domain: spi phys math info

text:
Jean Dezert
email: jean.dezert@onera.fr

Albena Tchamova
email: tchamova@bas.bg

Deqiang Han
email: deqhan@gmail.com

Jean-Marc Tacnet
email: jean-marc.tacnet@irstea.fr

Simplification of Multi-Criteria Decision-Making Using Inter-Criteria Analysis and Belief Functions

Keywords: Inter-Criteria Analysis, ICrA-BF, MultiCriteria Decision Making, MCDM, belief functions, information fusion. 2 Analytic Hierarchy Process 3 ELimination

In this paper we propose a new Belief Functionbased Inter-Criteria Analysis (BF-ICrA) for the assessment of the degree of redundancy of criteria involved in a multicriteria decision making (MCDM) problem. This BF-ICrA method allows to simplify the original MCDM problem by withdrawing all redundant criteria and thus diminish the complexity of MCDM problem. This is of prime importance for solving large MCDM problems whose solution requires the fusion of many belief functions. We provide simple examples to show how this new BF-ICrA works.

I. INTRODUCTION

In a Multi-Criteria Decision-Making (MCDM) problem we consider a set of alternatives (or objects) A {A 1 , A 2 , . . . , A M } (M > 2), and a set of criteria C {C 1 , C 2 , . . . , C N } (N ≥ 1). We search for the best alternative A given the available information expressed by a M × N score matrix (also called benefit or payoff matrix) S [S ij = C j (A i )], and (eventually) the importance factor w j ∈ [0, 1] of each criterion C j with N j=1 w j = 1. The set of normalized weighting factors is denoted by w {w 1 , w 2 , . . . , w N }. Depending on the context of the MCDM problem, the score S ij of each alternative A i with respect to each criteria C j can be interpreted either as a cost (i.e. an expense), or as a reward (i.e. a benefit). By convention and without loss of generality 1 we will always interpret the score as a reward having monotonically increasing preference. Thus, the best alternative A j for a given criteria C j will be the one providing the highest reward/benefit. The MCDM problem is not easy to solve because the scores are usually expressed in different (physical) units and different scales. This necessitates a choice of score/data normalization yielding rank reversal problems [START_REF] Barzilai | AHP rank reversal, normalization and aggregation rules[END_REF], [START_REF] Pavlicic | Normalization affects the results of MADM methods[END_REF]. Usually there is no same best alternative choice A for all criteria, so a compromise must be established to provide a reasonable and acceptable solution of the MCDM problem for decisionmaking support.

Many MCDM methods exist, see references in [START_REF] Dezert | A New Belief Function Based Approach for Multi-Criteria Decision-Making Support[END_REF]. Most popular methods are AHP 2 [START_REF] Saaty | The Analytic Hierarchy Process[END_REF], ELECTRE 3 [START_REF] Wang | Ranking irregularities when evaluating alternatives by using some ELECTRE methods[END_REF], TOPSIS 4 [6], [START_REF] Lai | TOPSIS for MODM[END_REF]. In 2016 and 2017, we did develop BF-TOPSIS methods [START_REF] Dezert | A New Belief Function Based Approach for Multi-Criteria Decision-Making Support[END_REF], [START_REF] Dezert | Multi-Criteria Decision-Making with Imprecise Scores and BF-TOPSIS[END_REF] based on Belief Functions (BF) to improve the original TOPSIS approach to avoid data normalization and to deal also with imprecise score values as well. It appears however that the complexity of these new BF-TOPSIS methods can become a bottleneck for their use in large MCDM problems because of the fusion step of basic belief assignments required for the implementation of the BF-TOPSIS. That is why a simplification of the MCDM problem (if possible) is very welcome in order to save computational time and resources. This is the motivation of the present work.

For this aim we propose a new Inter-Criteria Analysis (ICrA) based on belief functions for identifying and estimating the possible degree of agreement (i.e. redundancy) between some criteria driven from the data (score values). This permits to remove all redundant criteria of the original MCDM problem and thus solving a simplified (almost) equivalent MCDM problem faster and at lower computational cost. ICrA has been developed originally by Atanassov et al. [START_REF] Atanassov | Intercriteria Decision Making. A New Approach for Multicriteria Decision Making, Based on Index Matrices and Intuitionistic Fuzzy Sets[END_REF]- [START_REF] Atanassov | An approach to a constructive simplification of multiagent multicriteria decision making problems via intercriteria analysis[END_REF] based on Intuitionistic Fuzzy Sets [START_REF] Atanassov | On Intuitionistic Fuzzy Sets Theory[END_REF], and it has been applied in different fields like medicine [START_REF] Todinova | Blood Plasma Thermograms Dataset Analysis by Means of InterCriteria and Correlation Analyses for the Case of Colorectal Cancer[END_REF]- [START_REF] Zaharieva | InterCriteria decision making approach for Behterev's disease analysis[END_REF], optimization [START_REF] Pencheva | InterCriteria Analysis of Genetic Algorithm Parameters in Parameter Identification[END_REF]- [START_REF] Roeva | InterCriteria analysis of ACO and GA hybrid algorithms[END_REF], workforce planning [START_REF] Roeva | Intercriteria Analysis of ACO Performance for Workforce Planning Problem[END_REF], competitiveness analysis [START_REF] Atanassova | Discussion on the threshold values in the Inter-Criteria decision making approach[END_REF], radar detection [START_REF] Doukovska | InterCriteria Analysis approach in radar detection threshold analysis[END_REF], ranking [START_REF] Doukovska | InterCriteria Analysis Applied to EU Micro, Small, Medium and Large Enterprises[END_REF]- [START_REF] Krawczak | Application of the InterCriteria Decision Making Method to Universities Ranking[END_REF], etc. In this paper we improve ICrA approach thanks to belief functions introduced by Shafer in [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] from original Dempster's works [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. We will refer it as BF-ICrA method in the sequel.

