Abstract
Aggregation operators are useful tools for approaching situations in the realm of multiattribute decision-making (MADM). Among the most valuable aggregation strategies, the Hamy mean (HM) operator is designed to capture the correlations among integral parameters. In this article, a series of Hamy-inspired operators are used to combine 2-tuple linguistic Fermatean fuzzy (2TLFF) information. The new 2TLFF aggregation operators that are born from this adaptation include the 2-tuple linguistic Fermatean fuzzy Hamy mean (2TLFFHM) operator, 2-tuple linguistic Fermatean fuzzy weighted Hamy mean (2TFFWHM) operator, 2-tuple linguistic Fermatean fuzzy dual Hamy mean (2TLFFDHM) operator, and 2-tuple linguistic Fermatean fuzzy weighted Hamy mean (2TLFFWDHM) operator. Furthermore, various essential theorems are stated, and special cases of these operators are thoroughly examined. Then, a renewed multiattribute group decision-making (MAGDM) technique based on the suggested aggregation operators is provided. A practical example corroborates the usefulness and implementability of this technique. Finally, the merits of the proposed MAGDM method are demonstrated by comparing it with existing approaches, namely, it can deal with MAGDM problems by considering interactions among multiple attributes based on the 2TLFFWHM operator.
1. Introduction
Multiattribute decision-making (MADM) aims at evaluating a (typically finite) number of alternatives based on a set of criteria and designing operational strategies for picking a best alternative based on expert assessments of their levels of satisfaction of the criteria. But, because of the limitations of an individual’s knowledge or experience, it is difficult for a single decision maker (DM) to evaluate all important components of a situation. Therefore, MCGDM extends MADM with the incorporation of inputs from a group of experts. This approach is more suitable for solving complicated decision-making problems. DMs or experts give their preferences or views regarding the alternatives. These opinions are based on a fixed list of criteria. The final goal is to find a best option.
Zadeh [1] proposed the concept of fuzzy set (FS) which, for the first time, allowed the researchers to effectively describe imprecise and uncertain information through a numerical degree of association. Fuzzy set theory was further investigated, and many successful achievements in a variety of areas were obtained by a huge number of scholars. Because a fuzzy set only contains a membership component, it was quickly apparent that this might lead to significant information being overlooked in practical research. The reason is that the nonmembership scores of the alternatives under examination are implicitly assumed to be derived from their membership scores. Atanassov [2] introduced the intuitionistic fuzzy model in response to this setback. In an intuitionistic fuzzy set (IFS), all the objects are described by both their membership and nonmembership degrees, and it is assumed that their total sum is always bounded by 1. A number of scholars have investigated IFSs, their aggregation operators (AOs) and theoretical implications, and their applicability in a variety of MADM situations. For example, the IFWA operator, IFOWA operator, and IFHA operator were all investigated by Xu [3]. Xu and Yager [4] developed several fuzzy weighted geometric (IFWG) aggregating procedures based on IFSs. Hung and Yang [5] investigated IFS similarity metrics and Liu et al. [6] have studied centroid transformations of intuitionistic fuzzy values based on aggregation operators. However, if the experts produce estimates with a total larger than one in at least one situation, the IFSs will no longer be useful for decision-making. To address this shortcoming, Yager and Abbasov [7] proposed the Pythagorean fuzzy sets (PFS), which form a broader model. Many scholars have quickly taken notice of the PFS concept. Yager and Abbasov [7] looked at the relationships between Pythagorean membership grades (PMGs) and complex numbers. Khan et al. [8] investigated MADM issues in a Pythagorean hesitant fuzzy environment with insufficient information about weights. To further grasp PFSs, Peng and Yang [9] created the division and subtraction operations. Reformat and Yager [10] suggested a method based on Pythagorean fuzzy set to build the list of recommended movies from the Netflix competition database.
