In decision making, uncertainty is ubiquitous since objective things are uncertain and complex, and the managing and modelling of uncertain information are crucial for the acquisition of desirable solutions (Xu and Zhao, 2016). To date, a large number of tools have been proposed to model people’s imprecise decision information from different angles (Atanassov, 1986; Torra, 2010; Zhu et al., 2012; Zhou et al., 2013; Pang et al., 2016; Guan et al., 2017; Gou et al., 2017; Liao et al., 2017; Zhang, 2017; Zhao et al., 2018). Among them, to fully reflect the characteristics of affirmation and negation of human’s cognitive performance, Atanassov (1986) introduced intuitionistic fuzzy sets (IFSs), which assign to each element a membership degree and a non-membership degree. In order to portray the human’s hesitancy in the decision-making process, Torra (2010) introduced hesitant fuzzy sets (HFSs), which allocate to each element several different membership degrees. Afterwards, motivated by the ideas of IFSs and HFSs, Zhu et al. (2012) developed dual hesitant fuzzy sets (DHFSs) to manipulate the hesitation both for the membership degree and the non-membership degree by two functions from which two sets of membership and non-membership degrees can be respectively derived for each element. Obviously, DHFSs are able to express the human’s hesitancy in more detail and more comprehensively than IFSs and HFSs, and cover IFSs and HFSs as special cases. DHFSs are very suitable for group deciding making as they can well handle the situations where a decision organization consisting of several experts is not sure about a value, but hesitant among several possible values when it provides the membership degree or the non-membership degree of an element.

At present, the studies on theory and application of DHFSs have attracted great attention, and a lot of research results have been achieved (Farhadinia, 2014; Ju et al., 2014b; Li, 2014; Singh, 2014; Zhu and Xu, 2014; Zhang et al., 2016; Ren et al., 2017). Since the aggregation operators are very important for the application of DHFSs in decision making, many scholars have studied the aggregation operators of dual hesitant fuzzy information. Wang et al. (2014) and Yu et al. (2016) developed some fundamental dual hesitant fuzzy aggregation operators in terms of arithmetic and geometric operations. Afterwards, Yu (2014) and Wang and Li (2014) extended these aggregation operators (Wang et al., 2014), and presented a family of generalized dual hesitant fuzzy aggregation operators. Based on Hamacher t-norm and t-conorm, Einstein t-conorm and t-norm, Ju et al. (2014b) and Zhang et al. (2017a) developed a series of new dual hesitant fuzzy aggregation operators, respectively. For fusing the correlative dual hesitant fuzzy information, Ju et al. (2014a) developed some dual hesitant fuzzy Choquet aggregation operators; Wang et al. (2016) proposed a wide range of dual hesitant fuzzy power aggregation operators based on Archimedean t-conorm and t-norm; Zhang (2015) proposed the dual hesitant fuzzy Hamacher correlated geometric operator. In addition, considering that there may exist some interactions between the membership functions and the non-membership functions of different DHFSs, Xu et al. (2015) presented a class of dual hesitant fuzzy interaction operators. Analysing the above-mentioned dual hesitant fuzzy aggregation operators, we can find that these operators are all defined under the assumption that all attributes are at the same priority level. However, in many practical multi-attribute decision making (MADM) problems, the attributes are in different priority levels. For instance, a consumer who is unfamiliar with cars is selecting a car from five candidate cars. Due to his/her ignorance of cars, he/she invites three friends who are proficient in cars to evaluate these cars by some indices among which safety level and comfort level are important. Generally, the safety attribute has a higher priority than comfort, which is to say, it is not allowed to compensate a loss in safety by a benefit in comfort. When evaluating the safety level of a car, one of his/her friends thinks that its satisfaction degree is 0.5, while the others believe that it should be 0.7, and they can’t persuade each other. Meanwhile, one of his/her friends deems the dissatisfaction degree of safety as 0.1, another views it as 0.2, and the other considers it as 0.3, and they all strongly insist on their own opinions. Then the above problem is a dual hesitant fuzzy MADM problem with prioritized attributes, and the evaluation for safety level of the car can be represented by the dual hesitant fuzzy element (DHFE) { { 0.5 , 0.7 } , { 0.1 , 0.2 , 0.3 } }.

