Research on Two Dimensional Semantic Evidential Reasoning Decision Method in Different Frame

2018 International Conference on Applied Mechanics, Mathematics, Modeling and Simulation (AMMMS 2018) ISBN: 978-1-60595-589-6

Research on Two-Dimensional Semantic Evidential Reasoning Decision

Method in Different Frame

Hui WANG* and Hong-tao ZHANG

School of Science, Zhejiang Agriculture and Forestry University, Hangzhou, 311300, China

*Corresponding author

Keywords: Two-dimensional linguistic information; Multi-dimensional framework, Evidential reasoning.

Abstract. Two-dimensional linguistic information has some unique advantages in the expression on

the quality of decision-making information. In this paper, an extended 2-dimesional linguistic information decision making method is proposed, where the information is defined on different frameworks. Firstly, the second dimensional linguistic is defined as a fuzzy linguistic label, which can be quantified through the barycentre of the fuzzy number, and the two-dimensional linguistic information can be expressed by evidence body with weights. Then, two-dimensional linguistic information is considered as two-dimensional progressive framework, and the two-dimensional linguistic information in different frames is uniformed through the definition of framework equivalent and the method of uniformity of the evidence bodies in different frames. After then, multiple two-dimensional linguistic information in different frames can be integrated by the evidence reasoning combination rules. Finally, an example is given to test the effectiveness of the method.

Introduction

With the development of society and the progress of science and technology, knowledge and information content are increasing rapidly, the decision-making problem becomes more and more complicated. Under the circumstances, the decision information usually has the characteristics such as multi-source and heterogeneity, etc.

For the different quality of decision information, Zhu et al [1] first used a dimensional language information reflecting the confidence level of expert evaluation based on traditional language information, and proposed the concept of two-dimensional semantic evaluation information. On this basis, Zhu and Zhang et al. [2] used two-dimensional semantics for the evaluation of scientific fund projects, and constructed a research method based on two-dimensional semantics. This method took two-dimensional semantic information as a whole and represented it as evidence, and the multiple two-dimensional semantic information was integrated by the evidential reasoning operator. The advantage of the method is that the semantic representation is more accurate and the information loss is less, but there is also a certain subjective problem in the acquisition of evidence reliability. Liu, Yu, and Zhu et al. [3-7] gave the semantic representation of two-dimensional languages from another angle, where the first and second-dimensional languages in two-dimensional semantic information were both represented in language scales, and were aggregated by some proposed aggregation operators respectively. The advantage of this methods is that the semantic representation is clear and the calculation is simple; but there are also problems such as large loss of information in the calculation process and insufficient use of the second-dimensional language information.

information. On this basis, it is extended to the two dimensional semantic information of different frames. The two dimensional semantic information of different frames is converted into consistency by defining the equivalent of recognition frames and the equivalent of probability. Then the evidential reasoning operators are used to assemble and construct the two dimensional semantic information decision method.

Related Concepts of Two - Dimensional Semantics in Different Frameworks

Definition 1 For a judgment problem, suppose the possible results that people can recognize are represented by set , then any statement that people care about corresponds to a subset of , called

 frame of discernment.

Definition 2[8] Set  ₁, ₂, ,_nas n different identification frameworks, if these frameworks handle the same problem from different perspectives, they are called Parallel framework.

Definition 3[8] Set  ₁, ₂, ,_n as n different identification frames, if the elements in the latter identification frame of the n frame are used to modify the elements in the former identification frame, the first identification frame is used to describe the problem to be solved, then the n frame is called Progressive framework.

Definition 4[8] Set  ₁, ₂, ,_n as n different identification frames. If there are parallel frames and progressive frames in the n frames, the n frames are called Mixed framework.

Definition 5 For a decision problem, decision makers give two recognition framework as

1 2

{ , , , _N}

H H H H and S { ,S S_{1 2}, ,S_T}, using the I, II linguistic assessment information in two dimensions identified frame describe yourself for judging something, where the evaluation information H n_n( 1, 2, ,N) in the ith dimension identification framework is used to describe the attributes of the decision object, The evaluation information S t_t( 1, 2, , )T in the second dimension identification framework is used to reflect the attribute features of the decision-maker's knowledge evidence. The language evaluation information formed by two dimensions (H S_n, _t) is called two-dimension semantic evaluation information.

