sion problem based on imperfect information resulting from more or less reliable sources. Conclusions and perspectives are given in section IV.

## Ii. Methods For Multi-Criteria Decision Analysis And Imperfect Information

Information and decision are closely linked and different methods exist to take a decision on the basis of imperfect information. From one hand, main principles of multi-criteria decision analysis and existing theories to manage imperfect information are over-viewed. From the other hand, we then briefly analyze the characteristics and lacks of existing methods methods using both **MCDA** methods and theories for uncertainty management.

## A. The Analytic Hierarchy Process

Multi-criteria decision analysis aims to choose, sort or rank

![0_image_1.png](0_image_1.png)

alternatives or solutions according to criteria involved in the decision-making process. Main steps of a multi-criteria analysis consist in identifying decision purposes, defining criteria, eliciting preferences between criteria, evaluating alternatives or solutions and analyzing sensitivity with regard to weights, thresholds, . . . . Total aggregation methods such as the MultiAttribute Utility Theory (M.A.U.T.) [11], [13] synthesizes in a unique value the partial utility related to each criterion and chosen by the decision maker. Each partial utility function transforms any quantitative evaluation of criterion into an utility value. The additive method is the simplest method to aggregate those utilities (figure 3).

The Analytic Hierarchy Process (AHP) [12], [19], [20] is a single synthesizing criterion approach. This method is worldwide used in almost all applications related with decisionmaking [27]. AHP is a special case of complete aggregation method based on an additive preference aggregation and can be considered as an approximation of multi-attribute preference models [11]. Its principle is to arrange the factors considered as important for a decision in a hierarchic structure descending from an overall goal to criteria, sub-criteria and finally

![0_image_0.png](0_image_0.png)

alternatives in successive levels. It is based on three basic steps: decomposition of the problem, comparative judgments and hierarchic composition or synthesis of priorities (figure 4).

At each level of the hierarchy, a preference matrix is built up through pairwise comparisons using a semantic and ratio scale to assess the decision maker preferences between the criteria of the considered level. Through the AHP pairwise comparison process, weights and priorities are derived from a set of judgments that can be expressed either verbally, numerically or graphically. The original AHP process uses an additive preference aggregation and compares the solutions from one to each other in a so-called "Criterion-alternative approach". This implies to make pairwise comparisons between all the solutions or alternatives in order to obtain preferences levels between these alternatives. When dealing with great amount of data, this becomes quickly quite difficult. An other approach so-called "Criterion-index (or estimator)-alternative" is used in our developments (figure 5). Instead of comparing all the alternatives, the decision analyst evaluates, for each alternative, criterion according pre-existing classes. Each evaluation class corresponds to an increasing or decreasing level of satisfaction of a given criterion involved in the decision making. For example, the criterion *human vulnerability* exposed to natural hazards can be assessed according to three classes based on a number of existing and exposed buildings (figure 7). These classes code some kind of ordinal levels corresponding to a low, medium or strong contribution (or satisfaction) to (or of) the criterion. In that way, the AHP method, despite the known issues of complete aggregation methods, fits quite well to decision ranking problems where the alternatives are not all known.

B. Making a decision on the basis of imperfect information Several theories have been proposed to handle different kinds of imperfect information: *Fuzzy Sets Theory* for vague information [29], *Possibility Theory* for uncertain and imprecise information [9], [30] and *Belief Function Theory* that allows to consider uncertain, imprecise and conflicting information. In addition to original *Dempster-Shafer* theory