After a short presentation of basics of belief functions in section II, we present Atanassov's ICrA method in section III and discuss its limitations. In Section IV we present the new BF-ICrA approach based on a new construction of Basic Belief Assignment (BBA) matrix from the score matrix and a new establishment of Inter-Criteria belief matrix. In section V a method of simplification of MCDM using BF-ICrA is proposed. Examples are given in VI with concluding remarks in Section VII.

II. BASICS OF THE THEORY OF BELIEF FUNCTIONS

To follow classical notations of the theory of belief functions, also called Dempster-Shafer Theory (DST) [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], we assume that the answer (i.e. the solution, or the decision to take) of the problem under concern belongs to a known finite discrete frame of discernement (FoD) Θ = {θ 1 , θ 2 , . . . , θ n }, with n > 1, and where all elements of Θ are exclusive. The set of all subsets of Θ (including empty set ∅ and Θ) is the power-set of Θ denoted by 2 Θ . A BBA (or mass function) associated with a given source of evidence is defined [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] as the mapping m(•) : 2 Θ → [0, 1] satisfying m(∅) = 0 and A∈2 Θ m(A) = 1. The quantity m(A) is called the mass of A committed by the source of evidence. Belief and plausibility functions are usually interpreted respectively as lower and upper bounds of unknown (possibly subjective) probability measure [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. They are defined by5

Bel(A)

B⊆A,B∈2 Θ m(B), and P l(A) 1 -Bel( Ā). [START_REF] Barzilai | AHP rank reversal, normalization and aggregation rules[END_REF] If m(A) > 0, A is called a focal element of m(•). When all focal elements are singletons then m(•) is called a Bayesian BBA [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] and its corresponding Bel(•) function is homogeneous to a probability measure. The vacuous BBA, or VBBA for short, representing a totally ignorant source is defined as m v (Θ) = 1. The main challenge of the decision-maker consists to combine efficiently the possible multiple BBAs m s (•) given by s > 1 distinct sources of evidence to obtain a global (combined) BBA, and to make a final decision from it. Historically the combination of BBAs is accomplished by Dempster's rule proposed by Shafer in DST. Because Dempster's rule presents several serious problems (insensitivity to the level of conflict between sources in some cases, inconsistency with bounds of conditional probabilities when used for belief conditioning, dictatorial behavior, counter-intuitive results), many fusion rules have been proposed in the literature as alternative to Dempster's rule, see [START_REF] Smarandache | Advances and applications of DSmT for information fusion[END_REF], Vol. 2 for a detailed list of fusion rules. We will not detail here all the possible combination rules but just mention that the Proportional Conflict Redistribution rule no. 6 (PCR6) proposed by Martin and Osswald in [START_REF] Smarandache | Advances and applications of DSmT for information fusion[END_REF] (Vol. 3) is one of the most serious alternative rule for BBA combination available so far.

III. ATANASSOV'S INTER-CRITERIA ANALYSIS (ICRA)

Atanassov's Inter-Criteria Analysis (ICrA) approach is based on a M × N score matrix 6

S

[S ij = C j (A i ), i = 1, . . . , M, j = 1, . . . , N ], and intuitionistic fuzzy pairs [START_REF] Atanassov | On Intuitionistic Fuzzy Sets Theory[END_REF] including two membership functions µ(•) and ν(•). Mathematically, an intuitionistic fuzzy set (IFS) A is denoted by

A {(x, µ A (x), ν A (x))|x ∈ E}
, where E is the set of possible values of x, µ A (x) ∈ [0, 1] defines the membership of x to the set A, and ν A (x) ∈ [0, 1] defines the non-membership of x to the set A, with the restriction 0 ≤ µ A (x) + ν A (x) ≤ 1. The ICrA method consists to build an N ×N Inter-Criteria (IC) matrix from the score matrix S. The elements of the IC matrix consist of all intuitionistic fuzzy pairs (µ jj , ν jj ) whose components express respectively the degree of agreement and the degree of disagreement between criteria C j and C j for j, j ∈ {1, 2, . . . , N }. For a given column j (i.e. criterion C j ), it is always possible to compare with >, < and = operators all the scores S ij for i = 1, 2, . . . , M because the scores of each column are expressed in same unit. The construction of IC matrix can be used to search relations between the criteria because the method compares homogeneous data relatively to a same column. In [START_REF] Atanassov | Intercriteria analysis over normalized data[END_REF] Atanassov prescribes to normalize the score matrix before applying ICrA as follows

S norm ij = (S ij -S min j )/(S max j -S min j ) (2)
if one wants to apply it in the dual manner for the search of InterObjects analysis (IObA).