Definition 6 Suppose the two-dimensionality semantic recognition framework constructed by the

information source r (r1, 2, R) is { ₁, , }

r

r r r

N

H  H H and { ₁, ₂, , }

r

r r r r

T

S  S S S respectively,

Where the I dimension framework is the language set Hr for judging something, The II dimension framework Sr is the language set for quality evaluation of the evaluation information of the I dimension. There are differences in the number and semantics of language evaluation phrases in the recognition framework. The semantic information of the two dimensional language (H_nr,S_tr) is the semantic information of two dimensions of different frameworks.

Two - Dimensional Reasoning Decision Method with Different Frame

Semantic Representation of Two - Dimensional Meaning

Suppose { ₁, ₂, , }

r

r r r r

N

H  H H H and { ₁, ₂, , }

r

r r r r

T

S  S S S respectively is the I and II language evaluation framework given by the r th (r1, 2, R) information source, Er {(H_nr,S_tr) ,

(n1, 2, ,N_r; t1, 2, ,T_r)}is two language evaluation information for the r th information source, where the I dimension frameworkHr is a set of comments on the judgment of a certain thing, the II dimension framework r

S is the fuzzy language set for quality evaluation of the evaluation

information of the Ith dimension, Let { , r( ) [0,1]}

t

r

t S

S  x  x x ,where r( )

t

S x

 is a trig membership

Step1: Using the center of gravity of fuzzy Numbers, The II dimension evaluation information (reliability) is quantified, so reliability r

t

 is given as follows:

( )

( )

r t

r t

S r

t

S

x x dx

x dx

 









, (1)

where r t

 reflects the r n

H ’s credibility (reliability) of the evaluation object, and0_tr 1.

Step2: The II dimension evaluation information is used to weight the evidence body

As mentioned above, the evaluation information of the II dimension reflects the reliability of information, and the more reliable the evidence, the greater its weight in decision-making. For this purpose, the following weight acquisition methods are defined:

1

r t

r R _r

t r

w 



 



(2)

where w_ris therth(r1, 2, R) weight of evidence body.

Step3: The semantic information of the two dimensional languages is represented as evidence body with weight.

The information of the II dimension reflects the reliability, uncertainty and incompleteness of the I information. Therefore, the evidence body can reflect the evaluation information of the meaning of the two dimensional language more accurately with the weight of the evidence body.

According to the reliability r t

 of the II dimension evaluation information in step 1, the two dimensional semantic information is transformed into evidence body.

(Hnr,Str){( , ), ( ,1 ); }

r r r r

n t t r

H  H  w (3)

where(3)equation denotes the reliability that the rth(r1, 2, R)information source evaluates the evaluation object as r

n

H as r t

 , the reliability of which rating grade is 1 r t



 cannot be ascertained,the weight of this information in Rdecision information isw_r.

Consistent Transformation of Information in Two Decisional Languages under Different Frameworks

As the meaning of two decisions of different frames is essentially evidence bodies on multiple parallel frames, in order to facilitate the fusion of information on different identification frames, information on different identification frames is generally required to be converted to the same identification frame. Therefore, it is necessary to define the equivalent concept of different identification frames.

Identify the Equivalent Connotation of the Framework

Definition7 Set the two identification frames as { ,_n n1, ,N₁}and  { ,_n n1, ,N₂}, If

n



 , there's a unique _n equal to that, calls_n _n, On the other hand, if there is a unique

n

  for  _n , The identification frame  and  are equivalent, denoted as  .

Definition8 If _1,n (n 1, 2, ,N₁) , has _1,n {( _l, _2,l),l1, 2, ,N₂} , in reverse,if

1,n (n 1, 2, ,N2)



   ,has _1,n {( k, _1,k),k1, 2, ,N₁},the identification framework  and

 are denoted as

P  .

Different Identification Frame Information Consistency Transformation Methods

Set { ₁, ₂, , }

r

r r r r

N

H  H H H (r1, 2, R) as R parallel language recognition framework,

satisfying _{1 2}

r

r r r

N

H H H The utility of Hr element ₁, ₂, ,

r

r r r

N

H H H is ₁, ₂, ,

r

r r r

N

h h h satisfied

1 2

0 1

r

r r r

N

h h h

     . Suppose the basic probability distribution m_r on frame Hris:

{( r, r), 1, 2, , }

r n n r

m  H  n N , r1, 2, R (4)

Let H{H₁,H₂, ,H_N}be the public identification framework, H₁ H₂ H_N, where the

utility of N elements is h h₁, ₂, ,h_N, satisfying 0h₁ h₂  h_N 1. The public identification framework can be selected from the original R identification frameworks or reconstructed. In order to gather information of multiple recognition frameworks Hr (r1, 2, R), information on different recognition frameworks should be transformed to public recognition frameworks. The specific steps of consistency transformation are as follows:

Step 1 Converts the elements on the recognition framework r

H (r1, 2, R) to the common recognition framework.

