Applying ER-MCDA and BF-TOPSIS to Decide on Effectiveness of Torrent Protection

Experts take into account several criteria to assess the effectiveness of torrential flood protection systems. In practice, scoring each criterion is imperfect. Each system is assessed choosing a qualitative class of effectiveness among several such classes (high, medium, low, no). Evidential Reasoning for Multi-Criteria Decision-Analysis (ER-MCDA) approach can help formalize this Multi-Criteria Decision-Making (MCDM) problem but only provides a coarse ranking of all systems. The recent Belief Function-based Technique for Order Preference by Similarity to Ideal Solution (BF-TOPSIS) methods give a finer ranking but are limited to perfect scoring of criteria. Our objective is to provide a coarse and a finer ranking of systems according to their effectiveness given the imperfect scoring of criteria. Therefore we propose to couple the two methods using an intermediary decision and a quantification transformation step. Given an actual MCDM problem, we apply the ER-MCDA and its coupling with BF-TOPSIS, showing that the final fine ranking is consistent with a previous coarse ranking in this case.


Introduction
In mountainous areas, torrents put people and buildings at risk. Thousands of check dams, clustered in series, have been built to protect them. Risk managers must assess their effectiveness given several criteria such as their structural stability or their hydraulic dimensions. This is a Multi-Criteria Decision-Making (MCDM) problem. In practice, scoring each criterion is difficult and imperfect. Experts affect each check dam series to one of several qualitative evaluation classes of effectiveness (high, medium, low, no) [1]. Evidential Reasoning for Multi Criteria Decision Analysis (ER-MCDA) has been developed on the basis of fuzzy sets, possibility and belief function theories [2,3] to decide on such MCDM problems, taking into account imperfect assessment of criteria provided by several sources.
Given the final qualitative label for each check dam series, a coarse ranking of all of them can be provided, as shown in recent applications [1]. Nevertheless, risk managers need a finer ranking to choose the most effective one. To help it, the recent Belief Function-based Technique for Order Preference by Similarity to Ideal Solution (BF-TOPSIS) methods [4] are more robust to rank reversal problems than other classical decision-aid methods such as the Analytic Hierarchy Process (AHP) [5]. Nevertheless, the BF-TOPSIS methods are limited to MCDM problems with precise quantitative evaluation of criteria.
To help risk managers rank several check dam series according to their effectiveness, the BF-TOPSIS should take into account the initial imperfect assessment of criteria. Therefore, we propose to combine the ER-MCDA and BF-TOPSIS methods. We first detail the ER-MCDA process and apply it to an actual case with a final coarse ranking. We then combine ER-MCDA with BF-TOPSIS. Applying it to the same example, we finally show that the finer ranking result obtained is consistent with the previous coarse ranking result in this case.
Given Θ, numerous more or less effective rules allow combining several BBAs. Before their combination, each BBA m(·) can be differently discounted by the source reliability or importance [7]. The comparison of the combination rules is not the main scope of this paper, and hereafter we use the 6th Proportional Conflict Redistribution (PCR6) fusion rule, developed within the framework of Dezert-Smarandache Theory (DSmT) [8] (Vol. 3). The latter is a modification of belief function theory, designed to palliate the disadvantages of the classical Dempster fusion rule [9].
Given m(·), choosing a singletonθ ∈ Θ or subsetX ⊆ Θ is the decision issue. In general, it consists in choosingθ = θ k , k = 1, . . . , q with k arg max k C(θ k ), where C(θ k ) is a decision-making criterion. Among several C(θ k ), the most widely used one is the belief Bel (θ k ) m(θ k ) corresponding to a pessimistic attitude of the Decision-Maker (DM). On the contrary, the plausibility Pl (θ k ) X∩θ k =∅|X∈2 Θ m(X) is used for an optimistic attitude. Between those two extreme attitudes, an attitude of compromise is represented by the decision based on the maximum probability. For this, the BBA m(·) is transformed into a subjective probability measure P (·) through a probabilistic transformation such as the pignistic one [10], the normalized plausibility transformation [11], etc.
In some cases, taking into account non-singletons X ⊆ Θ is needed to make a decision. As shown in [12], the minimum of any strict distance metric d(m, m X ) between m(·) and the categorical BBA m X can be used in Eq. (1). If only singletons of 2 Θ are accepted, the decision is defined by Eq. (2).
Among the few true distance metrics 1 between two BBAs m 1 (·) and m 2 (·), the Belief Interval-based Euclidean d BI (m 1 , m 2 ) ∈ [0, 1] defined by Eq. (3) [13] provides reasonable results. It is based on the Wasserstein distance defined by Eq. (4) [14] with The quality indicator q(X) defined by Eq. (5) evaluates how good the decision X is with respect to other focal elements: the higher q(X) is, the more confident in its decisionX the DM should be. If only singletons of 2 Θ are accepted, q(X) = q({θ}) is defined by Eq. (6).
3 From ER-MCDA to decision-making

