter, we propose a methodology for approaching this phase of the modeling effort.

In Section 8.1.1, we define the term “consequence of an action” (Def. 8.1.1) to denote the various elements (effects, attributes, aspects,…) that can interact with the objectives or value system of an actor and affect how she builds, justifies, or transforms her preferences. Our methodology is designed to analyze and distinguish the various consequences according to their quantitative and qualitative influences on the comparison of actions. Before the modeling effort begins, these consequences are ill-defined and possess fuzzy boundaries. They stem from complex and highly interwoven entities. At this stage, we refer to the consequence cloud.

In Section 8.1.2, we show the breadth and general nature of the approach used to isolate and define what we call elementary consequences (Def. 8.1.2) and offer concrete examples and practical illustr ations. An elementary consequence usually points out the existence of an underlying dimension that reflects a preference shared among the different actors. This leads to the two basic concepts of Section 8.1.3: a preference scale (Def. 8.1.3) and a preference dimension (Def. 8.1.4). We present various examples and illustrations of these definitions in Section 8.1.3.2.

For a dimension to be operational, one must be able to map the impacts of a potential action on this dimension into a state or group of states with the help of some procedure. The procedure could be an empirical rule, a mathematical formula, a survey technique, or an experiment. This idea is the subject of Section 8.1.4 and leads to the concept of a state indicator (Def. 8.1.5) and the distinction between point and nonpoint state indicators. We end Section 8.1.4 with a discussion of the set of dimensions, which we call the consequence spectrum (Def. 8.1.6), that is used to describe the consequence cloud. Section 8.1.5 illustrates this first aspect of the methodology concerned with evaluating actions in the continuation of Examples 3, 5, and 6.

In Section 8.2.1, we discuss the deficiencies of using only point state indicators. These deficiencies are related to a lack of knowledge about the consequences of actions. The concept of a dispersion index is introduced to help model complementary information that can help portray the imprecision, uncertainty, and inaccurate determination associated with the consequences.

We introduce and illustrate the concept of dispersion thresholds in Section 8.2.2. We explain the difference between a nonpoint state indicator and a point state indicator with a threshold and define positive and negative dispersion thresholds associated with a point state indicator. We finish Section 8.2.2 by discussing the important difference between intrinsic and nonintrinsic dispersion thresholds.

The dispersion indicator that represents thresholds is in fact a special case of a category of dispersion indicators that we call modulation indicators. In Section 8.2.3, we illustrate four types of modulation indicators and provide a general definition (Def. 8.2.1).

Section 8.2.4 is devoted to a more general form of dispersion indicator: the referenced dispersion indicator.

We end the chapter with Section 8.2.5, where we summarize the importance of the different components of the evaluation model Г(A). We also summarize the principles that should guide determination of such a model for any problem.

In Section 9.1.1 we review the different meanings of the term “criterion” and mention that criterion and criterion function are usually synonymous in the field of decision aiding. We then define (Def. 9.1.1) and generalize this concept in Section 9.1.2 and add five remarks.

We address the development of criteria from action consequences that are modeled by state and dispersion indicators along various dimensions in Section 9.2. In Section 9.2.1 we discuss the case when criterion g is associated with a single dimension i that leads to a point evaluation. In this case, we show (Res. 9.2.1) that developing such a criterion leads to a state indicator encoding (Def. 9.2.1) and describe several examples. In Section 9.2.2, we discuss the case when the criterion g is associated with a single dimension i that leads to a nonpoint evaluation. Here, we distinguish between two cases. In the first case (Section 9.2.2.1), g is the only criterion that affects the evaluation along dimension i, and we say that there is point reduction in the dimension. We discuss and illustrate the principal point reduction techniques, especially the technique based on utility theory. In the second case (Section 9.2.2.2), g is not the only criterion that affects the evaluation along dimension i, and we say that the criteria split dimension i. We discuss the reasons for such splitting. Finally, in Section 9.2.3 we consider the case when a criterion g, affects all the dimensions in a subset I. When g_I, is conceived so as to contain all the evaluation information contained in this subset of dimensions, we say that there is subaggregation of these dimensions. We explore when developing such criteria makes sense and clarify terminology.

In Section 9.3 we investigate the limits of using the values of a criterion to form indifference or strict preference relations over pairs of actions. This leads us to propose the notion of discriminating power of a criterion in Section 9.3.1. We provide an operational meaning to this notion in Section 9.3.2 when we introduce the concepts of indifference and preference thresholds. In Section 9.3.3 we define the important concept of pseudo-criterion as a criterion function to which discrimination thresholds are added (Def. 9.3.2). We introduce specific cases where at least one of the two thresholds is empty (Def. 9.3.3). The structures of the resulting systems of preference relations correspond to structures already seen in Chapter 7 (Res. 9.3.1). We finish this subsection by discussing practical ways to determine discrimination thresholds.

