REGRESSION METHODS FOR HESITANT FUZZY PREFERENCE RELATIONS

. In this paper, we develop two regression methods that transform hesitant fuzzy preference relations (HFPRs) into fuzzy preference relations (FPRs). On the basis of the complete consistency, reduced FPRs with the highest consistency levels can be derived from HFPRs. Compared with a straightforward method, this regression method is more efficient in the Matlab environment. Based on the weak consistency, another regression method is developed to transform HFPRs into reduced FPRs which satisfy the weak consistency. Two algorithms are proposed for the two regression methods, and some examples are provided to verify the practicality and superiority of the proposed methods.


Introduction
Fuzzy preference relations (FPRs) are widely used in decision making, where consistency of FPRs is a major goal and interesting research topic (Herrera-Viedma et al. 2004, 2007Jiang, Fan 2008;Tanino 1984Tanino , 1988Wu et al. 2012;Wei et al. 2012;Stankevičienė, Mencaitė 2012;Baležentis et al. 2012). Recently, hesitant fuzzy sets (HFSs), originally introduced by Torra (2010), become a hot topic (Zhu et al. 2012a(Zhu et al. , b, 2013. HFSs can consider the degrees of membership by a set of possible values. The motivation to propose HFSs is that when defining the membership of an element, the difficulty of establishing the membership degree is not a margin of error (as in intuitionistic fuzzy sets ;Atanassov 1986), or some possibility distributions on the possible values (as in type 2 fuzzy sets; Zadeh 1975), but a set of possible values (Torra 2010).
With respect to the preference relations of HFSs, Xia and Xu (2013) defined hesitant fuzzy preference relations (HFPRs) and developed an approach to apply HFPRs to decision making. However, as a basic issue of HFPRs, the studies on consistency of HFPRs is not easy because the numbers of possible values in different hesitant fuzzy elements (HFEs) are often different. Since FPRs have been proven to be an effective tool used in decision making problems (Chiclana et al. 2001;Orlovsky 1978;Tanino 1984), we consider some techniques to transform HFPRs into FPRs based on their close relationship. Two regression methods are developed for the transformations based on the complete consistency and the weak consistency respectively.
The rest of this paper is organized as follows. Section 1 reviews some basic knowledge. In Section 2, we develop the regression methods, and illustrate their advantages with some examples. The final section ends the paper with some conclusions.

Preliminaries
This section introduces some concepts related to hesitant fuzzy sets (HFSs), fuzzy preference relations (FPRs), and hesitant fuzzy preference relations (HFPRs). Torra (2010) originally developed HFSs which cover arguments with a set of possible values.

Hesitant fuzzy preference relations
Definition 1 (Torra 2010). Let X be a fixed set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0,1].
To be easily understood, Xia and Xu (2011) expressed the HFS by a mathematical symbol: where ( ) h x is a set of some values in [0,1], denoting the possible membership degrees of the element x X ∈ to the set E. For convenience, Xia and Xu (2011) called h a hesitant fuzzy element (HFE).
For a HFE h , Xia and Xu (2011) developed some operations as follows: FPRs (Orlovsky 1978) are an effective tool in decision making. The definition is as follows.
Definition 2 (Orlovsky 1978). A fuzzy preference relation (FPR) P on a set of objectives, X, is a fuzzy set on the product set X X × , that is characterized by a membership function : When the cardinality of X is small, the fuzzy preference relation may be conveniently represented by a n n × matrix ( ) where: ( ) l ij σ γ is the lth largest element in ij h .

Consistency measures
The transitivity property is used to represent the idea that the preference degree obtained by directly comparing two objectives should be equal to or greater than the preference degree between those two objectives obtained using an indirect chain of objectives. This property is desirable to avoid contradictions reflected in preference relations. For the FPR ( ) ij n n P p × = , Tanino (1984) introduced an additive fuzzy transitivity property, or called the complete consistency: Tanino (1988) also introduced an additive fuzzy weak transitivity, or called the weak consistency: x is preferred to j x and j x is preferred to k x , then i x should be preferred to k x . This property verifies the condition that a logical and consistent person does not want to express his/her opinions with inconsistency, which guarantee the minimum requirement for consistency.

Regression methods for HFPRs
In this section, we develop two regression methods for HFPRs, which depend on the complete consistency and the weak consistency respectively.

