Optimal weighting method for interval-valued intuitionistic fuzzy opinions

In this work, we propose a method to achieve consensus in a group decision making situation, where the opinions are described by interval-valued intuitionistic fuzzy sets. Optimality is achieved by minimizing weighed incoherencies. An illustrative example is proposed.


Introduction
Since unanimity is rarely achieved in group decision making, a certain level of consensus might be acceptable. The achieved consensus must take into consideration human uncertainty, to do so, we model the expressed opinions by interval-valued intuitionistic fuzzy numbers. In the rest of this manuscript the needed background for fuzzy logic is presented in Section 2, while Section 3 encompasses the used algorithm with an illustrative example.

Preliminaries
In classical sets, each element either belongs to a certain set or not at all, while in fuzzy set theory a certain degree of membership is tolerated [13]. Let X be a set and F be a fuzzy set in X,where F is defined as follows: where µ F (x) is the degree of membership of x in F in the unity interval: Atanassov [1,2] extended the notion of fuzzy sets to intuitionistic fuzzy sets (IFS). An intuitionistic fuzzy set A is defined as follows: where µ A (x) and ν A (x) are respectively the membership function and the non-membership function, with the following conditions: The hesitancy function can be computed by the following formula: The fuzzy sets were presented in order to permit human uncertainty, while it is counterintuitive to demand an exact membership function and non-membership function. In that sense Atanassov and Gargov [4] extended the IFS to interval-valued intuitionistic fuzzy sets (IVIFS) fulfilling the following: are respectively the membership interval and the non-membership interval, and for these two intervals it holds that [4]: For convenience, we note an interval-valued fuzzy number as be a collection of interval-valued intuitionistic fuzzy numbers, the main aggregation operators are the interval-valued intuitionistic fuzzy weighting averaging IIFWA, and the interval-valued intuitionistic fuzzy weighting geometric IIFWG [11], hence the aggregated value according to IIFWA is: and w i are the weights of the respective β i .
The main question is how to attribute the correct weight to each decision.

Proposed method
Several method exists in the literature to attribute the correct weights [5,7,8,12,14]. Here we propose to follow the procedure proposed in [7] to the IVIFS. The desired consensus is achieved by minimizing the following function: , m is a positive integer (m > 1), S(β i , β) is the similarity between the i-th decision and the consensus, c is a real number (c > 1).
Several methods have been proposed to compute similarity from a distance [6,9,10], here we adopt the Hamming distance for IVIFS [3], and derive the similarity as by Santini and Jain [9] to ease computation S = 1 − D. Hence, the distance between two IVIFS β 1 and β 2 is:

Algorithm
Step 1: Each expert E i : 1 ≤ i ≤ n assesses each alternative using an IVIFS.
Step 3: Compute the aggregated consensus at Step l: Step 4: Let W l = w Step 5: If W l+1 − W l > ε, set l = l + 1 and go to Step 3. Else Stop. We choose m = 2, c = 1.5 and W 0 = (1, 0, 0). Table 1 resumes the evolution of weights in each iteration.

Conclusion
In this work, we adapted Lees algorithm to achieve group consensus in the interval-valued intuitionistic fuzzy context. We restricted ourselves to the interval-valued intuitionistic fuzzy weighting averaging operator to merge opinions, used the hamming metric to compute their distances and derived similarities as a distance dual. In future research, we will investigate different combinations of aggregation operators, similarities and distances that may be more appropriate in such situations.