Interaction in Choquet inegral model

µ 12 ≥ µ 1 ; µ 12 ≥ µ 2
µ 13 ≥ µ 1 ; µ 13 ≥ µ 3
µ 14 ≥ µ 1 ; µ 14 ≥ µ 4
µ 23 ≥ µ 2 ; µ 23 ≥ µ 3
µ 24 ≥ µ 2 ; µ 24 ≥ µ 4
µ 34 ≥ µ 3 ; µ 34 ≥ µ 4
µ 123 ≥ µ 12 ; µ 123 ≥ µ 13 ; µ 123 ≥ µ 23
µ 124 ≥ µ 12 ; µ 124 ≥ µ 14 ; µ 124 ≥ µ 24
µ 134 ≥ µ 13 ; µ 134 ≥ µ 14 ; µ 134 ≥ µ 34
µ 234 ≥ µ 23 ; µ 234 ≥ µ 24 ; µ 234 ≥ µ 34
µ 1234 ≥ µ 123 ; µ 1234 ≥ µ 124 ; µ 1234 ≥ µ 134 ; µ 1234 ≥ µ 234
µ 1234 = 1
ε ≥ 0
The linear program (P L 2 ) is feasible and optimal solution of (P L 2 ) is Z2 ∗ = 3.8 > 0,
then we can conclude that, the preference information {P, I} is representable by a general
Choquet integral model. Moreover, the results obtained by solving (P L 2 ) are given by
Tables 10 and 11.
S
µ(S)
∅, {1}, {3}, {4}, {1, 3}, {1, 4}, {3, 4} 0
{2}, {2, 3}, {2, 4}, {2, 3, 4} 0.9
{1, 2}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, N 1
Table 10: Capacity compatible with (P L 2 )
x A B C D E F G H I
C µ (x) 6 3 9 8.6 6 8.6 9.8 6.8 9
Table 11: General Choquet integral corresponding at previous capacity µ
Step 3 In order to know if the interaction within the subset of criteria {1, 2, 3} is
necessarily negative (resp. positive). We obtain the P L 123
NN (resp. P L123 NP ) by adding
at the previous linear program (P L 2 ) the constraints I µ 123 ≥ 0 (resp. I µ 123 ≤ 0) with
I µ 123 = µ 1234 +µ 123 −µ 124 −µ 134 −µ 234 −µ 12 −µ 13 +µ 14 −µ 23 +µ 24 +µ 34 +µ 1 +µ 2 +µ 3 −µ 4 .
• The linear program P L 123
NP is feasible and the optimal solution is Z∗ 3 = 3.8 > 0.
Then interaction within {Educational degree, Professional experience, Age} is not
necessary positive. Moreover, the results obtained by solving P L 123
NP
Tables 12 and 13 (with I µ 123 = −2.375 < 0).
25
are given by