Some Interval Neutrosophic Dombi Power Bonferroni Mean Operators and Their Application in Multi–Attribute Decision–Making

: The power Bonferroni mean (PBM) operator is a hybrid structure and can take the advantage of a power average (PA) operator, which can reduce the impact of inappropriate data given by the prejudiced decision makers (DMs) and Bonferroni mean (BM) operator, which can take into account the correlation between two attributes. In recent years, many researchers have extended the PBM operator to handle fuzzy information. The Dombi operations of T-conorm (TCN) and T-norm (TN), proposed by Dombi, have the supremacy of outstanding ﬂexibility with general parameters. However, in the existing literature, PBM and the Dombi operations have not been combined for the above advantages for interval-neutrosophic sets (INSs). In this article, we ﬁrst deﬁne some operational laws for interval neutrosophic numbers (INNs) based on Dombi TN and TCN and discuss several desirable properties of these operational rules. Secondly, we extend the PBM operator based on Dombi operations to develop an interval-neutrosophic Dombi PBM (INDPBM) operator, an interval-neutrosophic weighted Dombi PBM (INWDPBM) operator, an interval-neutrosophic Dombi power geometric Bonferroni mean (INDPGBM) operator and an interval-neutrosophic weighted Dombi power geometric Bonferroni mean (INWDPGBM) operator, and discuss several properties of these aggregation operators. Then we develop a multi-attribute decision-making (MADM) method, based on these proposed aggregation operators, to deal with interval neutrosophic (IN) information. Lastly, an illustrative example is provided to show the usefulness and realism of the proposed MADM method. The developed aggregation operators are very practical for solving MADM problems, as it considers the interaction among two input arguments and removes the inﬂuence of awkward data in the decision-making process at the same time. The other advantage of the proposed aggregation operators is that they are ﬂexible due to general parameter.


Introduction
While dealing with any real world problems, a decision maker (DM) often feels discomfort when expressing his\her evaluation information by utilizing a single real number in multi-attribute decision making (MADM) or multi-attribute group decision making (MAGDM) problems due to the intellectual fuzziness of DMs.For this cause, Zadeh [1] developed fuzzy sets (FSs), which are assigned by a For aggregating INNs, some AGOs are developed by utilizing different T-norms (TNs) and T-conorms (TCNs), such as algebraic, Einstein and Hamacher.Usually, the Archimedean TN and TCN are the generalizations of various TNs and TCNs such as algebraic, Einstein, Hamacher, Frank, and Dombi [47] TNs and TCNs.Dombi TN and TCN have the characteristics of general TN and TCN by a general parameter, and this can make the aggregation process more flexible.Recently, several authors defined some operational laws for IFSs [48], SVNSs [49], hesitant fuzzy sets (HFSs) [50,51] based on Dombi TN and TCN.In practical decision making, we generally need to consider interrelationship among attributes and eliminated the influence of awkward data.For this purpose, some researchers combined BM and PA operators to propose some PBM operators and extended them to various fields [52][53][54][55].The PBM operators have two characteristics.Firstly, it can consider the interaction among two input arguments by BM operator, and secondly, it can remove the effect of awkward data by PA operator.The Dombi TN and TCN have a general parameter, which makes the decision-making process more flexible.From the existing literatures, we know that PBM operators are combined with algebraic operations to aggregate IFNs, or IVIFNs, and there is no research on combining PBM operator with Dombi operations to aggregate INNs.
In a word, by considering the following advantages.(1) Since INSs are the more précised class by which one can handle the vague information in a more accurate way when compared with FSs and all other extensions like IVFSs, IFSs, IVIFSs and so forth, they are more suitable to describe the attributes of MADM problems, so in this study, we will select the INSs as information expression; (2) Dombi TN and TCN are more flexible in the decision making process due to general parameter which is regarded as decision makers' risk attitude; (3) The PBM operators have the properties of considering interaction between two input arguments and vanishes the effect of awkward data at the same time.Hence, the purpose and motivation are that we try to combine these three concepts to take the above defined advantages and proposed some new powerful tools to aggregate INNs.(1) we define some Dombi operational laws for INNs; (2) we propose some new PBM aggregation operators based on these new operational laws; (3) we develop a novel MADM based on these developed aggregation operators.
The following sections of this article are shown as follows.In Section 2, we review some basic concepts of INSs, PA operators, BM operators, and GBM operators.In Section 3, we review basic concept of Dombi TN and TCN.After that, we propose some Dombi operations for INNs, and discuss some properties.In Section 4, we define INDPBM operator, INWDPBM operator, INDPGBM operator and INWDPGBM operator and discuss their properties.In Section 5, we propose a MADM method based on the proposed aggregation operators with INNs.In Section 6, we use an illustrative example to show the effectiveness of the proposed MADM method.The conclusion is discussed in Section 7.