Because we focus on ICrA only, we don't need to apply a score matrix normalization because each column of the score matrix represents the values of a same criterion for different alternatives, and the criterion values are expressed with the same unit (e.g. m, m 2 , sec, Kg, or e, etc).

A. Construction of Inter-Criteria matrix

The construction of the N × N IC matrix, denoted7 K, is based on the pairwise comparisons between every two criteria along all evaluated alternatives. Let K µ jj be the number of cases in which the inequalities S ij > S i j and S ij > S i j hold simultaneously, and let K ν jj be the number of cases in which the inequalities S ij > S i j and S ij < S i j hold simultaneously. Because the total number of comparisons between the alternatives is M (M -1)/2 then one always has necessarily

0 ≤ K µ jj + K ν jj ≤ M (M -1) 2 (3)
or equivalently after the division by M (M -1)

2 > 0 0 ≤ 2K µ jj M (M -1) + 2K ν jj M (M -1) ≤ 1 (4)
This inequality permits to define the elements of

N × N IC matrix K = [K jj ] as intuitionistic fuzzy (IF) pairs K jj = (µ jj , ν jj ) where µ jj 2K µ jj M (M -1) and ν jj 2K ν jj M (M -1) (5)
µ jj measures the degree of agreement between criteria C j and C j , and ν jj measures their degree of disagreement. By construction the IC matrix K is always a symmetric matrix. The computation of K µ jj and K ν jj can be done explicitly thanks to Atanassov's formulas [START_REF] Atanassov | Intercriteria analysis over normalized data[END_REF]

K µ jj = M -1 i=1 M i =i+1 [sgn(S ij -S i j )sgn(S ij -S i j ) + sgn(S i j -S ij )sgn(S i j -S ij )] (6)
and

K ν jj = M -1 i=1 M i =i+1 [sgn(S ij -S i j )sgn(S i j -S ij ) + sgn(S i j -S ij )sgn(S ij -S i j )] (7)
where the signum function sgn(.) used by Atanassov is defined as follows

sgn(x) = 1, if x > 0 0, if x ≤ 0 (8)
Actually the values of K µ jj and K ν jj depend on the choice of sgn(x) function 8 . That is why in [START_REF] Roeva | Intercriteria Analysis of ACO Performance for Workforce Planning Problem[END_REF], [START_REF] Ikonomov | ICrAData software for InterCriteria analysis[END_REF], the authors propose different algorithms implemented under Java in an ICrA software yielding different K µ jj and K ν jj values for making the analysis and to reduce the dimension (complexity) of the original MCDM problem.

B. Inter-criteria analysis

Once the Inter-Criteria matrix K = [K jj ] of intuitionistic fuzzy pairs is calculated one needs to analyze it to decide which criteria C j and C j are in strong agreement (or positive consonance) reflecting the correlation between C j and C j , in strong disagreement (or negative consonance) reflecting non correlation between C j and C j , or in dissonance reflecting the uncertainty situation where nothing can be said about the non correlation or the correlation between C j and C j . If one wants to identify the set of criteria C j for j = j that are strongly correlated with C j then we can sort µ jj values is descending order to identify those in strong positive consonance with C j . In [START_REF] Bureva | Application of the InterCriteria decision making method to Bulgarian universities ranking[END_REF], [START_REF] Bureva | The InterCriteria decision making method to Bulgarian university ranking system[END_REF], the authors propose a qualitative scale to refine the levels of consonance and dissonance and for helping the decision making procedure. A dual approach based on ν jj values can be made to determine the set of criteria that are not correlated with C j . An other approach [START_REF] Atanassov | InterCriteria Analysis: Ideas and problems[END_REF], [START_REF] Krawczak | Application of the InterCriteria Decision Making Method to Universities Ranking[END_REF] proposes to define two thresholds α, β ∈ [0; 1] for the positive and negative consonance respectively against which the components µ jj and ν jj of K jj = (µ jj , ν jj ) will be compared. The correlations between the criteria C j and C j are called "positive consonance", "negative consonance" or "dissonance" depending on their µ jj and ν jj values with respect to chosen thresholds α and β, see [START_REF] Atanassova | Discussion on the threshold values in the Inter-Criteria decision making approach[END_REF] for details. More precisely, C j and C j are in

• (α, β) positive consonance (i.e. correlated):
If µ jj > α and ν jj < β. • (α, β) negative consonance (i.e. no correlated):

