1 Introduction
Preference relation has played a fundamental role in most decision processes Xu2007surverypr . According to previous studies, the preference relation can be divided into two categories. The first one is multiplicative preference relation herrera2001multiperson ; chiclana2004induced ; liu2012goal ; xia2014multiplicative which is subjected to the multiplicative reciprocal, i.e. . The second one is fuzzy preference relation Tanino1984fuzzypref ; gong2008least ; parreiras2012dynamic ; rezaei2013multi which is described by fuzzy pairwise comparison with an additive reciprocal, i.e. . As the basic element of many decision making methods especially in analytic hierarchy process (AHP) model saaty1980analytic ; chan2007global ; ishizaka2011selection ; ishizaka2013calibrated , the preference relation has attracted many interests fernandez2010solving ; evren2011multi ; xu2012error ; liu2012least ; xu2013distance ; xia2013preference ; Wang2014169 .
Fuzzy preference relations herrera2007consensus ; liu2012new provide a method to construct the decision matrices of pairwise comparisons based on the linguistic values given by experts. The value given by the experts represents the degree of the preference for the first alternative over the second alternative. Assume there are alternatives, a total of pairwise comparisons need to be answered for constructing a fuzzy preference relation. What’s more, there always exists a potential risk that the constructed fuzzy preference relation is inconsistent due to the inability of human beings to deal with overcomplicated objects chiclana2002note ; xu2013ordinal ; xia2013algorithms . In order to overcome the deficiencies, HerreraViedma et al. HerreraViedma2004someiss proposed consistent fuzzy preference relation (CFPR) to construct the pairwise comparison decision matrices based on additive transitivity property Tanino1984fuzzypref ; tanino1988fuzzy , i.e. . The merit of CFPR consists of two aspects. Firstly, only is a total of pairwise comparisons needed to construct a CFPR. Secondly, it is always consistent in a CFPR. Due to these merits, the CFPR is widely used in many fields wang2007applying ; chen2012supplier ; lan2012deriving .
Although the CFPR is so advantageous to express experts’ or decision makers’ preferences, however, the original CFPR is constructed on the foundation of complete and certain information. It is unable to deal with the cases involving incomplete and uncertain information. For example, an expert gives that the first alternative is more important than the second alternative . How to express the linguistic variable “more important”? Some one would say it means , some others will say it seems that . Or it is more reasonable that with a degree and with a degree , where . Besides, incomplete information means the preference values are not complete due to the lack of knowledge and the limitation of cognition. For example, an expert gives with a belief of 0.7, the remainder 0.3 belief can not be assigned to any linguistic values due to the lack of knowledge. With respect to these cases involving uncertain and incomplete entries, the conventional CFPR is incapable.
To overcome these weaknesses, D numbers Deng2012DNumbers ; Deng2014DAHPSupplier ; Deng2014EnvironmentDNs ; Deng2014BridgeDNs , a new model of expressing uncertain information, has been employed in the construction of CFPR. A new preference relation called D numbers extended consistent fuzzy preference relation (short for DCFPR) is proposed in this paper. The DCFPR uses D numbers to express the linguistic preference values given by experts or decision makers, and it can be reduced to classical CFPR. In contrast with the construction of CFPR, a method is proposed for constructing the DCFPR, and the proposed method is also effective to construct the CFPR. In addition, the priority weights and ranking of alternatives can be obtained from a DCFPR based on our previous work Deng2014DAHPSupplier . In Deng2014DAHPSupplier , we proposed a DAHP (D numbers extended AHP) model for multicriteria decision making (MCDM). The work in this paper provides a solution of constructing the preference matrix for DAHP. Based on these two studies, we systematically provide a novel solution for MCDM problems.
The rest of this paper is organized as follows. Section 2 gives a brief introduction about the consistent fuzzy preference relation. In Section 3, the proposed D numbers extended consistent fuzzy preference relation is presented. Some numerical examples are given in Section 4. Finally, concluding remarks are given in Section 5.