The MADM problem with prioritized attributes is an interesting and hot research topic and has received great attention from scholars. For example, Yager (2008; 2009) first investigated the MADM problems with prioritized attributes, and proposed the prioritized averaging (PA) operator and prioritized ordered weighted average (POWA) operator, in which the weight of an attribute with lower priority is dependent upon the satisfactions of the higher priority attributes. After that, Yu and Xu (2013) generalized the PA operator (Yager, 2008) to the intuitionistic fuzzy environment and developed several prioritized intuitionistic fuzzy aggregation (PIFA) operators in light of some basic intuitionistic fuzzy operations. Based on the POWA operator (Yager, 2009), Xu et al. (2011) proposed the intuitionistic fuzzy POWA operator. Yu (2012) formulated the generalized intuitionistic fuzzy prioritized weighted geometric operator based on Archimedean t-conorm and t-norm. Wei (2012) extended the prioritized operators (Yager, 2008) to the hesitant fuzzy environment and developed some hesitant fuzzy prioritized operators. Yu et al. (2013) generalized Wei (2012)’s prioritized operators and presented some generalized hesitant fuzzy prioritized operators. Zhou and Li (2014) developed two generalized hesitant fuzzy prioritized Einstein operators with the help of Einstein operations. By means of the softmax function, Torres et al. (2014) developed the hesitant fuzzy prioritized softmax average and hesitant fuzzy prioritized softmax geometric operators. Recently, based on a correctional score function and a dice similarity measure of DHFEs, Ren and Wei (2017) developed a method to solve the decision making problems in which there exists a prioritization relationship over attributes and the attribute values take the form of DHFEs.

It is worth pointing out that when we apply the PIFA operators (Yu and Xu, 2013) to solve the intuitionistic fuzzy MADM problems with prioritized attributes, the PIFA operators may not capture the prioritization relationship over attributes since there may exist compensation between attributes (for details, please refer to Section 5.1). Furthermore, in the process of generating weights for attributes (Yu and Xu, 2013; Xu et al., 2011; Yu, 2012; Wei, 2012; Yu et al., 2013; Zhou and Li, 2014; Torres et al., 2014; Ren and Wei, 2017), the concept of score function is used to transform the original decision information into single real numbers, which leads to distortion and loss of information, and further results in inaccurate decision results (for details, please refer to Section 5.1). Consequently, it is very necessary to develop new prioritized aggregation operators. Since both IFSs and HFSs are special cases of DHFSs, we just need to investigate the aggregation of dual hesitant fuzzy information in the MADM problems with prioritized attributes. Therefore, the main objective of this paper is to develop a family of prioritized dual hesitant fuzzy aggregation operators based on t-norms and t-conorms and explore their applications in the dual hesitant fuzzy MADM problems with prioritized attributes.

To do so, we organize the paper as follows: Section 2 reviews some basic knowledge that will be used in our work. In Section 3, we define dual hesitant fuzzy t-norm and t-conorm and survey a special class of DHF t-norms and t-conorms. In Section 4, based on dual hesitant fuzzy t-norms and t-conorms, we first develop the prioritized dual hesitant fuzzy weighted triangular operator, which preserves both the monotonicity and boundary conditions, and from which a family of prioritized dual hesitant fuzzy aggregation operators can be derived. Then, based upon the developed prioritized dual hesitant fuzzy aggregation operators, we propose a method to handle the dual hesitant fuzzy MADM problems with prioritized attributes and demonstrate the feasibility and applicability through a car selection example. In Section 5, two comparisons are implemented to illustrate the validity and superiority of the developed prioritized aggregation operators and the proposed decision making method. Section 6 ends the paper with some conclusions.

In this section, we review some preliminary knowledge that will be used thereinafter.

Atanassov (1986) first proposed the notion of intuitionistic fuzzy sets as follows:

Some operations on IFNs are presented as follows:

However, there exist some cases in which the difficulty of determining the membership degree of an element is not because we have a margin of error (IFSs), but because we have several possible values. In order to cope with such cases, Torra (2010) put forward the notion of hesitant fuzzy sets.

To be easily understood, Xia and Xu (2011) mathematically interpreted the HFS as: E = { ⟨ x , h E ( x ) ⟩ | x ∈ X } , where h E ( x ) is a set of several values in [ 0 , 1 ], representing the possible membership degrees of the element x ∈ X to the set E. For convenience, Xia and Xu (2011) called h = h E ( x ) a hesitant fuzzy element (HFE).

Meanwhile, Xia and Xu (2011) developed the following method to compare HFEs.

Recently, inspired by the ideas of IFSs and HFSs, Zhu et al. (2012) introduced dual hesitant fuzzy sets, which allow the membership and non-membership degrees of an element to a given set to be several different values in [ 0 , 1 ], respectively.

