Abstract
Induced aggregation operators are more suitable for aggregating the individual preference relations into a collective fuzzy preference relation. Therefore, in this paper, we introduce the notion of some new types of induced aggregation operators, based on the Aczel-Alsina operations. We construct some induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging/geometric (I-IVITrFAAOWA/G) operator, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging/geometric (I-IVITrFAAHA/G) operator. Moreover, some dominant properties of these developed operators are studied in detail. Based on these proposed approaches, a model is a build up for multicriteria decision making (MCDM), and their stepwise algorithm is being presented. Finally, in utilizing the developed approach, an illustrative example is solved with the help of proposed operators. In the end, we utilize an applicable example for supplier selection to prove the proposed methods and compare the result with existing methods, which shows the superiority, competence, and ability of the developed model.
1. Introduction
Multicriteria group decision-making (MCGDM) problems are of consequence in most kinds of fields such as engineering, economics, medical diagnosis, and management. Traditionally, it has been assumed that the information, which accesses the alternatives in terms of criteria, and weight are expressed in real numbers. Multicriteria group decision making (MCGDM) has develop into tremendously precious in preceding few decades in the learning of decision support systems [1–3]. The convolution of DM problems increases time to time in this world of global antagonism as the socioeconomic structure becomes more dynamic. Therefore, making sensible and successful decisions in this situation is very complicated for a single decision expert. In this real world, group DM models are often used to combine the opinions of team of experienced experts in order to generate highly accurate and optimal values. Hence, MCGDM have an excellent ability and systematic method for improving and evaluating various competing criteria in all aspect of DM in order to achieve more appropriate and possible DM outcomes. In DM issues, the knowledge based about even a fact is often unclear, making the decision-making task more complicated and ambiguous. To resolve this restriction, Zadeh [4] in 1965 first established the idea of fuzzy set (FS) by considering the membership degree MD. This concept has been investigated in a various real-world problems, including clustering analysis [5], decision-making problems (DMPs) [6], and medical diagnosis [7]. Atanassov [8] developed the intuitionistic fuzzy set (IFS) as an extension of the FS, and its constraint is that the sum of the MD and the N-MD will be less than or equal to one. IFS has been a hot research topic for researchers who examined its hybrid structure in a variety of ways. The idea of intuitionistic fuzzy weighted averaging (IFWA) aggregation operators was first investigated by Xu [9]. Xu and Yager [10] developed the definition of intuitionistic fuzzy weighted geometric (IFWG) aggregation operators. The graphical techniques for rating score and accuracy functions were created by Ali et al. [11]. He et al. [12] explored the concept of intuitionistic fuzzy neutral averaging operators (IFNAO). He et al. [13] invented the notions of geometric relationship averaging operator and proposed its implementation in DM. Also Atanassov [14] presented preference relation based on IVIF. Further, Wan and Dong [15] work on extension of best-worst method based on IFS. Zaho et al. [16] were the first to apply the generalized intuitionistic fuzzy weighted averaging (GIFWA), generalized intuitionistic fuzzy order weighted averaging (IFOWA), and generalized intuitionistic fuzzy hybrid averaging (GIFHA) operators to DM. However, it was discovered that these aggregation process was performed by using Archimedean t-norms and t-conorm by operators. As an equivalent to algebraic product and sum, Einstein-based t-norm and t-conorm provides the best estimate for product and sum of intuitionistic fuzzy numbers (IFNs). Wang and Liu and Wang and Zhang [17, 18] provided IF Einstein weighted averaging (IFEWA) and IF Einstein weighted geometric (IFEWG) operators by using the concept of Einstein operation. After that, Abbas et al. [19] developed the idea of intuitionistic fuzzy rough Einstein weighted averaging (IFREWA), intuitionistic fuzzy rough Einstein hybrid averaging (IFREHA), and intuitionistic fuzzy rough ordered weighted averaging (IFREOWA) aggregation operators based on the TOPSIS method. Yager and Filev [20, 21] developed an expansion of the OWA operator, called the induced ordered weighted