Because for any r r n

H H , its utility r n

h satisfies 0h_nr 1, The utility of any element in H_kH

in the public identification framework also satisfies 0hk 1. Therefore, for any element

r r

n

H H

in frame r

H , there must be two adjacent elements H_k,H_k₁H in common frame H , so that

1

[ , ]

r

n k k

h  h h . Then the element HnrHr is equivalent to the set {( , , ), ( 1, 1, )}

r r

k k n k k n

H  H _  _ in

probability, that is:

, 1 1,

{( , ), ( , )}

r r r

n k k n k k n

H H  H _  _ (5)

where 1 ,

1

,

r

r k n

k n k k h h h h       1, 1 r

r n k

k n k k h h h h  _   

 ,obvious , 1, 1

r r

k n k n

  _  . when r

n k

h h , have_{k n}r_, 1 ,

1, 0

r

k n

   ;when 1

r

n k

h h ,have , 0

r k n

  ,_kr_1,n 1.

Step 2 Maps the focal element of Hrequally to the common identification framework The basic probability distribution on frame r

H is (H_nr,_nr) inm_r {(H_nr,_nr),n1, 2, ,N_r}, Then, by Hnr {(Hk,k), (Hk₁,k₁)}, the evaluation ( , )

r r

n n

H  can be transformed into the evaluation information in the public framework that is:

, 1 1,

(H_nr,_nr) {(H_k,_{k n}r ), (H_k,_kr_{ n})} (6)

where _{k n}r_,  _k _nr,_kr_1,n _k1_nr,obviously_{k n}r_, _kr_1,n _nr.

Step 3 Merges the reliability of the same focal element in the common recognition framework After transforming each (Hnr,nr) in {( , ), 1, 2, , }

r r

r n n r

m  H  n N into the information on the public identification frame, and combining the reliability on the same element (focal element), the basic probability distribution M_r (m_r M_r) equivalent to m_r on the public identification frame can be obtained. that is:

{(Hnr,nr),n1, 2, ,Nr} {( , ), 1, 2, , }

r

k k

H  k  N (7)

where _,

1

r

N

r r

k k n

n

 







,Mr {(Hk,kr),k1, 2, ,N}is identifying the basic probability distribution mr

in frame r

H into the information on common frame H. By repeating the above steps, multiple parallel frames r

Two - Dimensional Semantic Information Aggregation Methods for Several Different Frameworks

According to the semantic representation of two-dimensionality semantic information in section 3.1, two-dimensionality semantics of different frameworks are represented as evidence body

{( , ), ( ,1 ); }

r r r r r

n t t r

E  H  H  w on different recognition frameworks with weights, Use the method in section 3.2 to transform the evidence body on different recognition to the same recognition frame, then the two-dimensionality semantic information of multiple different frames is converted to {(H_k,_kr),k 1, 2, ,N} ( r1 2, , ,R ). Where {H ,k_k 1 2, , ,N} is the public identification framework, r

k

 rated the evaluation object as H_k reliability for the r (r1, 2, R) source. Then, on the framework of public recognition, multiple evidential information can be gathered using the evidential reasoning method [9].

Let r( )

n i

m a indicate the degree to which the information source r( )

i

E a supports the alternative

plan a_i and is evaluated as H_n, which is called basic credibility. mHr ( )ai indicate unassigned

trustworthiness, That is, information source r( )

i

E a supports the remaining reliability after

alternative plan a_i is evaluated as level H H₁, ₂, ,H_N. It meets:

( ) ( ), 1, 2, ,

r r

n i r n i

m a w a n N (8)

1 1

( ) 1 ( ) 1 ( )

N N

r r r

H i n i r n i

n n

m a m a w  a

 

 



 



(9)

Decompose r H

m into two parts, r H

m and r H

m :

( )

r

H i

m a r ( ) r ( )

H i H i

m a m a

  (10)

where

( )

r

H i

m a  1 w_r,

1

(1 )

( ) ( )

N r

r n

r

H i i

n

m a w  a



 



(11)

The basic reliability allocation of R information sources can be fused by the following algorithm:

1 1

{ }: ( ) [ ( ( ) ( )) ( )]

R R

r r r

n n i n i H i H i

r r

H  a k m a m a m a

 





 



_{, n1, 2, ,N (12)}

1 1

{ }: ( ) [ ]

R R

r r

H i H H

r r

H  a k m m

 









₍₁₃₎

where

1

1 1 1 1

[ ( ( ) ( )) ( 1) ( ) ( )]

R R R

N

r r r r

n i H i H i H i

n r r r

k m a m a N m a m a 

   





  







(14)

In the formula, _n( )a_i and _H( )a_i respectively represent the total reliability after integrating R information sources. After synthesis is still a confidence vector: S a( )i {(Hn,n( )),ai n1, 2, , }N .

