Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment

In real applications, most decisions are fuzzy decisions, and the decision results mainly depend on the choice of aggregation operators. In order to aggregate information more scientifically and reasonably, the Heronian mean operator was studied in this paper. Considering the advantages and limitations of the Heronian mean (HM) operator, four Heronian mean operators for bipolar neutrosophic number (BNN) are proposed: the BNN generalized weighted HM (BNNGWHM) operator, the BNN improved generalized weighted HM (BNNIGWHM) operator, the BNN generalized weighted geometry HM (BNNGWGHM) operator, and the BNN improved generalized weighted geometry HM (BNNIGWGHM) operator. Then, their propositions were examined. Furthermore, two multi-criteria decision methods based on the proposed BNNIGWHM and BNNIGWGHM operator are introduced under a BNN environment. Lastly, the effectiveness of the new methods was verified with an example.


Introduction
In the real world, there is lots of uncertain information in science, technology, daily life, and so on.Particularly under the background of big data, the uncertainty of information is more complex and diverse.Now, how to make use of mathematical tools to deal with the uncertain information is an urgent problem for researchers.In order to describe uncertain information, Zadeh [1] put forward the concept of fuzzy sets.Considering the complexities and changes of uncertainty in the real environment, there was a certain limit on fuzzy sets to describe complex uncertainty; then, some extension theories [2][3][4] were put forward.Afterword, the neutrosophic set (NS) containing three neutrosophic components and the single-valued neutrosophic set were proposed by Smarandache [5], and the single-valued neutrosophic set was also mentioned by Wang and Smarandache [6].Wang and Zhang [7] put forward an interval neutrosophic set (INS) theory.Furthermore, an n-value neutrosophic set [8] theory was proposed by Smarandache.The fuzzy set theory changed the binary view of people, but ignored the bipolarity of things.Under the background of big data, the confliction between data became more and more obvious.Traditional fuzzy sets could not do well in analyzing and handing uncertain information with incompatible bipolarity; this phenomenon was identified in 1994.For the first time, Zhang [9] introduced incompatible bipolarity into the fuzzy set theory, and put forward the bipolar fuzzy set (BFS).The founder of the fuzzy set theory, Zadeh, also affirmed that the bipolar fuzzy set theory was a breakthrough in traditional fuzzy set theory [10].Then, Zemankova et al. [11] discussed a more generalized multipolar fuzzy problem, and pointed out that the multipolar fuzzy problem can be divided into multiple bipolar fuzzy problems.Chen et al. [12] studied m-polar fuzzy sets.Bosc and Pivert [13] introduced a study on fuzzy bipolar relational algebra.Manemaran and Chellappa [14] gave some applications of bipolar fuzzy groups.Zhou and Li [15] introduced some applications of bipolar fuzzy sets in semiring.Deli et al. [16] put forward a bipolar neutrosophic set (BNS), which can describe bipolar information.Later, some studies about BNS were put forward [17][18][19][20].In this paper, we propose four Heronian mean operators for bipolar neutrosophic number (BNN).Compared with the literature [17][18][19], the HM operator can embody the interaction between attributes to avoid unreasonable situations in information aggregation.Compared with the literature [20], the Bonferroni mean (BM) aggregation operator not only neglects the relationship between each attribute and itself, but also considers the relationship between each attribute and other attributes repeatedly.However, the BM aggregation operator has large computational complexity, but the Heronian mean (HM) can overcome these two shortcomings.
The remaining sections are organized as follows: some related concepts are reviewed in Section 2. The four operators are defined and their properties are investigated in Section 3; these four operators are BNN generalized weighted HM (BNNGWHM), BNN improved generalized weighted HM (BNNIGWHM), BNN generalized weighted geometry HM (BNNGWGHM), and BNN improved generalized weighted geometry HM (BNNIGWGHM).Multi-criteria decision-making (MCDM) methods based on the BNNIGWHM and BNNIGWGHM operators are established in Section 4. A numerical example is provided and the effects of parameters p and q are analyzed in Section 5.The conclusion of this paper is given in Section 6.

BNN and Its Operational Laws
Definition 1 [16].Let U = {u 1 , u 2 , . . ., u n } be a universe; a BNS Γ in U is defined as follows: mean, respectively, the truth membership, false membership, and indeterminate membership to some implicit counter-property corresponding to a BNS Γ.

GWHM Operators for BNNs
Definition 10.Let t, s ≥ 0, and t + s = 0, a collection then, we define the BNNGWHM operator as follows: where According to Definitions 3 and 10, the following theorem can be attained: of BNNs, using the BNNGWHM operator; then, the aggregation result is still a BNN, which is given by the following form: where Proof. (1) (2) This proves Theorem 1.