averaging (IOWA) operator. It takes the argument in pairs (OWA pairs), in which one component is used to induce an ordering over the second component and then aggregates. Wei [22] developed two new aggregation operators: the induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and the induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator. Su et al. [23] also constructed induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator and induced interval-valued intuitionist fuzzy hybrid averaging (I-IIFHA) operator. Shakeel at al [24] developed the induced Pythagorean trapezoidal fuzzy Einstein ordered weighted averaging operator and the induced Pythagorean trapezoidal fuzzy Einstein hybrid averaging (I-PTFEHA) operator. In [25], Klement et al. discussed the triangular norms, and in [26, 27], Aczel-Alsina (AA) presented t-norm as well as t-conorm. After that, some new idea of AA was developed by Senapati et al. in [28]. Senapati et al. introduced the IF Aczel-Alsina weighted averaging (IFAAWA) operator, the IF Aczel-Alsina orders weighted averaging (IFAAOWA) operator, and the IF Aczel-Alsina hybrid averaging (IFAAHA) operator. Senapati et al. [29, 30] developed interval-valued intuitionistic fuzzy aggregation operators, such as the IVIF Aczel-Alsina weighted averaging/geometric operator, the IVIF Aczel–Alsina order weighted averaging/geometric operator, and the IVIF Aczel–Alsina hybrid averaging/geometric operator and built up several features of such operators. Senapati [31] also defined picture fuzzy aggregation operator such as the PF Aczel–Alsina weighted average (PFAAWA) operator, the PF Aczel–Alsina order weighted average (PFAAOWA) operator, and the PF Aczel–Alsina hybrid average (PFAAHA) operator with their properties. Later on, Senapati et al. [32] originated a few new aggregation operators for aggregating hesitant fuzzy information, namely, the HF Aczel-Alsina weighted averaging (HFAAWA) operator, HF Aczel-Alsina ordered weighted averaging (HFAAOWA) operator, HF Aczel-Alsina hybrid averaging (HFAAHA) operator, HF Aczel-Alsina weighted geometric (HFAAWG) operator, HF Aczel-Alsina ordered weighted geometric (HFAAOWG) operator, HF Aczel–Alsina hybrid geometric (HFAAHG) operator, and HF Aczel–Alsina weighted Bonferroni mean (HFAAWBM) operator.
After that, Ban [33, 34] initiated the concept of trapezoidal fuzzy numbers (TrFNs) and interval-valued trapezoidal fuzzy numbers (I-VTrFNs). Later on, Wang and Zhang [35] expanded the definition of intuitionistic trapezoidal fuzzy numbers (ITrFNs) and interval-valued intuitionistic trapezoidal fuzzy numbers (IVITrFNs). Wei [36] developed the intuitionistic trapezoidal fuzzy ordered weighted averaging (ITrFOWA) operator and the intuitionistic trapezoidal fuzzy hybrid aggregation (ITrFHA) operator. Wan and Dong [37] defined the anticipation and expectant score of intuitionistic trapezoidal fuzzy numbers from the geometric angle. As for our information and based from the above review, no implementation of the MCGDM approach to review of induced interval-valued intuitionist trapezoidal fuzzy Aczel-Alsina (I-IVITrFAA) averaging/geometric aggregation operators. We proposed the average/geometric aggregation operators such as I-IVITrF Aczel-Alsina ordered weighted averaging (I-IVITrFAAOWA) operator, I-IVITrF Aczel-Alsina hybrid averaging (I-IVITrFAAHA) operator, I-IVITrF Aczel-Alsina ordered weighted geometric (I-IVITrFAAOWG) operator, and I-IVITrF Aczel-Alsina hybrid geometric (I-IVITrFAAHG) operator for solving MCGDM. To solve a MCGDM process, the weight of the attributes plays an important role in making decisions under the aggregation approaches.
The paper provided the following: in the next section, we briefly discuss some fundamental ideas of IFSs, I-VIFS, TrFNs, IVITrFNs, IFAA, and I-VIFAA aggregation operators. In Section 3, we outline AA operation laws for the I-VITrFNs. In Section 4, we develop I-IVITrFAAOWA operator, I-IVITrFAAHA operator, I-IVITrFAAOWG operator, and I-IVITrFAAOWG operator. In Section 5, we apply the I-IVITrFAAOWA operator to build up certain methodologies for managing MCGDM issues. Section 6 provides an example of choosing greatest alternative of the proposed method. In Section 7, we compare the developed method with the existing methods. At end of this paper, a few conclusions and upcoming research are mentioned in Section 8.
2. Preliminaries
In this section, we will define basic definition, results, and operational laws which are important for the improvement of this paper.
2.1. IFSs
Atanassov [8] introduced the idea of IFSs which is a generalization of FSs. FSs discussed only M-D where IFSs discussed M-D and NM-D. Moreover, the sum of M-D and NM-D is less than or equal to 1.