2 Consistent fuzzy preference relations (CFPR)
Fuzzy preference relations Xu2007surverypr ; Tanino1984fuzzypref ; herrera2007consensus ; HerreraViedma2004someiss enable an expert or a decision maker to give linguistic values for the comparison of alternatives or creteria. The preference values employed in a fuzzy preference relation are real numbers belonging to . A reciprocal fuzzy preference relation on a set of alternatives is represented by a fuzzy set on the product set , and is characterized by a membership function Xu2007surverypr ; herrera2007consensus ; HerreraViedma2004someiss
(1) 
when the cardinality of is small, the preference relation may be conveniently represented by an matrix , being , namely
(2) 
where (1) ; (2) ; (3) . denotes the preference degree of alternative over alternative .
HerreraViedma et al. HerreraViedma2004someiss proposed the consistent fuzzy preference relation (CFPR) for the construction of pairwise comparison decision matrices based on additive transitivity property Tanino1984fuzzypref ; tanino1988fuzzy . A reciprocal fuzzy preference relation is called a consistent fuzzy preference relation if and only if . For a CFPR , the following two equations are satisfied HerreraViedma2004someiss :
(3) 
(4) 
The biggest advantage of CFPRs is that it is reciprocal and consistent. Based on the results presented in Eq.(3) and Eq.(4), a CFPR can be constructed from the set of values . That means only pairwise comparisons are required in the process of constructing a CFPR. Compared with the construction of the ordinary fuzzy preference relation which requires pairwise comparisons, the rest of pairwise comparisons are computed by using additive transitivity property in the construction of CFPRs. It is noted that the values in the generated CFPR may do not fall in the interval , but fall in an interval , . In such a case, the values in the obtained CFPR need to be transformed by using a transformation function that preserves reciprocity and additive consistency. The transformation function is defined as HerreraViedma2004someiss :
(5) 
3 Proposed D numbers extended consistent fuzzy preference relations (DCFPR)
3.1 D numbers
D number Deng2012DNumbers ; Deng2014DAHPSupplier ; Deng2014EnvironmentDNs ; Deng2014BridgeDNs is a new model of representing uncertain information. It has extended the DempsterShafer theory Dempster1967 ; Shafer1976 . DempsterShafer theory is with advantages to handle uncertain information liu2006analyzing ; schubert2011conflict ; jousselme2012distances ; yang2013evidential ; Deng2014PAontheaxi ; dubois2012conditioning ; lefevre2013preserve ; dezert2014validity , and is extensively used in many fields, such as risk assessment zhang2012assessment ; bolar2013condition , expert systems yang2006belief ; zhou2013bi , classification and clustering denoeux2004evclus ; bi2008combination
, parameter estimation
denoeux2013maximum , decision making casanovas2012fuzzy ; mokhtari2012decision ; yager2013decision , and so forth srivastava2009representation ; klein2012belief ; cuzzolin2010geometry ; cuzzolin2012relative ; kang2012evidential ; wei2013identifying ; Deng2013TOPPER . However, there are some weaknesses in DempsterShafer theory. D numbers overcome a few of existing deficiencies (i.e., exclusiveness hypothesis and completeness constraint) in DempsterShafer theory and appear to be more effective in representing various types of uncertainties. Some basic concepts about D numbers are given as follows.Let be a nonempty set satisfying if , , a D number is a mapping formulated by
(6) 
with
(7) 
where is the empty set and is a subset of .
If , the information expressed by the D number is said to be complete; if , the information is said to be incomplete. The degree of information’s completeness in a D number is defined as below.
Let be a D number on a finite nonempty set , the degree of information’s completeness in is quantified by
(8) 
For the sake of simplification, the degree of information’s completeness of a D number is called as its value.
For a discrete set , where and if , a special form of D numbers can be expressed by Deng2014DAHPSupplier ; Deng2014EnvironmentDNs
(9) 
or simply denoted as , where if , and . Some properties of this form of D numbers are introduced as follows.
Permutation invariability. If there are two D numbers that and , then .
For a D number , the integration representation of is defined as
(10) 
where , and . For the sake of simplification, the integration representation of a D number is called as its value.
