1. Introduction
Decision-making is an important phenomenon to obtain the best-suited alternative among the available ones. In it, a number of researchers have presented a variety of concepts to reach the correct decisions. In primitive times, decisions were framed on the basis of crisp numbered data sets, but they were led to inadequate results having less applicability towards the real-life operational situations. However, with the passage of the time and due to the increase of the complexities in the system, it is difficult for the decision maker to handle the uncertainties in the data and hence the decisions under the traditional approach are unable to identify the best alternative. Thus, the researchers have represented the information in terms of fuzzy sets (FSs) [
1], interval-valued fuzzy sets (IVFSs) [
2], intuitionistic fuzzy sets (IFSs) [
3], interval-valued intuitionistic fuzzy sets (IVIFSs) [
4]. During the last decades, the researchers are paying more attention to these theories and have successfully applied it to the various situations in the decision-making process. Among these, an aggregation operator is an important part of the decision-making which usually takes the form of mathematical function to aggregate all the individual input data into a single one. For instance, Xu and Yager [
5] developed some geometric aggregation operators to aggregate the different preferences of the decision-makers in the form of the intuitionistic fuzzy numbers (IFNs). Later on, Wang and Liu [
6] extended these operators by using Einstein norm operations. Garg [
7] had presented generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein norm operations for aggregating the different intuitionistic fuzzy information. Garg [
8], further, proposed some series of interactive aggregation operators for intuitionistic fuzzy numbers (IFNs). Garg [
9] presented generalized intuitionistic fuzzy aggregation operators under the intuitionistic multiplicative preference relation instead of intuitionistic fuzzy preference relations. Garg [
10] extended the theory of the IFSs to the Pythagorean fuzzy sets and presented their generalized averaging aggregation operators. Wang and Liu [
11] presented some hybrid weighted aggregation operators using Einstein norm operators while Garg [
12] presented some improved interactive aggregation operators. However, apart from that, some other authors have presented different methods such as ranking functions [
13,
14,
15,
16,
17], aggregation operators [
18,
19,
20,
21,
22,
23,
24,
25] to solve the decision-making problems.
As the above aggregation operators have widely been used by the researchers during the decision-making (DM) process in which they have highlighted the importance of each factor or its ordered position but cannot reflect the interrelationships of the individual data. On the other hand, in our real-life situation, there always exists a situation in which a relationship between the different criteria such as prioritization, support, and impact each other plays a dominant role during an aggregation process. For handling it and to incorporate into the DM analysis, Yager [
26] introduced the power average (PA) aggregation operator which allows argument values to support each other in the aggregation process. Further, Xu and Yager [
27], Yu [
28] investigated the prioritized averaging and geometric aggregation operators under IFS environment. Also, Yager [
29] proposed the concept of the Bonferroni Mean (BM) [
30] whose main characteristic is its capability to capture the interrelationship between the input arguments. Beliakov et al. [
31] introduced the generalized Bonferroni mean to overcome the drawback of BM. Xu and Yager [
32] developed an intuitionistic fuzzy Bonferroni mean to aggregate the intuitionistic fuzzy information. Xu and Chen [
33] extended these mean operators to the IVIFSs environment. Xia et al. [
34] proposed the generalized intuitionistic fuzzy BMs. Liu et al. [
35] presented the partitioned BM operators under IFSs environment. Shi and He [
36] threw light on optimizing BMs with their applications to various decision-making processes. Garg and Arora [
37] presented BM aggregation operator under intuitionistic fuzzy soft set environment.