Definition 1 (see [8]). Let a set be fixed. An in is an object having the form where and represent the M-D and N-MD such that for each in is called the degree of indeterminacy of to .
Definition 2. Atanassov [8] let and are two IFSs over the universe , then the operational laws between and are defined as (i) if and (ii)(iii)(iv)(v)(vi)(vii)
Definition 3 (see [26]). Let and be any two IFSs, then the generalized intersection and the generalized union are defined in the following ways: where presents a t-norm, and presents a t-conorm.
2.2. I-VIFSs
Definition 4 (see [38]). Let be an I-VIFNs, where and represent an I-VIFNs, hence and , such that .
Definition 5. Let be any two , numbers, and Then, (1)(2)(3)(4)
Example 6. Let be any three I-VIFNs, and ; then, we verify the above results such that, (1)(2)(3)(4)
Definition 7 (see [39]). Let be an , a can be defined as follows:
Example 8. Let .
be . Then, we verify the above results such that,
In Figure 1, we supposed some interval-valued of intuitionistic fuzzy numbers and apply them score function to find out the highest score values for ranking process.
Definition 9 (see [39]). Let be an ; an accuracy function can be defined as follows:
Example 10. Let be . Then, we verify the above results such that,
In Figure 2, we supposed some interval-valued of intuitionistic fuzzy numbers and apply their accuracy function to find out the highest accuracy values for ranking process.
2.3. ITFNs and I-VITFNs
Definition 11 (see [40]). Let be ITFNs, its membership function Its nonmembership function is where Then, is called ITFNs. For convenience, .
Definition 12 (see [40]). Let be the two ITFNs, and . Then, (1)(2)(3)(4)
Definition 13 (see [40]). Let be any three I-VITrNs, and Then, (1)(2)(3)(4)
Example 14. Let be any three I-VITrNs, and . We verify the above operational laws such that,
Definition 15 (see [40]). Let be ; a score function can be defined as follows:
Definition 16 (see [40]). Let be ; an accuracy function can be defined as follows:
2.4. The Aczel-Alsina Operations of IFNs
In this section, we will present A-A operations on IFNs [28]. Let presents a t-norm and presents a t-conorm be A-A product and A-A sum , respectively. A-A product and A-A sum over two IFSs and defined in the following way :
Definition 17 (see [28]). Let and be any three , , and Then, the A-A t-norm and t-conorm operation of IFNs are defined as (i)(ii)(iii)(iv)
3. The Aczel-Alsina Operations of I-VITrFNs
In this section, we will present the operational laws of I-VITrFNs based on A-A aggregation operators.
Definition 18. Let be any three I-VITrFNs, Then, the Aczel-Alsina operation laws on I-VITrFNs are defined as (1)(2)(3)(4)
Example 19. Let be any three I-VITrFNs; then, using Aczel-Alsina operation on I-VITrFNs as defined in above operation laws for and we get (1)(2)(3)(4)
Theorem 20. Let be any three I-VITrFNs, . Then, we define the following results. (1)(2)(3), (4)(5),
Proof. For the three ,, and , and as in Definition 18 such that (1)(2)(3)(4)(5)
4. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Ordered Weighted Averaging (I-IVITrFAAOWA) Operator
Definition 21. Let be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging operator of dimension is mapping , such that
be the weighting vectors of with and .
Also, is the value of the I-VIFTrNs, and pairs having the largest and in is referred to as the order inducing variable and as the interval-valued intuitionistict trapezoidal fuzzy argument variable.
Theorem 22. Let be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging operator of dimension is mapping : with the corresponding vector is the weight vector such that and , such that, Also, is the value of the I-VITrFNs, and pairs having the largest and in is referred to as the order inducing variable and as the interval-valued intuitionistic trapezoidal fuzzy argument variable.
Proof. By mathematical induction, we can prove that, Equation (30) holds for all positive integer . First, we show that Equation (22) holds for , since
Based on Definition 18, we get
Hence, (22) is true for .
Let, we want to show that Equation (30) is true for , then we have
Now, for , we get
Thus, (30) is legitimate for .
As consequences, we might come to conclusion that Equation (30) holds for all .
Theorem 23 (Commutativity). Let where is any permutation of
Proof. As we know that since is any permutation of
Theorem 24 (Idempotency). If all are equal, for all , then
Proof. Since for all , we have Thus,
Theorem 25 (Monotonicity). If for all , then
Proof. Let Since for all it follows that then
Example 26. Consider the following collection of I-IVITrFNs in pairs form such that Performing the ordering of the I-IVITrF pairs with respect to the first component, we have This ordering includes the ordered I-VITrF arguments. If the associated weighting vector is and then, we get an aggregate value by Equation (9).