If µ jj < β and ν jj > α. • (α, β) dissonance (i.e. full uncertainty): Otherwise. At the beginning of ICrA development it was not very clear how these intuitionistic fuzzy (IF) pairs (µ jj , ν jj ) had to be used and that is why Atanassova [START_REF] Atanassova | Interpretation in the Intuitionistic Fuzzy Triangle of the Results Obtained by the InterCriteria Analysis[END_REF], [START_REF] Atanassova | Traversing and ranking of elements of an intuitionistic fuzzy set in the intuitionistic fuzzy interpretation triangle[END_REF] proposed to handle both components of the IF pair. For this, she interpreted pairs (µ jj , ν jj ) as points located in the elementary T F U triangle, where the point T of coordinate (1, 0) represents the maximal positive consonance (i.e. the true consonance), the point F with coordinate (0, 1) represents the maximal negative consonance (i.e. the falsity), and the point U with coordinates (0, 0) represents the maximal dissonance (i.e. the uncertainty). From this interpretation it becomes easy to identify the top of consonant IF pairs (µ jj , ν jj ) that fall in bottom right corner of (T F U ) triangle limited by vertical line from x-axis x = α, and horizontal line from y-axis y = β. The set of consonant IF pairs are then ranked according to their Euclidean distance d T Cj C j with respect to T point of coordinate (1, 0) defined by

d T Cj C j = d((1, 0), (µ jj , ν jj )) = (1 -µ jj ) 2 + ν 2 jj (9)
In the MCDM context only the criteria that are negatively consonant (or uncorrelated) must be kept for solving MCDM and saving computational resources because they have no (or only very low) dependency with each other, so that each uncorrelated criterion provides useful information. The set of criteria that are positively consonant (if any), called the consonant set, indicates somehow a redundancy of information between the criteria belonging to it in term of decisional behavior. Therefore all these positively consonant criteria must be represented by only one representative criterion that will be kept in the MCDM analysis to simplify MCDM problem. Also all the criteria that are deemed strongly dissonant (if any) could be taken out of the original MCDM problem because they only introduce uncertainty in the decision-making.

C. General comments on ICrA

Although appealing at the first glance, the classical ICrA approach induces the following comments:

1) The IF values µ jj and ν jj can be easily interpreted in the belief function framework. Indeed, the belief and plausibility of (positive) consonance between criteria C j and C j can be directly linked to the values µ jj and ν jj by taking Bel jj (θ) = µ jj and P l jj (θ) = 1 -ν jj . Moreover U jj (θ) = P l jj (θ) -Bel jj (θ) = 1 -ν jjµ jj represents the dissonance (the uncertainty about the correlation) of the criteria C j and C j . Here the proposition θ means: the criteria C j and C j are totally positively consonant (i.e. totally correlated) and the frame of discernment is defined as Θ {θ, θ}, where θ means: the criteria C j and C j are totally negatively consonant (uncorrelated). From this, one can define any BBA m jj (θ), m jj ( θ) and

m jj (θ ∪ θ) of 2 Θ by m jj (θ) = µ jj (10) m jj ( θ) = ν jj (11) m jj (θ ∪ θ) = 1 -µ jj -ν jj (12)
2) The construction of µ jj and ν jj proposed in the classical ICrA is disputable because it is only based on counting the valid ">" or "<" inequalities but it doesn't exploit how bigger and how smaller the scores values are in each comparison done in the construction of the Inter-Criteria Matrix K. Therefore the construction of µ jj and ν jj is actually only a very crude method to estimate IF pairs. 3) The construction of the Inter-Criteria Matrix K is in fact not unique as reported in [START_REF] Ikonomov | ICrAData software for InterCriteria analysis[END_REF]. This will yield different results in general. 4) The exploitation of the ICrA method depends on the choice of α and β thresholds that will impact the final result.

m ij (A i ) = Bel ij (A i ) (13) m ij ( Āi ) = Bel ij ( Āi ) = 1 -P l ij (A i ) (14) m ij (A i ∪ Āi ) = P l ij (A i ) -Bel ij (A i ) (15)
Assuming A j max = 0 and A j min = 0, we take9

Bel ij (A i ) Sup j (A i )/A j max ( 16
)
Bel ij ( Āi ) Inf j (A i )/A j min ( 17
)
where A j max max i Sup j (A i ) and A j min min i Inf j (A i ) and with

Sup j (A i ) k∈{1,...M }|S kj ≤Sij |S ij -S kj | ( 18
)
Inf j (A i ) - k∈{1,...M }|S kj ≥Sij |S ij -S kj | (19)
The entire justification of these formulas can be found in our previous works [START_REF] Dezert | A New Belief Function Based Approach for Multi-Criteria Decision-Making Support[END_REF]. For example, consider the j-th column corresponding to a criterion C j of a score matrix S = [S ij ] with seven rows given by s j = [10, 20, -5, 0, 100, -11, 0] T , where T denotes the transpose. Then based on above formula we get the BBA values listed in Table I. For another criterion C j and the j -th column of the score matrix we will obtain another set of BBA values m ij (•). Applying this method for each column of the score matrix we are able to compute the BBA matrix

Table I BBAS CONSTRUCTED FROM SCORE VALUES. m ij (A i ) m ij ( Āi ) m ij (A i ∪ Āi ) A 1 0.
M = [m ij (•)] whose each component is in fact a triplet (m ij (A i ), m ij ( Āi ), m ij (A i ∪ Āi )) of BBA values in [0, 1] such that m ij (A i ) + m ij ( Āi ) + m ij (A i ∪ Āi )) =
1 for all i = 1, . . . , M and j = 1, . . . , N .

