**7. Discussion**

To date, various soft set-based techniques have been applied to solve decision-making problems. Some of them have proposed novel methodology to find the solution [18,22,27,28], while some authors have made effort to adapt the well-known decision-making methods, such as SAW, TOPSIS, entropy, OWA, and OWG to the soft set theory [15,16,21,40,41]. However, a technique to solve decision-making problems based on m-polar fuzzy soft information has not been studied yet. Thus, new methodologies are proposed to handle the consensus stage and selection stage of MAGDM problem with M-pFS inputs.

For aggregating input data which take their values from [0, <sup>1</sup>]*<sup>m</sup>*, some new m-polar aggregating methods, called M-pFSMWM operator, M-pFSIOWA operator, and M-pFSIOWG operator, are developed in Sections 3 and 5. The properties comparison of these operators are summarized in Table 9. It can be seen that the most interesting property of M-pFSMWM operator is it is sensitive for different scenarios of a partial agreemen<sup>t</sup> at the consensus degree *α*. This characteristic makes the M-pFSIOWA operator more adaptable for MAGDM problem in which not only the number of individuals satisfying an alternative is important but also the weight of decision makers who agree with this decision affect the final output. Moreover, by changing the value of consensus degree *α* different cases of agreemen<sup>t</sup> are obtained. In particular, when *α* → *K*, the partial agreemen<sup>t</sup> becomes a full agreement.


**Table 9.** Properties comparison of different aggregation operators.

To reach the process of selection, we propose two procedures in Section 4.2, Algorithm 1 where the consensus stage is reached based on the new M-pFSMWM operator, and Section 5, Algorithm 2 in which the consensus stage is obtained by M-pFSIOWA or M-pFSIOWG operators are extended based on IOWA and IOWG, respectively. To reach the selection stage a new score value function, described by Equation (9), is applied. The main advantage of the proposed formulation is to rank and compare objects based on a collective m-polar fuzzy soft preference relationship. This allows us to have a ranking system of alternatives, from the most preferred element to the least preferred element, which may include some incomparable objects because of preference relationships nature.

Illustrative examples, given in Examples 2 and 3, show the application of Algorithm 1 to analyze MAGDM problems with multi-polar fuzzy soft information. The obtained results are then compared with the m-polar fuzzy soft extensions of two well-known aggregation operators IOWA and IOWG, i.e., M-pFSIOWA and M-pFSIOWG, in Example 4. Table 10 makes a comparison of the preference orders of the alternatives for methods using different aggregation operators including the new proposed M-pFSMWM method, the m-polar fuzzy soft induced ordered weighted average (M-pFSIOWA) method, and the m-polar fuzzy soft induced ordered weighted geometric (M-pFSIOWG) method. As can be seen, Hotel *h*10 is the best option for staying based on the all discussed methods except the M-pFSMWM-based method, where *h*10 is considered as the best second option for accommodation. According to final preference order obtained based on the M-pFSMWM operator, Hotel *h*1, which has the second place based on the other methods, is the best option accepted by the majority, i.e., 60%, of decision makers. Hotel *h*5 is the best place to stay in terms of all decision makers. The analysis derived in Table 10 shows a good agreemen<sup>t</sup> among thees methods, however the number of computational steps in M-pFSMWM-based algorithm is *CK*,*<sup>α</sup>* + 1 in comparison with *K* + 2 stages in Algorithm 2. On the other hand, the main disadvantage of M-pFSIOWA and M-pFSIOWG methods is that there is no unique approach to determine the associated weighting vector *w* related to the aggregation operators M-pFSIOWA and M-pFSIOWG. Finally, in Figure 4a, the overall scores of alternatives based on the different methods for case *α* = 3 are shown. Figure 4b shows the scores of alternatives obtained by using different methods for case *α* = 5. Note that, using some relative preference matrices to find the scores of alternatives in all methods, leads to record a similar trend in Figure 4b.


**Table 10.** Comparison of the alternatives' preference orders for different methods.

(**a**) The overall scores of alternatives for different methods when *α* = 3 (**b**) The overall scores of alternatives for different methods when *α* = 5

**Figure 4.** The effect of different methods on scores of alternatives for different consensus degrees.