Following this line of thought, Senapati and Yager [11] proposed the concept of Fermatean fuzzy sets (FFSs) as a further expansion of both IFSs and PFSs. The cubic sum of an object’s membership and nonmembership values is bounded by 1 in a FFS. Senapati and Yager [12] defined some new operations on Fermatean fuzzy numbers and used them to tackle MADM issues. With respect to aggregation, operators like the FFAOs and FFFOs were presented by Senapati and Yager [13]. Garg et al. [14] proposed a technique for selecting the most appropriate laboratory for COVID-19 tests in a Fermatean fuzzy environment. Also, in this setting, Akram et al. [15] used a MADM technique to demonstrate the benefits of a sanitizer in COVID-19. Shahzadi and Akram [16] developed the concept of Fermatean fuzzy soft AOs and used this tool to pick an antiviral mask in the realm of group decision-making. Aydemir and Gunduz [17] described the Fermatean fuzzy TOPSIS (FF-TOPSIS) approach, which uses the Dombi AOs. Other related models have become popular in recent years while gaining further insight into the accurate manipulation of vague information. For example, Akram et al. [18] proposed a model for group decision-making under FF soft expert knowledge. Feng et al. [19] set forth some novel score functions of generalized orthopair fuzzy membership grades with applications in MADM. Concerning the Hamacher-type aggregation operators, Waseem et al. [20] used them to aggregate data in an -polar fuzzy setting, and under FFE, Shahzadi et al. [21] proposed Hamacher interactive hybrid weighted averaging operators. In addition, Akram et al. [22] developed a new hybrid model with applications under complex Fermatean fuzzy -soft sets.
Another breakthrough in information retrieval was made by Herrera and Martínez [23] who proposed the 2TL representation model. Its basic component consists of a linguistic term and a numeric value, based on the concept of symbolic translation. It has precise linguistic information processing abilities, and it may successfully avoid data loss and misinterpretations, which used to occur in previous linguistic modelizations. Experts prefer this model to operate in many practical decision-making situations. Herrera and Martínez [23] demonstrated that a 2TL information processing method may successfully minimize information loss and distortion. Herrera and Herrera-Viedma [24] came up with a few 2-tuple arithmetic aggregating operators. A group DM model was developed by Herrera et al. [25] for controlling nonhomogeneous data processing. Consensus support was introduced by Herrera-Viedma et al. [26] with the help of multigranular linguistic preference relations. The linguistic information processing model was adopted by Liao et al. [27] to decide on an ERP system. To cope with unbalanced linguistic data, Herrera et al. [28] presented a fuzzy linguistic technique. Liu et al. [29] proposed dependent interval 2TL aggregation operators and their application to multiple-attribute group decision making. To set interval numerical scales of 2TL word sets, Dong and Herrera-Viedma developed the consistency-driven automated methods [30]. Qin and Liu [31] proposed the 2TL Muirhead mean operators for multiattribute group decision-making (MAGDM). Also, in the context of 2TL MAGDM, but in the presence of inadequate information about weights, Zhang et al. [26] developed a consensus reaching model. Liu et al. [32] devised linguistic -rung orthopair fuzzy generalized point weighted aggregation operators which were applied to MAGDM issues. Jan et al. [33, 34] developed new decision-making methods.
Although certain correlations between arguments are intrinsic to some actual MADM problems, the aggregation operators discussed above do not take these relationships into account. The HM [35] operator can adequately assimilate the interaction among arguments, thus it is no surprise that its popularity is rising among a significant number of scholars. Li et al. [36] constructed several intuitionistic fuzzy Dombi Hamy operators on the basis of IF information and used these aggregation operators for car supplier selection. Li et al. [35] devised several PF Hamy operations to identify the most attractive green supplier in order to reduce the limitations of IF sets. Wei et al. [37] developed dual hesitant PF Hamy mean operators and used them to tackle MADM. Wang [38] developed some q-rung orthopair fuzzy Hamy mean operators in MADM and shown their application to the problem of enterprise resource planning systems selection. Deng et al. [39] defined a 2TLPFS by combining the 2TLS and the PFS and then presented several Hamy operators in a 2-tuple linguistic Pythagorean fuzzy environment. Liu and Liu [40] proposed linguistic intuitionistic fuzzy Hamy mean operators and their application to multiple-attribute group decision making. Liu and You [41] suggested several linguistic neutrosophic Hamy operators for MADM issues on the basis of the linguistic neutrosophic set. Wang et al. [42] proposed multicriteria group decision-making method based on interval 2-tuple linguistic information and Choquet integral aggregation operators.
