TOPSIS Technique of MCDM under Cubic Intuitionistic Fuzzy Soft Set Environment

A powerful tool for dealing with ambiguity, attributive values, fuzziness, and inconsistency is the cubic intuitionistic fuzzy soft set. In this paper, multi-criteria group decision-making (MCGDM) technique called extended TOPSIS based on distance and similarity measures is presented. Furthermore, a real-world problem is resolved by applying CIFS-set to support the theoretical visualization. Arguably, the proposed technique works well and has practical uses.


INTRODUCTION
Taking the best decision requires taking into account a range of factors due to increasing complexities in business, engineering, scientific, and technological environments. The most effective way to select the ideal alternative among all possible options is decision-making. Since generalized variants are frequently used in decision-making, nearly every other problem has a considerable range of requirements. Such requirements frequently come into conflict with each other and no solution may ever be able to fully satisfy all of them. To address such challenges, decisionmakers need to overcome the MCGDM problem. For this purpose, many mathematical theories have been expounded such as the fuzzy set theory, intuitionistic set theory, and interval-valued intuitionistic set theory. In 1965, fuzzy set theory was introduced by Zadeh [1]. This theory deals with membership value over closed interval [0,1] and overcomes the vague and ambiguous environment. However, due to the increased complexities in the environment, a decision-maker faces difficulties to express their idea in the The technique known as the order of preference by similarity to ideal solution (TOPSIS) was introduced by Lai et al. [6] and remains a widely known method. The purpose of this method is to identify the longest path from the negative ideal alternative (NIA) as well as the shortest path from the positive ideal alternative (PIA).
Since this technique has been introduced, various researchers have used the TOPSIS method to resolve decision-making problems in an ambiguous environment. Interval-valued intuitionistic fuzzy set and its fundamental properties was proposed by [7]. Jahanshahloo et al. [8] presented the TOPSIS method for DM under a fuzzy environment, Chu et al. [9] suggested the TOPSIS method for robot selection, and Zulqarnain et al. [10] generalized the fuzzy TOPSIS for MCGDM. Saqlain et al. [11] discussed the use of the TOPSIS method for the selection of a smartphone. Shen et al. [12] extended the application of this technique under intuitionistic fuzzy environment, Chang et al. [13] discussed the distance approaches using the TOPSIS method, Li et al. [14] suggested the Pythagorean fuzzy TOPSIS based on similarity measures, and Gupta et al. [15] discussed the extended TOPSIS under interval-valued intuitionistic fuzzy environment.
All these theories involve uncertainty to some extent as they do not deal with the membership and non-membership values simultaneously along with fuzziness. To deal with such kind of situations, Garg [16] proposed the TOPSIS method for MCGDM under a cubic environment which deals with both the intuitionistic value and fuzziness, simultaneously. Pramanik et al. [17] defined the TOPSIS method for neutrosophic cubic information, Jun et al. [18] suggested the cubic fuzzy set (CFS), while Garg [19] discussed the cubic intuitionistic fuzzy set (CIFS) and its fundamental properties. The soft set theory with its fundamentals was proposed by Maji et al. [20]. The theory of soft set was merged with cubic set and is known as cubic intuitionistic fuzzy soft set (CIFSS). It was proposed by Saqlain  Cubic intuitionistic fuzzy set (CIFS) is considered highly effective for decision-making problems as it involves interval-valued intuitionistic fuzzy number (IVIFS) and intuitionistic fuzzy number (IFS) over the interval. Still, this theory does not deal with multi-attributive values. Considering the soft set, this paper attempts to define the multi-attributive decision-making problem under the cubic intuitionistic fuzzy soft environment, where each element is defined by the CIF-number (CIFN). A methodology which utilizes the extended TOPSIS method is also proposed. Furthermore, some distance and similarity measure formulas have been defined which evaluate the positive ideal alternative (PIA) and negative ideal alternative (NIA).

PRELIMINARY SECTION
In this section, some basic concepts of intuitionistic fuzzy set (IF-set), interval-valued intuitionistic fuzzy set (IVIF-set), soft set, intuitionistic fuzzy soft set (IFS-set), CF-set, CIF-set, and CIFS-set are defined.
This pair is denoted as ӡ = (ʯ, ѵ) and termed as a cubic intuitionistic fuzzy set.
This pair is denoted as ӡ = (ʯ , ѵ ) and termed as cubic intuitionistic fuzzy soft set.

Definition2.7 [18]. Consider
Then, for ≥ 1, the distance measures are defined as where n represents the mean of attributive values.

