Bipolar Neutrosophic Dombi Aggregation Operators With Application in Multi-Attribute Decision Making Problems

In this paper, Dombi t-norm (TN) and Dombi t-conorm (TCN) are used to generate more complex, flexible and feasible operation rules by managing a parameter in bipolar neutrosophic fuzzy (BNF) environment. We introduce the notion of bipolar neutrosophic Dombi weighted geometric aggregation (BNDWGA) and bipolar neutrosophic Dombi ordered weighted geometric aggregation (BNDOWGA) operators. We discuss their different properties along with proofs and also investigate multi-attribute decision making (MADM) methods on basis of propose aggregation operators under bipolar neutrosophic fuzzy (BNF) environment. We propose an algorithm of selection of cultivating crop to explain the proposed methods under bipolar neutrosophic fuzzy (BNF) environment. The effect of parameter on our proposed bipolar neutrosophic Dombi aggregation operators are discussed graphically. Moreover, the comparison of our proposed methods with existing methods to test their suitable preferences are discussed in detail.


I. INTRODUCTION
In fuzzy theory, a newly defined model generally overcome drawbacks of previously defined models. Due to vagueness and uncertainties issues in many daily life problems, the routine mathematics is not always available. To deal with such issues, various procedures such as hypothesis of probability, rough set hypothesis, and fuzzy set hypothesis have been considered as alternative models and to avoid vulnerabilities as well. Unfortunately, most of the alternative such mathematics have their own down sides and drawbacks. For instance, most of the words like, genuine, lovely, best, renowned are not measurable and are, in fact, ambiguous. The criteria for words like wonderful, best, renowned etc., fluctuate from individual to individual. To handel such type of ambiguous and uncertain information, Zadeh [1] initiated the study of possibility based on participation function that assigns an enrollment grade in [0,1]. FSs have only one membership degree and The associate editor coordinating the review of this manuscript and approving it for publication was Xuefeng Zhang . cannot handle complex problem. The concept of intuitionistic fuzzy set (IFS) was introduced by Atonasso [2]. IFS is used for a highly flexible description of uncertain information. IFS includes both membership degree and non-membership degree. And thereafter, the concept of interval-valued intuitionistic fuzzy set (IVIFS) was introduced by Atanassov and Gargov [3]. IVIFS is a generalization of IFS. In practical decision-making problems, IFS and IVIFS can not represent inconsistent information. The concept of neutrosophic set (NS) was introduced by Smarandache [4]. NS includes truth membership degree, indeterminacy membership degree, and falsity membership degree to characterize incomplete, inconsistent and uncertain information. Wang et al. [5], [6] introduced the concept of interval neutrosophic set(INS) and the concept of single-valued neutrosophic set (SVNS) to apply NS in practical decision-making problems.
In practical decision-making problems, when we want to take decision then we think about positive and negative effect. Positive information explains why is acceptable, permitted, possible, described, or satisfactory. On contrary, negative information explains why is rejected, forbidden, or impossible discussed by Gruber in [7]. Satisfactory or acceptable perceptions are positive preferences, while unsatisfactory or unacceptable perceptions are negative preferences. Positive preferences relate to desires, while negative preferences relate to the constraints investigated by Bistarelli et al. [8]. For example, when the decision maker (DM) examines an object, he/she may explain why they considers (less or more) satisfactory or acceptable; on contrary, he/her may also explain why they considers unsatisfactory or unacceptable because of different constraints discussed by Wang et al. [9]. Zhang [10] introduced the concept of bipolar fuzzy set (BFS) consisting of positive membership degree and negative membership degree to describe the above information. Due to the advantages of BFS and NS, Deli et al. [11] introduced the concept of bipolar neutrosophic set (BNS) which describes fuzzy, bipolar, inconsistent and uncertain information. Dey et al. [12] proposed the bipolar neutrosophic TOPSIS(BN-TOPSIS) method to solve MADM problems under bipolar neutrosophic fuzzy environment. Wang et al. [9] proposed the methods based on the Frank choquet Bonferroni Mean Operators to solve MADM problems under bipolar neutrosophic fuzzy environment. In [32]- [34] discussed some aggregation operators on different models.
The MADM model refers to make decisions when there are multiple but a finite list of alternatives and attributes. Dombi [13] introduced the new triangular norms which are Dombi TN and Dombi TCN. Dombi TN and Dombi TCN show the good flexibility with operational parameter. Until now, Dombi operations have not extended to aggregate bipolar neutrosophic fuzzy environment. In this paper, we investigate some new aggregation operators based on the combination of bipolar neutrosophic numbers(BNNs) and Dombi operations. We have proposed some bipolar neutrosophic Dombi aggregation operators to aggregate bipolar neutrosophic fuzzy information, and developed MADM methods based on BNDWGA and BNDOWGA operators to solve MADM problems with bipolar neutrosophic fuzzy information.
The remainder of the paper is as follows. In Section 2, the review of some basic definitions and concepts are given to be used in this paper. In Section 3, we investigate the notion of Dombi operations of bipolar neutrosophic numbers (BNNs). In Section 4, we propose BNDWGA and BNDOWGA operators and their properties in detail. In Section 5, the comprehensive MADM methods based on proposed bipolar neutrosophic Dombi aggregation operators are proposed. In Section 6, provides a numerical example of selection of cultivating crop. In Section 7, we discuss the parametric analysis and comparative analysis with existing methods.

