The BF-TOPSIS Approach for Solving Non-classical MCDM Problems

. In this paper we show how the Belief-Function based Technique for Order Preference by Similarity to Ideal Solution (BF-TOPSIS) approach can be used for solving non-classical multi-criteria decision-making (MCDM) problems. We give simple examples to illustrate our presentation.


Introduction
Classical Multi-Criteria Decision-Making (MCDM) consists in choosing an alternative among a known set of alternatives based on their quantitative evaluations (numerical scores) obtained with respect to different criteria. A typical example could be the selection of a car to buy among a given set of cars based on different criteria (cost, engine robustness, fuel economy, CO 2 emission, etc.). The classical MCDM problem, although easily formulated, have no solution at all in general due to the fact that no alternative exists that optimizes all criteria jointly. Thus MCDM problems are generally not solved, but a decision is found by means of ranking, compromises etc. The difficulty of MCDM problems is also because the scores are usually expressed in different (physical) units with different scales which generally necessitates an ad-hoc choice of a normalization step that may lead to many problems, e.g. rank reversal.
1 TOPSIS 3 [8,9] methods are the most well-known and widely used MCDM methods in applications. These methods have already been extended in the belief function framework in our previous works [2,10,11] to take into account epistemic uncertainty, missing scores' values as well as conflicting information between sources 4 . In this work, we show how the BF-TOPSIS methods proposed recently in [11] (with application in [12]), can be directly used for solving also nonclassical multicriteria decision-making problems where not only alternatives are scored (with possibly missing values), but also any element of the power set of alternatives.
In the sequel, we assume the reader to be familiar with the theory of belief functions [13] and its definitions and notations, mainly the basic belief assignment (BBA) m(·), the belief function Bel(·) and the plausibility function Pl(·) defined with respect to a discrete finite frame of discernment (FoD).

Non-classical MCDM Problem Formulation
We consider a given set of alternatives A {A 1 ,A 2 ,...,A M } (M>2) representing the FoD of our problem under consideration, and we denote 2 A the power set 5 of A. In our approach, we work with Shafer's classical model of FoD and we do not allow the empty set to be a focal element 6 because in our opinion it does not make sense to compare an alternative with respect to the empty set from the decision-making standpoint. The cardinality of the (non empty) elements of the power set varies from 1 to 2 M − 1. We also consider a given set of criteria C {C 1 ,C 2 ,...,C N } (N ≥ 1), where each criterion C j is characterized by a relative importance weighting factor w j ∈ [0, 1], j =1 ,...,N such that N j=1 w j = 1. The set of normalized weighting factors is denoted by w = {w 1 ,w 2 ,...,w N }. The score 7 value is a number S ij = S j (X i ) related to the evaluation of an element X i ∈ 2 A \ {∅} from a given criterion C j . If the score value S j (X i ) is not available (or missing), we denote it by the "varnothing" symbol ∅. The non-classical MCDM problem can be formulated as follows in the worst case (i.e. when scores apply to all elements of 2 A ): given the (2 M − 1) × N score matrix S =[ S j (X i )] whose elements take either a numerical value or a ∅ value (if the value is not available) and knowing the set w of the relative importance weights of criteria, how to rank the elements of 2 A \ {∅} to make the final decision?
3 Technique for order preference by similarity to ideal solution. 4 In the MCDM context, a source of information consists in the list of scores values of alternatives related to a given criterion. 5 The power set 2 A is the set of all subsets of A,e m p t ys e t∅ and A included. 6 As proposed in Smets Transferable Belief Model for instance. 7 Depending on the context of the MCDM problem, the score can be interpreted either as a cost/expense or as a reward/benefit. In the sequel, by convention and without loss of generality, we will interpret the score as a reward having monotonically increasing preference. Thus, the best alternative with respect to a given criterion will be the one providing the highest reward/benefit.
Example: Let us consider the ranking of five students A 1 , ..., A 5 basedont w o criteria C 1 and C 2 . The criterion C 1 is their long jump performance (in meters), and the criteria C 2 is a realization of a small project to collect funds (in euros) to help a bigger nature protection project. Highest scores values mean better results in this particular context. Let us assume that students were allowed to realize their project in joint collaboration (no more than three students are allowed in a group), or alone. At the end term of the project, suppose that one has the two following evaluations (scoring) The scores' values listed in S C1 indicate in fact that the student A 2 has not been able to pass the long jump test for some reason (medical, familial or whatever), so his score is missing. The scores' values listed in S C2 indicate that A 5 did choose to realize his project alone with a pretty good performance, and the project realized by the collaboration of students A 3 with A 4 has obtained the best performance (the highest amount of collected funds). In this very simple example, one sees that the score evaluation can be done not only on single alternatives (as for criterion C 1 ) but also on a subset of elements of 2 A (as for criterion C 2 ). All the elements having a score are called scoring focal elements. In general, these focal elements can differ from one criterion C j to another criterion C k for k = j and the score matrix cannot be built by a simple (horizontal) stacking of scoring lists. In general, one must identify all focal elements of each scoring list to determine the minimum number of rows necessary to define the scoring matrix. As mentioned, we use the symbol ∅ to identify all values that are missing in the scoring matrix. Note that we do not set missing values to zero number (or any other chosen number) to make explicit distinction between the known precise numerical value zero and a missing value. In this example, the scoring matrix will be defined as The question we want to address is how to rank the students based on such a kind of scoring information including disjunctions of alternatives and missing values, taking into account the relative importance weight of each criterion. Is it possible to solve such type of non-classical MCDM problems, and how? 3

