FOMBA Soumana, Toulouse University, France and Bamako University, Mali ZARATE Pascale, Toulouse University, France KILGOUR Marc, Wilfrid Laurier

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1 FOMBA Soumana, Toulouse University, France and Bamako University, Mali ZARATE Pascale, Toulouse University, France KILGOUR Marc, Wilfrid Laurier University, Waterloo, Canada 1

2 Introduction Aggregation Operators Determination of a Fuzzy Measure Choquet Integral Implementation Recommender System: Web Platform Conclusion 2

3 Decision Deck: Multcriteria DSS Web Services Multicriteria Recommender System Web based Generation of Total Order Performances Matrix Partial Order 3

4 Notations X = {a, b, } set of alternatives N = {1,, n} set of indices of criteria a b : a is preferred to b a ~ b iff a b and b a : a is indifferent to b 4

5 ψ a 1,..., a n = w i a i n i=1 where, for i = 1, 2,, n, w i [0, 1] represents the weight of criterion i and n i=1 w i = 1 5

6 OWA w (a 1,..., a n ) = n i=1 w i a ( i ) where W = (w1,...,wn) is a weight vector satisfying wi [0, 1] for all i and n i=1 w i = 1 6

7 Definition: A fuzzy measure μ on N is a function μ: 2 N [0, 1] which is monotonic, that is, μ(s) μ (T) whenever S T, and satisfies the limit conditions μ( ) = 0 and μ(n) = 1 Let μ be a fuzzy measure on N. The Choquet integral of x R n with respect to μ is defined by n C μ(x) := i=1 x i [μ(a i ) - μ(a i+1 )] The index of importance or Shapley s value for criterion i with respect to μ is defined by Φ(µ, i) = T N\i (n t 1)! t! [µ(t i) µ(t)] n! If μ is additive, we have μ(t i) μ(t) = μ(i) The index of interaction I(μ,ij) is in the range [-1, 1] for all i, j N. If the index is positive, then there is a synergy between these two criteria. Conversely, if the index of interaction is negative, the criteria are called redundant n t 2! t! I μ, ij = ( n 1! ij μ)(t) T N\ij ( ij μ)(t) := μ(t ij) μ(t i) μ(t j) + μ(t) 7

8 Identification of Choquet Integral capacity Respect to capacity representing Decision Maker preferences Notations Subset O X of alternatives of interest Partial preorder O on the subset O Partial preorder N on the set of criteria, N a O b is equivalent to C μ (a) > C μ (b) a ~ O b is equivalent to C μ (a) = C μ (b) where μ is the capacity to be determined Similarly, i N j can be taken to be equivalent to Φ(μ, i) Φ(μ, j) on the set of criteria N and i ~ N j to Φ(μ, i) = Φ(μ, j) on the same set 8

9 Translating all the preferences expressed by the decision-maker using the rules above produces an optimization problem whose solution is the fuzzy measure μ on N Min or Max μf(. S.. i) μ S 0, i N, S N \i, μ = 0, μ N = 1, Subject to C μ(a) C μ(b) δc, Φ(μ, i) Φ(μ, j) δsh where F is an objective function that depends on the method of identification chosen. Among the main methods are: Approaches based on least squares Approaches based on linear programming Method of minimum variance 9

10 Kappalab package: Minimum variance method JRI (Java program within R Interface) JDK (Java Development Kit) J2EE application librairies 10

11 1. Define the set of criteria used for the decision problem, entered by the user 2. Define the performance of each alternative for each criterion. This is the performance matrix, entered by the user 3. Establish a partial order on the subset of alternatives specified by the user. Preference is defined for a pair of alternatives by preference value, which must be one of 1, if the first alternative is preferred to the second 1, if the second alternative is preferred to the first 0, if both alternatives are indifferent or equivalent 4. Using the preference table, create an R matrix containing all preferential information 5. Use the function mini.var.capa.ident of package Kappalab to determine the capacity corresponding to the preferential information 6. Recover the resulting capacity from the R platform 11

12 Kappalab package: determine a final ranking 1. Calculate the Choquet integral value of each alternative 2. Sort the alternatives based on the overall score of each one Note that Step 1 can be simplified if the capacity determined above is 2-additive 12

13 Example from Rolland A., Ah-Pine J. and Mayag B. (2015) We want to evaluate four chefs based on their ability to prepare three dishes Frog legs (CG) Steak tartare (ST) Scallops (SJ) Evaluation of the 4 cookers a, b, c, d on a scale 0 to 20 is given CG ST SJ a b c d Reasoning of the 18 decision maker When a chef is known for his preparation of Scallops, it is more important that he prepares frog legs well, as compared to steak tartare; Conversely, when a cook does not do a good job preparing scallops, it is more important that he prepares steak tartare well, as compared to frog legs. Thus, we can conclude that the decision maker thinks : a b c d And the implemented system must obtained the same result 13

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15 Total order generation using a fuzzy measure Webplatform under development Using other operators and decision aid concepts (bicapacity) Automatic adjustement of aggregation technics depending of user s profile and context of use 15

16 Labex CIMI : Marc Kilgour, Wilfrid Laurier University, Waterloo, Canada Scientific Expert Toulouse University, France May-June