Preliminaries
In this part, some basic definitions, properties about INSs, BM operators and PA operators are discussed.

The INSs and Their Operational Laws
Definition 1.Let Ω be the domain set [8,9], with a non-specific member in Ω expressed by v.A NS NS in Ω is expressed by where, t NS v , i NS v and f NS v respectively express the TMD, IMD and FMD of the element v ∈ U to the set NS.For each point v ∈ U, we have, t NS v , i NS v f NS v ∈ ]0 − , 1 + [ and 0 The NS was predominantly developed from philosophical perspective, and it is hard to be applied to engineering problems due to the containment of subsets of ]0 − , 1 + [.So, in order to use it more easily be any two INNs [12], and ζ > 0. Then the operational laws of INNs can be defined as follows: (1) In order to compare two INNs, the comparison rules were defined by Liu et al. [36], which can be stated as follows.

Definition 6.
Let be any two INNs [42].Then we have: (1) If S in 1 > S in 2 , then in 1 is better than in 2 , and denoted by in 1 > in 2 ; (2) If S in 1 = S in 2 , and A in 1 > A in 2 , then in 1 is better than in 2 , and denoted by in 1 > in 2 ; (3) If S in 1 = S in 2 , and A in 1 = A in 2 , then in 1 is equal to in 2 , and denoted by in 1 = in 2 .

Definition 7.
Let be any two INNs [15].Then the normalized Hamming distance between n 1 and n 2 is described as follows.

The PA Operator
The PA operator was first presented by Yager [36] and it is described as follows.

The BM Operator
The BM operator was initially presented by Bonferroni [38], and it was explained as follows: Definition 9.For non-negative real numbers ℘ h (h = 1, 2, . . ., l), and x, y ≥ 0 [38], the BM operator is described as Symmetry 2018, 10, 459 6 of 32 The BM operator ignores the importance degree of each input argument, which can be given by decision makers according to their interest.To overcome this shortcoming of BM operator, He et al. [52] defined the weighted Bonferroni mean (WBM) operators which can be explained as follows: Definition 10.For positive real numbers ℘ h (h = 1, 2, . . ., l) and x, y ≥ 0 [52], then the weighted BM operator (WBM) is described as where κ = ( κ 1 , κ 2 , . . . ,κ l ) T is the importance degree of every ℘ h (h = 1, 2, . . ., l).
Similar to BM operator, the geometric BM operator also considers the correlation among the input arguments.It can be explained as follows: Definition 11.For positive real numbers ℘ h (h = 1, 2, . . ., l) and x, y ≥ 0 [53], the geometric BM operator (GBM) is described as GBM x,y (℘ 1 , ℘ 2 , . . ., ℘ l ) = The GBM operator ignores the importance degree of each input argument, which can be given by decision makers according to their interest.In a similar way to WBM, the weighted geometric BM (WGBM) operator was also presented.The extension process is same as that of WBM, so it is omitted here.
The definition of power Bonferroni mean (PBM) and power geometric Bonferroni mean (PGBM) operators are given in Appendix A. Definition 12. Let and ℵ be any two real numbers [47].Then the Dombi TN and TCN among and ℵ are explained as follows:
According to the Dombi TN and TCN, we develop a few operational rules for INNs.The GBM operator ignores the importance degree of each input argument, which can be given by decision makers according to their interest.In a similar way to WBM, the weighted geometric BM (WGBM) operator was also presented.The extension process is same as that of WBM, so it is omitted here.
The definition of power Bonferroni mean (PBM) and power geometric Bonferroni mean (PGBM) operators are given in Appendix A.

