3.3.2. Sensitivity to Weights and Data Variation

In order to assess the sensitivity of the tool to the weights' variation, two steps have been followed. Initially, the weights have been considerably varied to assess whether the final ranking is affected by such variations and consequently assess the efficiency of the tool. To this end, the scenarios described in Section 3.3.1 have been considered. The AHP pairwise comparisons were completed by the authors based on their knowledge and expertise in the field. In each of the said scenarios, one criterion is strongly prioritized over the other two criteria. A scenario assuming an equal weighting among the criteria has also been included. The pairwise comparisons of the aforementioned scenarios and the resulting weights are demonstrated in Table 7. All scenarios were checked for consistency, indicating a consistency ratio value below the threshold value of 0.1. The consistency ratio is a metric that indicates the consistency between pairwise comparisons. The rankings obtained from the aforementioned scenarios are presented in Table 8. The min-max normalization was considered for the normalization of the initial data. The results suggested different rankings for the different scenarios considered, and thus, the proposed method was found to be sensitive to the variations of the weight derived from considerable changes in the decision maker's judgments.


**Table 7.** Pairwise Comparisons and Resulting Weights for Different Scenarios.

In the second step of the sensitivity analysis, the rank stability of the MCDM tool was evaluated by adding noise to the criteria weights. To this end, the scenario for which environmental impact was prioritized (Scenario 2) was taken as the reference scenario, and minor adjustments to the user judgments were made. Therefore, based on the AHP scale of Table 1, three alternatives to the reference scenario were considered, for which one scale above or below the reference judgments was accounted for. The pairwise comparisons of the aforementioned scenarios are presented in Table 9. The results showed that the considered minor weight adjustments did not alter the ranking order (except for an exchange between two places of the alternative scenario 3), and hence, the proposed method does not appear to be affected by such minor weight adjustments.


**Table 8.** Ranking Obtained from The Different Weighting Scenarios of Table 6.

**Table 9.** Pairwise Comparisons for The Assessment of Small Weights Variations.


In order to test the stability of the method to small changes in the values of the initial data, 1000 perturbated samples of the original data were simulated in each of the following cases. It is assumed that the errors that perturb the initial data follow a normal distribution with zero mean and standard deviation 0.01, 0.05, 0.1, 0.25, and 0.5 of the standard deviation of the corresponding indices of the original data. The simulated samples were normalized with the three normalization methods (min-max, z-score and proportionate), ranked with respect to the overall sustainability index, and the mean rank of each material was calculated for each normalization method and for each selected value of the standard error. In Table 10, the mean ranking of each material based on the 1000 simulated samples for the Min–Max and z-score normalization methods are presented for all the selected values of errors' standard deviation. In Table 11, the corresponding mean ranks for the proportional normalization method are presented. The proposed method is fairly stable in terms of the mean rank for each normalization method and for a relatively large value of the standard deviation of the errors.


**Table 10.** Mean Ranking for The Min–Max and Z-score Normalization Methods.

**Table 11.** Mean Ranking for The Proportionate Normalization Method.


In Figure 1, the mean ranks and an interval of ±one standard deviation of the rankings are presented for the different levels of noise variation and the three studied normalization methods. As it is observed in Figure 1, the larger the standard deviation of the errors, the larger the variability of each material ranking. Despite the increase in the variation of the rankings, the mean rankings seem to converge to the initial ranking.

**Figure 1.** Mean Ranks ± One Standard Deviation of The Rankings for The Different Levels of Noise Variation (percSD).