B. Construction of Inter-Criteria Matrix from BBA matrix

The next step of BF-ICrA approach is the construction of the

N × N Inter-Criteria Matrix K = [K jj ] from M × N BBA matrix M = [m ij (•)]
where elements K jj corresponds to the BBA (m jj (θ), m jj ( θ), m jj (θ ∪ θ)) about positive consonance θ, negative consonance θ and uncertainty between criteria C j and C j respectively. The principle of construction of the triplet K jj = (m jj (θ), m jj ( θ), m jj (θ ∪ θ)) is based on two steps that will be detailed in the sequel:

• Step 1: For each alternative A i , we first compute the BBA (m i jj (θ), m i jj ( θ), m i jj (θ∪ θ)) for any two criteria j, j ∈ {1, 2, . . . , N }.

• Step 2: The BBA (m jj (θ), m jj ( θ), m jj (θ ∪ θ)) is then obtained by the combinations of the M BBA m i jj (.). Construction of BBA m i jj (.) The mass of belief m i jj (θ) represents the degree of agreement between the BBA m ij (•) and m ij (•) for the alternative A i , and m i jj ( θ) represents the degree of disagreement between m ij (•) and m ij (•). The mass m i jj (θ ∪ θ) is the degree of uncertainty about the agreement (or disagreement) between m ij (•) and m ij (•) for the alternative A i . The calculation of m i jj (θ) could be envisaged in several manners. The first manner would consist to consider the degree of conflict [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]

k i jj X,Y ⊆Θ|X∩Y =∅ m ij (X)m ij (Y ) and consider the Bayesian BBA m i jj (θ) = 1 -k i jj , m i jj ( θ) = k i
jj and m jj (θ ∪ θ) = 0. Instead of using Shafer's conflict, the second manner would consist to use a normalized distance d i jj = d(m ij , m ij ) to measure the closeness between m ij (•) and m ij (•), and then consider the Bayesian BBA modeling defined by m i jj (θ) = 1-d i jj , m i jj ( θ) = d i jj and m jj (θ∪ θ) = 0. These two manners however are not very satisfying because they always set to zero the degree of uncertainty between the agreement and disagreement of the BBA, and the second manner depends also on the choice of the distance metric. So, we propose a more appealing third manner of the BBA modeling of m i jj (θ), m i jj ( θ), and m i jj (θ ∪ θ). For this, we consider two sources of evidences (SoE) indexed by j and j providing the BBA m ij and m ij defined on the simple FoD {A i , Āi } and denoted

m ij = [m ij (A i ), m ij ( Āi ), m ij (A i ∪ Āi )] and m ij = [m ij (A i ), m ij ( Āi ), m ij (A i ∪ Āi )].
We also denote Θ = {θ, θ} the FoD about the relative state of the two SoE, where θ means that the two SoE agree, θ means that they disagree and θ ∪ θ means that we don't know. Then the BBA modeling is based on the important remarks • Two SoE are in total agreement if both commit their maximum belief mass to the element A i or to element Āi . So they perfectly agree if

m ij (A i ) = m ij (A i ) = 1, or if m ij ( Āi ) = m ij ( Āi ) = 1.
Therefore the pure degree of agreement10 between two sources is modeled by

m i jj (θ) = m ij (A i )m ij (A i ) + m ij ( Ā)m ij ( Ā) (20)
• Two SoE are in total disagreement if each one commits its maximum mass of belief to one element and the other to its opposite, that is if one has m ij (A i ) = 1 and

m ij ( Āi ) = 1, or if m ij ( Āi ) = 1 and m ij (A i ) = 1.
Hence the pure degree of disagreement 11 between two sources is modeled by

m i jj ( θ) = m ij (A i )m ij ( Āi ) + m ij ( Āi )m ij (A i ) (21)
• All possible remaining products between components of m ij and m ij reflect the part of uncertainty we have about the SoE (i.e. we don't know if they agree or disagree).

Hence the degree of uncertainty between the two sources is modeled by

m i jj (θ∪ θ) = mij(Ai)m ij (Ai∪ Āi)+mij( Āi)mij (Ai∪ Āi) + mij(Ai ∪ Āi)mij (Ai) + mij(Ai ∪ Āi)mij ( Āi) + mij(Ai ∪ Āi)mij (Ai ∪ Āi) (22)
By construction m i jj (•) = m i j j (•), hence this BBA modeling permits to build a set of M symmetrical Inter-Criteria Belief Matrices (ICBM)

K i = [K i jj ] of dimension N × N relative to each alternative A i whose components K i jj correspond to the triplet of BBA values m i jj = (m i jj (θ), m i jj ( θ), m i jj (θ ∪ θ)
) modeling the belief of agreement and of disagreement between C j and C j based on A i . One has also 12

m i jj (θ), m i jj ( θ), m i jj (θ∪ θ) ∈ [0, 1] and m i jj (θ)+m i jj ( θ)+ m i jj (θ∪ θ) = 1.
This BBA construction can be easily extended for modeling the agreement, disagreement and uncertainty of n > 2 criteria C j1 , . . . , C jn altogether if needed by taking

m i j1...jn (θ) = n k=1 m ij k (A i ) + n k=1 m ij k ( Āi ) m i j1...jn ( θ) = Xj 1 ,...,Xj n ∈{Ai, Āi} Xj 1 ∩...∩Xj n =∅ n k=1 m ij k (X j k ) m i j1...jn (θ ∪ θ) = 1 -m i j1...jn (θ) -m i j1...jn ( θ) Construction of BBA m jj (.)
Once all the BBAs m i jj (.) (i = 1, . . . , M ) are calculated one combines them to get the component

