**7. Discussion**

In this section, a detailed analysis about the above introduced algorithm and the obtained result based on the algorithm is carried through. The weighted quadripartitioned similarity measure between the alternatives (A*<sup>j</sup>* , *j* = 1, 2, 3, 4) and the positive ideal quadripartitioned bipolar neutrosophic solution is shown in Table 2 and the same with the negative ideal quadripartitioned bipolar neutrosophic solution is given in Table 2. The main objective of the proposed algorithm i.e., the average ideal solution is obtained. The ranking results based on these similarity measures is highlighted in the final Table (Table 3).

Deli et al. [28] and Sahin et al. [27], in their study on bipolar neutrosophic sets, obtained the ranking result without considering the negative ideal solution. From their algorithm, it can be seen that if the decision maker is asked to choose an alternative emphasizing more on the satisfaction degree of the given criteria than the satisfaction degree of the counter-property of the criteria, the similarity measure with the positive ideal solution can be used to get the most suitable alternatives. In the reverse case, the similarity measure with negative ideal solution will be more fruitful.

In the case of QSVBNS, our algorithm follows the same footstep as Deli et al. [28]; however, the major difference is that we have used the average of the measure values of positive ideal solution and negative ideal solution. From the proposed algorithm, it seems that the ranking results to choose the best alternatives using a positive ideal solution and a negative ideal solution will give exactly the opposite ranking, but this is not the case, as can be seen in Table 2. The similarity index *p* in the weighted quadripartitioned similarity measure also has a big role to play. From the study, we have found that as the similarity index starts to take more higher values, the formula predicts more accurately, as the difference between the first two choices of the alternatives starts increasing for higher values of *p*. From Figure 2, it is observed that for *p* = 1, 2, the alternative values are quite close and A<sup>3</sup> comes out as the best alternative with a very small margin from A1. For *p* = 3, A<sup>1</sup> overtakes A<sup>3</sup> as the best alternative and as *p* stars increasing, A<sup>1</sup> remains the best among the four alternatives and the margin of the second best choice A<sup>3</sup> starts increasing. The differences between the alternative values is shown in Figure 2 for *p* = 1, 2, 3. In Figure 3, it can be observed that for *p* = 4, 5, 6, A<sup>1</sup> enlarge by a

bit more margin than A3. In Figure 4, the comparison between the best two alternatives A<sup>1</sup> and A<sup>3</sup> is shown. It can be seen that for a very large value of *p*, A<sup>1</sup> − A<sup>3</sup> ≈ 0.1.


**Table 2.** Similarity measures for different values of *p*.

**Figure 2.** Comparison between the alternatives for different similarity indexes.

**Figure 3.** Comparison between the alternatives for different similarity indexes.

**Figure 4.** Comparison between the best two alternatives A<sup>1</sup> and A<sup>3</sup> for some higher similarity index.

From the above analysis it is clear that A<sup>1</sup> is the most suitable alternative to the decision maker. The proposed method can get the better of the decision making problem with quadripartitioned bipolar neutrosophic information. Also, the higher similarity index produces a more accurate method by indicating clear differences between the alternatives. A comparison of the above discussed MCDM problem is made in a fuzzy system and a bipolar fuzzy system, where it is observed that in a fuzzy system, the same conclusion is reached at *p* = 5 and in bipolar fuzzy system the result fluctuate between A<sup>1</sup> and A3. So, we think that the proposed method can serve immensely in decision making purpose.


**Table 3.** Ranking results.