II. PRELIMINARIES
In this section, we present a brief survey of few fundamentals of different sorts of sets which will be utilized in sequel.

A. BIPOLAR NEUTROSOPHIC SET AND BIPOLAR NEUTROSOPHIC NUMBER
Definition 1 [11]: Let U be a fixed set. Then, BNS N can be defined as follows are the truth membership, indeterminacy membership degree and falsity membership degree of an element u ∈ U corresponding to BNS N and the negative membership degrees denote the truth membership degree, indeterminacy membership degree and falsity membership degree of an element u ∈ U to some implicit counter property corresponding to a BNS N.
In particular, if U has only one element, then [11] defined the algebraic operations of BNNs which are as follows: Then, algebraic operations of BNNs are defined as follows: . For comparing two BNNs, Deli et al. [11] developed a comparison method which consists of the score function, accuracy function and certainty function.

B. DOMBI OPERATIONS
Dombi product and Dombi sum are special cases of TN and TCN respectively, and are given in the following definition. Definition 5 [13]: Let ℵ 1 and ℵ 2 be any two real numbers, then Dombi TN and Dombi TCN are defined in the following expressions: and Some special cases can be easily proved.
. Dombi sum and Dombi product are reduced to simple max-operator and simple min-operator, respectively.

III. DOMBI OPERATIONS OF BNNs
In this section, we define the Dombi operations of BNNs and discuss their properties in detail.
As Theorem1 can be easily verified. Therefore, the proof is omitted here. The properties of Dombi operations of BNNs are defined as follows.

IV. BIPOLAR NEUTROSOPHIC DOMBI AGGREGATION OPERATORS
In this section, we will propose the bipolar neutrosophic Dombi weighted geometric aggregation(BNDWGA) and the bipolar neutrosophic Dombi ordered weighted geometric aggregation(BNDOWGA) operators and discuss different properties of these aggregation operators (AOs).

A. BIPOLAR NEUTROSOPHIC DOMBI WEIGHTED GEOMETRIC AGGREGATION OPERATOR
. . , r) be a family of BNNs and ψ = (ψ 1 , ψ 2 , . . . , ψ r ) T be the weight vector of‫א‬ α , ψ α ∈ [0, 1] and α=1 ψ α = 1. Then, the value aggregated by using BNDWGA operator is still a BNN, which is calculated by using the following formula Proof: If r = 2 based on the Definition 7 of BNNs for Dombi operations, the following result can be obtained: If r = s is based on the Equation (6), then we have got the following equation: If r = s + 1, then there is following result: Hence, Theorem 3 is true for r = s+1. Thus, the Equation (6) holds for all r.
The property is obvious based on the Equation (6).

