Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators

In this paper, we extended the Hamy mean (HM) operator, the Dombi Hamy mean (DHM) operator, the Dombi dual Hamy mean (DDHM), with the intuitionistic fuzzy numbers (IFNs) to propose the intuitionistic fuzzy Dombi Hamy mean (IFDHM) operator, intuitionistic fuzzy weighted Dombi Hamy mean (IFWDHM) operator, intuitionistic fuzzy Dombi dual Hamy mean (IFDDHM) operator, and intuitionistic fuzzy weighted Dombi dual Hamy mean (IFWDDHM) operator. Following this, the multiple attribute group decision-making (MAGDM) methods are proposed with these operators. To conclude, we utilized an applicable example for the selection of a car supplier to prove the proposed methods.

Dombi [54] proposed the operations of the Dombi T-norm and T-conorm.Following this, Liu et al. [55] proposed the Dombi operations with IFNs.Chen and Ye [56] proposed the Dombi weighted arithmetic average and geometric average fuse the single-valued neutrosophic numbers (SVNNs).Wei and Wei [57] gave some dombi prioritized weighting aggregation operators with single-valued neutrosophic numbers.
Through existing studies, we can see that the combination Hamy mean (HM) operator [58,59] and Dombi operations are not extended to IFNs so far.In order to develop Hamy mean operators and Dombi operations for IFNs, the main purposes of this study are (1) to develop some Dombi Hamy mean aggregating operators for IFNs and to investigate their properties, and (2) to propose two models to solve the MADM problems based on these operators with IFNs.
To do so, the rest of this paper is organized as follows.In the next section, we introduce some basic concepts of IFSs, Dombi operations and HM operators.In section three we propose some intuition fuzzy Hamy mean operators based on Dombi T-norm and T-conorm.In section four, we have applied these operators to solve the MAGDM problems with IFNs.In section five, a practical example for the selection of a car supplier is given.In section six, we conclude the paper and give some remarks.

Preliminaries
In this section, we introduce the concept of IFS, HM operator, and Dombi T-conorm and T-norm.

Intuitionistic Fuzzy Sets
Definition 1.Let X be a fixed set, with a generic in X denoted by x.An intuitionistic fuzzy set (IFS) I in X is following [1,2]: where µ i (x) is the membership function, and ν i (x) is the non-membership function.For each point x in X, we have µ i (x), ν i (x) ∈ [0, 1] and 0 ≤ µ i (x) + ν i (x) ≤ 1.
Definition 4. Let k = (µ k , ν k ) be an IFN, then an accuracy function H of k can be defined as follows [61]: where H(k) ∈ [0, 1], for the difference µ k + ν k , the larger the H(k) is, the greater the IFN k is.Xu and Yager [5] develop a comparison method of IFNs.

HM Operator
Definition 6.The HM operator is defined as follows [58]: where x is a parameter and The properties of the operator are shown as follows: Two particular cases of the HM operator are given as follows.

Dombi T-Conorm and T-Norm
Dombi operations involve the Dombi product and Dombi sum, which are special cases of T-norms and T-conorms, respectively.Definition 7. Suppose M = { x, µ M (x), ν M (x) } and N = { x, µ N (x), ν N (x) } are any two IFNs, then the generalized intersection and generalized union are proposed as follows [54]: where T denotes a T-norm and T * denotes a T-conorm.
Dombi proposed a generator to produce Dombi T-norm and T-conorm which are shown as follows.

Intuition Fuzzy Hamy Mean Operators Based on Dombi T-Norm and T-Conorm
In this section, we propose the intuitionistic fuzzy Dombi Hamy mean (IFDHM) operator and intuitionistic fuzzy weighted Dombi Hamy mean (IFWDHM) operator.

The IFDHM Operator
a collection of IFNs, then we can define IFDHM operator as follows: where x is a parameter and ) be a collection of IFNs, then the aggregate result of Definition 8 is still an IFN, and have Proof.

1.
First of all, we prove (10) is kept.According to the operational laws of IFNs, we have Next, we prove ( 10) is an IFN. Let Then we need to prove that the following two conditions which are satisfied, We get 0 ≤ a + b ≤ 1, so the aggregated result of Definition 8 is still an IFN.Next we will discuss about some of the properties of the IFDHM operator.

