A New TOPSIS Approach Using Cosine Similarity Measures and Cubic Bipolar Fuzzy Information for Sustainable Plastic Recycling Process

Algorithm 2

Extension of TOPSIS towards CBF information.

	Step 1. Identify MCGDM problem:
	Suppose that is the set of DMs, is the set of alternatives, and is the set of criterion.
	Step 2. Construct the weighted criterion matrix with the help of linguistic variables that are given in Table 3. ,
	where is the fuzzy weight assigned by the decision maker to the criterion by considering the values of linguistic variables.
	Step 3. Obtain normalized weighted criterion matrix ,
	where .
	Step 4. Construct weight vector as follows: .
	Step 5. Construct CBF-decision matrices ,
	where is a CBFS element which is assigned by the decision maker . The unanimous CBF decision matrix can be obtained as follows ,
	where .
	Step 6. In many decision-making problems, the criteria under consideration are of two types, i.e., cost and benefit. Therefore, we need to normalize our unanimous CBF decision matrix ,
	The complement can be computed by using Definition 2.9.
	Step 7. Construct weighted normalized CBF decision matrix as follows: ,
	where .
	Step 8. Obtain the CBF-PIS and CBF-NIS given by the following formulas: ,
	.
	Step 9. Compute the cosine similarity measure between each alternative and PIS and the cosine similarity measure between each alternative and NIS by utilizing equations (3)–(12).
	If we use equation (3), then
	If we use equation (7), then
	If we use equation (12), then
	Step 10. Calculate the value of closeness coefficient of each alternative to ideal solution by using the following:
	If we use , then
	If we use , then
	If we use , then
	Step 11. Rank the alternatives by arranging the values of closeness coefficients in the descending order. The best alternative has the maximum value of closeness coefficient.