#### Abstract

An -polar fuzzy set is a powerful mathematical model to analyze multipolar, multiattribute, and multi-index data. The -polar fuzzy sets have appeared as a useful tool to portray uncertainty in multiattribute decision making. The purpose of this article is to analyze the aggregation operators under the -polar fuzzy environment with the help of Dombi norm operations. In this article, we develop some averaging and geometric aggregation operators using Dombi -norm and -conorm to handle uncertainty in -polar fuzzy (, henceforth) information, which are Dombi weighted averaging () operator, Dombi ordered weighted averaging () operator, Dombi hybrid averaging () operator, Dombi weighted geometric () operator, Dombi weighted ordered geometric operator, and Dombi hybrid geometric () operator. We investigate properties, namely, idempotency, monotonicity, and boundedness, for the proposed operators. Moreover, we give an algorithm to solve multicriteria decision-making issues which involve information with and operators. To prove the validity and feasibility of the proposed model, we solve two numerical examples with our proposed models and give comparison with -ELECTRE-I approach (Akram et al. 2019) and Hamacher aggregation operators (Waseem et al. 2019). Finally, we check the effectiveness of the developed operators by a validity test.

#### 1. Introduction

Multicriteria decision making (MCDM) is performing a vital role in different areas, including social, physical, medical, and environmental sciences. MCDM methods are not only used to determine a suitable object but also used to rank the objects in an appointed problem. To solve different uncertain problems for decision making, Atanassov [1] presented the concept of intuitionistic fuzzy set (IFS) which considers both membership and nonmembership parts, an extension of fuzzy set [2] in which simple membership part is characterized.

Aggregation operators (AOs) perform an important role in order to combine data into a single form and solve MCDM problems. For example, Yager [3] introduced weighted AOs. Xu [4] proposed some new AOs under IFSs. Xu and Yager [5] developed certain new geometric AOs and solved some real-world MCDM problems. From the inspection of an object, it can be easily seen that there exist two properties of the object which are opposite to each other. With this perspective, Zhang [6] presented the idea of bipolar fuzzy set (BFS). BFSs provide generalized structure as compared to fuzzy sets [2] whose memberships belong to . Bipolarity plays an important role in different research areas and provides more flexibility as compared to the fuzzy methods. In the last decades, a lot of researchers, attracted by this efficient concept, applied it to aggregate bipolar information using different -norms and their corresponding conorms, including Hamacher and Dombi -norms and their corresponding conorms. For example, Wei et al. [7] developed some bipolar fuzzy Hamacher weighted averaging and geometric AOs. By combining Hamacher operations and prioritized AOs, Gao et al. [8] proposed dual hesitant bipolar fuzzy Hamacher prioritized weighted AOs and applied the proposed methodologies to an MCDM problem. Liu [9] utilized interval-valued intuitionistic fuzzy numbers with Hamacher AOs and developed multicriteria methods for group decision making. Jana et al. [10] applied weighted, ordered weighted, and hybrid average and geometric AOs for the aggregation of bipolar fuzzy information using Dombi -conorm and -norm. They also proposed bipolar fuzzy Dombi prioritized AOs in [11]. He [12] developed hesitant fuzzy Dombi AOs and investigated typhoon disaster assessment using proposed theory. Xu and Wei [13] introduced different dual hesitant bipolar fuzzy AOs to solve MCDM problems. Xu [14] proposed intuitionistic fuzzy power AOs for multiattribute group decision making. Xiao [15] constructed induced interval-valued intuitionistic fuzzy Hamacher AOs and discussed their application to MCDM. Chen and Ye [16] discussed MCDM problem under Dombi operations in single-valued neutrosophic situation. Garg [17] presented some generalized interactive AOs under Einstein operations in Pythagorean fuzzy environment and discussed a decision-making issue. Akram et al. [18] proposed different Pythagorean Dombi fuzzy AOs and studied their applications in MCDM. Shahzadi et al. [19] introduced Pythagorean fuzzy Yager AOs for decision making. Peng and Yang [20] investigated different basic properties of interval-valued Pythagorean fuzzy AOs. Wang et al. [21] introduced some new types of -rung orthopair fuzzy Hamy mean AOs to handle MCDM situations. Arora and Garg [22] proposed robust AOs with an intuitionistic fuzzy soft environment. Wang and Li [23] developed Pythagorean fuzzy interaction power Bonferroni mean AOs and discussed their applications to MCDM. Chiclana et al. [24] introduced some ordered weighted geometric operators and solved a decision-making problem. Liang et al. [25] developed Pythagorean fuzzy Bonferroni mean AOs.