Zhu et al. (2012) called the component d ( x ) = ( h A ( x ) , g A ( x ) ) a dual hesitant fuzzy element (DHFE), denoted by d = ( h , g ). Apparently, for the DHFE d = ( h , g ), if # h = # g = 1 ( # h and # g denote the numbers of elements in h and g, respectively), then d is simplified to an IFN; if h ≠ ϕ and g = ϕ (ϕ denotes the null set), then d is simplified to an HFE. In addition, when h ≠ ϕ and g ≠ ϕ, the complement of d is given by d c = ( g , h ) (Zhu et al., 2012). Since DHFEs are the basic units of a DHFS, for the purpose of simplicity, we thereinafter will only focus on the DHFEs.

For convenience, we give the following two definitions:

Farhadinia (2014) proposed a linear order to implement the comparison of two DHFEs under the assumption that the DHFEs have the same numbers of membership degrees and non-membership degrees, respectively.

Below we show the above comparison method through an example:

For comparing two DHFEs effectively, motivated by the comparison method for IFNs (Xu and Yager, 2006), we adjust the comparison principle in Definition 9 as:

For the DHFEs d 1 and d 2 in Example 1, we have d 1 ∘ = − 0.5333 < − 0.2333 = d 2 ∘ . Thus, d 1 is inferior to d 2 , which is in line with human’s intuition.

The aim of this section is to define dual hesitant fuzzy t-norms and t-conorms and study a special class of dual hesitant fuzzy t-norms and t-conorms in detail, which are very helpful for the development of prioritized dual hesitant fuzzy aggregation operators.

From Definition 6, we know that for two different DHFEs, the numbers of membership degrees (or non-membership degrees) may be different. Hence, a normalization process should be considered. Inspired by the normalization method for HFEs (Xu and Xia, 2011), below we normalize them by adding membership degrees (or non-membership degrees):

Thus, for two DHFEs with different numbers of membership degrees (or non-membership degrees), we may employ the parameter τ to normalize them such that they have the same number of membership degrees (or non-membership degrees). For example, for two DHFEs d 1 = { { 0.1 , 0.2 } , { 0.4 , 0.5 , 0.6 } } and d 2 = { { 0.2 , 0.3 , 0.4 , 0.6 } , { 0.3 , 0.4 } }, it is obvious that # h 1 < # h 2 and # g 2 < # g 1 . In Definition 11, let τ = 1, then in this case, h 1 ‾ = 0 × σ h 1 ( 1 ) + 1 × σ h 1 ( # h 1 ) = 0.2 and g 2 ‾ = 1 × σ g 2 ( 1 ) + 0 × σ g 2 ( # g 2 ) = 0.3. Thus, we can extend d 1 by adding 0.2 into its membership degree set h 1 and extend d 2 by adding 0.3 into its non-membership degree set g 2 . As a result, d 1 becomes { { 0.1 , 0.2 , 0.2 , 0.2 } , { 0.4 , 0.5 , 0.6 } } and d 2 becomes { { 0.2 , 0.3 , 0.4 , 0.6 } , { 0.3 , 0.3 , 0.4 } }. In general, the parameter τ is determined by the decision maker used to reflect his/her risk attitude. Clearly, if τ = 1, then the largest membership degree and the smallest non-membership degree are added, respectively, which means that the decision maker is optimistic. For other cases, for example, τ = 0 indicates that the decision maker is pessimistic, and τ = 0.5 indicates that the decision maker is indifferent. In this paper, we only consider the case in which the decision makers are optimists, and other cases can be studied in a similar way.

For facilitating the definitions of dual hesitant fuzzy t-norms and t-conorms, a partial order “ ⩽ D ” on the set of all DHFEs D is defined as follows:

Now with the help of the bounded lattice ( D , ⩽ D ), we can straightforwardly extend the definitions of t-norms and t-conorms to the dual hesitant fuzzy setting.

A dual hesitant fuzzy t-conorm (DHF t-conorm) is a mapping D S : D 2 → D possessing the following properties: (i)

Commutativity: D S ( d 1 , d 2 ) = D S ( d 2 , d 1 );

The associativity property of DHF t-norms and t-conorms permits us to extend them to any number of dual hesitant fuzzy arguments, i.e. given n DHFEs d i , i = 1 , 2 , … , n, we have D T ( d 1 , … , d n ) = D T ( D T ( d 1 , … , d n − 1 ) , d n ) and D S ( d 1 , … , d n ) = D S ( D S ( d 1 , … , d n − 1 ) , d n ). Besides, we get the notable properties: D T ( d 1 , … , d n , d n + 1 ) ⩽ D D T ( d 1 , … , d n ) and D S ( d 1 , … , d n ) ⩽ D D S ( d 1 , … , d n , d n + 1 ) for n ⩾ 2.