The IFWDHM Operator
The weights of attributes play an important role in practical decision making, and they can influence the decision result.Therefore, it is necessary to consider attribute weights in aggregating information.It is obvious that the IFWDHM operator fails to consider the problem of attribute weights.In order to overcome this defect, we propose the IFWDHM operator.
then we can define IFWDHM operator as follows: a group of IFNs, and their weight vector meet ω i ∈ [0.1] and n ∑ i=1 ω i = 1 then the result from Definition 9 is still an IFN, and have or Proof.(1) First of all, we prove that ( 20) and ( 21) are kept.For the first case, when (1 ≤ x < n), according to the operational laws of IFNs, we get For the second case, when (x = n), we get (2) Next, we prove the (20) and (21) are IFNs.For the first case, when 1 Then we need prove the following two conditions.
(II) Since 0 ≤ a + b ≤ 1, we can get the following inequality.
For the second case, when x = n, we can easily prove that it is kept.So the aggregation result produced by Definition 9 is still an IFN.Next, we shall deduce some desirable properties of IFWDHM operator.
(2) For the second case, when x = n, which proves the idempotency property of the IFWDHM operator.
If µ i j ≥ µ θ j , ν i j ≤ ν θ j for all j, and weight vector meets the k and π are equal, then we have Similarly, we have and S(k), S(π) be the score values of a and π respectively.Based on the score value of IFN in (2) and the above inequality, we can imply that S(k) ≥ S(π), and then we discuss the following cases: (1) If S(k) > S(π), then we can get Since µ i j ≥ µ θ j ≥ 0, ν θ j ≥ ν i j ≥ 0, and based on the Equations ( 2) and (3), we can deduce that Therefore, it follows that H(k) = H(π), the IFWDHM we can prove it in a similar way.
Definition 10.The DHM operator is defined as follows [59]: where x is a parameter and x = 1, 2, . . ., n, i 1 , i 2 , . . ., i n are x integer values taken from the set {1, 2, . . . ,n} of n integer values, C x n denotes the binomial coefficient and C In this section, we will propose the intuitionistic fuzzy Dombi dual Hamy mean DHM (IFDDHM) operator.
a collection of IFNs, then we can define IFDDHM operator as follows: where x is a parameter and x = 1, 2, . . ., n, i 1 , i 2 , . . ., i n are x integer values taken from the set {1, 2, . . . ,n} of n integer values, C x n denotes the binomial coefficient and C ) be a collection of the IFNs, then the aggregate result of Definition 10 is still an IFNs, and have Proof.
(1) First of all, we prove ( 42) is kept.According to the operational laws of IFNs, we get x Moreover, Furthermore, (2) Next, we prove ( 42) is an IFN.Let Then we need to prove that the following two conditions which are satisfied.

The IFWDDHM Operator
The weights of attributes play an important role in practical decision making, and they can influence the decision result.Therefore, it is necessary to consider attribute weights in aggregating information.It is obvious that the IFWDDHM operator fails to consider the problem of attribute weights.In order to overcome this defect, we propose the IFWDDHM operator.
then we can define IFWDDHM operator as follows: a group of IFNs, and their weight vector meet ω i ∈ [0, 1], then the result from Definition 12 is still an IFN, and has or Proof. x x Moreover, For the second case, when (x = n), we get Next, we prove the ( 52) and ( 53) are IFNs.For the first case, when 1 ≤ x < n, Then we need prove the following two conditions.
Therefore, 0 ≤ a ≤ 1.Similarly, we can get For the second case, when x = n, we can easily prove that it is kept.So the aggregation result produced by Definition 9 is still an IFN.Next, we shall deduce some desirable properties of IFWDDHM operator.
(2) For the second case, when x = n, which proves the idempotency property of the IFWDDHM operator.
for all j, and weight vector meets the k and π are equal, then we have IFWDDHM Since µ i j ≥ µ θ j ≥ 0, ν θ j ≥ ν i j ≥ 0, and based on the Equations ( 2) and (3), we can deduce we can prove it in a similar way.