Nowadays, experts believe that multipolarity performs a vital role in many practical situations. Due to the presence of multipolar data in different daily life problems of science and technology, Chen et al. [26] initiated the notion of set theory as generalization of fuzzy and bipolar fuzzy sets. Waseem et al. [27] studied recently MCDM problems based on Hamacher AOs. Khameneh and Kilicman [28] proposed soft weighted AOs and applied these AOs in decision making. In view of the fact that sets have an efficient strength to handle vague data which arise in several real-life problems, in this paper, we generalize Dombi AOs to aggregate the information. The study of AOs under Dombi operations is very popular. Thus, an efficient research topic is how to aggregate numbers with Dombi operations. To tackle this dilemma, in this article, we present some Dombi AOs on the ground of classical geometric, arithmetic, and Dombi operations. For more information and terminologies on AOs, the readers are referred to [29–45].

An model is more general than the fuzzy sets and BFSs due to the wider range of applicability over different complex problems. The sets can handle much more details about an element and can explain uncertainties concurrently more precisely than the other existing methods, like fuzzy set and BFS. The motivation of developed AOs is summarized as below.(1)A very difficult MCDM problem is the estimation of the supreme option in an environment due to the involvement of several imprecise factors. Assessment of information in different MCDM techniques is simply depicted through fuzzy and bipolar fuzzy numbers which may not consider all the data in a real-world problem.(2)As a general theory, numbers describe efficient execution in the assessment process about uncertain, imprecise, and vague multipolar information. Thus, theory provides an excellent approach for the assessment of objects under multinary data.(3)In view of the fact that Dombi AOs are simple but provide a pioneering tool for solving MCDM problems when combined with other powerful mathematical tools, this article aims to develop Dombi AOs in an environment to handle complex problems.(4)An model is different from the mathematical tools like fuzzy sets and BFSs because the fuzzy set and BFS can only handle one-dimensional data and two-dimensional data, respectively, which may prompt a loss in data. Nevertheless, in many daily life problems, we handle the situations having higher dimension to sort out all the attributes and their subcharacteristics.(5)The Dombi AOs employed in the construction of Dombi AOs are more suitable than all other aggregation approaches to tackle the MCDM situations as developed AOs have ability to consider all the information within the aggregation procedure.(6)Dombi AOs make the optimal outcomes more accurate and definite when utilized in practical MCDM problems under environment.(7)The proposed Dombi operators handle the drawbacks of existing AOs, including bipolar fuzzy Dombi AOs [10].

Therefore, some Dombi AOs are developed to choose the best option in different decision-making situations. The developed operators have some advantages over other approaches which are given as follows:(1)Our proposed methods explain the problems more accurately which involve multiple attributes because they consider numbers.(2)The developed AOs are more precise and efficient with single attribute.(3)To solve practical problems by using Dombi AOs with numbers is very significant.

The rest of this article is structured as follows. Section 2 recalls some fundamental definitions and operations of the numbers (). Section 3 presents , , , , , and operators. Section 4 develops a methodology of these AOs to model MCDM problems. Section 5 discusses two applications: first for the selection of best agricultural land and second for the selection of best commercial bank. Section 6 provides comparative analysis of developed approaches with ELECTRE-I model [46] and Hamacher AOs [27]. Section 7 discusses the conclusions and future directions.

#### 2. Preliminaries

*Definition 1. *(see [26]). An set on a universal set is a mapping . The membership of every object is described by where is the -th projection mapping.

Let be an , where , . We define the score and accuracy functions of , respectively, as follows.

