Multiple Attribute Decision Making Based on Generalized Aggregation Operators under Dual Hesitant Fuzzy Environment

We investigate the multiple attribute decision making (MADM) problems with dual hesitant fuzzy information. We first introduce some basic concepts and operations on dual hesitant fuzzy sets.Then, we develop some generalized dual hesitant fuzzy aggregation operators which encompass some existing operators as their particular cases and discuss their basic properties. Next, we apply the generalized dual hesitant fuzzy Choquet ordered aggregation (GDHFCOA) operator to deal with multiple attribute decision making problems under dual hesitant fuzzy environment. Finally, an illustrative example is given to show the developed method and demonstrate its practicality and effectiveness.

Recently, Zhu et al. [18] introduced a dual hesitant fuzzy set (DHFS) which is another new extension of FSs.It is a comprehensive set containing FSs, IFSs, FMSs, and HFSs as special cases under certain conditions.By several possible values for the membership and nonmembership degrees, respectively, DHFSs can take much more information given by decision makers into account in multiple attribute decision making.In their work, some basic operations and properties for DHFSs were investigated.Then Wang et al. [19] investigated the multiple attribute decision making (MADM) problem based on the aggregation operators with dual hesitant fuzzy information.They also developed some aggregation operators for aggregating dual hesitant fuzzy information including dual hesitant fuzzy weighted average (DHFWA) operator, dual hesitant fuzzy weighted geometric (DHFWG) operator, dual hesitant fuzzy ordered weighted average (DHFOWA) operator, dual hesitant fuzzy ordered weighted geometric (DHFOWG) operator, dual hesitant fuzzy hybrid average (DHFHA) operator, and dual hesitant fuzzy hybrid geometric (DHFHG) operator.
However, the existing dual hesitant fuzzy aggregation operators above only consider situations where all the attributes in the dual hesitant fuzzy set are independent.Nevertheless, attributes in DHFSs are usually correlative in real life.Incidentally, the Choquet integral [20] can characterize the correlations of the decision data.Motivated by this idea, we propose a dual hesitant fuzzy Choquet ordered aggregation (DHFCOA) operator, whose prominent characteristic is that it can consider both the importance of the attributes and the correlations of the attributes.It is worth mentioning that DHFCOA can be regarded as an extension of DHFWA and DHFOWA.Then, we also generate DHFCOA operator to GDHFCOA.
To do so, the remainder of this paper is organized as follows.In the next section, we introduce some basic concepts related to dual hesitant fuzzy sets, as well as the existing dual hesitant fuzzy aggregation operators.Some generalized aggregation operators for DHFSs are proposed and their properties are studied in Section 3. In Section 4, we discuss the families of GDHFCOA operators.In Section 5, we develop an approach to multiple attribute decision making problems based on GDHFCOA operator under dual hesitant fuzzy environment.An illustrative example is also given to show the effectiveness of the developed approach in Section 6.We conclude the paper and give some remarks in Section 7.
Definition 1 (see [21]).Let  be a reference set; then we define hesitant fuzzy set on  in terms of a function that when applied to  returns a sunset of [0, 1].
To be easily understood, Xia and Xu [22] express the HFS by a mathematical symbol:  = (⟨, ℎ  ()⟩ |  ∈ ), where ℎ  () is a set of some values in [0, 1], denoting the possible membership degree of the element  ∈  to the set .For convenience, Xia and Xu [22] call ℎ = ℎ  () a hesitant fuzzy element (HFE) and  the set of all HFEs when there is no confusion.
In order to compare two dual hesitant fuzzy elements, corresponding score function is defined as follows.
Definition 3 (see [18]).Let  1 = {ℎ 1 ,  1 } and  2 = {ℎ 2 ,  2 } be any two DHFEs; then the score function of and the accuracy function of where (ℎ  ) and (  ) are the numbers of the elements in ℎ  and   , respectively.Then Besides, some new operations on the DHFEs ,  1 , and  2 are also introduced in [18]: Wang et al. [19] developed some aggregation operators for dual hesitant fuzzy information such as DHFOWA and DHFOWG operators.
In order to weight the elements in , a fuzzy measure  is defined as follows.
If all the elements in  are independent, then we have The discrete Choquet integral is a linear expression up to a reordering of the elements, which is defined as below.

