MCGDM based on VIKOR and TOPSIS methods using spherical interval valued fuzzy soft with aggregation operators

: Spherical interval valued fuzzy soft set (SIVFS set) has much stronger ability than Pythagorean interval valued fuzzy soft set and interval valued intuitionistic fuzzy soft set. Now, we talk about aggregated operation for aggregating SIVFS decision matrix. TOPSIS and VIKOR methods are strong point of view for multi criteria group decision making (MCGDM), which is a various extensions of interval valued fuzzy soft sets. We talk through a score function based on aggregating TOPSIS and VIKOR method to the SIVFS-positive ideal solution and the SIVFS-negative ideal solution. Also TOPSIS and VIKOR methods are provides the weights of decision makings. To ﬁnd out the optimal alternative under closeness is introduced.


Introduction
Decision making (DM) problem indicates the finding of best optional alternatives. Hwang and Yoon [6] was discussed by multiple criteria decision making (MCDM) methods. The matrix form of MCDM problem as: Here A 1 , A 2 , ..., A n are called possible alternatives means which decision makers have to choose, B 1 , B 2 , ..., B m are called criteria means which alternative effecting are calculated and x ij means estimate of A i with respect to B j . These two methods (TOPSIS and VIKOR) for DM problems have been studied by Adeel et al. [1], Akram et al. [2], Boran et al. [4], Eraslan et al. [5], Peng et al. [17], Xu et al. [22] and Zhang et al. [27]. In 2021, Zulqarnain et al. discussed the TOPSIS extends to interval valued intuitionistic fuzzy soft sets (shortly IVIFSS). He also discussed a new type of correlation coefficient under IVIFSS's [28]. In TOPSIS method consists of distances to positive ideal solution(PIS) and negative ideal solution(NIS), and calculate a preference order is ranked under relative closeness, and finding a combination of these two distance measures. In VIKOR method focal point on ranking and selecting from a set of alternatives, and compute compromise solutions for a problem with inconsistent criteria, which can help the decision makers to get a final decision [14,15]. Opricovic et al. [16] discussed VIKOR method using fuzzy logic. Tzeng et al. [19] discussion about comparison of VIKOR with TOPSIS methods using public transportation problem.
Definition 2.2 Let U and E be the universe and set of parameter respectively. The pair (Υ, X) or Υ X is called a SIVFS set on U if X E and Υ : X → SIV F U , where SIV F U is denote the set of all spherical interval valued fuzzy subsets of U . (ie) : e ∈ X, u ∈ U .
: u ∈ U . This collection is called aggregate spherical interval valued fuzzy set of SIVFS set Υ X . The positive membership function δ θ * .., m . This matrix is called SIVFS aggregate matrix of SIV F S agg ( cΥ X , Υ X ) over U . ∀i .
Proof. The proof of Theorem 3.4 by Definition 3.1 and Definition 3.2.
We can make a MCGDM based on SIVFS set by the following algorithms: Algorithm-I Step 1: Form SIVFS set Υ X over the universal U.
Step 2: Calculate the cardinalities and cardinal set cΥ X of Υ X .
Step 4: Find the score function S c (u) = Step 5: Find the best alternative by max i S c (u i ).
Example 3.5 An automobile company produces ten different types of scooter U = {S 1 , S 2 , ..., S 10 } and five parameters namely E = {e 1 , e 2 , ..., e 5 } consists of fuel tank capacity, better style, better price, more mileage, more durable respectively. Suppose that a customer has to establish which scooter to be purchased ? Each scooter is evaluated and which is a subset of parameters. That is X = {e 1 , e 2 , e 3 , e 4 } ⊆ E . We appeal to algorithm-I as follows.
Step-1: Form SIVFS set Υ X of U is defined below:  .
Step-4: The score function S c (S i ) as follows.

Scooter
Sc Step 5: Since max i S c (S i ) = −0.00094 . Hence the customer to be purchased by the scooter S 3 .

Algorithm-II
Step-1: Form spherical interval valued fuzzy soft matrix (SIVFS matrix) on the basis of the parameters.
Step-2: Case-I Obtain the choice matrix for the positive, neutral and negative-membership of SIVFS matrix (weights are equal).
Case-II Find the choice matrix for the positive, neutral and negative-membership of SIVFS matrix (weights are unequal). Step-3: Step-4: Find the best alternative by max i S(u i ).
Case-I: By Example 3.5,

Algorithm-III
Step-1: Find the spherical interval valued fuzzy weighted averaging numbers(SIVFWANs) Step-3: Find the best alternative by max i S c (u i ).

Comparison Analysis for SIVFS-Methods:
Comparison analysis of final ranking as follows: M ethods Ranking of alternatives Optimal alternatives Therefore the customer to be purchased by the scooter S 3 .

MCGDM based on SIVFS-TOPSIS aggregating operator
Algorithm-IV (SIVFS-TOPSIS) Step-1: Suppose that the finite decision makers namely D = {D i : i ∈ N} and the finite collection of alternatives namely C = {c i : i ∈ N} and finite family of parameters namely D = {e i : i ∈ N} .
Step-2: Form a linguistic variable with weighted parameter matrix Here the weight ω ij means D i to P j by considering linguistic variables.
Step-8: Find the closeness of ideal solution by C L * (c j ), Step-9: Find the rank of alternatives using closeness coefficients under the order of decreasing (or) increasing.

MCGDM based on SIVFS-VIKOR aggregating operator
Algorithm-V (SIVFS-VIKOR) Step-1: Suppose that the finite decision makers namely D = {D i : i ∈ N} and the finite collection of alternatives namely C = {c i : i ∈ N} and finite family of parameters namely D = {e i : i ∈ N} .
Step-2: Form a linguistic variables with obtain weighted parameter matrix Here the weight ω L ij , ω U ij means that D i to P j .
Step-3: Form weighted normalized decision matrix  Step-4: Form decision SIVFS matrix ..., c L+ l ,c U + l = ∨ k c L jk ,c U jk , ∧ k c L jk ,c U jk , ∧ k c L jk ,c U Figure 2 Graphical representation using MCGDM based on VIKOR.

Comparison and discussion
These two methods are assume a scalar component for each criterion and these two methods are different from normalization approach. In TOPSIS utilize to vector normalization approach and VIKOR utilize to linear normalization approach. The major difference between two methods looks in the aggregation function. We can finding ranking of values using an aggregating function. The best ranked alternative by VIKOR is closest to the ideal solution. However, the best ranked alternative by TOPSIS is the best using ranking index, but doesn't closest to the ideal solution. Hence advantage of VIKOR gives to be compromise solution.

Conclusion:
In this present communication, the first three algorithms follow by MCGDM under SIVFS and last two algorithms follow by SIVFS linguistic TOPSIS and VIKOR approaches under aggregation operator. Again we interact SIVFS aggregation operator and score function values based on some technique. Also we have inserted various sorts of statistical charts to image the rankings of alternatives under consideration.

Conflicts of Interest
The authors declare no conflict of interest.