   
According to the Dombi TN and TCN, we develop a few operational rules for INNs.(1)

Definition
(2) Symmetry 2018, 10, 459 8 of 32 (3) Now, based on these new operational laws for INNs, we develop some aggregation operators to aggregate IN information in the preceding sections.

The INPBM Operator Based on Dombi TN and Dombi TCN
In this part, based on the Dombi operational laws for INNs, we combine PA operator and BM to introduce interval neutrosophic Dombi power Bonferroni mean (INDPBM), interval neutrosophic weighted Dombi power Bonferroni mean, interval neutrosophic Dombi power geometric Bonferroni mean (INDPGBM) and interval neutrosophic weighted Dombi power Bonferroni mean (INWDPGBM) operators and discuss some related properties., (i = 1, 2, . . ., l), be a group of INNs, and x, y ≥ 0.
then I NDPBM x,y is said to be IN Dombi power Bonferroni mean (INDPBM) operator, where T in z = l ⊕ s=1,s =z Sup in z , in s .Sup in z , in s is the support degree for in z from in s , which satisfies the following In order to simplify Equation (25), we can give Symmetry 2018, 10, 459 9 of 32 and call Λ = (Λ 1 , Λ 2 , . . . ,Λ l ) T is the power weight vector (PWV), such that Λ z ≥ 0, turns Equation ( 25) into the following form , (i = 1, 2, . . ., l) be a group of INNs.
Then the value obtained by utilizing Equation ( 25) is expressed as In order to simplify Equation ( 25), we can give and call   ,...., .
..., ) be a group of INNs.Then the value obtained by utilizing Equation ( 25) is expressed as , ,...., Proof.Proof of Theorem 6 is given in Appendix B. □ In order to determine the PWV  , we firstly need to determine the support degree among INNs.In general, the similarity measure among INNs can replace the support degree among INNs.That is, Proof.Proof of Theorem 6 is given in Appendix B.
In order to determine the PWV Λ, we firstly need to determine the support degree among INNs.In general, the similarity measure among INNs can replace the support degree among INNs.That is, Step 1. Determine the supports Sup in i , in j , i, j = 1, 2, 3 by using Equation (29), and then Step 2. Determine the PWV Because T in z = ...., ).
which is calculated as follows: Step 1. Determine the supports Sup in in i j  by using Equation (29), and then we get ,, Step 3. Determine the comprehensive value , , , , , by using Equation ( 28), we have then Step ...., ).
which is calculated as follows: Step 1. Determine the supports Sup in in i j  by using Equation (29), and then we get by using Equation ( 28), we have Similarly, we can get n = [0.2590, Proof.From Definition 14, we have , and Hence, I NDPBM x,y in 1 , in 2 , . . . ,in l = I NDPBM x,y in 1 , in 2 , . . ., in l .
be a group of INNs, and in Proof.Proof of Theorem 9 is given in Appendix D. Now, we shall study a few special cases of the I NDPBM x,y with respect to x and y. (1) When y → 0, γ > 0, then we can get ..., , , ,..., , ,..., .
, ,..., , ,..., ..., ) be a group of INNs, and Proof.Proof of Theorem 9 is given in Appendix D. □ Now, we shall study a few special cases of the

INDPBM in in in
In the INDPBM operator, we can only take the correlation among the input arguments and cannot consider the importance degree of input arguments.In what follows, the INWPDBM operator shall be proposed to overcome the shortcoming of the INDPBM operator.
..., ) , be a group of INNs, then the INWDPBM operator is defined as , ..., )  (35) In the INDPBM operator, we can only take the correlation among the input arguments and cannot consider the importance degree of input arguments.In what follows, the INWPDBM operator shall be proposed to overcome the shortcoming of the INDPBM operator.

be a group of INNs, then the INWDPBM operator is defined as
Symmetry 2018, 10, x 13 of 32 In the INDPBM operator, we can only take the correlation among the input arguments and cannot consider the importance degree of input arguments.In what follows, the INWPDBM operator shall be proposed to overcome the shortcoming of the INDPBM operator.
, ,..., , ,..., ..., ) be a group of INNs, and Proof.Proof of Theorem 9 is given in Appendix D. □ Now, we shall study a few special cases of the Proof.Proof of Theorem 10 is similar to Theorem 6. □ Similar to the INDPBM operator, the INWDPBM operator has the properties of boundedness, idempotency and commutativity.