V. MODEL FOR MADM USING BIPOLAR NEUTROSOPHIC FUZZY INFORMATION
In this section, we discuss two comprehensive MADM methods are extended based on the proposed BNDWGA and BNDOWGA operators.
An algorithm is constructed based on proposed bipolar neutrosophic Dombi aggregation operators which solves MADM problems.

Algorithm
Step 1 Collect information on the bipolar neutrosophic evaluation.
Step 2 Calculate score and the accuracy values of collected information.
The score values s(‫א‬ αβ ) and accuracy values a(‫א‬ αβ ) of alternatives A α can be calculated by using Equations (1) and (2).
Step 3 The compressions method in Definition 4 to reorder information on evaluation under each attribute.
The comparison method is used to reorder‫א‬ αβ .

VI. NUMERICAL EXAMPLE
A numerical example of the selection of cultivating crop taken from Zhang [10] is provided here. Furthermore, parametric analysis and comparative analysis confirm the flexibility and effectiveness of the proposed methods. In ordered to increase the production of agriculture, an agriculture department considers the selecting a crop for cultivating in the farm. Four cultivating crops A 1 , A 2 , A 3 , and A 4 are selected for further evaluation through preliminary screening. The agriculture department decided to invite four group experts to evaluate information. The expert group consists of investment experts, land experts, weather experts and labour experts. The four cultivating crops are evaluated by experts on the basis of four attributes or criteria: cost (C 1 ), nutrition of land (C 2 ), effect of weather (C 3 ), labor (C 4 ). These attributes are interactive and interlinked.

Algorithm
Step 1 Collect information on bipolar neutrosophic evaluation.
The information collected from expert discussion on evaluation is given in Table 1.
Step 2 Calculate score and accuracy values of collected information.
For each alternative A α under attribute C β , the score values s(‫א‬ αβ ) and accuracy values a(‫א‬ αβ ) can be calculated based on Equations (1) and (2). The score values s(‫א‬ αβ ) and accuracy values a(‫א‬ αβ ) are shown in Tables 2 and 3, respectively.  Step 3 Reorder information on evaluation under each attribute.
The comparison method based on Definition 4 is used to reorder‫א‬ αβ .
Method 1 The following score values are obtained by using the BNDWGA operator. Step 6 Rank all the alternatives A α (α = 1, 2, . . . , s) and select favorable one(s).
The alternative can be ranked in descending order based on the comparison method, and favorable alternative can be selected.
Method 1 The following ranking order based on score values is obtained by using BNDWGA operator: Method 2 The following ranking order based on the score values is obtained by using BNDOWGA operator: Step 7 End.

VII. PARAMETRIC ANALYSIS AND COMPARATIVE ANALYSIS
This section describes effect of parametric λ on decision making results and comparison between proposed methods and existing methods.

A. ANALYSIS ON THE EFFECT OF PARAMETER λ ON DECISION MAKING RESULTS
This subsection discusses the effect of parameter λ in detail. First, effect of parameter λ on the proposed operators is as follows. Table 5 shows that the corresponding ranking orders with respect to the BNDWGA operator are changed as the value of λ changing from 1 to 10.    Table 5 shows that the ranking order is slightly different when the value of λ is changed for the BNDWGA operator, but the corresponding favorable alternative remains identical. For 1 ≤ λ ≤ 10, the ranking orders are shown as A 3 A 4 A 1 A 2 and A 3 A 1 A 4 A 2 . As a result, the favorable alternative is A 3 as shown in shown in Figure 1. Table 6 shows that the corresponding ranking orders with respect to the BNDOWGA operators are changed as the value of λ changing from 1 to 10.     Table 6 shows that ranking order is stable and the corresponding favorable alternative remains identical, when the value of λ is changed for BNDOWGA operator. For, 1 ≤ λ ≤ 10, the corresponding ranking order is A 3 A 4 A 2 A 1 , then favorable alternative is A 3 . As a result, favorable stable alternative is A 3 as shown in Figure 2.

