An Approach for the Assessment of Multi-National Companies Using a Multi-Attribute Decision Making Process Based on Interval Valued Spherical Fuzzy Maclaurin Symmetric Mean Operators

: Many fuzzy concepts have been researched and described with uncertain information. Collecting data under uncertain information is a difﬁcult task, especially when there is a difference between the opinions of experts. To deal with such situations, different types of operators have been introduced. This paper aims to develop the Maclaurin symmetric mean (MSM) operator for the information in the shape of the interval-valued spherical fuzzy set (IVSFS). In this article, a family of aggregation operators (AOs) is proposed which consists of interval valued spherical fuzzy Maclaurin symmetric mean operator (IVSFMSM), interval valued spherical fuzzy weighted Maclaurin symmetric mean (IVSFWMSM), interval valued spherical fuzzy dual Maclaurin symmetric mean (IVSFDMSM), and interval valued spherical fuzzy dual weighted Maclaurin symmetric mean (IVSFDWMSM) operators. In this paper, we studied an elucidative example to discuss the evaluation of multi-national companies for the application of the proposed operator. Then the obtained results from the proposed operators are compared. The results obtained are graphed and tabulated for a better understanding.

The score function for IVSFVs is given below.

(2)
Definition 4: [11] Let L 1 = f in f (v) be two IVSFVs, then we can define the following operations. Note that ⊗ and ⊕ denote the multiplication and addition of two IVSFVs.
Definition 5: [35] Let A i = (i = 1, 2, 3, . . . , r) be a collection of positive real numbers. Then is called MSM. Where (c 1 , c 2 , . . . , c Y ) convert all the l-tuple combinations of (1, 2, . . . , r) and C Y r is the binomial coefficient. Now we discuss the main work regarding this article.

Interval Valued Spherical Fuzzy Maclaurin Symmetric Mean (IVSFMSM) Operator
In this section, we developed the concept of the IVSFMSM and IVSFWMSM operators.
and be two IVSFVs. Then, the IVSFMSM operator is given by (3) be a collection of IVSFVs. Then, using IVSFMSM operator, we get as we know, an aggregation operator fulfills the criteria of three properties (boundedness, idempotency, and monotonicity). So, IVSFMSM operators satisfied these properties as given below

collection of IVSFVs and let A in f and A sup denote the smallest and the greatest IVSFVs respectively. Then
be a collection of IVSFVs and ω i be the weight vector of A i such that ∑ n i=1 ω i = 1 . Then, the IVSFWMSM operator is defined by (4) be a collection of IVSFVs. Then, using IVSFWMSM operator, we get Proof: Proof is skipped.

Interval-Valued IVSFDMSM Operator
The main purpose of this part of the paper is to develop the ideas of IVSFDMSM and IVSFDWMSM by using MD, NMD, and AD.
denote the collection of IVSFVs. Then, by using IVSFDMSM operators, we have IVSFDMSM operators also satisfied the aggregation properties (Boundedness, Idempotency, and Monotonicity).

Property 5: (Boundedness Property) Let
be a collection of IVSFVs and let A in f and A sup denote the smallest and the greatest IVSFVs, respectively. Then be a collection of IVSFVs. Then, the IVSFDWMSM operator is given as

Theorem 4: Let
denote the collection of IVSPFNs. Then, by using IVSFDWMSM operators, we have

Proof:
The proof is the same as in Theorem 3 above. We will discuss a numerical example to illustrate the calculation process. We take the values For MD,

Special Cases Analysis
In this section, we observe the changes in the proposed operators in different frameworks. To begin, take the IVSFMSM abstinence values, which are zero, and convert them into IVPyFMSM operators such as g in f if we set g iτ = 0 then IVSFMSM and IVSFDMSM change into IVPyFSMSM and IVPyDMSM operators.
when we take AD as zero, then the interval-valued spherical dual fuzzy MSM (IVSFDMSM) operator changes into the interval-valued Pythagorean fuzzy dual MSM (IVPyFDMSM) operator.