*Definition 2. *(see [27]). For an , we define a score function as follows:

*Definition 3. *(see [27]). For an , an accuracy function is defined asFrom Definitions 2 and 3, it can be readily seen that for any , and . Notice that represents the accuracy degree of . Thus, a higher value of represents a higher accuracy degree for .

Using Definitions 2 and 3, we now give the following ordered relation criteria for any two .

*Definition 4. *(see [27]). Let and be two . Then,(1).(2).(3).(4).(5).Some basic operations for are given by [27](1).(2).(3).(4).(5).(6), if and only if .(7).(8).

Theorem 1. *(see [27]). For two and with , we have*(1)*.*(2)*.*(3)*.*(4)*.*(5)*.*(6)*.*(7)*.*

Dombi [47] proposed operations, namely, Dombi sum and Dombi product , which are, respectively, -conorm and -norm given bywhere and .

#### 3. Dombi AOs

In this section, we first give Dombi operations for via Dombi -conorm and Dombi -norm and then we present Dombi arithmetic and geometric AOs. Let and be . We give some fundamental Dombi operations of as follows:where .

##### 3.1. Dombi Arithmetic AOs

We present Dombi arithmetic AOs as follows.

*Definition 5. *For a collection of where , a mapping from to is called an operator, which is given bywhere denotes the weights of , and with .

We give the following theorem, which is used to apply the Dombi operations on .

Theorem 2. *For a collection of where , an accumulated value of these using the operators is defined as*

*Proof. *We utilize the induction approach to show it.*Case 1*. For , by equation (6), we obtainHence, equation (6) satisfies when .*Case 2*. Now, we presume that equation (6) satisfies for ; here is an arbitrary natural number; then,For ,Therefore, equation (6) satisfies for . Hence, we deduce that equation (6) satisfies for every natural number .

*Example 1. *Let , and be 3FNs and be weights related to these 3FNs. Then, for ,We now explore some useful laws of operators as follows.

Theorem 3. *(idempotent law). Let be a family of “” , which are equal, i.e., ; then,*

*Proof. *Since , where , then by equation (6),Hence, holds if , for all “” varies from 1 to .

Theorem 4. *(bounded law). Let be a collection of “” , , and ; then,*

Theorem 5. *(monotonic law). For two collections of and , if ,**Now, we present operator.*

*Definition 6. *For a collection of , , an operator is a function : , which is given bywhere denotes the weights and with . represents the permutation, for which .

Theorem 6. *For a collection of where , an accumulated value of these using the operators is defined as*

*Example 2. *Let , and be 4FNs with weights . Then, for , we compute the score values asSince ,Then, from Definition 6,

*Remark 1. *Note that operators satisfy properties, namely, idempotency, boundedness, and monotonicity, as described in Theorems 3, 4, and 5.

Theorem 7. *(commutative law). For any two collections of and , we getwhere is any permutation of .**We see that and operators aggregate weighted and their ordering, respectively. Now, we develop a novel operator called operator, which obtains the properties of both and operators.*

*Definition 7. *For a family of , , an operator is defined aswhere denotes the weights corresponding to the with the conditions , is the biggest , , and is a vector having weights, with .

Notice that when , operator converts into operator. If , then operator becomes operator. Thus, operator is a generalization for both operators, and , which describes the degrees and ordering of .

The following theorem can be readily showed by same steps as in Theorem 2.

Theorem 8. *For a collection of where , an accumulated score of these using the operators is defined as*

*Example 3. *Let , , and be 3FNs with , a weight vector corresponding to given 3FNs, and a vector having weights. Then, by Definition 7, for ,Similarly,Then, scores of for are calculated asSince ,Then, from Theorem 8,

##### 3.2. Dombi Geometric AOs

We now propose different types of Dombi geometric AOs with , namely, operator, operator, and operator.

*Definition 8. *For a family of , , a mapping : is called operator, which is given bywhere represents the weights, with .

Theorem 9. *For a collection of where , an accumulated score of these using the operators is defined by*

*Proof. *Its proof is identical to Theorem 2.

*Example 4. *Let , and be 3FNs with weights . Then, for ,