*7.3. Comparative Studies*

In order to analyze the alignment of the proposed approach's results with the existing theories and to validate our proposed results, the score values corresponding to different operators are given in Table 6. The operators in the considered existing theories are: probabilistic dual hesitant fuzzy weighted average (PDHFWA) by Hao et al. [42], hesitant probabilistic fuzzy Einstein weighted average and Einstein weighted geometric (HPFEWEA, HPFEWEG) by Park et al. [50] and hesitant probabilistic fuzzy weighted average (HPFWA), hesitant probabilistic fuzzy weighted geometric (HPFWG), hesitant probabilistic fuzzy ordered weighted average (HPFOWA) , hesitant probabilistic fuzzy ordered weighted geometric (HPFOWG) aggregation operators by Xu and Zhou [48]. Noticeably, the approach outlined by Hao et al. [42] by utilizing PDHFWA operator figures out *A*2 as the best alternative and the least preferred alternative *A*4 remains same as that of our proposed approach. However, if we consider only the probabilistic hesitant fuzzy information and ignores the non-membership probabilistic hesitant values, then the best alternative starts fluctuating among *A*1 and *A*3 by varying the different aggregation operators and the least preferred alternative remains same as *A*4, which coincides the outcomes of our proposed approach. This variation is due to the negligence of the non-membership values and their corresponding probabilities. Thus, the proposed approach is advantageous among the traditional approaches because it remains firm on the same output ranking for different operators. Moreover, the best alternative chosen by the proposed approach remains the same as that with that of the existing approaches signifies that the proposed approach is the valid one.

Further, a deep insight into the comparison of our method with the existing ones is given by comparing the characteristics of all the approaches with the proposed one. In Table 7, it can be seen that the approaches put-forth by Hao et al. [42] and Xu and Zhou [48] considers multiple experts in analysis process whereas Park et al. [50] does not consider the multi-expert problems. All the existing approaches are the probabilistic approaches so they consider probabilities corresponding to their considered membership or non-membership values. Moreover, it is analyzed that the method proposed by [42] considers the non-membership probabilistic information but the rest two only considers the hesitant values and their probabilities. In all the three existing approaches, the weights are not derived by using any non-linear technique such as maximum deviation method for determination of weights but the weights corresponding to two different distance measures are considered in the proposed methodology.

In addition to above comparison studies, we elicit some characteristic comparison of our approach with existing DM methods proposed in [42,48,50] which are tabulated in Table 7.

In Table 7, the symbol ' -' describes that the corresponding DM approach considers more than one decision maker, handles probabilities, accounts for non-membership entities and has weights derived by the non-linear approach, whereas the symbol '×' means that the associated method fails. The symbols tabulated in Table 7 depicts that the MCDM mentioned in [42] as well as [48] consider multiple multiple decision-makers whereas the approach utilized by [50] consists of preference evaluations through single expert. It is seen that all the three considered approaches considers the probabilities along with their respective fuzzy environments whereas only [42] considers only the

non-membership values along with the membership ones while the other two considers only the membership value ratings. On the other hand, none of the existing approach among the specified ones, adopt a non-linear weight determination technique. Thus, it is analyzed that our proposed approach consists of all the four said characteristics and thus it deals with the real life situations, more efficiently as compared to the existing approaches [42,48,50].


**Table 6.** Comparison of overall rating values and ranking order of alternatives.

**Table 7.** Characteristic comparison of the proposed approach with different methods.

