#### Abstract

This research article is devoted to presenting the concept of 2-tuple linguistic -polar fuzzy sets (2 TL FSs) and introducing some fundamental operations on them. With 2 TL FSs, we shall be able to capture imprecise information with high generality. With the appropriate operators, we shall be able to apply 2 TL FSs in decision-making efficiently. The aggregation operators that we propose are the 2 TL F Hamacher weighted average (2 TL FHWA) operator, 2 TL F Hamacher ordered weighted average (2 TL FHOWA) operator, 2 TL Hamacher hybrid average (2 TL FHHA) operator, 2 TL F Hamacher weighted geometric (2 TL HWG) operator, 2 TL Hamacher ordered weighted geometric (2 TL HOWG) operator, and 2 TL F Hamacher hybrid geometric (2 TL FHHG) operator. We investigate their properties, including the standard cases of monotonicity, boundedness, and idempotency. Then we develop an algorithm to solve multicriteria decision-making problems formulated with 2 TL F information. The 2 TL F data in multiattribute decision-making are merged with the help of aggregation operators, and we consider the particular instances of the 2 TL FHA and 2 TL FHG operators. The influence of the parameters on the outputs is explored with a numerical simulation. Moreover, a comparative study with existing methods was performed in order to show the applicability of the proposed model and motivate the discussion about its virtues and advantages. The results confirm that the model here developed is reliable for decision-making purposes.

#### 1. Introduction

With its large display of different approaches, multiattribute decision-making (MADM) is a collective enterprise that aspires to deal with complex situations in the presence of multiple attributes. The choice of a decision-making approach plays a vital role in the selection of desirable alternatives. Also, the representation of the information is crucial because the formulation of problems with real-valued attribute endowments is a rarity in decision sciences. For this reason, Zadeh [1] introduced the idea of fuzzy sets (FSs), a mathematical tool that easily tackles MADM, being a mutated form of crisp set theory. Yager [2, 3] presented the less stringent concept of Pythagorean fuzzy sets (PFSs) for which the condition imposed by IFSs is relaxed to . FS has a pathbreaking structure that allows it to account for vagueness for the first time. But it is not a totally general framework for the mathematical treatment of partial knowledge. To enlarge its scope, Atanassov [4] introduced intuitionistic fuzzy sets theory (IFSs), a simple modification that can tackle uncertain and vague data more precisely.

Aggregation operators (AOs) play a crucial role in converting different datasets into a single result and dealing with collective decision-making problems. A very popular tool for aggregating data was introduced by Yager [5] under the name of ordered weighted averaging aggregation (OWA) operators. Yager [6] contributed to quantifier guided aggregation using OWA operators. Xu [7] first gave intuitionistic fuzzy set-based aggregation operators. Then Xu and Yager [8] studied geometric aggregation operators and produced some real-life applications. Alcantud et al. [9] first produced AOs that operate on infinitely many intuitionistic fuzzy sets. Improvements to IFSs appeared, and AOs were produced in these new models too. Since there is often a counterpart for each attribute of the alternatives, bipolarity has been used as a conceptual tool for the representation of dual attributes. Wei et al. [10] came forward with hesitant bipolar fuzzy weighted aggregation operators as arithmetic and geometric operators. With the help of hesitant bipolar fuzzy weighted aggregation operators and geometric operators, Xu and Wei [11] gave dual hesitant bipolar fuzzy weighted aggregation operators and geometric operators. In other frameworks, AOs have been studied too. For example, Garg [12] gave a framework for linguistically prioritized aggregation operators.