From the above existing literature, we can see that all the existing studies mainly focus on the fuzzy set, interval fuzzy set, IFS, IVIFS, and their corresponding applications. Later on, Jun et al. [
38] introduced the concepts of the cubic sets (CSs) by the combination of both interval-valued fuzzy numbers and fuzzy number and defined some logic operations of the cubic sets. Under this set, Khan et al. [
39] presented some cubic aggregation operators while Mahmood et al. [
40] introduced the concepts of the cubic hesitant fuzzy sets and their aggregation operators in the decision-making process. However, above theories contain only the information in the form of membership intervals and do not stress on the non-membership portion of the data entities, which also play an equivalent role during assessing the alternative in the decision-making process. On the other hand, in the real world, it is often difficult to express the value of a membership function by an exact value in a fuzzy set. In such cases, it may be easier to describe vagueness and uncertainty in the real world using an interval value and an exact value, rather than unique interval/exact values. Thus, the hybrid form of an interval value and an exact value may be a very useful expression for a person to describe certainty and uncertainty due to his/her hesitant judgment in complex decision-making problems. For this purpose, we present the concept of the cubic intuitionistic fuzzy set (CIFS) which is described by two parts simultaneously, where one represents the membership degrees by an interval-valued intuitionistic fuzzy value and the other represents the membership degrees by intuitionistic fuzzy value. Hence, a CIFS is the hybrid set combined by both an IVIFN and an IFN. Obviously, the advantage of the CIFS is that it can contain much more information to express the IVIFN and IFN simultaneously. On the other hand, the CIFS contains much more information than the general intuitionistic set (IVIFS/IFS) because the CIFS is expressed by the combined information of both the sets. Hence, CIFS its rationality and effectiveness when used for evaluating the alternatives during the decision-making process since the general decision-making process may either use IVIFSs or IFSs information which may lose some useful evaluation information, either IVIFSs or IFSs, of alternatives, which may affect the decision results. Currently, since there is no study on aggregation operators which reflect the relationship between the different criteria of the decision-making process having cubic intuitionistic fuzzy information.
In the present communication, motivated by the concept of the Bonferroni mean and by taking the advantages of the CIFS to express the uncertainty, we propose some new aggregation operators called the cubic intuitionistic fuzzy Bonferroni mean (CIFBM), as well as weighted cubic intuitionistic fuzzy Bonferroni mean (WCIFBM) operator to aggregate the preferences of decision-makers. Various desirable properties of these operators have also been investigated in details. The major advantages of the proposed operator are that they have considered the interrelationships of aggregated values. Further, we examine the properties and develop some special cases of proposed work. Some of the existing studies have been deduced from the proposed operator which signifies that the proposed operators are more generalized than the others. Finally, a decision-making approach has been given for ranking the different alternatives based on the proposed operators.
The remainder of the article is organized as follows.
Section 2 briefly describes some concepts of IFSs, IVIFSs, and CSs.
Section 3 presents cubic intuitionistic fuzzy sets and the new aggregation operators called the cubic intuitionistic fuzzy Bonferroni mean (CIFBM) and weighted cubic intuitionistic fuzzy Bonferroni mean (WCIFBM) operators and discuss its particular cases. Some properties of these operators are also discussed here. In
Section 4, a decision-making approach has been established, based on proposed operators, to solve the multi-attribute decision-making (MADM) problems. A numerical example is presented in
Section 5 to illustrate the proposed approach and to demonstrate its practicality and effectiveness. The paper ends in
Section 6 with concluding remarks.
4. Proposed Decision-Making Approach Based of Cubic Intuitionistic Fuzzy Bonferroni Mean Operator
In this section, we shall utilize the proposed Bonferroni mean aggregation operator to solve the multi-attribute decision making under the cubic intuitionistic fuzzy sets environment. For it, the following assumptions or notations are used to present the MADM problems for evaluating these with a cubic intuitionistic fuzzy set environment. Let be the set of m different alternatives which have to be analyzed under the set of ‘n’ different criteria . Assume that these alternatives are evaluated by an expert which give their preferences related to each alternative under the CIFSs environment, and these values can be considered as CIFNs where , , represents the priority values of alternative given by decision maker such that , and , for . Let be the weight vector of the criteria such that and . Then, the proposed method has been summarized into the various steps which are described as follows to find the best alternative(s).
- Step 1:
Collect the information rating of alternatives corresponding to criteria and summarize in the form of CIFN
,
,
:
;
. These rating values are expressed as a decision matrix
D as
- Step 2:
Normalize these collective information decision matrix by transforming the rating values of cost type into benefit type, if any, by using the normalization formula:
and hence summarize it into the decision matrix
.
- Step 3:
Aggregate the different preference values
of the alternatives
into the collective one
by using WCIFBM aggregation operator for a real positive number
as
- Step 4:
Compute the score value of the aggregated CIFN
by using Equation (
4) as
- Step 5:
Rank the alternative with the order of their score value .