4.1. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel Alsina Hybrid Averaging (I-IVITrFAAHA) Operator
Definition 27. Let be a collection of I-VITrFNs; then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging operator of dimension is mapping -, such that where is the weighted interval-valued intuitionistic trapezoidal fuzzy values, and of the interval-valued intuitionistic trapezoidal fuzzy ordered weighted averaging pair having the largest and in is referred to as the order inducing variable and as the interval-valued intuitionistic trapezoidal fuzzy argument variable. is the weighting vector of with and .
Theorem 28. Let be a collection of I-VITrFNs. then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid averaging operator is defined as
Theorem 29. The I-VITrFAAWA operator is the special case of the I-IVITrFAAHA operator.
Proof. Let , then
Theorem 30. The I-IVITrFAAOWA operator is the special case of the I-IVITrFAAHA operator.
Proof. Let then
4.2. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Ordered Weighted Geometric (I-IVITrFAAWG) Operator
Definition 31. Let be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina weighted geometric operator of dimension is mapping , such that
be the weighting vectors of with and .
Also is the value of the I-VIFTrNs, and pairs having the largest and in is referred to as the order inducing variable and as the interval-valued intuitionistict trapezoidal fuzzy argument variable.
Theorem 32. Let be a collection of I-VITrFNs. Then, induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina ordered weighted averaging operator of dimension is mapping , with the corresponding vector is the weight vector such that and , such that, Also is the value of the I-VITrFNs, and pairs having the largest and in is referred to as the order inducing variable and as the interval-valued intuitionistic trapezoidal fuzzy argument variable.
Proof. By mathematical induction, we can prove that Equation (56) holds for all positive integer . First, we show that Equation (56) holds for , since
Based on Definition 18, we get
Hence, (17) is true for .
Let we want to show that Equation (17) is true for , then we have
Now, for , we get
Theorem 33 (Commutativity). Let where is any permutation of
Proof. As we know that since is any permutation of
Theorem 34 (Idempotency). If all are equal, for all , then
Proof. Since for all we have Thus,
Theorem 35 (Monotonicity). If for all , then
Proof. Let Since for all it follows that then
4.3. Induced Interval-Valued Intuitionistic Trapezoidal Fuzzy Aczel-Alsina Hybrid Geometric (I-IVITrFAAHG) Operator
Definition 36. Let be a collection of I-VITrFNs, then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid geometric operator of dimension is mapping , such that where is the weighted interval-valued intuitionistic trapezoidal fuzzy values and .
of the interval-valued intuitionistic trapezoidal fuzzy ordered weighted averaging pair having the largest and in is referred to as the order inducing variable and as the interval-valued intuitionistic trapezoidal fuzzy argument variable. and is the weighting vector of with and .
Theorem 37. Let , be a collection of I-VITrFNs. then induced interval-valued intuitionistic trapezoidal fuzzy Aczel-Alsina hybrid geometric operator is defined as
5. MCGDM Techniques on I-IVITrFAAWA Operator
In terms of I-VITrF informations, we suggest an approach for solving MCGDM problems. A decision matrix can be used to express MCGDM, where the attributes (criteria) are expressed by columns, and alternatives are expressed by row. Suppose we have a set which consists number of alternatives represented by and a set consists number of attributes (criteria) represented as Let be the set of decision makers and be the weighting vector of DMs and DMs provided their assessment report for each alternative against their attributes (criteria) . Let represent the weight vector for attribute such that and .The I-IVITrF decision matrix is assumed to be denoted by where M-D and N-MD consider by decision makers such that,
The following steps are developed for solving MCGDM problem with I-IVITrFAAOWA operators.
Step 1. In this step, we construct the I-IVITrF matrix in which the decision makers give their opinion related to each alternative with respect to each criteria.
Step 2. In this step, we perform the ordering of the decision matrix with respect to the first component.
Step 3. We utilize the decision information given in the I-IVITrF matrix . Assume that , and apply the I-IVITrFAAOWA operator to derive the individual overall preference values of the alternative , and be the weight vector attribute , such that,
Step 4. We utilize the I-VITrFAAOWA operator to aggregate all the individual overall preference values into the collective overall preference values, where is weighting vector of decision makers such that and .