A MAGDM Approach Based on the Proposed Operators
In this section, we will apply the proposed IFWDHM (IFWDDHM) operator to cope with the MAGDM problem with IFNs.Let X = {x 1 , x 2 , • • • , x m } be a set of alternatives, and C = {c 1 , c 2 , • • • , c n } be a set of attributes, the weighting vector of attributes be ω ={ω 1 , ω There are experts Y = {y 1 , y 2 , • • • y z } who are invited to give the evaluation information, and their weighting vector is w The expert y t evaluates each attribute c j of each alternative x i by the form of IFN a

and then the decision matrix
The ultimate goal is to give a ranking of all alternatives.Then, we will give the steps for solving this problem.Step1: Calculate the collective evaluation value of each attribute for each alternative by Step2: Calculate the overall value of each alternative with the IFWDHM (IFWDDHM) operator Step3: Calculate the S( a) and H( a).

An Illustrate Example
In this section, we give an example to explain the proposed method.A transportation company wants to pick a car and there are four cars as candidates M i = (M 1 , M 2 , M 3 , M 4 ).We evaluate each supplier from four aspects E i = (E 1 , E 2 , E 3 , E 4 ), which are "production price", "production quality", "production's service performance", and "risk factor".The weight vector of attributes is ω = (0.1, 0.4, 0.3, 0.2) T .There are four experts, and the weight vector of the experts is (0.3, 0.4, 0.2, 0.1) T .Then the decision matrix R t = a t ij 4×4 (t = 1, 2, 3, 4) are shown in Tables 1-4, and our goal is to rank four cars and select the best one.

Decision-Making Processes
Step 1: Since the four attributes are of the same type, thus, we don't need to normalize the matrix R 1 ∼ R 4 .
Step 4: Obtain the score values.
Step 5: Rank all alternatives.a 2 a 3 a 1 a 4 , then the best choice is a 2 .
Considering the different parameter values of an IFWDHM operator that may have an impact on the ranking results, we calculated the scores produced from the different x and the results are listed in Table 7. Considering the different parameter values of an IFWDDHM operator that may have an impact on the ordering results, we calculated the scores with different x and the results are listed in Table 8. x Score of S(a i ) Ranking x = 1 S(a 1 ) = 0.0437, S(a 2 ) = 0.0915, S(a 3 ) = 0.0548, S(a 4 ) = 0.0110.a 2 a 3 a 1 a 4 x = 2 S(a 1 ) = 0.7907, S(a 2 ) = 0.9369, S(a 3 ) = 0.8497, S(a 4 ) = 0.6399.a 2 a 3 a 1 a 4 x = 3 S(a 1 ) = 0.2059, S(a 2 ) = 0.3597, S(a 3 ) = 0.2397, S(a 4 ) = 0.0320.a 2 a 3 a 1 a 4 x = 4 S(a 1 ) = 0.2398, S(a 2 ) = 0.3695, S(a 3 ) = 0.2477, S(a 4 ) = 0.0424.a 2 a 3 a 1 a 4 From Tables 7 and 8, we get following conclusions.When x = 1, the sorting of alternatives is a 2 a 3 a 1 a 4 , and the best choice is a 2 .When x = 2, 3, 4, the sorting of alternatives is a 2 a 3 a 1 a 4 , and the best choice is a 2 .Although there is the same best selection, the ranking is different.When x = 1, the interrelationship between the attributes is not considered, and when x = 2, 3, 4, we can consider the interrelationship for different number of attributes.So these results are reasonable for these two conditions.

Comparative Analysis
Following this, we compare the proposed method with IFWA operator [4], IFWG operator [5], IFWMM operator [62], and IFDWMM operator [62] and the comparative results are depicted in Table 9.From above analysis, we arrived at the same results.However, the existing operators, such as IFWA operator and IFWG operator do not consider the relationship between arguments, and thus cannot eliminate the corresponding influence of unfair arguments on decision result.The IFWMM operator, IFDWMM operator, IFWDHM and IFWDDHM operators consider the relationship among the arguments.
and weight vector meets ω i ∈ [0, 1] and k ∑ i=1 ω i = 1, the k and π are equal, then we have

Table 5 .
The collective decision matrix R.

Table 6 .
The collective decision matrix R.

Table 8 .
Score and order of the alternatives with different parameter values x.

Table 9 .
Ordering of the green suppliers.IFWA a 2 a 3 a 1 a 4 IFWG a 2 a 3 a 1 a 4 IFWMM a 2 a 3 a 1 a 4 IFDWMM a 2 a 3 a 1 a 4