K jj = (m jj (θ), m jj ( θ), m jj (θ ∪ θ)) of the Inter-Criteria Belief matrix (ICBM) K = [K jj ]
. This fusion step can be done in many ways depending on the combination rule chosen by the user. If the number of alternatives M is not too large we recommend to combine the BBAs m i jj (.) with PCR6 fusion rule [START_REF] Smarandache | Advances and applications of DSmT for information fusion[END_REF] (Vol. 3) because of known deficiencies of Dempster's rule. If M is too large to prevent PCR6 working on computer, we can just use the simple averaging rule of combination in these high dimensional MCDM problems.

V. SIMPLIFICATION OF ORIGINAL MCDM

Once the global Inter-Criteria Belief Matrix K = [K jj = (m jj (θ), m jj ( θ), m jj (θ ∪ θ))] is calculated, we need to identify and cluster the criteria that are in strong agreement, in strong disagreement, and those on which we are uncertain. For identifying the criteria that are in very strong agreement, we evaluate the distance of each component of K jj with the BBA representing the best agreement state and characterized by the specific BBA 13 m T (θ) = 1. From a similar approach we can also identify, if we want, the criteria that are in very strong disagreement using the distance of m jj (•) with respect to the BBA representing the best disagreement state characterized by the specific BBA m F ( θ) = 1. As alternative of Jousselme's distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF], we use the d BI (., .) distance based on belief interval [START_REF] Han | New Distance Measures of Evidence based on Belief Intervals[END_REF] because it is a good method for measuring the distance d(m 1 , m 2 ) between the two BBAs14 m 1 (•) and m 1 (•) over the same FoD. It is defined by

d BI (m 1 , m 2 ) N c • X∈2 Θ d 2 W (BI 1 (X), BI 2 (X)) (23)
where the Belief-Intervals are defined by BI 1 (X) [Bel 1 (X), P l 1 (X)] and BI 2 (X) [Bel 2 (X), P l 2 (X)] and computed from m 1 (.) and m 2 (.) thanks to formula [START_REF] Barzilai | AHP rank reversal, normalization and aggregation rules[END_REF].

d W (BI 1 (X), BI 2 (X)) is Wassertein's distance between in- tervals calculated by d W ([a 1 , b 1 ], [a 2 , b 2 ]) = a 1 + b 1 2 - a 2 + b 2 2 2 + 1 3 b 1 -a 1 2 - b 2 -a 2 2 2 and N c = 1/2 |Θ|-1 is a factor to get d BI (m 1 , m 2 ) ∈ [0, 1].
Because all criteria that are in strong agreement somehow contain redundant (correlated) information and behave similarly from decision-making standpoint, we propose to simplify the original MCDM problem by keeping in the MCDM only criteria that are non redundant The remaining criteria can be eventually weighted by their degree of importance reflecting the number of different criteria that are in agreement through this BF-ICrA approach.

For instance, if one has a seven criteria MCDM problem and if criteria C 1 , C 2 and C 3 are in strong agreement we will only select one remaining criterion among {C 1 , C 2 , C 3 } and we give it a weight of w 1 + w 2 + w 3 . Moreover if C 4 and C 5 are in strong agreement also we will only select one remaining criterion among {C 4 , C 5 } and we give it a weight of w 4 + w 5 , and we will use the weight w 6 for C 6 , and w 7 for C 7 . Hence the original MCDM problem will reduce to a four simplified MCDM problem that can be solved using BF-TOPSIS method already presented in details in [START_REF] Dezert | A New Belief Function Based Approach for Multi-Criteria Decision-Making Support[END_REF] and in [START_REF] Dezert | Multi-Criteria Decision-Making with Imprecise Scores and BF-TOPSIS[END_REF], or with AHP [START_REF] Saaty | The Analytic Hierarchy Process[END_REF] if one prefers, or with any other chosen method that the system-designer may prefer.

The strategy for selecting the most representative criterion among a set of redundant criteria is not unique and depends mainly on the cost necessary (i.e. human efforts, data mining, computational resources, etc) for getting the values of the score matrix of the problem under concern. The least costly criteria may be a good option of selection. In the next section we provide simple examples for BF-ICrA and, for simplicity, we will select the representative criterion as being the one with smallest index. So in the aforementioned example the simplified MCDM problem will reduce to a M × 4 MCDM problem involving only four criteria C 1 , C 4 , C 6 and C 7 .