According to a review of the literature, the Fermatean fuzzy set is a useful tool for depicting imprecise and ambiguous information, and the HM operators may explore the interaction between any number of combined arguments. By inspiration of the classical HM operator and the FF sets, we combine 2-tuple linguistic sets with Fermatean fuzzy sets and construct 2TLFFHM aggregation operators in this work. The motivation of the present contribution is summarized as follows:(1)In classical FFS, the membership and nonmembership degrees are given by numerical values that lie within the interval , while in 2TLFFS, the membership and nonmembership degrees are given by the 2TL model. This is more useful to tackle those real-life MAGDM problems in which experts express their opinion through linguistic labels.(2)The proposed operators are very general. They perform excellently, not only for 2TLFF information but also for 2TLIF and 2TLPF data. Thus, they overcome the drawbacks and limitations of the existing operators.(3)The proposed operators produce more exact findings when applied to real-life MAGDM problems based on 2TLFF data, because these operators have the ability of accounting for correlated arguments.
The following is a summary of primary contributions of this article:(1)The concept of 2TLFFS is explained with certain basic operations and properties. The score and accuracy functions of 2TLFFFSs are discussed. These tools are used for providing a verifiable ordering of 2TLFFFSs.(2)The concepts of 2TLFFHM operator, 22TLFFWHM operator, 2TLFFDHM operator, and 2TLFFDWHM operator are proposed. Several significant properties of these operators are studied and verified.(3)A mathematical model for MAGDM based on 2TLFF data is presented to choose the optimal alternative from a finite number of alternatives. An example is fully solved based on the proposed methodology. This exercise evaluates the superiority and applicability of our proposal.(4)Finally, the effectiveness and authenticity of the suggested aggregation operators are demonstrated by a comparison analysis.
Thus, the fundamental goal of this paper is to present a more acceptable aggregation operator for multiple-attribute decision-making issues, as well as a more scientific and effective manner to communicate assessment information. Furthermore, we may dynamically alter the parameter to generate various decision-making results under different risk scenarios by taking into account the decision maker’s risk attitude.
The rest of this work is organized as follows. In Section 2, some basic definitions are reviewed, which are helpful for further development. In Section 3, the 2TLFFS model, some operations on 2TLFFSs, and the score and accuracy functions of 2TLFFS are discussed. The 2TLFFHM operator, 2TLFFWHM operator, 2TLFFDHM operator, and 2TLFFDWHM operator are proposed, and their properties are studied. In Section 4, we propose a model for MAGDM problems with 2-tuple linguistic Fermatean fuzzy information based on the 2TLFFWHM and 2TLFFWDHM operators. In Subsection 4.1, we present a numerical example of selection of technique for reducing the smog with 2TLFF information, in order to illustrate the method proposed in this paper. We conclude the paper with some remarks in Section 5.
2. Preliminaries
In this section, we review basic definitions that are necessary for this paper.
Definition 1 (see [43]). Let there exist , a linguistic term set, with odd number of terms, where indicates a possible linguistic term for a linguistic variable. For instance, a linguistic term set having seven terms can be described as follows:
S = \{ = none, = very poor, = poor, = fair, = good, = very good, = perfect\}.
If , then the linguistic term set meets the following characteristics:(i)Ordered set: , if and only if (ii)Max operator: , if and only if (iii)Min operator: , if and only if (iv)Negative operator: Neg such that
Definition 2 (see [44]). Let be a set of linguistic term. The 2TL model is a pair , where are linguistic terms and is a numeric value, called symbolic translation that indicates the translation of fuzzy membership function, which represents the closest term, . The value of is defined as
Definition 3 (see [44]). Let be a linguistic term set and be a set of 2-tuple linguistic terms. The function is defined bywhere(1)i = round ,(2),where round(.) is the usual round operation.
