Interaction in Choquet inegral model

The candidates, evaluated by the executive manager and their scores for every criterion
on a [0, 10] scale are presented in Table 9. We suppose that the criteria have to be
maximized.
Now, suppose that the executive manager (the DM) on the basis of her preference structure
is able to give only the following partial information on the reference actions X ′ = {Arthur,
Bernard, Charles, Germaine, Irene}:
(a) The candidate Charles (C) is preferred to candidate Bernard (B);
(b) The candidate Germaine (G) is preferred to candidate Arthur (A);
(c) The candidates Charles (C) and Irene(I) are indifferent.
A B C D E F G H I
8 3 10 5 8 5 8 5 0
6 1 9 9 0 9 10 7 10
7 10 0 2 8 4 5 9 2
5 10 5 9 6 7 7 4 8
Table 9: Evaluation matrix.
Step 1 Linear program (P L 1 ) with nonnegative slack variables α + CB , α− CB , α+ GA , α− GA , α+ CI
and α − CI .
Minimize Z 1 = α + CB + α− CB + α+ GA + α− GA + α+ CI + α− CI .
Subject to
C µ (C) − C µ (B) + α + CB − α− CB ≥ ε
C µ (G) − C µ (A) + α + GA − α− GA ≥ ε
C µ (C) − C µ (I) + α + CI − α− CI = 0
C µ (A) = 5 + µ 123 + µ 13 + µ 1
C µ (B) = 1 + 2µ 134 + 7µ 34
C µ (C) = 5µ 124 + 4µ 12 + µ 1
C µ (D) = 2 + 3µ 124 + 4µ 24
C µ (E) = 6µ 134 + 2µ 13
C µ (F ) = 4 + µ 124 + 2µ 24 + 2µ 2
C µ (G) = 5 + 2µ 124 + µ 12 + 2µ 2
C µ (H) = 4 + µ 123 + 2µ 23 + 2µ 3
C µ (I) = 2µ 234 + 6µ 24 + 2µ 2
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