The BF-ICrA method proposed in this work allows also, in principle, to make a refined analysis (if necessary) based on IC matrices K i jj about the origin of disagreement between criteria with respect to each alternative A i in order to identify the potential inconsistencies in original MCDM problem. This aspect is not developed in this paper and has been left for future investigations. It is worth mentioning that the analysis of the number of redundant criteria versus time improvements that could be proposed as an effective measure of performance of this approach depends highly of the application under consideration and the difficulty (and cost) to get the value of each criteria. For convenience the Figure 1 shows the flow chart of BF-ICrA to help the reader to have a better understanding of this new proposed method. Here we compare the construction of the global IC matrix K based on Atanassov ICrA and our new BF-ICrA approach. For this, we use the 5×4 MCDM example given in [START_REF] Ikonomov | ICrAData software for InterCriteria analysis[END_REF] based on the following score matrix (called sample data matrix in [START_REF] Ikonomov | ICrAData software for InterCriteria analysis[END_REF]). Each row of S corresponds to an alternative, and each column to a criterion. In [START_REF] Ikonomov | ICrAData software for InterCriteria analysis[END_REF], the authors use rows for criteria and columns for alternatives so they work with S T . Based on Atanassov's ICrA method (using unbiased algorithm presented in details in [START_REF] Ikonomov | ICrAData software for InterCriteria analysis[END_REF]) we will get the following 4 × 4 global Inter-Criteria K µ and K ν matrices

S = [S ij ] =      
K µ = [K µ jj ] =   
0.9 0 0.5 0.5 0 0.9 0.5 0.3 0.5 0.5 1 0.5 0.5 0.3 0.5 1 As we see from this D θ BI matrix, the distance of the intercriteria BBA for C 1 and C 2 with respect to the total agreement state m T (θ) = 1 is very large (i.e. 0.9018) which means that C 1 and C 2 strongly disagree in this example as we expect from a more intuitive reasoning based on K ν 12 = 0.8 value. Similar analyses can be done for all (non diagonal) elements = K PCR6 32 = (0.18, 0.58, 0.24). Based on D θ BI matrix [START_REF] Doukovska | InterCriteria Analysis Applied to EU Micro, Small, Medium and Large Enterprises[END_REF] it is obvious that no criteria strongly agree in this example so that no judicious MCDM simplification is recommended according to BF-ICrA.

   K ν = [K ν jj ] =    0

B. Example 2 (MCDM simplification)

Here we consider a more interesting example showing how an MCDM simplification is possible. We consider a 6 × 5 MCDM problem with the following score matrix.

S = [S

ij ] =     
    
It is not very obvious to identify the closeness of these criteria (if any) to know if there is some underlying relationship between them. For the analysis, we apply the BF-ICrA approach proposed in this work. After applying all derivations (similarly to those presented in Example 1), we finally get the following Does the BF-ICrA make sense in this example? The answer is positive because it suffices to remark that the columns of the score matrix are not totally independent because C 2 (A i ) = 2 • C 1 (A i ) + 3, C 3 (A i ) = C 2 (A i ) + ( being a small contamination noise), and C 5 (A i ) = 0.1 • C 4 (A i ) -5. Hence the decision based either on C 1 , C 2 or C 3 will be very close, as well as the decision based on C 4 or C 5 . Therefore the result of BF-ICrA makes sense and the expected simplification of MCDM is well obtained from BF-ICrA. If we apply AHP, which is nothing but the weighted arithmetic average and we use the normalized score matrix based on (2), or BF-TOPSIS methods to solve original MCDM (assuming equal importance of criteria), or if we apply simplified MCDM based on BF-ICrA, we will get same preference order A 1 A 2 A 4 A 5

A 6 A 3 . So, the best decision to make is A 1 in this example.

VII. CONCLUSION

In this paper we have proposed a new method called BF-ICrA to simplify (when it is possible) Multi-Criteria Decision-Making problems based on inter-criteria analysis and belief functions. This method is in the spirit of Atanassov's method but proposes a better construction of Inter-Criteria Matrix that fully exploits all information of the score matrix, and the closeness measure of agreement between criteria based on belief interval distance. This BF-ICrA approach for simplifying MCDM could deal also with imprecise or missing score values using the technique presented in [START_REF] Dezert | Multi-Criteria Decision-Making with Imprecise Scores and BF-TOPSIS[END_REF]. An application of BF-ICrA for GPS surveying problem is presented in [START_REF] Fidanova | Inter-Criteria Analysis Based on Belief Functions for GPS Surveying Problems[END_REF], and applications of BF-ICrA for simplifying and solving real MCDM problems for the prevention of natural risks in mountains will be the object of forthcoming investigations.

5 )

 5 The classical ICrA method cannot deal directly with imprecise or missing score values. IV. A NEW ICRA METHOD BASED ON BELIEF FUNCTIONS In this paper we propose a new ICrA method, called BF-ICrA for short, based on belief functions that circumvents most of the aforementioned drawbacks of classical ICrA. Here we show how to get more precisely the Inter-Criteria Belief Matrix and how to exploit it for MCDM simplification. A. Construction of BBA matrix from the score matrix From any non-zero score matrix S = [S ij ], we can construct the M × N BBA matrix M = [m ij (•)] as follows

Figure 1 .