3.2 DCFPR: D numbers extended CFPR
As mentioned above, the CFPR provides an option to establish the decision matrix which only requires pairwise comparisons. Moreover, the reciprocity and additive consistency have been preserved in a CFPR. However, the original CFPR is constructed on the foundation of complete and certain information. It is unable to deal with the cases involving incomplete and uncertain information. This deficiency also has existed in the fuzzy preference relation. For example, assume there are experts who were invited to evaluate alternatives and . Consider these cases.
Case 1: experts evaluate that is preferred to with a degree , the remainder experts evaluate that is preferred to with a degree , where and .
Case 2: experts evaluate that is preferred to with a degree , the remainder experts do not give any evaluations due to the lack of knowledge, where and .
Obviously, both the original CFPR and fuzzy preference relation are incapable of representing and handling the aforementioned cases. In Deng2014DAHPSupplier we studies the deficiency in the situation of fuzzy preference relations, and proposed the concept of D numbers preference relations which extends the fuzzy preference relations by using D numbers in order to overcome this deficiency. In this paper, we concentrate on the deficiency in the situation of CFPRs. The D numbers extended CFPR, shorted for DCFPR, is proposed to strengthen CFPR’s ability of expressing uncertain information by using D numbers. the DCFPR is formulated by
(11) 
where , , and , , . Obviously, in .
In Eq.(11), is called a DCFPR because it is constructed based on pairwise comparisons denoted as which is a set of D numbers. Here, a method is proposed to implement the construction of DCFPRs.
At first, for the elements in the set of in which , is given by
(12) 
in which every component is obtained by
(13) 
(14) 
where , and is the th component of , is the th component of , , is the th component of .
At second, for the rest of entries in the DCFPR, they can be calculated based on the reciprocal property, namely , .
According to the aforementioned two steps, a DCFPR can be constructed based on pairwise comparisons . For the generated DCFPR, it is possible that some values of s in these D numbers do not fall in the interval , but fall in an interval , . In such a case, the values of all s in every D numbers need to be transformed by using a transformation function which is given in Eq.(5). The transformation function works as normalization, which transforms the values of s from to the interval .
In summary, the proposed method to construct a DCFPR on alternatives from preference values is implemented as the following steps:

Start. Let as input.

Calculate the set of preference values as
,
.

. For , if the values of all s in each D numbers fall in the interval , the DCFPR is obtained as , go to step 6; otherwise, go to next step.

, , .

The DCFPR is obtained as such that
,
, , .

End.
Up to now, the method to construct a DCFPR is totally presented. It should be pointed that the DCFPR will reduce to the classical CFPR if the D numbers based preference values are substituted by real numbers. And the method to construct a DCFPR is completely suitable for the construction of CFPRs. The proposed DCFPR is an extension of the classical CFPR.
3.3 Solution for the DCFPR
Once a DCFPR has been constructed, another key problem is aroused that how to obtain the ranking and priority weights of alternatives based on the DCFPR. In Deng2014DAHPSupplier , we studied the solution for D numbers preference relations which extends the fuzzy preference relations by using D numbers. Differ from common D numbers preference relations, the DCFPR is obtained based the additive transitivity property of CFPRs. The proposed DCFPR in the paper essentially is an special case of D numbers preference relations. Therefore, the solution for D numbers preference relations is also suitable for DCFPRs. The procedure of the solution for DCFPRs is shown in Figure. 1. At first, the DCFPR is converted to an values matrix by using Eq.(10
). At second, construct a probability matrix
based on the values matrix to represent the preference probability between pairwise alternatives. At third, a triangular probability matrix can be obtained in terms of probability matrix with the aid of local information which contains the preference relation of pairwise alternatives. According to , the ranking of alternatives is determined. At fourth, a triangulated values matrix is generated based on and , and the weights of alternatives can calculated through . Please refer to literature Deng2014DAHPSupplier for more details.3.4 Inconsistency for the DCFPR
The classical CFPR is totally consistent due to it is constructed based on the additive transitivity property from preference values. As an extension of CFPRs, the DCFPR is also totally consistent when it has reduced to the classical CFPR. When the DCFPR is with entries which contain uncertain or incomplete information, it is not totally consistent. In order to measure the inconsistency of DCFPRs, an inconsistency degree defined for D numbers preference relations Deng2014DAHPSupplier is utilized to express such inconsistency. The inconsistency degree is on the basis of the triangular probability matrix .