The BF-TOPSIS Approach
The BF-TOPSIS approach has been proposed recently in [11] in a classical MCDM context where the focal elements of the scoring function S j (·) (j =1 ,...,N) are only the singletons A i (i =1 ,...,M) of the frame of discernment A. BF-TOPSIS is initially based on belief functions for MCDM support which exploits only the M × N score matrix S =[S j (A i )] and the relative importance weighting factors of criteria. The first main step of BF-TOPSIS is the construction of an M × N BBA matrix M =[m ij (·)] from the score matrix S, and then the combination of components of M to make a final decision thanks to the Euclidean belief interval distance, denoted by d BI , defined in [14,15].
In fact, the BF-TOPSIS approach can also be directly applied to solve the non-classical MCDM problems because the belief interval [Bel ij (X i ),Pl ij (X i )] of each proposition (i.e. each focal element which is not necessarily a singleton) X i based on a criteria C j can be established in a consistent manner 8 from the where the Sup j (X i )a n dInf j (X i ) are computed from the score matrix S by Sup j (X i ) is called the "positive support" of X i because it measures how much X i is better than other propositions according to criterion C j ,andInf j (X i ) is called the "negative support" of X i because it measures how much X i is worse than other propositions according to criterion C j . The length of interval [0,Sup j (X i )] measures the support in favor of X i as being the best proposition with respect to all other ones, and the length of [Inf j (X i ), 0] measures the support against X i based on the criterion C j . The denominators involved in (3), are defined by X j max max i Sup j (X i ) and X j min min i Inf j (X i ), and they are supposed different from zero 9 .F r o m the belief interval [Bel ij (X i ); Pl ij (X i )], we obtain the BBA m ij (·) defined by 8 Indeed, Belij(Xi)a n dBelij(Xi)( w h e r eXi is the complement of Xi in the FoD A) belong to [0, 1] and they are consistent because the equality Plij(Xi)=1−Belij(Xi) holds. The proof is similar to the one given in [11]. 9 If X j max =0t h e nBelij(Xi) = 0, and if X j min =0t h e nPlij(Xi) = 1, so that Belij(Xi)=0.
If a numerical value S j (X i ) is missing in the score matrix S (it is equal to ∅), one chooses m ij (·) equals (0, 0, 1), i.e., one takes a vacuous belief assignment. In [11], we have proposed four methods (called BF-TOPSIS1, ..., BFTOPSIS4) to make a decision from the BBA matrix M =[ m ij (·)]. Due to space restriction constraint, we just recall the principle of the BF-TOPSIS1 method because it is the simplest one. Applications of BFTOPSIS2-BFTOPSIS4 methods to non-classical MCDM problems is also possible without difficulty. The proposed transformation of score values to BBAs and basis of BF-TOPSIS method are theoretically justified in [11].
Before presenting succinctly the BF-TOPSIS1 method, we need to recall the definition of Belief Interval-based Euclidean distances d BI (m 1 ,m 2 ) introduced in [14] between two BBAs m 1 (·)a n dm 2 (·) defined on a same FoD Θ. Mathematically, d BI (m 1 ,m 2 ) is defined by where N c =1/2 |Θ|−1 is a normalization factor to have d BI (m 1 ,m 2 ) ∈ [0, 1], and d W (BI 1 (X),BI 2 (X)) is the Wassertein distance [16] between belief intervals In [14], we have proved that d BI (x, y) is a true distance metric.
Principle of BF-TOPSIS1: From the BBA matrix M and for each proposition (focal element) X i , one computes the Belief Interval-based Euclidean distances d BI (m ij ,m best ij ) defined in (9) between the BBA m ij (·) and the ideal best BBA defined by m best ij (X i ) = 1, and the distance d BI (m ij ,m worst ij ) between m ij (·)and the ideal worst BBA defined by m worst ij (X i )=1. Then, one computes the weighted average of d BI (m ij ,m best ij ) values with relative importance weighting factor w j of criteria C j . Similarly, one computes the weighted average of d BI (m ij ,m worst ij ) values. More specifically, one computes The relative closeness of the proposition X i with respect to ideal best solution X best defined by is used to make the preference ordering according to the descending order of C(X i ,X best ) ∈ [0, 1], where a larger C(X i ,X best ) value means a better proposition X i . Note that once the BBA matrix is computed from Eqs. (6), (7) and (8), we can also apply (if we prefer) BF-TOPSIS2, BF-TOPSIS3 or BFTOPSIS4 methods to make the final decision. Their presentation is out of the scope of this paper.