The INDPGBM Operator and INWDPGBM Operator
In this subpart, we develop INDPGBM and INWDPGBM operators.
Then,  in defined in Definition (7).
In order to simplify Equation (38), we can describe and call
Similar to the INDPBM operator, the INWDPBM operator has the properties of boundedness, idempotency and commutativity.

The INDPGBM Operator and INWDPGBM Operator
In this subpart, we develop INDPGBM and INWDPGBM operators.
Then the INDPGBM operator is defined as Then, I NDPGBM x,y is said to be an interval neutrosophic Dombi power geometric Bonferroni mean (INDPGBM) operator.Where T in z = l ∑ s=1,s =z Sup in z , in s , Sup in z , in s is the support degree for n z from n s , which satisfies the following axioms: In order to simplify Equation (38), we can describe and call Λ = (Λ 1 , Λ 2 , . . . ,Λ l ) T is the power weight vector (PWV), such that Then Equation ( 38) can be written as follows: , (i = 1, 2, . . ., l) be a group of INNs.
..., ) be a group of INNs.Then the aggregated result from Equation ( 45) is expressed as

MADM Approach Based on the Developed Aggregation Operator
In this section, based upon the developed INWDPBM and INWDPGBM operators, we will propose a novel MADM method, which is defined as follows.
Assume that in a MADM problem, we need to evaluate u alternatives   is an INN for the alternative g M with respect to the attribute (46)

MADM Approach Based on the Developed Aggregation Operator
In this section, based upon the developed INWDPBM and INWDPGBM operators, we will propose a novel MADM method, which is defined as follows.
Assume that in a MADM problem, we need to evaluate u alternatives M = M 1 , M 2 , . . ., M u with respect to v attributes C = C 1 , C 2 , . . ., C v , and the importance degree of the attributes is represented by = ( 1 , 2 , . . . ,v ) T , satisfying the condition h ∈ [0, is an INN for the alternative M g with respect to the attribute C h , (g = 1, 2, . . ., u; h = 1, 2, . . ., v).Then the main purpose is to rank the alternative and select the best alternative.
In the following, we will use the proposed INWDPBM and INWDPGBM operators to solve this MADM problem, and the detailed decision steps are shown as follows: Step 1. Standardize the attribute values.Normally, in real problems, the attributes are of two types, (1) cost type, (2) benefit type.To get right result, it is necessary to change cost type of attribute values to benefit type using the following formula: Step 2. Calculate the supports where, D d gh , d gl is the distance measure defined in Equation (10). Step Step 5. Determine the score values, accuracy values of R g (g = 1, 2, . . ., u), using Definition 5.
Step 6. Rank all the alternatives according to their score and accuracy values, and select the best alternative using Definition 6.
Step 7. End.This decision steps are also described in Figure 1.
Step 6. Rank all the alternatives according to their score and accuracy values, and select the best alternative using Definition (6).
Step 7. End.This decision steps are also described in Figure 1.

Illustrative Example
In this part, an example adapted from [42] is used to illustrate the application and effectiveness of the developed method in MADM problem.
An investment company wants to invest a sum of money in the best option.The company must invest a sum of money in the following four possible companies (alternatives): (1) car company M 1 ; (2) food company M 2 ; (3) Computer company M 3 ; (4) An arm company M 4 , and the attributes under consideration are (1) risk analysis C 1 ; (2) growth analysis C 2 ; (3) environmental impact analysis C 3 .The importance degree of the attributes is = (0.35, 0.4, 0.25) T .The four possible alternatives M g (g = 1, 2, 3, 4) are evaluated with respect to the above attributes C h (h = 1, 2, 3) by the form of INN, and the IN decision matrix D is listed in Table 1.The purpose of this decision-making problem is to rank the alternatives.

The Decision-Making Steps
Step 1.Since C 1 , C 2 are of benefit type, and C 3 is of cost type.So, C 3 will be changed into benefit type using Equation (47).So, the normalize decision matrix D is given in Table 2. C will be changed into benefit type using Equation (47).So, the normalize decision matrix D is given in Table 2.