Application to MADM
In this section, we use the proposed operators in the MADM process. MADM is a process by which we can choose the better option in an uncertain situation [42][43][44]. In the MADM process, we take limited alternatives that depend on limited attributes [45][46][47]. In this way, the experts express their opinions in the form of interval-valued spherical fuzzy numbers (IVSFVs) and prefer different sources according to their thinking. The best option is chosen by basing the aggregation operators on expert recommendations. SupposeP = P 1 ,P 2 , . . . ,P n are the set of attributes that can choose the best option based on observations, and that the set of alternatives isǩ = ǩ 1 ,ǩ 2 , . . . ,ǩ n . The data is given in the form of IVSFVs under uncertain information, where we have four facts from an expert's opinion that can be aggregated for all the alternatives. In the proposed work, experts expressed their opinions in the form, "We use MSM operators to aggregate the result in the form of IVSFMSM and IVSFDMSM operators". The details of this procedure are as follows: Step 1. The experts use uncertain data in the form of MD, NMD, and AD restrictions for IVSFS. The results of alternativesP =P i concerning attributesǩ =ǩ i are given in the form IVSFV.
Step 2. In this step, we deal with two types of attributes (benefit and cost). To deal with the cost type of attributes, we use the process of normalization. In which we change all cost types of attributes into benefit types of attributes.
Step 3. After using the process of normalization, we apply the two aggregation operators IVSFMSM and IVSFDMSM to the given uncertain data.
Step 4. After getting aggregated results by applying the proposed operators, we use Definition 3 to get the score function value of those results.
Step 5. By using score values, we get a ranking of the obtained results.
Example 1: Consider the problem of evaluating of the progress of multinational companies. Suppose the multinational companies are evaluated based on some attributes as given below: 1.ǩ 1 for stock purchases; 2.ǩ 2 for stock award; 3.ǩ 3 for the charge of control; 4.ǩ 4 for the bonus of the company.
Consider four companies P 1 ,P 2 ,P 3 ,P 4 be evaluated based on attributes ǩ 1 ,ǩ 2 ,ǩ 3 ,ǩ 4 . For this reason, the expert gave his opinion in the form of IVSFV, where k 1 represents stock purchases; k 2 shows stock awards; k 3 shows the charge of control, and k 4 shows the bonus of the company. In this scheme, each attribute is given the weight vector w iτ = (0.2, 0.3, 0.1, 0.4) T such that ∑ n i=0 w i = 1. The process of the multinational companies is given below. In this example, the values of r are 4 and the parameter Y = 3.
In Table 1, we use undefined data in the form of IVSFVs as the opinions of experts of the company.
In Table 2, the experts use proposed operators to aggregate the given undefined data, however, we provided the aggregated values obtained by the IVSFMSM operator. In the next step, we must find the score values of the alternatives.
In Table 3, we use Definition 1 of the score function. The score values show that all the operators are much more fruitful, but IVSFDMSM is the most fruitful compared to other operators because it shows much more accuracy. IVSFDWMSM shows negative score values. The stepwise procedure of the evaluation of the multinational companies is given below: The decision panel of the company gave opinions in the form of IVSFV; Step 1. Then we aggregated the obtained information with proposed operators IVSFMSM, IVSFWMSM, IVSFDMSM, and IVSFDWMSM; Step 2. In this step, we make a ranking of proposed operators by using score functions; Step 3. This is the end of the procedure with the ranking of the companies.
From Table 4, it is shown that the finest multinational company is P 4 using IVSFMSM and IVSFDWMSM. P 3 is the better option for the IVSFWMSM and IVSFDMSM operators. Table 1. Values obtained from experts regarding companies in form of IVSFV.      Table 4. Ranking of the score values of the aggregated results from Table 2.

Operators Ranking Values
Now we show the ranking values in Figure 1. Figure 1. From Figure 1, we can see the behavior of the score values obtained as the result of all the developed AOs. In Figure 1, all the results are shown that are described in Table 1. Now, we discuss sensitivity analysis, in which we compare the accuracy of the proposed operators by taking different values of parameter Y, as in Table 5. Table 5. Sensitivity analysis of the proposed operators.

Now we show the ranking values in
Operators l = 1 l = 2 l = 3 l = 4 From Table 5 above, we show that the proposed operators (IVSFMSM, IVSFWMSM, IVSFDMSM, and IVSFDWMSM) show different accuracy by taking different values of parameter Y. At Y = 1, all the alternatives show the same result for IVSFMSM and IVSFDMSM. In IVSFWMSM P 3 is much more accurate and in IVSFDWMSM P 1 is much more effective. At Y = 2, P 3 is much more effective in IVSFMSM and IVSFWMSM. P 4 is much more accurate in IVSFDMSM and IVSFDWMSM. At Y = 3, P 3 is much more accurate in IVSFWMSM and IVSFDMSM operators. P 4 is effective in IVSFMSM and IVSFDWMSM operators. At Y = 4, P 4 is effective in IVSFMSM and IVSFWMSM also P 2 is much more accurate in IVSFDMSM and IVSFDWMSM operators.

Comparative Study
In this part of the article, we compare traditionally used aggregation operators with those proposed. As we know, SFS eradicates uncertainty due to its extended limits. In this section, we compare IVSFMSM operators with existing operators' interval-valued spherical fuzzy weighted averaging (IVSFWA), interval-valued SF Hamacher weighted averaging (IVSFHWG), interval-valued SF weighted geometric (IVSFWG), interval-valued SF Dombi weighted averaging (IVSFDWA), and interval-valued SF Dombi weighted geometric (IVSFDWG) operators.
In Table 6, we show the ranking of various defined operators with proposed operators by using the score values of those operators. We show that P 1 is much more accurate in IVSFHWA, IVSFHWG, IVSFWA, IVFWG, IVSFDWA, IVSFMSM, and IVSFDMSM. P 2 is much more fruitful in IVSFWMSM and IVSFDWMSM. IVSFMSM P 4 < P 3 < P 2 < P 1 IVSFWMSM P 3 < P 2 < P 4 < P 1 IVSFDMSM P 3 < P 2 < P 4 < P 1 IVSFDWMSM P 4 < P 3 < P 2 < P 1 From Figure 2. in a comparative analysis, it is clear thatP 4 is a much more effective proposed operator than other defined operators. The basic benefit of the MSM operator is to aggregate the interrelated data in fuzzy theory. Hamacher AOs and Dombi AOs are also very effective in fuzzy theory for aggregating undefined data. In Figure 2, we see thatP 4 is the most fruitful option.

Conclusions
In this paper, we proposed the MSM operator by using IVSF information. The main advantage of these proposed operators is that IVSFMSM gave more accuracy than other operators (IVFMSM). After proposing the operator, we investigated the properties (Boundedness, Monotonicity, and Idempotency) of each AO. The progress of the multinational companies is evaluated with the help of the MADM procedure. Then we compared the results obtained with Dombi weighted aggregation (DWAOs), Dombi weighted geometric mean operators (DWGOs), and Hamacher weighted aggregation operators (HWAMOs). We can see that the proposed operator is much more effective because IVSFMSM minimizes the loss of information under uncertain conditions. However, the IVSF information is also limited because sometimes the sum of the upper values of the intervals may exceed one. Hence, the developed AOs are also limited to only IVSF information and can be further extended to any other framework. In the future, we aim to extend the developed approach to the framework defined in [52].