A regression method for HFPRs based on the complete consistency
Herrera-Viedma et al. (2007) developed a method with error analysis to measure the consistency levels of FPRs. Motivated by this method, and based on the complete consistency and error analysis, we develop a regression method to transform HFPRs into FPRs.
Given a HFPR, represented by a matrix  is a fixed set of objectives. According to the definition of the complete consistency, the possible preference degrees over the paired objectives ( , ) i k represented by a HFE ik h ( i k ≠ ) can be estimated using an intermediate objective j where: j ik h can be called an estimated HFE, and the operations " +  " and " =  " are efined as follows.

Definition 4. Let h , 1
h and 2 h be three HFEs, and a be a real number, then we define In order to use Eq. (4) to estimate j ik h , the objectives ( 1,2, , )  should generally be classified into several sets defined as follows: where: B is a set of all paired objectives; ). Based on the discussions above and according to Eq.(4), we can get all the estimated HFE . To select the optimal preference degree from j ik h ( 1,2, ; , ) j n j i k = ≠  , we calculate an average estimated preference degree defined as follows: where: the coefficient 2 / 3 is used to make sure each value of the error belongs to the unit interval [0,1]. If there exists a preference degree * * ( ) ik ik ik h h h ∈ that corresponds to the minimum value of the error ik h ε satisfying: With respect to one objective i x , the consistency level is defined as follows.   (12) and (13).
Step 3. Repeat Steps 1 and 2 until all HFEs have been located, then turn to the next Step.
Step 4 Step 1 Step 2. According to Eq. (11), we have: By Eqs. (12) and (13), we can get: Step 4. Collecting * Step 5 Step 6. End. For the HFPR ( ) ij n n H h × = , since each preference degree in ij h is a possible value, H can be directly separated into all possible FPRs. Then based on some existing consistency measure methods, the FPR with the highest consistency level can be found out. In order to compare this straightforward method with our method, we give the following example.  So the results are the same by the regression method and the straightforward method. Considering the efficiency, the number of operations of our regression method and the straightforward method are 2 ( 1) 1 n n n − + + and ( ( 1) 1) m n n n − + + (m is the number of all possible FPRs separated from a HFPR), respectively. Since 2 m ≥ (at least two FPRs can be separated from a HFPR), we have ( ( 1) 1) 2 ( 1) 1 m n n n n n n − + + > − + + . So our method is simpler. Moreover, the bigger the value m, the simpler the regression method.

A regression method for HFPRs based on the weak consistency
For the decision making problems in practical applications, the complete consistency is sometimes not necessary due to the complicated environment and the cognitive diversity of humans. But, the weak consistency is essential because a contradictory HFPR doesn't make sense. On the basis of the weak consistency, we now develop another regression method to get reduced FPRs satisfying the weak consistency. In what follows, we begin with some necessary definitions and discussions.
where: ij s is called a hesitant preference element (HPE), satisfying: Then ( ) ij n n M m × = is called a hesitant preference relation (HPR). According to graph theory (Bondy, Murty 1976), the relationship included in the HPR can be described by a directed graph which can be called a hesitant fuzzy preference graph. In such a graph, each node stands for an objective, and each directed edge stands for a preference relation. If 1 ij m = , then there is a directed edge from a node i to a node j, which represents that the objective i is superior to the objective j.
Step 6. End. In practical applications, Algorithms I and П can be combined to obtain reduced FPRs from HFPRs, where the obtained reduced FPR can not only satisfy the weak consistency but also have the highest confidence level.
For example, we replace Step 5 in Example 4 by Algorithm I. Then we can obtain a reduced FPR, denoted by * 2 H , with the highest consistency level 95.56% :

Conclusions
For a hesitant fuzzy preference relation (HFPR), it is not easy to deal with its consistency due to different numbers of possible values in hesitant fuzzy elements (HFEs). In this paper, we have developed two regression methods to transform HFPRs into reduced fuzzy preference relations (FPRs). Based on the complete consistency, we use error analysis to select the optimal preference degree for each paired objectives in HFPRs to produce reduced FPRs. The step by step procedure of this regression method is shown in Algorithm I. On the basis of the weak consistency, we have defined a hesitant preference relation (HPR) and a circular triad power to find circular triads of objectives in HFPRs. Then we have given Theorems 1 and 2 to identify the weak consistency of HFPRs. With these definitions and methods, we have given Algorithm П to transform HFPRs into FPRs that satisfy the weak consistency.