A Large-Scale Group Decision-Making Consensus Model considering the Experts’ Adjustment Willingness Based on the Interactive Weights’ Determination
Algorithm 2
The determination method of the DMs’ weights and the attributes’ weights.
Input: The values of ,,, and , and the subgroups G1, G2, …, GK.
Output: The final values of the DMs’ weights and the attributes’ weights.
Step 1. Calculate the evaluation information for each subgroup, k = 1, …, K, where
Step 2. Compute the evaluation information differences between each DM and his or her subgroup by using equation (4). For instance, the difference between the DM em and the subgroup Gk is
Step 3. The initial weight values for each DM are given. Note that the evaluation difference of the DM and his or her subgroup is an important basis for determining the initial value of the DM’s weight. The higher the difference between them, the lower the initial value of the DM’s weight. Conversely, if the DM’s opinion or preference are closer his or her subgroup, a higher weight should be given initially. Therefore, the determination of the initial weight value is given as follows.
Step 4. Calculate the weighted attributes’ weight values for each subgroup. The calculation formula is
Step 5. Calculate the distances of weight value between each DM and his or her subgroup. For instance, if the distance dH(em, Gk) between the DM em and the subgroup Gk is larger, the weight value of the DM em in the subgroup Gk should be decreased to a certain extent. On the contrary, the value of dH(em, Gk) is lower, it indicates that the subjective attitude of the DM em is closer to the collective attitude of the subgroup Gk, the weight value of the DM em in the subgroup Gk should be improved to some extent. dH(em, Gk) can utilize the Hamming distance given by
Step 6. Updating the weight value for each DM. The updating formula is:
where represents the importance of adjusting DMs’ weight for each time, the higher the value of , the larger the importance of updating DMs’ weight in this time, and is the deviation proportion of the DM em in the subgroup Gk, and the compute formula is shown as follows.
Step 7. Computing the proportion of the allowed modification range, and update the weight value for each DM. The greater the proportion of the allowed modification range + , the more concessions the DM makes in order to obtain the subgroup consensus, and the larger the proportion should be given when adjusting the DMs’ weight. The calculation and modification equations are performed so that
where represents the proportion of the allowed modification value of the DM em in the subgroup Gk, and meets condition 0 ≤ ≤ 1. The normalization process should be then carried out.
Step 8. Recalculate the weighted attributes’ weight values for each subgroup by using equation (9), and the calculation results are presented that
Step 9. Show the weight values of the aggregated attributes in this time to each DM, and then obtain the satisfaction degree provided by em. The values of provided by em are chosen from the set {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}, which means {extremely dissatisfied, very dissatisfied, dissatisfied, slightly dissatisfied, average, slightly satisfied, satisfied, very satisfied, extremely satisfied}.
Step 10. Re-update the weight value for each DM through the satisfaction degree . The greater the values of , the more satisfied the DM em is with the current weight value, the higher the enthusiasm for the current decision results, and the weight value of the DM em in the subgroup Gk should be improved to some extent. Conversely, if the value of is smaller, the weight value of the DM em in the subgroup Gk should be reduced appropriately. The updating formula of the DMs’ weights is that:
The weight value in this time is the final weights for each DM. The final weight value of the DM em in the subgroup Gk is denoted as , that is,
Step 11. Compute the aggregated attributes’ weight values for each subgroup according to the DMs’ final weight values by using equation (9), that is,
The weight value in this time is the final attributes’ weights. The final weight value for each attribute in the subgroup Gk is denoted as , that is,