(15) 
4 Numerical examples
In this section, some numerical examples are given to show the construction of DCFPRs.
4.1 Example 1: preference values with certain information
This example is from literature HerreraViedma2004someiss . Suppose there is a set of four alternatives where we have certain knowledge to assure that alternative is weakly more important than alternative , alternative is more important than and finally alternative is strongly more important than alternative . Suppose that this situation is modelled by the preference values .
By applying the method proposed in HerreraViedma2004someiss , a CFPR is obtained:
(16) 
Howver, if the preference values are seen as a set of D numbers , a DCFPR can be obtained by using the proposed method in this paper as follows.
due to
,
.
due to
,
.
due to
,
.
, , , , , , and therefore:
(17) 
For , the values of all s in each D numbers fall in the interval , so the final DCFPR . DCFPR is identical with CFPR shown in Eq.(16).
Finally, the method proposed in literature Deng2014DAHPSupplier is utilized in order to obtain the priority weight of each alternative. The calculating process is omitted. Some important intermediate results are given as follows.
(18) 
(19) 
(20) 
(21) 
The inconsistency degree is . The obtained Priority weights and ranking of alternatives are shown in Table 1.
Alternatives  Priority(different credibility of preference values)  Ranking  
High  Medium  Low  Interval range  
0.338  0.294  0.272  (0.250, 0.409]  1  
0.312  0.281  0.266  (0.250, 0.364]  2  
0.237  0.244  0.247  [0.227, 0.250)  3  
0.112  0.181  0.216  [0.000, 0.250)  4 
4.2 Example 2: preference values with uncertain and incomplete information
In this example, we make a change to the preference values from the above example so that the preference values are with uncertain and incomplete information. Assume this situation is modelled by a set of D numbers .
In this case, the preference values involve incomplete entry (i.e., ) and uncertain entry (i.e., ). The classical CFPR can not deal with this case, but the proposed DCFPR is effective for this case. A DCFPR can be obtained as follows.
due to
,
;
due to
,
;
,
;
due to
,
.
,
.
, , , , , , and therefore:
(22) 
Due to the preference values in do not totally fall in the interval , but fall in an interval , the transformation function shown in Eq.(5) will be used to obtain the final DCFPR . The result is given as below.
(23) 
Similar with Example 1, in order to obtain the priority weight of each alternative, the method proposed in literature Deng2014DAHPSupplier is employed. Some important intermediate results are given as follows.
(24) 
(25) 
(26) 
(27) 
In this example, the inconsistency degree is . The obtained Priority weights and ranking of alternatives are shown in Table 2.
Alternatives  Priority(different credibility of preference values)  Ranking  
High  Medium  Low  Interval range  
0.327  0.289  0.269  (0.250, 0.402]  1  
0.309  0.280  0.265  (0.250, 0.366]  2  
0.241  0.246  0.248  [0.232, 0.250)  3  
0.123  0.186  0.218  [0.000, 0.250)  4 
5 Concluding remarks
In this paper, we have studied the consistent fuzzy preference relation (CFPR) by combining with D numbers. The proposed new preference relation is called D numbers extended consistent fuzzy preference relation, shorted for DCFPR. A method is proposed to implement the construction of DCFPR based on a set of preference values which are expressed by D numbers. In comparison with CFPR, DCFPR is able to deal with the case that preference values involve incomplete and uncertain information. DCFPR can be reduced to the classical CFPR when the preference values expressed by D numbers have degenerated to real numbers. What’s more, based on our previous study in Deng2014DAHPSupplier , the priority weights and ranking of alternatives can be obtained given a DCFPR. In Deng2014DAHPSupplier , the hierarchical structure of DAHP model has been established. In this paper, the proposed DCFPR can be employed to construct the preference matrix for DAHP. Based on these two studies, the DAHP model has been systematically built. In the future, we will focus on the application of proposed DCFPR and DAHP model.
Acknowledgements
The work is partially supported by National Natural Science Foundation of China (Grant nos. 61174022 and 71271061), National High Technology Research and Development Program of China (863 Program) (Grant no. 2013AA013801), R & D Program of China (2012BAH07B01).
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