More sophisticated aggregation operators were developed to improve the accuracy of the subsequent applications. For example, based on algebraic and Einstein -conorm and -norm [13], Hamacher -conorm and -norm [14] were developed to aggregate data for decision making. Peng and Luo [15] presented decision-making for China’s stock market bubble warning. Aggregation operators based on Hamacher operations produce a transparent result in decision-making. Thus, inspired by these operators, Wei et al. [16] developed some induced geometric aggregation operators with intuitionistic fuzzy information and showed their applications to group decision making. Liu [17] proposed aggregation operators for interval-valued intuitionistic fuzzy fields. Further contributions were made by Akram et al. [18] with the introduction of a decision-making model using complex intuitionistic fuzzy Hamacher aggregation operators. Huang [19] put forward the idea of intuitionistic fuzzy Hamacher aggregation operators and illustrated their application in multiattribute decision making. After that, Hamacher aggregation operators were extended so that they could operate on Pythagorean fuzzy sets. Wu and Wei [20] presented the Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Akram et al. [21] used these operators in the complex fuzzy field and developed the idea of a hybrid method for complex Pythagorean fuzzy decision making. Akram et al. [22] designed a decision-making model with the help of complex picture fuzzy Hamacher aggregation operators. After that, bipolar fuzzy Hamacher arithmetic and geometric operators were also developed by Wei et al. [23]. Akram et al. [24] developed q-rung orthopair fuzzy graphs under Hamacher operators.

As many real-life situations contain multiinformation, Chen et al. [25] developed the -polar fuzzy ( F) set, which enables decision-makers to manipulate multipolar data for the purpose of decision-making. Further, Jana and Pal [26] presented the -polar fuzzy operators and their application in the multiple-attribute decision-making process. Hwang and Yoon [27] presented the concept of multi-objective decision making-methods with applications. Nevertheless, Hamacher operations were not originally designed to collect information in the form of intuitionistic fuzzy numbers (IFNs), Pythagorean fuzzy numbers (PFNs), bipolar fuzzy numbers [28], or F numbers ( FNs). Waseem et al. [29] used Hamacher operators to aggregate data in a -polar fuzzy setting. Akram et al. [30, 31] adapted respective mathematical models to approach decisions in -polar fuzzy environments.

Most people want to express themselves with common terms like “magnificent,” “superb,” “best,” “better,” “poor,” and “worst” to gauge some attributes. These assessments of an object’s properties should then be used in MADM. Thus, the aggregation operators that collect information in a linguistic form are crucially essential. By using the 2-tuple linguistic (2 TL) tool, we can prevent the loss of data and get more transparent results in decision making. Firstly, Herrera and Martinez [32] introduced the idea of 2 TL representation, which is the most successful tool to take on linguistics decision-making issues. For further notions and applications, the readers are suggested to [33–40]. The main goal of this article is the aggregation of 2-tuple linguistic information by using Hamacher operators and their application in decision-making.

##### 1.1. Motivation and Contribution

The motivation of this work is described as follows:(i)The justification of any reliable choice in a problem formulated with 2 TL F information is a highly complicated MADM issue. Nevertheless, the proposed MADM model provides significant results through convincing arguments.(ii)The field of application of 2 TL FS is enormous, as this model combines the benefits of both 2 TL and -polar fuzzy sets. However, the treatment of linguistic techniques with multipolar fuzzy situations, particularly the 2 TL F MADM approach, remains a challenge that we take up in this article.(iii)The toolbox that helps us for this purpose includes aggregation operators. Taking this into consideration, aggregation operations are capable of providing valid data combinations in the form of 2 TL Fs.(iv)Hamacher aggregation operators are a straightforward tool, easy to apply to real-life MADM problems based on the 2 TL F environment.(v)The previously existing techniques, which are designed to take over MCDM problems, are restricted to dealing only with the -polar fuzzy information. These techniques are unable to take into account linguistic information. So, this may cause a loss of information, which typically leads to undesired results. Thus, existing technical hindrances can be sorted out by using the newly proposed work.

Thus, to choose the best alternative, our 2 TL F methodology relies on Hamacher AOs. As compared to other plans of action, the developed operators have three major advantages. Firstly, we can make use of 2 TL F information, which is an asset in decision-making problems as explained above. Secondly, a single parameter suffices to make the methodology flexible while preserving its transparency and simplicity. Further, the decision-making issues are not affected by varying the parameters. Thirdly, the use of Hamacher aggregation operators for 2 TL F information in MADM produced significant results. To operate in complicated situations with a real background, as in the case of the selection of the best place for a thermal power station, the proposed operators are very affordable.