In Section 9.4, we examine when the difference g(a′) — g(a) can be used to reflect the qualitative importance of the difference in actions a′ and a according to the criterion g. After motivating the general problem in Section 9.4.1, we present the two important definitions of gradation (Def. 9.4.1) and gradable criterion (Def. 9.4.2) in Section 9.4.2. We devote Sections 9.4.3 and 9.4.4 to criteria that can be called measures. In Section 9.4.3, we present the basic definition (Def. 9.4.3), provide examples, and discuss properties (Res. 9.4.2). In Section 9.4.4, we discuss the special case of the von Neumann-Morgenstern expected utility criterion. We specify the axiomatic basis (Axioms 9.1–9.4) that defines this criterion up to a positive affine transformation (Res. 9.4.3) and discuss the conditions under which it is a measure. These conditions are closely related (Axiom 9.5 or 9.6) to those that allow differences in criterion values to reflect importance.

The model Г(a) presented in Section 8.2.5 does not generally allow the comparison of two actions. Therefore, we use the techniques presented in Chapter 9 to synthesize Г(a) into a criterion family F. In Section 10.1 we show that for both theoretical and practical reasons there are no set rules for automatically deducing F from Г(A). However, the analyst must respect some logical requirements, which then lead to the definitions of exhaustiveness, cohesiveness, and nonredundancy that characterize the concept of a coherent criterion family.

In Section 10.2, we introduce the performance tableau which indicates the performance level for each criterion member g_j of a coherent family F for each action a in a subset A′ of A. Indifference and preference thresholds associated with the criteria can also be included in the tableau. In a description problematic, the performance tableau usually represents the final product of the study. We highlight the types of fruitful discussions these tableaus can engender and caution against their common misuses.

In Section 10.3 we discuss several forms of dependence among criteria and place them in two main categories. We also discuss these types of dependence in the context of the two major approaches to preference modeling. Section 10.3.1 presents these descriptive and constructive approaches. The former is based on the existence of a rational decision maker with a coherent and stable SPR that is to be described as reliably as possible. The latter pays special attention to the conflicting and unstable nature of preference judgments and emphasizes the importance of significance axes for facilitating discussion of these preferences and constructing one or several SPR’s. In Section 10.3.2 we show how the components of the criterion supports — the state indicators, dispersion indicators, and factors used to define them — can cause dependence among criteria and emphasize the often contingent nature of the set A. Although in a descriptive approach this form of dependence will lead to a desire to reduce the number of criteria, it is not considered a weakness in a constructive approach. We turn our attention in Section 10.3.3 to dependence stemming from value systems. Such dependence can be characterized by the fact that one cannot reason on the basis “all other things considered equal.” One type of this dependence is related to the very notion of a criterion, which implies a certain ability to consider the criterion in isolation from others in the family F and gives meaning to the idea of preferences restricted to a significance axis of the criterion. Such a dependence, called utility dependence, is extremely troublesome in a descriptive approach. In a constructive approach, utility dependence is considered to be the sign of a missing criterion. We then introduce a second way of reasoning based on “all other things being equal” for a subfamily J of F. This “preference independence” of J in F allows the possibility of replacing the multiple criteria of J by a single criterion.

In Section 10.4, we contrast multicriteria and single criterion analysis. Multicriteria analysis is based on value systems that make explicit a family F of n (n > 1) unanimous, clear, and exhaustive criteria. Single criterion analysis avoids such explicitness by amalgamating, often prematurely, two types of information — information related to the consequences of actions and intercriteria information that is strongly influenced by the actors’ value systems. To finish, we discuss the notions of dominance, substitution rate, concordance, discordance, and veto in the context of interpreting performance tableaus in δ-problematics.

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)…, g_n(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.

In the first section we discuss issues related to the choice of operational approach. Just as when choosing the problematic, the analyst may often hesitate among several possibilities. We propose that neither the choice of problematic nor the characteristics of A should systematically influence the choice of the operational approach. The influence will more likely come from the general environment surrounding the problem, the analyst’s relation with the other actors, how the analyst will fit in the decision process, and the analyst’s training.

We summarize general paths the analyst might take to overcome difficulties arising from: dependence among actions (e.g., exclusions, redundancies, complementarities, synergies) in Section 12.2; multiple scenarios in Section 12.3; conflicting value systems among actors in Section 12.4; hesitations on the roles that the various criteria play in directing a strategic decision in Section 12.5; and poorly defined actions with difficult to evaluate performance levels in Section 12.6. Although there could be other difficulties in practical applications, these common ones indicate the complexity of the problem. No decision aiding methodology can provide a recipe of steps. Its contribution will rather come from guiding the thought process by placing the various factors and relations in a systematic framework, thereby giving the decision maker and other actors improved insight into the problem at hand.