Apply BF-TOPSIS to Non-classical MCDM Problems
Due to space limitation restriction, we present the results of the BF-TOPSIS1 method only for two simple non-classical MCDM problems. Example 1: This example is given by the score matrix of Eq. (2). We consider the relative importance weights w 1 =1 /3a n dw 2 =2 /3 of criteria C 1 and C 2 respectively. Applying BBA construction formulas (6), (7) and (8) From this matrix M, we compute the distances d BI (., .) with respect to ideal best and worst solutions shown in Table 1. Table 2 provides d best (X i ), d worst (X i ) and C(X i ,X best ) values computed from the formulas (11), (12) and (13). Based on C(X i ,X best ) values sorted in descending order, we finally get the preference order ( If we restrict the preference order to only singletons, we will get A 5 ≻ A 4 ≻ A 1 ≻ A 3 (i.e. student A 5 is the best one). Note that student A 2 alone cannot be ranked with respect to the other students, which is normal based on the non-specific input (scoring) information one has for him. Of course ad-hoc ranking solutions to rank all five students can always be developed 11 , but without necessarily preserving the compatibility with the rank obtained previously.
Example 2: In mountains, protecting housing areas against torrential floods is based on a lot of alternatives at the watershed scale such as check dams' series, sediment traps, dikes, and individual protections [12]. Moreover, alternatives can be the maintenance of existing structures or the construction of new ones to 10 When a score value is missing for some proposition Xi (i.e. if Sj(Xi)=∅), then we take the vacuous BBA mij(Xi ∪Xi)=1. 11 For instance by normalizing the C(Xi,X best ) values (the most right column of Table 2) and interpret it as a BBA, and then apply a decision method described in [15]. increase the protection level. Final propositions generally involve several of previous individual alternatives. We propose here a simplified case of application. Within a given watershed, a check-dams' series already exists. Older than one century years old, its maintenance (alternative A 1 ) is questioned. Some experts propose to abandon it and to build a sediment trap upstream the alluvial fan (alternative A 2 ) or to limit damage on buildings through individual protections (alternative A 3 ). The Decision-Maker (DM), here the local municipality, must decide the best proposition taking into account several criteria: the investment cost (C 1 in e, in negative values), the risk reduction in 50 years between the current situation and the expected situation after each proposition implementation (C 2 in e), the impact on environment (C 3 is a grade from 1 to 10), and the land-use areas needed in privates (C 4 in m 2 , in negative values). For each criterion, the higher is the score, the better is the proposition. The DM gives the same importance weight to C 1 and C 2 (w 1 = w 2 =0.33), but they are more important than C 3 (w 3 =0 .20) which is more important than C 4 (w 4 =0 .14).
The score matrix is given in Eq. (15). In this case, the problem is not to have no knowledge on some scores but is that they are not cumulative in the same way for each criterion. For C 1 and C 4 , the score of the disjunction of two alternatives is the sum of individual scores whereas it is not the case for C 2 and C 3 .
−150000 100000 10 0 −550000 250000 10 −5000 −700000 250000 10 −5000 The BBA matrix based on S using (3), (4), (5), (6), (7) and (8) The weighted distances to the ideal best and worst solutions and the relative closeness indicator are listed in Table 3. Based on relative closeness indicator sorted in descending order, the final preference order is ( : maintaining the existing check dams' series and implementing individual protections is the best option. If the preferences are restricted to single alternatives, one will get as final preference order A 3 ≻ A 1 ≻ A 2 , i.e. option A 3 (only individual protections) should be preferred by the DM.

Conclusions
In this paper, we have shown how the BF-TOPSIS approach can be exploited to solve non-classical MCDM problems. This method is relatively easy to use. It does not require the normalization of data and offers a consistent construction of basic belief assignments from the available scoring values. It can also deal with missing scoring values and different criteria weights as well. In this paper only the BF-TOPSIS1 method has been presented, but other more sophisticate BF-TOPSIS methods could be also used to solve non-classical problems, but at the price of a higher complexity. The application of this new BF-TOPSIS approach to solve non-classical MCDM problems for natural risk prevention is currently under evaluation, and it will be reported in a forthcoming publication.