 1 Figure 1. Flow chart of BF-ICrA method.



  Regrouping these two matrices into one matrixK = [K jj ] with components K jj = (K µ jj , K ν jj , 1 -K µ jj -K ν jj ), one gets the following global Inter-Criteria matrix K According to this K matrix it appears intuitively that none of the criterion is in strong agreement with others. We observe that criteria C 1 and C 2 are in relatively strong disagreement because K ν 12 = K ν 21 = 0.8 which is quite close to one. Criteria C 2 and C 4 are in relatively medium disagreement because K ν 24 = K ν 42 = 0.6. In this example no MCDM simplification is prescribed based on Atanassov's ICrA. To get a more precise evaluation of degree of agreement between criteria based on Atanassov's ICrA we apply formula[START_REF] Doukovska | InterCriteria Analysis approach in radar detection threshold analysis[END_REF] to get the D θ BI distance matrix from each component of K to the total agreement state m T = [m(θ), m( θ), m(θ ∪ θ)] = [1, 0, 0].

  D θ BI distance matrix from each component of K PCR6 to the total agreement state m T = [m(θ), m( θ), m(θ ∪ θ)] = [1, 0, 0] From the analysis of upper off-diagonal components of D θ BI (put in boldface for convenience) it is clear that criteria C 1 , C 2 and C 3 are in almost total agreement because their distance is close to zero. Also we can observe from D θ BI that criteria C 4 and C 5 are also very close. So the original 6× 5 MCDM problem in this example can be simplified into a 6×2 MCDM problem considering only the simplified score matrix involving only C 1 and C 4 because C 2 and C 3 behave similarly to C 1 for decision-making, and C 5 behaves similarly to C 4 . Then the simplified MCDM will have to be solved by any preferred technique.

because it suffices to multiply the scores values by -1 to reverse the preference ordering.

where the symbol means equal by definition.

called index matrix by Atanassov in[START_REF] Atanassov | Index Matrices: Towards an Augmented Matrix Calculus[END_REF].

We use K because it corresponds to the first letter of word Kriterium, meaning criteria in German. The letter C is being already in use.

for instance if we use sgn(x) = 1 if x ≥ 0 and sgn(x) = 0 if x < 0, we will obtain, in general, other K µ jj and K ν jj values.

If A j max = 0 then Bel ij (A i ) = 0, and if A j min = 0 then P l ij (A i ) = 1.

or positive consonance according Atanassov's terminology.

or negative consonance according Atanassov's terminology.

because (m ij (A i )+m ij ( Āi )+m ij (A i ∪ Āi ))(m ij (A i )+m ij ( Āi )+ m ij (A i ∪ Āi )) = 1 • 1 = 1.

We use the index T in the notation m T (•) to refer that the agreement is true, and F in m F (•) to specify that the agreement is false.

Here m 1 (•) = m jj (.), and m 2 (•) = m T (•) or m 2 (•) = m F (•)

of D θ BI to identify which criteria are in strong agreement, or not (if any).

Based on our new BF-ICrA method we first compute the 5 × 4 BBA matrix M = [m ij (•)] from the score matrix S based on formulas ( 13)- [START_REF] Zaharieva | InterCriteria decision making approach for Behterev's disease analysis[END_REF]. We get (all the values of results have been rounded at their second digit) M ≈ (0.5, 0.08, 0.42) (0.71, 0.05, 0.24) (0.18, 0.35, 0.47) (0, 1, 0) (0.25, 0.33, 0.42) (0.71, 0.05, 0.24) (0.09, 0.53, 0.38) (0.1, 0.6, 0.3) (0, 1, 0) (1, 0, 0) (0.30, 0.24, 0.46) (0.3, 0.3, 0.4) (1, 0, 0) (0, 1, 0) (1, 0, 0) (0.6, 0.1, 0.3) (0.5, 0.09, 0.41) (0.14, 0.62, 0.24) (0, 1, 0) (1, 0, 0)

The construction of Inter-Criteria Matrices K i = [K i jj ] (for i = 1, . . . , 5) from the BBA matrix M based on formulas (20)-( 22) yields the following five matrices

3) (0.6, 0.1, 0.3) (0.1, 0.6, 0.3) (0.6, 0.1, 0.3) (0.370.12, 0.51) K 5 ≈ (0.26, 0.08, 0.66) (0.12, 0.32, 0.56) (0.08, 0.5, 0.42) (0.5, 0.08, 0.42) (0.12, 0.32, 0.56) (0.40, 0.18, 0.42) (0.62, 0.14, 0.24) (0.14, 0.62, 0.24) (0.08, 0.5, 0.42) (0.62, 0.14, 0.

.5, 0.08, 0.42) (0.14, 0.62, 0.

The componentwise PCR6 fusion of all five K i matrices provides the following global Inter-Criteria matrix K PCR6 KPCR6 ≈ (0.90, 0.02, 0.08) (0.06, 0.83, 0.11) (0.55, 0.15, 0.30) (0.49, 0.25, 0.26) (0.06, 0.83, 0.11) (0.95, 0.01, 0.04) (0.18, 0.58, 0.24) (0.08, 0.80, 0.12) (0.55, 0.15, 0.30) (0.18, 0.58, 0.24) (0.89, 0.02, 0.09) (0.22, 0.48, 0.30) (0.49, 0.25, 0.26) (0.08, 0.80, 0.12) (0.22, 0.48, 0.30) (0.90, 0.02, 0.08)

Applying formula [START_REF] Doukovska | InterCriteria Analysis approach in radar detection threshold analysis[END_REF]

We see that K and K PCR6 are different specially K 23 = K 32 = (0.5, 0.4, 0.1) with respect to K PCR6