Parameterize a territorial risk evaluation scale using multiple experts knowledge through risk assessment examples

Evaluating and comparing the threats and vulnerabilities associated with territorial zones according to multiple criteria (industrial activity, population, etc.) can be a time-consuming task and often requires the participation of several experts and decision makers. Rather than a direct evaluation of these zones, building a risk evaluation scale and using it in a formal procedure permits to automate the assessment and therefore to apply it in a repeated way and in large-scale contexts and, provided the chosen procedure and scale are accepted, to make it objective. One of the main difficulty of building such a formal evaluation procedure is to account for the multiple experts knowledge and decision makers preferences. The procedure used in this article, ELECTRE TRI, uses the performances of each territorial zone on multiple criteria, together with preferential parameters, to qualitatively assess their associated risk level. The preferential parameters to be determined are the category limits, i.e. the limits on each criterion of the performance range associated with a given risk level, and the weights associated with the different criteria. To obtain these parameters with no direct questioning of the stakeholders, tools based on mathematical programming have been developed to deduce these preferential parameters from assessment examples. In this article, several such tools are applied together in order to build a complete example leading to the definition of a territorial risk evaluation scale taking into account the preferences of multiple stakeholders. 1 RISK ASSESSMENT MODELLED AS MULTICRITERIA SORTING MODELS Assessing the risk levels associated with geographical zones can be a difficult task. It involves multiple, and often conflicting, point of views: a zone may have a low risk according to one criterion while being exposed to a critical risk according to an other one. Associating one risk level to a zone involves aggregating these point of views. This article suggests to use the tools developed in the domain of multicriteria decision aid, which enable a formal approach to that aggregation problem when assessing risk. The paper is structured as follows. This section describes the relation between a risk assessment problem and a multicriteria sorting problem. Section 2 describes a specific multicriteria sorting method, ELECTRE TRI (Figueira et al. 2005, Mousseau et al. 2000, Roy 1991), and discusses existing methodologies to elicit an ELECTRE TRI based risk model with experts. Section 3 suggests a strategy to build a risk evaluation model as an ELECTRE TRI model using indirect parameters elicitation. An illustrative example is given in Section 4. The algorithm used for the inference is detailed in Section 5. 1.1 MCDA problem statements Real-world decision problems such as territorial risk evaluation usually involve multiple points of view on evaluation of zones they concern. These points of view are relevant for a Decision Maker (DM), or expert (who is either a single person or a collegial body). They are formally represented by real-valued functions called criteria. Multicriteria (MC) decision aiding aims at recommending a decision which is consistent with a value system of the DM. Various methodologies have been proposed to support DMs facing a MC decision problem (Keeney and Raiffa 1976, Roy 1996, Bouyssou et al. 2006). Three main problematics structure the MC decision aiding field: choice, ranking or sorting (Tsoukiàs 2007). Given a finite set A of alternatives, the choice problematic consists in selecting a subset of alternatives (as small as possible) being judged as the most satisfactory. The ranking problematic consists in establishing a preference pre-order (either partial or complete) on the set of alternatives. These two problematics are said to be comparative, as they require to compare alternatives one to another in order to set up the choice set or the preference pre-order. The sorting problematic concerns ordinal classification of alternatives and consists in assigning each alternative to one of some pre-defined and ordered categories. The assignment of an alternative to an appropriate category relies on the alternative’s intrinsic value (and not on the comparison to others). In this paper, we consider the multicriteria (MC) sorting problematic to represent qualitative risk assessment models. 1.2 Defining a qualitative risk assessment scale Most concepts used in a risk assessment problem easily map to concepts used in the MC sorting problematic. What is called a category in such a problematic represents the set of possible risk levels that the territorial zones must be mapped into. They can be e.g.: {Critical risk, Medium risk, Low risk}. Ideally these risk levels should be defined according to some precautionary measures associated with them, to give these categories a precise meaning. The point of views involved in a risk evaluation problem correspond to the criteria in a MC sorting problem. Examples include the presence of a school or the percentage of vulnerable persons. Building an evaluation scale thus amounts to build a MC sorting model. 1.3 Building a sorting model Each zone being described by a vector of risk factors associated with the point of views involved in the problem, the task at hand consists in assigning these zones to a set of risk categories. This is done using a MC sorting preference model. This model contains a set of subjective data representing the preferences of the considered DM with regard to, e.g., the relative importance of each of the criteria. These objective and subjective data together with a sorting method allow to aggregate the different point of views to assess the risk level of each considered zone. These preferential parameters may be elicited in a direct way, but this is often difficult as it requires the DM to undertand the fine details of their use in the considered MC sorting method. That is why it has been suggested to deduce the preferential parameters in an inverse way, by asking the DM examples of alternatives, or zones, and the category, or risk level, they would consider appropriate for these. A supplementary difficulty arises when the evaluation method to be defined involve multiple DMs, as different stakeholders may favor different subjective parameter values. Applying inverse elicitation in a multiple DMs context amounts to ask each DM for a set of examples, which may be conflicting, and deduce preferential parameter values that may be either entirely shared by the DMs, or shared for a part of the parameters, and individual for other values. This is the approach we use in this article. 2 ELECTRE TRI FOR RISK EVALUATION The MC sorting method used here is a simplified version of ELECTRE TRI. It is appropriate for a risk evaluation setting as it only requires ordinally evaluated performances on the different criteria. The version considered here is very close to the version studied by Bouyssou and Marchant (2007a, 2007b). 2.1 Sorting procedure ELECTRE TRI requires, as a definition of the preferences of a DM, criteria importance parameters and category limits separating the categories. The criteria importance parameters include a weight for each of the criteria and a majority threshold that defines when a set of criteria is good enough to be decisive. The category limits separate, for each criterion, two consecutive risk levels. Consider a finite set of territorial zones A, a set of category limits B = {b0, . . . bk}, and a finite set of criteria indices J . A criterion gj (j ∈ J) is a function from A ∪ B to R where gj(a) denotes the performance of the zone a on the criterion gj . The zones have to be sorted in k risk levels, c1, . . . , ck, ordered by their desirability. c1 is the worst (i.e. highest) risk level, and ck is the best (the lowest) one. Each risk level ch is defined by the performances of its lower frontier, or category limit, bh−1 and its upper frontier bh of B. The performances are here supposed to be such that a higher value denotes a better performance (i.e. associated with less risks) and the performances on the frontiers are non-decreasing, i.e. ∀j ∈ J,1 ≤ h ≤ k : gj(bh−1) ≤ gj(bh). To sort the zones, ELECTRE TRI uses the concept of outranking relation. The assignment rule used here, known as the pessimistic rule, assigns a zone a to the highest possible risk level ch such that the zone outranks the category’s lower frontier bh−1.A zone a outranks a frontier bh−1 if and only if there is a sufficient coalition of criteria supporting the assertion “a is at least as good as bh−1”, and no criterion strongly opposes (vetoes) that assertion. To compute this, preferential parameters given by a DM are used.The coalition of criteria in favor of the outranking, ∀a ∈ A,1 ≤ h ≤ k, is defined as