B. COMPARATIVE ANALYSIS
In this subsection, a comparative analysis of the proposed methods based on proposed bipolar neutrosophic Dombi aggregation operators with existing methods are discuss in detail. The ranking orders obtained by the proposed methods show that A 3 is favorable alternative.
In similar to A ψ , G ψ , BN-TOPSIS(ψ 1 ) and BN-TOPSIS(ψ 2 ) methods, the proposed method based on proposed BNDWGA operator does not consider the interaction and interrelation among attributes. In contrast to A ψ , G ψ , BN-TOPSIS(ψ 1 ) and BN-TOPSIS(ψ 2 ) method, the proposed BNDWGA operator can choose the appropriate parameters according to the preferences of the DMs. In similar to FBNCWBM (s, t = 1) and FBNCGBM (s, t = 1) operators, the proposed BNDWGA operator can select the appropriate parameters according to preferences of the DMs. In contrast to FBNCWBM (s, t = 1) and FBNCGBM (s, t = 1) operators, the proposed BNDWGA operator does not consider the interaction and interrelation among attributes. Some MADM problems are independent of attributes. Then, DMs can use the proposed method based on the proposed BNDWGA operator to solve the MADM problems which are independent of attributes. In contrast to A ψ , G ψ , BN-TOPSIS(ψ 1 ) and BN-TOPSIS(ψ 2 ) methods, the proposed method based on proposed BNDOWGA operator considers the interaction and interrelationship among the attributes by using ordered weighted geometric aggregation operator and can choose proper parameters according to preferences of DMs. In similar to FBNCWBM (s, t = 1) and FBNCGBM (s, t = 1) operators, the proposed BNDOWGA operator can choose proper parameters according to preferences of DMs and can consider the interaction and interrelationship among the attributes. Some MADM problems are dependent of attributes. Then, DMs can use the proposed method based on the proposed BNDOWGA operator to solve these MADM problem which are dependent of attributes. Thus, the proposed methods based on the proposed bipolar neutrosophic Dombi aggregation operators are more reliable and flexible. The DMs can use these proposed methods based on the proposed bipolar neutrosophic Dombi aggregation operators according to their requirements in practical MADM problems.

VIII. CONCLUSION
BNSs describe fuzzy, bipolar, inconsistent and uncertain information. BNDWGA and BNDOWGA operators were proposed based on Dombi operations to make sure that the bipolar neutrosophic Dombi aggregation operators are reliable and flexible. Furthermore, a numerical example was given to verify proposed methods. The reliability and flexibility of the proposed methods were further illustrated through a parameter analysis and a comparative analysis with existing methods.
The participation of this paper is as follows. First, BNSs were used to present the decision-making information based on evaluation. Second, Dombi operations were put forward to bipolar neutrosophic fuzzy environments. Third, BNDWGA and BNDOWGA operators are proposed under the bipolar neutrosophic fuzzy environment. Fourth, MADM methods based on proposed bipolar neutrosophic Dombi aggregation operators were developed. Finally, the flexibility and reliability of proposed methods were verified by the numerical example. In future, BNDWGA and BNDOWGA operators can be put forward to other fuzzy environments such as bipolar neutrosophic soft expert sets and bipolar interval neutrosophic sets. We can develop Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making and some new aggregation operators under singlevalued neutrosophic 2-tuple linguistic environment and their applications to multiattribute group decision making in further work.
MUHAMMAD KHALID MAHMOOD received the M.Sc., M.Phil., and Ph.D. degrees from the University of the Punjab, Pakistan. Since 2001, he has been working with the Mathematics Department, University of the Punjab. His educational background has a silver medal, a gold medal and national talent scholarship, and the President UGC scholarship. He has a vast teaching and research experience. His research interests include number theory, mathematical modeling, combinatorics, and fuzzy theory.