Multi-Criteria Decision-Making problems
In a MCDM problem, the DM compares alternatives represents the DM preferences between criteria. The AHP process helps extract it, comparing criteria pairwisely [5]. The DM gives an M ×N score matrix S = [S ij ] in Eq. (7). S ij is a score value of A i according to the scoring scale of the criterion C j . In practice, S ij for each alternative A i is given in hazardous situations, with no sensor and in a limited amount of time. The sources of information can therefore be imprecise, epistemically uncertain, incomplete and possibly conflicting.
we consider two different decision-making assessments (DMA1 and DMA2). For each criterion C j , a possibility distribution π ij [15] is provided by an expert through intervals F αι , ι = 1, . . . , ι max with a confidence level. This represents the imprecise scoring of S ij of each alternative A i . The mapping [2] of each possibility distribution into q fuzzy sets θ k , k = 1, . . . , q [16] provides each BBA m Θ ij (·) on 2 Θ for each A i , i = 1, . . . , M and C j , j = 1, . . . , N in the BBA matrix M Θ .
The algorithm of the geometric mapping process is detailed in [2]. A BBA m Xj ij (·) is first extracted from each π ij : the FoD is the scoring scale X j of the criterion C j ; focal elements are the intervals F αι , ι = 1, . . . , ι max . Then each interval F αι is mapped into each fuzzy set θ k to obtain its geometric area Aι .
• Step 2 old (DMA1): We refer the reader to [3] for details. Each BBA m Θ ij (·) is discounted by the importance weight w j of each criterion C j . For each A i ∈ A, the N BBAs m Θ ij (·) are combined 2 with importance discounting [3] to obtain the BBA m Given a preference ranking of the q elements of Θ, comparing all theθ(A i ) chosen for each A i helps rank the A i alternatives. Nevertheless, it is not necessarily a strict ranking since the labelθ(A i ) may be the same for several A i .

BF-TOPSIS methods for the DMA2 given precise S ij
Four BF-TOPSIS methods were developed to decide on the corresponding M ×N matrix S = [S ij ] (Eq. (7)), with the precise score value S ij . Details are given in [4]. All To help rank all alternatives A i ∈ A, the main idea of BF-TOPSIS methods is to compare each A i with the best and worst ideal solutions. It is directly inspired by the technique for order preference by similarity to the ideal solution (TOPSIS) developed in [17]. The four BF-TOPSIS methods differ from each other in how they process the M ×N matrix M A with an increasing complexity and robustness to rank reversal problems. In this paper, we focus on BF-TOPSIS3 (the 3rd BF-TOPSIS method using the PCR6 fusion rule) [4]. 3. The relative closeness of each alternative A i with respect to an unreal ideal best solution defined by A best is given by

For each
On the contrary, if d worst (A i ) = 0, then C(A i , A best ) = 0, meaning that alternative A i coincides with the ideal best solution A worst . Thus, the preference ranking of all alternatives A i ∈ A is made according to the descending order of C(A i , A best ).