Generalized Aggregation Operators for DHFS
In this section, by introducing parameter , we will propose some generalized aggregation operators for DHFSs.
According to the operational laws of DHFEs, we can get the theorem below.Theorem 9. Let   = {ℎ  ,   } ( = 1, 2, . . ., ) be a collection of DHFEs; then their aggregated value by using the GDHFWA operator is also a DHFE, and Proof.By using mathematics inductive method, we prove Theorem 9 as follows.
For  = 2 by the operational laws of DHFEs, we can get If Theorem 9 holds for  = , that is, then, when  =  + 1, by the operational laws for DHFEs, we have Then, we get That is, Theorem 9 holds for  =  + 1.Thus, by the principle of mathematical induction Theorem 9 holds for all .
By Theorem 9, we can prove that the GDHFWA operator has the following properties.
Remark 11.Theorem 10 shows that idempotency of GDHFWA operator usually does not hold.When  = 1, GDHFWA degenerates to DHFWA, so Theorem 10 also indicates that the Theorem 2 in [19] is not correct.To further clarify this, we give a concrete example in the following.
According to the operational laws of DHFEs, we can get the theorem below.As its proof is similar to Theorem 9, we omit it for simplicity.Theorem 16.Let   ( = 1, 2, . . ., ) be a collection of DHFEs; then their aggregated value by using the GDHFWA operator is also a DHFE, and Then we will discuss some properties of GDHOWA operator, as their proofs are parallel to Theorems 10, 13, and 14, and we need not to prove them.
Theorem 18 (boundedness).Let   = {ℎ  ,   } ( = 1, 2, . . ., ) be a collection of DHFEs, and  − ,  + are defined as before; then  The DHFCOA operator can be easily transformed into the following form by induction on : Obviously, this aggregated value is still a dual hesitant fuzzy element.
Step 1. Confirm the fuzzy measures  of attributes of G and attributes sets of G.
Step 2. Utilize the decision information given in matrix  and the GDHFCOA operator to derive the overall preference values d ( = 1, 2, . . ., ) of the alternative   .
Remark 32.The advantages of the generalized dual hesitant fuzzy Choquet ordered aggregation (GDHFCOA) operator lie in four aspects.
First, there is a fuzzy measure  in the GDHFCOA operator, which can be regarded as an extension of the weight vector.Sometimes, some attributes may have little importance, respectively, but when they gather together, they become very important.In order to deal with this situation, we can use the fuzzy measure to define a weight on not only each attribute but also each combination of attributes.
Second, our method does not assume the independence of one attribute from another and it can deal with the situation where the attributes are correlative.Traditional additive aggregation operators, such as DHFWA operator and DHFOWA operator, are all based on the assumption that the attributes are independent, and each attribute is given a fixed weight representing its importance during the decision process.Nevertheless, they do not consider the addition of the importance of individual attribute, and as a result, they cannot get reasonable results when the attributes are correlative.In real decision problems, since there are often interdependent or interactive phenomena among attributes, the overall importance of an attribute is not only determined by itself, but also by other attributes.So our method is a good choice to solve the real decision making problems.
Third, the GDHFCOA operator can accommodate situations in which the input arguments are dual hesitant fuzzy information.As dual hesitant fuzzy set is a comprehensive set containing FSs, IFSs, FMSs, and HFSs as special cases, our method can be widely used.
Fourth, the GDHFCOA operator has an additional parameter  which controls the power.If the parameter takes different values, the proposed operators can be evolved into many special aggregation operators, which make decision making more flexible and can meet the different needs of different decision makers.That is to say, the decision makers can choose the value of the parameter according to their preferences and interests.

The Decision Making
Steps.Next, we apply the developed approach to evaluate these theses with dual hesitant fuzzy information.
Step 1.We use the decision information given in matrix  and the GDHFCOA operator to obtain the overall preference values d of the thesis   ( = 1, 2, 3, 4, 5).Take thesis  1 , for example, we have (take  = 1)  (2) Tables 2 and 3 have some common scores such as ( d1 ) = 0.119733.This indicates that DHFOWA operator is a special case of GDHFCOA operator under certain conditions, which has been pointed out in Theorem 30.
(3) We find that the rankings in Table 2 are quite different from Tables 3 and 4. The reason may be that there are interdependent or interactive phenomena among attributes in this numerical example.From another perspective,  ( 1 ) +  ( 2 ) +  ( 3 ) +  ( 4 ) = 0.30 + 0.35 + 0.30 + 0.22 > 1 =  ( 1 ,  2 ,  3 ,  4 ) (52) also tells us that the attributes are correlative.The GDHFCOA operator can perform aggregation of attributes when they are correlative.However, DHFOWA and DHFOWG operators always suppose that the attributes are independent, and each attribute is given a fixed weight subjectively.So the GDHFCOA operator is a better choice here.
(4) When we change the parameter , we get different rankings in Table 2.This indicates that the GDHF-COA operator have an additional parameter , which makes decision making more flexible and can meet the needs of different types of decision makers.

Conclusion
In this paper, we have investigated the multiple attribute decision making (MADM) problem based on the GDHFCOA operator with dual hesitant fuzzy information.Firstly, some operational laws of dual hesitant fuzzy elements and score function of dual hesitant fuzzy elements as well as existing aggregation operators have been introduced.Then, motivated by the ideal of Choquet integral, the generalized dual hesitant fuzzy Choquet ordered aggregation (GDHFCOA) operator has been developed.Its advantage is that it can consider the importance of the attributes as well as the correlation among the attributes, which makes it more feasible and practical.At the same time, we have introduced several generalized aggregation operators for DHFS such as GDHFWA, and discussed their basic properties.As different parameters can be chosen in these generalized aggregation operators, the decision becomes more flexible.Furthermore, we have discussed the families of GDHFCOA operator.Next, we have applied the GDHFCOA operator to multiple attribute decision making problems with dual hesitant fuzzy information.Finally, an illustrative example for evaluation of theses has been given to demonstrate its practicality and effectiveness.In the future, we will consider the monotonicity of GDHFCOA operator and apply the dual hesitant fuzzy multiple attribute decision making to other domains.

Table 2 :
Scores of the dual hesitant fuzzy values obtained by GDHFCOA operator.