(b)
Determine the comprehensive value of every alternative using the INWDPGBM operator, that is Equation ( 51), (Assume that x = y = 1; γ = 3), we have

Step 5. (a)
Determine the score values of R g (g = 1, 2, 3, 4) by Definition 5, we have Determine the score values of R g (g = 1, 2, 3, 4) by Definition 5, we have Step 6.(a) According to their score and accuracy values, by using Definition 6, the ranking order is So the best alternative is M 2 , while the worst alternative is M 1 .

(b)
According to their score and accuracy values, by using Definition 6, the ranking order is So the best alternative is M 2 , while the worst alternative is M 1 .
So, by using INWDPBM or INWDPGBM operators, the best alternative is M 2 , while the worst alternative is M 1 .

Effect of Parameters γ, x and y on Ranking Result of this Example
In order to show the effect of the parameters x and y on the ranking result of this example, we set different parameter values for x and y, and γ = 3 is fixed, to show the ranking results of this example.The ranking results are given in Table 3.
As we know from Tables 3 and 4, the score values and ranking order are different for different values of the parameters x and y, when we use INWDPBM operator and INWDPGBM operator.We can see from Tables 3 and 4, when the parameter values x = 1 or 0 and y = 0 or 1, the best choice is M 4 and the worst one is M 1 .In simple words, when the interrelationship among attributes are not considered, the best choice is M 4 and the worst one is M 1 .On the other hand, when different values for the parameters x and y are utilized, for INWPBM and INWDPGBM operators, the ranking result is changed.That is, from Table 4, we can see that when the parameter values x = 1, y = 1, the ranking results are changed as the one obtained for x = 1 or 0 and y = 0 or 1.In this case the best alternative is M 2 while the worst alternative remains the same.

Parameter Values INWDPBM Operator
Ranking Orders Table 4. Ranking orders of decision result using different values for x and y for INWDPGBM.

Parameter Values INWDPGBM Operator
Ranking Orders Symmetry 2018, 10, 459 21 of 32 From Tables 3 and 4, we can observe that when the values of the parameter increase, the score values obtained using INWDPBM decrease.While using the INWDPGBM operator, the score values increase but the best choice is M 2 for x = y ≥ 1.
From Table 5, we can see that different ranking orders are obtained for different values of γ.When γ = 0.5 and γ = 2, the best choice is M 4 by the INWPBM operator; when we use the INWPGBM operator, it is M 2 .Similarly, for other values of γ > 2, the best choice is M 2 while the worst is M 1 .
Table 5. Ranking orders of decision result using different values for γ.

Parameter Values INWDPBM Operator INWDPGBM Operator
Ranking Orders

Comparing with the Other Methods
To illustrate the advantages and effectiveness of the developed method in this article, we solve the above example by four existing MADM methods, including IN weighted averaging operator, IN weighted geometric operator [12], the similarity measure defined by Ye [15], Muirhead mean operators developed by Liu et al. [42], IN power aggregation operator developed by Liu et al. [37].
From Table 6, we can see that the ranking orders are the same as the ones produced by the existing aggregation operators when the parameter values x = 1, y = 0, γ = 3, but the ranking orders are different when the interrelationship among attributes are considered.That is why the developed method based on the proposed aggregation operators is more flexible due the parameter and practical as it can consider the interrelationship among input arguments.Table 6.Ranking order of the alternatives using different aggregation operators.

Yes
Symmetry 2018, 10, 459 22 of 32 From the above comparative analysis, we can know the proposed method has the following advantages, that is, it can consider the interrelationship among the input arguments and can relieve the effect of the awkward data by PWV at the same time, and it can permit more precise ranking order than the existing methods.The proposed method can take the advantages of PA operator and BM operator concurrently, these factors makes it a little complex in calculations.
The score values and ranking orders by these methods are shown in Table 6.