BF-TOPSIS coupled with ER-MCDA to deal with imperfect S ij
To deal with the DMA2 and imperfect information, we propose to couple (mix) BF-TOPSIS with ER-MCDA according to the following steps: • Step 3 new (ranking alternatives): We use q(θ(i, j)) as the reliability factor to discount each BBA m A ij (·) using the Shafer discounting method [6]. For each A i , we combine them with the PCR6 rule to obtain the BBA m A i (·), taking into account the importance factor w j of each criterion C j [7]. As explained in points 2 and 3 of subsection 3.4, the relative closeness factors C(A i , A best ) are calculated, from which the preference ranking of all A i is deduced.

Effectiveness of torrential check dam series
To reduce potential damage on at-risk housing, each torrential check dam series stabilizes the torrent's longitudinal profile to curtail sediment release from the headwaters. Their effectiveness in achieving this function depends on N = 7 technical criteria C j with their importance weights w j , as shown in Figure 1. An expert assesses M = 4 check dam series A i according to their effectiveness given an imperfect evaluation of each C j and using ER-MCDA step 1. After this common step, ER-MCDA step 2 is used to assess (DMA1) the effectiveness of each A i expressed by four qualitative labels (levels) in Θ = {high, medium, low, no} [1]. Then steps 2 and 3 of the method based on BF-TOPSIS3 developed in section 3.4 are used to rank all A i and to choose the most effective one, A i (DMA2).
• Step 1 new =Step 1 old (M Θ construction): The expert evaluates each criterion C j for each A i through possibility distributions. N = 7 fuzzy scales are specified, each one gathering the q = 4 fuzzy sets θ k , k = 1, . . . , q. The BBA matrix M Θ = [m Θ ij (·)] obtained for Θ = {high, medium, low, no} is given in Table 1.    Table 1, the column d min BI in Table 2 lists the minimal value obtained for d BI (m Θ i , m {θ k } ) defined by Eq. (3), for each A i ∈ A. The best labelθ(A i ) is chosen for each A i . Three check dam series A 1 , A 2 , and A 4 are declared as medium, and A 3 is declared as low. The DM coarsely has A 1 A 3 , A 2 A 3 and A 4 A 3 . • DMA2 (based on Step 2 new and Step 3 new described in section 3.4): given M Θ in Table 1, for each A i and C j , one computes arg min k=1,...,q d BI (m Θ ij , m {θ k } ) between each m Θ ij (·) and the categorical BBA m {θ k } (·), with Θ = {θ 1 = high, θ 2 = medium, θ 3 = low, θ 4 = no}. The linear quantitative transformation: θ 1 = 4, θ 2 = 3, θ 3 = 2, θ 4 = 1 is assumed to establish the matrix S = [S ij ] in Table 3. For each A i and C j , the quality factor q(θ(A i , C j )) is also computed in Table 3 applying Eq. (5) with m = m Θ ij . Table 3. Sij and q(θ(Ai, Cj)) (= q(i, j)) provided by Step 2new from M Θ . After the reliability discounting of BBAs from Table 1 by the factors q(θ(A i , C j )) from Table 3, one obtains M A = [m A ij (·)] for A = {A 1 , A 2 , . . . , A M }. After applying BF-TOPSIS3, we obtain the relative closeness C(A i , A best ) values in Table 4. The ranking of all A i according to their effectiveness is consistent with the DMA1 results: A 2 A 4 A 1 A 3 . The most effective check dam series is A 2 .

Conclusion
The ER-MCDA helps provide a coarse ranking of torrential check dam series according to their effectiveness, taking into account several imperfectly scored criteria. Given the same imperfect MCDM problem, risk managers may need a finer ranking. For this purpose, we suggested coupling the ER-MCDA and BF-TOPSIS methods. We have shown the consistency of coarse and finer ranking results for only one example. Further studies are needed to determine whether such consistency holds in general or for certain classes of examples. Moreover, an intermediary decision step and a quantitative transformation are needed to meet this goal. The sensitivity of results to their definition is under evaluation and will be reported in forthcoming publications.