Step 5. In this step, we calculate the scores function to aggregate the value of each alternative .
Step 6. In ascending order, the alternative with the greatest value is our best choice.
6. Numerical Example
This example is taken from [38], with the help of this example we will explain our proposed method. A company wants to select a supplier and now there are four suppliers as candidates , We evaluate each supplier from four aspects which are the following:
is the production cost, is the production quality, is the supplier’s service performance, and is the risk factor.
The weight vector of attributes is There are four experts, and the weight vector of the experts is Then, the decision matrix are shown in Tables 1–4, and our goal is to rank four suppliers and select the best one.
Step 1. The decision makers give their decision in the following tables.
Perform the ordering of the decision matrix with respect to the first component in the following tables.
From Tables 5–8, we introduced induced aggregation operators perform the ordering of the decision matrix with respect to the first component.
Step 3. We utilize the decision information given in the I-VITrF matrix and the I-IVITrFAAOWA operator to derive the individual overall preference values of the alternative .
In Table 9, we apply IVITrFAAOWA operators to find out overall preference values.
Step 4. We utilize the IVITrFAAOWA operator to aggregate all the individual overall preference values into the collective overall preference values. Consider that .
Table 10 shows that the collective values by applying the IVITrFAAOWA operators for the ranking process.
Step 5. Calculate score values of alternatives
Step 6. Ranking the score values
In Figure 3, we represent the graphical approach of proposed aggregation operator and using the score function to find out the best alternative.
7. Comparative Analysis
To expand on the advantages of the developed methods, we compare them with the existing methods by solving the same example with same weight , such as the interval-valued intuitionistic fuzzy weighted average [39] operator, the intuitionistic fuzzy weighted average [39] operator, the intuitionistic fuzzy Einstein weighted average [41] operator, the intuitionistic fuzzy weighted geometric [10] operator, and the intuitionistic trapezoidal fuzzy weighted geometric [41] operator.
But in our paper, we take the data in the form of I-IVITrF information and used basic concept of the Aczel-Alsina aggregation operators like, I-IVITrFAAOWA operator, I-IVITrFAAHA operator, I-IVITrFAAOWG operator, and I-IVITrFAAHWG operator to aggregate the MCGDM issues. Furthermore, we have used these aggregation operators for solving the MCGDM in form of I-IVITrFNs environment. If we compare the score values of proposed aggregation operators with existing aggregation operators. We can see that our proposed work is more accurate and more applicable than existing methods due to highest score value of the proposed method. Applying the above mentioned methods, we obtain the comparison results shown in Table 11.
From Table 11, we observe that the optimal ranking results are different of the different methods, even though the score functions are different in different methods. Therefore, the novel methods proposed by us are reasonable and valid. Moreover, compared the proposed I-IVITrFAAOWA method with the existing methods, there is difference between the score functions obtained from the existing methods such that the highest score value of the proposed method is 0.4400 which is shown in Table 11. Therefore, decision-making processing based on the novel methods can be performed in an evaluation system, which implies that the propose methods have greater advantages than the existing methods.
In Figure 4, we compared all the results of proposed method with existing methods with the help of graphical approach.
8. Conclusions
The objective of this paper is to present some induced operators based on I-VITrFAA aggregation operators and apply them to the MCGDM problems. The MCGDM has an enormous prospective and restraint development used for improving and assessing different contradictory criteria in all aspects of DM in order to achieve more suitable and pragmatic DM result. In DM issues, the effectual erudition concerning a fastidious fact makes the decision-making task more complicated and dynamic. Induced interval-valued intuitionistic fuzzy sets are general mathematical method that can easily handle ambiguous and imprecise knowledge. In this article we developed operation laws based on the Aczel-Alsina (A-A) and introduced four new types of aggregations operators such as I-IVITrFAAOWA operator, I-IVITrFAAHA operator, I-IVITrFAAOWG operator and I-IVITrFAAHG operator. Finally, examples are given to show the technique’s prospective application and improvement. Furthermore, the established approach can be expanded for future research by incorporating other existing fuzzy sets and applying them to various MCGDM problems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
Authors will pay all the outstanding dues after the acceptance of this manuscript. Yong-Long Wang, Ebenezer Bonyah, Mashael Khayyat, Zubair Ahmad, Muhammad Shakeel, and Waris Khan contributed equally to this work.
Acknowledgments
The authors from King Khalid University extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under Grant No. RGP.2/229/43.