The major contributions of this research paper are(i)The generalization of F Hamacher operators to 2 TL F Hamacher operators. Some fundamental properties are given, including their proofs and explanations. These operators are more flexible and produce transparent results by aggregating -polar fuzzy data with linguistic information.(ii)In order to undertake decision-making problems, an algorithm is developed for 2 TL multipolar information.(iii)Lastly, the validity, versatility, and traits of the proposed operators are investigated by a comparative study with existing techniques.

##### 1.2. Structure of Paper

The structure of this research work is as follows: in Section 2, we perform a basic revision of some concepts about 2 TL and F sets, which are properly described. Section 3, contains the study of some operators, namely 2 TL FHWA, 2 TL FHOWA, and 2 TL FHHA, with some basic properties. In Section 4, we investigate additional operators like 2 TL FHWG, 2 TL FHOWG, and 2 TL FHHG. Again, their fundamental properties are studied, and examples are given. In Section 5, a procedure is developed to tackle multicriteria decision-making issues that involve 2 TL F information. The procedure takes advantage of the 2 TL FHA and 2 TL FHG operators. In the next Section 6, numerical work is done for the selection of the best location for a thermal power station by using 2 TL FHA and 2 TL FHG operators. It also includes a study about the influence of parameters on the decision. Section 7 contains a comparative study with previously existing methods, which shows the applicability and strength of our method. It also outlines the advantages and limitations of the proposed work. Section 8 contains some concluding remarks with future research directions. The structure of the proposed research article is displayed in Figure 1.

#### 2. Preliminaries

This section reviews some basic definitions that are necessary for this paper.

*Definition 1. *(see [32, 34]). Let a set of odd numbers of linguist terms, where indicates the probable linguistic term for the linguistic variables. For instance, a linguistic term set having seven terms can be described as follows:

S = \{ = none, = very low, = low, = medium, = high, = very high, = perfect\}.

If , then the set S meets with 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)Negation: Neg such that t .Herrera and Martinez [32], introduced 2 TL representation model based on the idea of symbolic translation, which is useful for representing the linguistic assessment information by means of a 2-tuple .

where(i) is a linguistic label for a predefined linguistic term set .(ii) is called symbolic translation and .

*Definition 2. *(see [32]). Let be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set , i.e., the result of a symbolic aggregation operation, , where is the cardinality of . Let and be two values, such that, and then is called a symbolic translation.

*Definition 3. *(see [32]). Let be a linguistic term set and be a number value representing the aggregation result of linguistic symbolic. Then the function used to obtain the 2-tuple linguistic information equivalent to is defined as

*Definition 4. *(see [32]). Let be a linguistic term set and be a 2-tuple, there exists a function that restores the 2-tuple to its equivalent numerical value , where

*Definition 5. *(see [32, 34]). Let us consider and be two 2 TL values. Then,(1)For , we have, is less than (2)If *k* = l, then(i)For , implies that and are same.(ii)For , implies that is less than (iii)For , implies that is greater than Chen et al. [25] first considered the notion of -polar fuzzy sets. The membership grade of -polar fuzzy set belongs to the interval , and it stands for different divisions of an attribute.

*Definition 6. *(see [25]). An set on a nonempty set is defined as a mapping . The representation of membership value for every element is denoted asHere is the th projection mapping.

Notice that, (-th power of [0, 1]) is a Poset with the pointwise order , where is an arbitrary ordinal number (we make the convention that when ), is defined by for each , and is the th projection mapping . Where in , the greatest value is and the smallest value is . For convenience, is the representation of mF number.

*Definition 7. *(see [25]).The accuracy function of an , is defined asThus, arbitrarily, for any -polar fuzzy numbers , .

*Definition 8. *(see [25]). Let , and be two -polar fuzzy numbers. Then(1) , if .(2) , if .(3) , If and .(4) , if , but .(5) , if , but .The nomenclature of the proposed research terms is given in Table 1.

#### 3. 2 TL F Hamacher Aggregation Operators

We first define the concept of 2-tuple linguistic -polar fuzzy sets and some basic operations.

*Definition 9. *A 2TLmFSs on a nonempty set is defined aswhere , represent the membership degrees, with the conditions , , . For convenience, we say , a 2-tuple linguistic -polar fuzzy number, where .