Let d i = ( h i , g i ), i = 1 , 2 be two DHFEs with # h 1 = # h 2 = k and # g 1 = # g 2 = l, then some typical examples of DHF t-norms are listed as follows:

1) Dual hesitant fuzzy minimum (DHFM) t-norm: (2) D T T M , S M ( d 1 , d 2 ) = d 1 ∧ d 2 = { { σ h 1 ( t ) ∧ σ h 2 ( t ) | t = 1 , 2 , … , k } , { σ g 1 ( t ) ∨ σ g 2 ( t ) | t = 1 , 2 , … , l } } .

2) Dual hesitant fuzzy product (DHFP) t-norm: (3) D T T P , S S ( d 1 , d 2 ) = d 1 ⊗ d 2 = { { σ h 1 ( t ) σ h 2 ( t ) | t = 1 , 2 , … , k } , { σ g 1 ( t ) + σ g 2 ( t ) − σ g 1 ( t ) σ g 2 ( t ) | t = 1 , 2 , … , l } } .

3) Dual hesitant fuzzy Lukasiewicz (DHFL) t-norm: (4) D T T L , S L ( d 1 , d 2 ) = { { ( σ h 1 ( t ) + σ h 2 ( t ) − 1 ) ∨ 0 | t = 1 , 2 , … , k } , { ( σ g 1 ( t ) + σ g 2 ( t ) ) ∧ 1 | t = 1 , 2 , … , l } } . Three typical examples of DHF t-conorms are presented as follows:

1) Dual hesitant fuzzy maximum (DHFM) t-conorm: (5) D S S M , T M ( d 1 , d 2 ) = d 1 ∨ d 2 = { { σ h 1 ( t ) ∨ σ h 2 ( t ) | t = 1 , 2 , … , k } , { σ g 1 ( t ) ∧ σ g 2 ( t ) | t = 1 , 2 , … , l } } .

2) Dual hesitant fuzzy probabilistic sum (DHFPS) t-conorm: (6) D S S S , T P ( d 1 , d 2 ) = d 1 ⊕ d 2 = { { σ h 1 ( t ) + σ h 2 ( t ) − σ h 1 ( t ) σ h 2 ( t ) | t = 1 , 2 , … , k } , { σ g 1 ( t ) σ g 2 ( t ) | t = 1 , 2 , … , l } } .

3) Dual hesitant fuzzy Lukasiewicz (DHFL) t-conorm: (7) D S S L , T L ( d 1 , d 2 ) = { { ( σ h 1 ( t ) + σ h 2 ( t ) ) ∧ 1 | t = 1 , 2 , … , k } , { ( σ g 1 ( t ) + σ g 2 ( t ) − 1 ) ∨ 0 | t = 1 , 2 , … , l } } .

To date, we have introduced the definitions of DHF t-norms and t-conorms, which are consistent with those of traditional t-norms and t-conorms (similar work can be found in Deschrijver et al., 2004; Dimuro et al., 2011). In what follows, we research a special class of DHF t-norms and t-conorms, which are constructed by traditional t-norms and t-conorms and will play a crucial role in the development of prioritized dual hesitant fuzzy aggregation operators.

Theorem 1 can be easily proven by the definitions of t-norms and t-conorms. Obviously, the DHF t-norms and t-conorms respectively presented by Eqs. (2)–(4) and Eqs. (5)–(6) can be formed through Theorem 1. Property 1.

Let D T T , S and D S S , T be a DHF t-norm and t-conorm constructed through Theorem 1, respectively, then for any two DHFEs d i , i = 1 , 2, we have (i)

D T T , S ( 0 D , d 1 ) = 0 D and D S S , T ( 1 D , d 1 ) = 1 D ;

In this section, we focus on the resolution of dual hesitant fuzzy MADM problems in which there exists a prioritization relationship over attributes. We first describe this kind of problems as follows: suppose that there are several attributes c j ( j = 1 , 2 , … , m ) according to which a decision organization containing several experts assesses several feasible alternatives x i ( i = 1 , 2 , … , n ) by using the DHFEs d j ( x i ). Assume that there exists the prioritization relationship over the attributes: c 1 ≻ c 2 ≻ ⋯ ≻ c m , i.e. the attribute c r has a higher priority than the attribute c j if r < j. The goal of the decision maker is to get the ranking of all feasible alternatives.

A simple method to solve the aforesaid dual hesitant fuzzy MADM problems with prioritized attributes is to conduct an aggregation for the dual hesitant fuzzy evaluation values of each alternative under all attributes to obtain a collective value, and then to compare the collective values of all alternatives to obtain a ranking of alternatives. However, from the introduction, we note that no studies have been done on the aggregation of dual hesitant fuzzy information in the MADM problems with prioritized attributes. Therefore, below we attempt to develop prioritized dual hesitant fuzzy aggregation operators based on the DHF t-norms and t-conorms defined in the former section.