Improved Generalized Weighted HM Operators for BNNs
Definition 11.Let t, s ≥ 0, and t + s = 0, a collection of BNN; then, we define the BNNIGWHM operator as follows: where According to Definitions 3 and 11, the following theorem can be attained: of BNNs, using BNNIGWHM operator; then, the aggregation result is still a BNN, which is given by the following form: The proof of Theorem 3 can be achieved according to the proof of Theorem 1; thus, we omit it here.
The proof of Theorem 5 is similar to Theorem 2; thus, we omit it.

GWGHM Operators of BNNs
Definition 12. Let t, s ≥ 0, t + s = 0, a collection of BNNs; then, we define the BNNGWGHM operator as follows: where According to Definitions 3 and 12, the following theorem can be attained: of BNNs, using the BNNGWGHM operator; then, the aggregation result is still a BNN, which is given by the following form:

Theorem 8. (Monotonicity).
Set The proofs of theorems about BNNGWGHM are similar to those about BNNGWHM; thus, we omit them.

Theorem 12. (Boundedness). Set a collection τ
, and The proofs of theorems about BNNIGWGHM are similar to those about BNNIGWHM; thus, we omit them.

MCDM Methods Based on the BNNIGWHM and BNNIGWGHM Operator
We applied the BNNIGWHM and BNNIGWGHM operator to manage MCDM problems within BNN information in this section.
Step 2: According to Definition 11 or Definition 13, calculate τ i .
Step 4: According to Definition 5, rank all the alternatives corresponding to the values of s(τ i ).

Illustrative Example
In this section, we used a numerical example adapted from the literature [16].A woman wants to buy a car.Now, four kinds of cars Γ 1 , Γ 2 , Γ 3 , and Γ 4 are taken into account according to gasoline consumption (Φ 1 ), aerodynamics (Φ 2 ), comfort (Φ 3 ), and safety performances (Φ 4 ).The importance of these four attributes is given as ε = (0.5, 0.25, 0.125, 0.125) T .Then, she evaluates four alternatives under the above four attributes in the form of BNNs.

The Decision-Making Process Based on the BNNIGWHM Operator or BNNIGWGHM Operator
Step 1: Establish the BNN decision matrix (τ ij ) 4×4 provided by customer, as shown in Table 1.Step 2: According to Definition 11 (suppose p = q = 1) and ε of attributes, calculate τ i (i = 1, 2, 3, 4): IGWHM and IGWGHM aggregation operators can take into account the correlation between attribute values and can better reflect the preferences of decision-makers and make the decision results more reasonable and reliable.A BNS has two fully independent parts, one part has three independent positive membership functions and the other has three independent negative membership functions, which can deal with uncertain information containing incompatible polarity.Here, we used the BNNIGWHM and BNNIGWGHM operators to solve real problems and analyze the influences of parameters p and q on the results of decisions, using different parameter values for sorting and comparing the corresponding results.Then, it could be found that the influences of parameters p and q on the results of decisions were small in these both methods.Comparing the results of the two methods, it can be found that their results were consistent; therefore, the proposed methods in this paper have feasibility and generality.

Comparison with Related Methods
In this section, we compared the methods proposed in this paper with other related methods proposed in the literature [16,19].Table 4 lists the ranking results.

Aggregation Operator Score Value Ranking
The bipolar neutrosophic weighted average operator (Aw) and bipolar neutrosophic weighted geometric operator (Gw) proposed in Reference [16] σ(τ 1 ) = 0.50, σ(τ 2 ) = 0.52, σ(τ 3 ) = 0.56, σ(τ 4 ) = 0.54 The Similarity measures of bipolar neutrosophic sets proposed in Reference [19] with the following variables: In Table 4, we can see that the ranking results were different; Γ 3 was obtained as the optimal alternative except the method in Reference [19] with λ = 0.9.Compared with these related methods, the BNNIGWHM and BNNIGWGHM operators considered the correlation between attribute values and could better reflect the preferences of decision-makers and make the decision results more reasonable and reliable while dealing with uncertain information containing incompatible polarity.Thus, we think the proposed methods in this paper are more suitable to handle these decision-making problems.

Conclusions
This paper firstly proposed the BNNGWHM, BNNIGWHM, BNNGWGHM, and BNNIGWGHM operators for BNNs and discussed the related properties of these four operators.Furthermore, we developed two methods of MCDM in a BNN environment based on the BNNIGWHM and BNNIGWGHM operators.Finally, these two methods were used for a numerical example to establish their effectiveness and application.Dealing with the calculation, we took different values for p and

Table 4 .
Decision results based on four aggregation operators.