*Definition 10. *The *score function* of a 2 TL -polar fuzzy number , is defined as

*Definition 11. *The *accuracy function* of a -polar fuzzy number , is defined as

*Definition 12. *Let , and , be two 2-tuple linguistic -polar fuzzy numbers. Then we define operations on 2-tuple linguistic -polar fuzzy numbers as follows:(1) (2) (3) , ,(4) , ,(5) , ,(6) , if and only if ,(7) ,(8) .We now define Hamacher operations for 2-tuple linguistic -polar fuzzy numbers.

*Definition 13. *Let , and be 2-tuple linguistic -polar fuzzy numbers. Then, the basic Hamacher operations for 2-tuple linguistic -polar fuzzy numbers with is defined as(1) ,(2) ,(3) , (4) ,

*Example 14. *Let and be 2-tuple linguistic 3-polar fuzzy numbers. Then for ,Thus, is again a 2-tuple linguistic 3-polar fuzzy number. So, the closure law is satisfied.

Thus, in a similar pattern, the closure law is verified for all the above-defined Hamacher operations for 2-tuple linguistic -polar fuzzy numbers.

*Definition 14. *Let be a set of 2 TL F numbers, where . Then, an 2 TL F Hamacher weighted average operator is a mapping m , whose domain is the family of 2 TL F numbers , which is defined as,where is the weight vector representation for , for each ‘j’, , with and .

*Example 1. *Let , , and be 2-tuple linguistic 4-polar fuzzy numbers with a weight vector . Then, for ,

Theorem 1. *Let**be a set of 2 TL**F numbers, where**. The assembled values of these 2 TL**F numbers using the 2 TL**FHWA operator is also 2 TL* *F numbers, given as*

*Proof. *We use mathematical induction to prove it.

*Case 1. *Let us take *n* = 1, by using *Equation* (4), we obtained

Thus, (10) holds for *n* = 1.

*Case 2. *Next, we suppose that the result is true for *n* = k, where (N: natural numbers), we obtain

for ,

,

Thus, (10) holds for . Conclusively, the result holds for any .

*Remark 1. *For , 2TLmFHWA operator reduces to 2 TL F weighted averaging (2 TL FWA) operator given as follows:2. For , 2 TL FHWA operator reduces to 2TLmF Einstein weighted averaging (2 TL FEWA) operator as follows:

Theorem 2. *Idempotency. Let**be a set of 2 TL**F numbers, where**. If all these numbers are equal, that is,**,**’j’ varies 1 to n, then we have*

*Proof. *Since , where , then by using *equation* (4), we get

Hence, holds only if we use where

Further, we will discuss the remaining properties, namely, boundedness and monotonicity, and their proofs are directly followed by definitions.

Theorem 3. *Let**, be a set of 2 TL**F numbers, where ‘j’ varies from 1 to n,**and**, then*

Theorem 4. *Let**and**,**be the two sets of 2 TL**F numbers. If**, then**We now propose the 2 TL F Hamacher ordered weighted average (2 TL FHOWA) operator.*

*Definition 15. *Let be the set of 2 TL F numbers where . Then, a 2 TL FHOWA operator is a mapping with a weight vector

, and . Then,where is the permutation of the indices , for which ,

.

Theorem 6. *Let**be a set for 2 TL**F numbers, where ‘j’ varies from 1 to n. Then the assembled values of these 2 TL**F numbers using the 2 TL**FHOWA operator is again 2 TL* *F numbers, given as*

*Proof. *The proof of the theorem is directly followed by the similar arguments as used in Theorem 3.9, as mentioned above.

*Remark. 1. *For , 2 TL FHOWA operator reduce to 2 TL FOWA operator as follows:2. For , 2 TL FHOWA operator reduce to 2 TL FEOWA operator given as follows:

Theorem 7. *Idempotency. Let us consider**a collection of 2 TL**F numbers, where**. For the equality of all these numbers, in other words,**, where,**, then we have*

*Proof. *Since , where . Then by using *Equation* (11), we obtain

Hence, holds only if ,