**2. Methodology**

The method of plithogenic SWARA-TOPSIS is used to find the ranking of the alternatives. The steps involved are as follows:

Step 1: The initial decision-making matrix of order *u* � *v* with *u* alternatives and *v* criteria is constructed from the expert's perspective. This matrix consists of the criterion satisfaction by the alternatives, and the representation is made by using linguistic variables such as very high, high, moderate, low, and very low. The linguistic terms are not confined to these values alone. In general, a minimum of two expert's opinions is considered in framing the initial decision-making matrix. The aggregate expert's opinion is obtained using plithogenic intersection operators based on the representations (Fuzzy/intuitionistic/neutrosophic) of the linguistic variables.

Plithogenic Fuzzy Intersection *a* V *<sup>F</sup> b:*

Plithogenic Intuitionistic Intersection ð Þ *a*1, *a*<sup>2</sup> ∧ *IFS*ð Þ¼ *b*1, *b*<sup>2</sup> ð Þ *a*<sup>1</sup> ∧ *<sup>F</sup> b*1, *a*<sup>2</sup> ∨ *<sup>F</sup> b*<sup>2</sup> Plithogenic Neutrosophic Intersection ð Þ *a*1, *a*2, *a*<sup>3</sup> ∧ *<sup>P</sup>*ð Þ¼ *b*1, *b*2, *b*<sup>3</sup> *a*<sup>1</sup> ∨ *<sup>F</sup> b*1, <sup>1</sup> <sup>2</sup> ½ � ð Þþ *a*<sup>2</sup> ∧ *<sup>F</sup> b*<sup>2</sup> ð Þ *a*<sup>2</sup> ∨ *<sup>F</sup> b*<sup>2</sup> , *a*<sup>3</sup> ∧ *<sup>F</sup> b*<sup>3</sup> � �

$$a \bigwedge\_F b = ab, a \lor\_F b = a + b - ab$$

Step 2: The criterion weights are obtained by the method of SWARA, which are as follows:


$$w\_h = \frac{q\_h}{\sum\_{k=1}^{v} q\_k}$$

where *qj*, the recalculated weight

$$q\_h = \begin{cases} \mathbf{1} \, h = \mathbf{1} \\ \frac{q\_{h-1}}{k\_h} \quad h > \mathbf{1} \text{ and } k\_h = \begin{cases} & \mathbf{1} \, h = \mathbf{1} \\\\ s\_{h+1} \, h > \mathbf{1} \end{cases} \end{cases}$$

Step 3: After finding the criterion weights by the method of SWARA, the aggregate normalized weighted matrix *D* = (*dih*) is determined by using any of the normalization techniques before which the criteria are classified as benefit criteria and cost criteria, where the former must be maximized and the latter to be minimized. The four normalization techniques are shown in **Table 2**.

Step 4: The positive ideal solution *D*<sup>+</sup> = *d*<sup>þ</sup> <sup>1</sup> , *d*<sup>þ</sup> <sup>2</sup> , *d*<sup>þ</sup> <sup>3</sup> , … *d*<sup>þ</sup> *v* � � = max (*dih*) for benefit criteria and min (*dih*) for cost criteria. The negative ideal solution

*D*� = *d*� <sup>1</sup> , *d*� <sup>2</sup> , *d*� <sup>3</sup> , … *d*� *v* � � = min (*dih*) for benefit criteria and max (*dih*) for cost criteria.


**Table 2.** *Normalization techniques.*

Step 5: *F*<sup>þ</sup> *<sup>i</sup>* , the distance between the alternatives and the positive ideal solution *F*� *<sup>i</sup>* is the distance between the alternatives and the negative ideal solution is calculated as follows:

$$F\_i^+ = \sqrt{\sum\_{h=1}^v \left(d\_h^+ - d\_{ih}\right)^2}; i = 1, 2, \dots u$$

$$F\_i^- = \sqrt{\sum\_{h=1}^v \left(d\_h^- - d\_{ih}\right)^2}; i = 1, 2, \dots u$$

Step 6: The relative closeness to the ideal solution *Ri* <sup>¼</sup> *<sup>F</sup>*� *i F*þ *<sup>i</sup>* �*F*� ð Þ*<sup>i</sup>* is determined and the preferential ranking of the alternatives is made by the values of *Ri*. The alternatives with high scores are ranked from high to low.