EXTENDED TOPSIS TECHNIQUE
In this section, based on the suggested distance measure, a TOPSIS (technique for order of preference by similarity to ideal solution) technique for tackling MAGDM problems in the form of CIFS sets is proposed.

Description of the Problem
A recently opened restaurant is hiring a new head chef to take charge of the kitchen. For hiring a head chief, they have published the advertisement in the newspaper and different applicants have applied in response. Assume that there is a set of m applicants (alternative) ʓ = {ʓ 1 , ʓ 2 , ʓ 3 , … , ʓ } chosen for the interview. The restaurant has gathered decision-makers { 1 , 2 , 3 , 4 } and assigned them the duty to identify the best head chef for the restaurant. The selection committee has decided to assess the applicants ʓ = {ʓ 1 , ʓ 2 , ʓ 3 , … , ʓ } based on n different criteria ɗ = {ɗ 1 , ɗ 2 , ɗ 3 , … , ɗ }.
So, Ӽ + and Ӽ − complement each other.

=1
The stepwise algorithm of TOPSIS is presented below.
Step 5: Rank the alternatives based on the descending values of ℂ s

Figure 1. TOPSIS Algorithm
Example: To demonstrate the above-mentioned approach depicted in Figure 1, an example is discussed below.

Case Study
A recently opened restaurant is hiring a new head chef to take charge of the kitchen. For hiring a head chef, they have published the advertisement in the newspaper and different applicants have applied in response. A total School of Science Volume 7 Issue 1, 2023 of four individuals ʓ ; = 1,2,3,4 have been chosen for the interview. The restaurant has gathered decision-makers { 1 , 2 , 3 , 4 } and given them the responsibility to identify the best head chef for the restaurant. The selection committee has decided to assess the applicants ʓ ; = 1,2,3,4 based on four criteria ɗ = {ɗ 1 , ɗ 2 , ɗ 3 , ɗ 4 } defined as ɗ 1 : ɗ 2 : , ɗ 3 : , ɗ 4 : ℎ .
For the assessment, they conducted group discussion with all the applicants and the results are formulated by a panel in the form of an IVIFS set. From lots of applicants appearing for group discussion, only four applicants are shortlisted for the interview. At this stage, the results are recorded in the form of an IFS set. Then, the following steps of the proposed approach are executed to find the best head chef for the kitchen.
Scientific Inquiry and Review

Step 2: Computing CIFS-PIA and CIFS-NIA for Each Decision-maker.
Computing the CIFS-PIA and CIFS-NIA for each decision-maker is depicted below in Table 5.
Using the Eq.
Using Eqs. Solving these separation measures corresponding to each decisionmaker, we get Table 6. Evaluation of Separation Measures The outcome is manifested in Table 7. Step 5: Ranking of Alternatives. Using the relative closeness coefficient, the ranking of alternatives corresponding to each decisionmaker is manifested in Table 8.

DISCUSSION
The problem of the selection of the head chef at a famous restaurant has been solved by applying the extended TOPSIS method based on the suggested distance and similarities measure. Based on the proposed method, four decision-makers were selected to select the best head chef to take charge of the kitchen. Each decision-maker assigned the rating values that corresponded to each attribute and alternative, as shown in tables 1-4. Using CIF-PIA and CIF-NIA, the separation measure between the alternatives was computed, as shown in tables 5-6. Then, the last relative closeness coefficient was calculated, as shown in Table 7. Each value of the relative closeness coefficient was arranged in descending order and all alternatives were ranked corresponding to each decision-maker, as shown in Table 8 and Figure 2. According to (1) decision-maker, ʓ 2 remains the PIA for the selection of head chef. According to (2) decision-maker, ʓ 3 is the PIA for the selection of head chef. According to (3) decision-maker, ʓ 3 is the PIA for the selection of head chef and according to (4) decision-maker, ʓ 4 is the PIA for the selection of head chef. However, if only one head chef needs to be selected, then the highest ranking PIA should be arranged in descending order and the best head chef should be chosen to take charge of the kitchen.

5.1.Conclusion
In this project, the issue of MCGDM under the CIFS-set environment has been discussed. An adaptation of the TOPSIS method has been illustrated to show the effectiveness of the proposed operators. According to the findings, these decision-making techniques can represent uncertainty more effectively than the current approaches and provide us with a comprehensive understanding of real-life scenarios. The problem of the selection of head chef for the newly opened restaurant has been dealt with in this research. The proposed approach of TOPSIS yields the best head chef. The findings of this research can be applied in the future to hypersoft set, interval-valued soft set, bi-polar soft set, and other ambiguous environments.