Conclusions
The PBM operator can take the advantage of PA operator, which can eliminate the impact of awkward data given by the predisposed DMs, and BM operator, which can consider the correlation between two attributes.The Dombi operations of TN and TCN proposed by Dombi have the edge of good flexibility with general parameter.In this article, we combined PBM with Dombi operation and proposed some aggregation operators to aggregate INNs.Firstly, we defined some operational laws for INSs based on Dombi TN and TCN and discussed some properties of these operations.Secondly, we extended PBM operator based on Dombi operations to introduce INDPBM operator, INWDPBM operator, INDPGBM operator, INWDPGBM operator and discussed some properties of these aggregation operators.The developed aggregation operators have the edge that they can take the correlation among the attributes by BM operator, and can also remove the effect of awkward data by PA operator at the same and due to general parameter, so they are more flexible in the aggregation process.Further, we developed a novel MADM method based on developed aggregation operators to deal with interval neutrosophic information.Finally, an illustrative example is used to show the effectiveness and practicality of the proposed MADM method and comparison were made with the existing methods.The proposed aggregation operators are very useful to solve MADM problems.
In future research, we shall define some distinct aggregation operators for SVHFSs, INHFSs, double valued neutrosophic sets and so on based on Dombi operations and apply them to MAGDM and MADM.
Author Contributions: Q.K. and P.L. visualized and work together to achieve this work.Q.K. and P.L. wrote the paper.T.M., F.S. and K.U. offered plenty of advice enhancing the readability of work.All authors approved the publication work.
Funding: National Natural Science Foundation of China: 71771140, 71471172, Special Funds of Taishan Scholars Project of Shandong Province: ts201511045.

Acknowledgments:
We, the authors of this project are deeply grateful for the valuable suggestions and comments provided by the respected reviewers to improve the quality and value of this research.

Conflicts of Interest:
No potential conflict of interest was reported by authors.
In Definitions A1 and A2, T(℘ i ) = l ∑ j=1,j =i supp(℘ i , ℘ j ), and supp(℘ i , ℘ j ) is the SPD for ℘ i from ℘ j satisfying the axioms as; (1) where D(℘ i , ℘ j ) is the distance measure among ℘ i and ℘ j .

b y l b x l c y l c x l d y l d x l g y l g x l h y l h
;

b y l b x l c y l c x l d y l d x l g y l g x l h y l h
 Moreover, we have

b y l b x l c y l c x l d y l d x l g y l g x l h y l h
x l a y l a x l a y l a x l b y l b x l b y l b x l c y l c x l c y l c x l d y l d x l g y e g x l g y l g x l h y l h x l h y lv h

So, we can have
Symmetry 2018, 10, x 25 of 32 x l a y l a x l a y l a x l b y l b x l b y l b x l c y l c x l c y l c x l d y l d x l g y e g x l g y l g x l h y l h x l h y lv h , in Equation ( 54), we can get Symmetry 2018, 10, x 26 of 32 l l i j i j i i j j i i j j i j i j l l x y l l x y l l x y l a l a x y l b l b x y , Now, put in Equation ( 54), we can get Now, put in Equation (A3), we can get Symmetry 2018, 10, l l i j i j i i j j i i j j i j i j l l x y l l x y l l x y l a l a x y l b l b x y ,1 in Equation ( 54), we can get     This is the required proof of the Theorem 6. □ This is the required proof of the Theorem 6.
Based on Dombi TN and TCN Dombi TN and TCN Dombi operations consist of the Dombi sum and Dombi product.

2 be
any three INNs and Φ > 0.Then, based on Dombi TN and TCN, the following operational laws are developed for INNs.
INNs and 0  .Then, based on Dombi TN and TCN, the following operational laws are developed for INNs.

4. 1 . 14 .
The INDPBM Operator and INWDPBM Operator Definition Let in i = in b is the distance measure between INNs in a and in b defined in Definition 7.

3 .
Determine the comprehensive value in = TR L is the distance measure between INNs a in and b in b , in which D(n a , n b ) is the distance measure between INNs in a and in b defined in Definition 7.

Figure 1 .
Figure 1.Flow chart for developed approach.Figure 1. Flow chart for developed approach.

Figure 1 .
Figure 1.Flow chart for developed approach.Figure 1. Flow chart for developed approach.

Table 1 .
The IN decision matrix D.

Table 2 .
The Normalize IN decision matrix D. Determine the supports Supp d gh , d gl , (g = 1, 2, 3, 4; h, l = 1, 2, 3) by Equation (48) (for simplicity we denote Supp d gh , d gl with S Since 1 C , 2 C are of benefit type, and 3 C is of cost type.So, 3

Table 3 .
Ranking orders of decision result using different values for x and y for INWDPBM.