Definition 4 (see [44]). Let be a linguistic term set and . Then, a function is defined by
Definition 5 (see [44]). Let and be two 2-tuple linguistic values. Then,(1)If , then (2)If k = l, then(1)If , then (2)If , then (3)If , then
Definition 6 (see [11]). A Fermatean fuzzy set , on universe of discourse , is an object having the formwhere and , including the condition , for all . The numbers and , respectively, the membership degree, and the nonmembership degree of the element x are in the set .
Definition 7 (see [45]). The HM operator is defined as follows:where x is a parameter and are x integer values taken from the set of k integer values, denotes the binomial coefficient, and .
The HM operator satisfies the properties of idempotency, monotonicity, and boundedness. The two special cases of HM operators are given as follows:(i)If , then it reduces to arithmetic mean operator.(ii)If , then it reduces to geometric mean operator.
Definition 8 (see [45]). The DHM operator is defined as follows:where is a parameter and are integer values taken from the set of integer values, denotes the binomial coefficient, and . The list of nomenclature is given in Table 1.
3. 2-Tuple Linguistic Fermatean Fuzzy Hamy Mean Operators
We first introduce the concept of 2TLFFS
Definition 9. Let be linguistic term set with odd cardinality and let be a set of all 2-tuple linguistic terms defined on . A 2TLFFS is defined aswhere , , satisfying the condition and represent the membership degree and nonmembership degree by 2-tuple linguistic terms.
Definition 10. Letbe set of all 2TLFFNs and let be any two 2TLFFNs in , , then some basic operations on them are defined as follows:(i),(ii),(iii),(iv).
Definition 11. Let be a 2TLFFNs in . Then, the score function of P is defined as follows:The accuracy function of P is defined as follows:
Definition 12. Let be any two 2TLFFNs based on score function and accuracy function ; we give an order relation between two 2TLFFNs, which is defined as follows:(1)If , then (2)If , then (3)If , , then (4)If , , then (5)If , , then We now introduce the concept of 2TLFFHM operator.
Definition 13. Let be a group of 2TLFFNs. The 2TLFFHM operator is defined as follows:
Theorem 1. Let be a collection of 2TLFFNs. The 2TLFFHM operator is also a 2TLFFN, where
Proof. From the basic operation on 2-TLFFN 3.2, we can getTherefore,Now, we need to prove that 2TLFFHM is also a 2TLFFN. For this, we need to show the following two relations:(1)(2)LetSince , we get and This means that . Similarly, we can have . Sincewe can get
.
This implies that .
Example 1. Let , and are four 2TLFFNs, and assume , then from equation (2), we haveThe 2TFFHM operator has three properties.
Theorem 2. If are same, then
Proof. Since , then
Theorem 3. Let and be two sets of 2TFFNs. If and holds for all j, then
Proof. Let and ; suppose thatWe can obtainFurthermore,This means that . Similarly, we can get . Thus, if and , thenIf and , then
Theorem 4. Let be a set of 2TFFNs. If and , thenFrom Theorem 2,From Theorem 3,
3.1. 2-Tuple Linguistic Fermatean Fuzzy Weighted Hamy Mean Operator
Now, we propose 2-tuple linguistic Fermatean fuzzy weighted Hamy mean (2TLFFWHM) operator.
Definition 14. Let be a collection of 2TLFFNs with weight vector , thereby satisfying and . The 2TLFFWHM operator is defined as follows:
Theorem 5. Let be a collection of 2TLFFNs. The 2TLFFWHM operator is also a 2TLFFN, where
Proof. From the basic operation on 2-TLFFN 3.2, we can getTherefore,Now, we need to prove that 2TLFFWHM is also a 2TLFFN. For this, we need to show the following two relations:(1)(2)LetSince , we getThis implies that . Similarly, we can have . Sincewe can get
.
This means that .
Example 2. Let , and be four 2TLFFNs, , and assume , then from equation (3), it follows thatWe discuss two properties of 2TLFFWHM operator.
Theorem 6. Let and be two sets of 2TFFNs. If and hold for all j, thenThe proof is similar to 2TLFFWHM operator; it is omitted.
Theorem 7. Let