Analysis

first 3 experts give the evaluation framework of I and II dimensions respectively, 1

H and 1

S , the last 2 experts gives the evaluation framework of I and II dimensions respectively 2

H and 2

S , as follows:

1

H  { 1 1 H , 1

2 H , 1

3 H , 1

4

H }={ best, good, moderate, worst}, H2 { 2 1 H , 2

2 H , 2

3 H , 2

4 H , 2

5 H }

[image:6.595.73.522.190.281.2]

={optimal, best, good, moderate, worst}, S1 S2 {S₁ ,S₂,S₃ } ={Familiar, more familiar, part familiar }. The two dimensional semantic evaluation information of 4 scientific and technological projects by 5 experts were:

Table 1. Two-dimensional semantic evaluation information on the project.

project expert E1 expert E2 expert E3 expert E4 expert E5

A1 ₍ 1

2

H ,S₂) (H₁1,S₁ ) (H₂1,S₁ ) (H₂2,S₁ ) (H₁2,S₂)

A2 ₍ 1

3

H ,S₃) (H₃1,S₂) (H₁1,S₁ ) (H₂2,S₁ ) (H₁2,S₂)

A3 ₍ 1

1

H ,S₃) (H1₂,S₂) (H₂1,S₃ ) (H₃2,S₁ ) (H₂2,S₂)

A4 ₍ 1

1

H ,S₂) (H1₂,S₂) (H₂1,S₂) (H₃2,S₂) (H₂2,S₃)

Suppose the utility of the evaluation scale of dimension I language is: 1 1

( ) 0.95

u H  , 1 2

( ) 0.7 u H  ,

1 3

( ) 0.45

u H  , u H( 1₄)0.2; u H( ₂1) 1 , u H( ₂2)0.8, u H( ₃2)0.6, u H( ₄2)0.4, u H( ₅2)0.2. The second dimension language scale S₁ ,S₂ ,S₃ is defined as triangular fuzzy number, and its

subordinate functions are: S₁(0.7, 0.9,1) ,S₂ (0.5, 0.7, 0.9), S₃ (0.3, 0.5, 0.7) . According to equation (2), the quantized values of the scale reliability of dimension II language can be obtained as:

1 0.867

  ,₂ 0.7,₃ 0.5. According to equation (3), the semantic information of each two dimensions can be weighted. Taking project A1 as an example, the quantitative reliability of 5 experts is respectively 1 1

1 0.7, 2 0.867

    , 1 1 1

3 0.867, 4 0.867, 5 0.7

      from the 2d evaluation information. Then the weights of each expert on project A1 are respectively 1

1 0.175 w  , 1

2 0.217 w  ,

1 1 1

3 0.217, 4 0.217, 5 0.175

w  w  w  . Similarly, the weight of 5 experts on other items can be calculated.Then, from equation (4), the above two-dimensionality semantic evaluation information can be transformed into evidence with weight information, for example( 1

2

H ,S₂ ){( 1 2 H ,0.7), ( 1

H ,0.3); 1

1 0.175

w  }. Similarly, other two dimensional semantic evaluation information can be transformed.

Because of two identification frames H1 and H2 differences, in order to facilitate the aggregation of multiple evidences, the evidences in different identification frames need to be converted to the same identification frame. Here, choose H H2 {H₁ ,H₂,H₃,H₄,H₅} ={ optimal, best, good, moderate, worst } as the public identification framework, Then, the above evidence body is transformed into the evidence body on the public identification framework by the method in section 3.2, and the evaluation of 5 experts in each project is integrated by using equation (8) - (14) of the evidence reasoning method. The results are shown in table 2.

Table 2. Comprehensive evaluation of 4 projects.

project H1 H2 H3 H4 H5 H Expected utility interval A1 0.2609 0.4292 0.1496 0 0 0.1602 [0.73,0.85] A2 0.3133 0.2815 0.0477 0.1468 0 0.2107 [0.67,0.84] A3 0.0514 0.2985 0.3902 0 0 0.2599 [0.58,0.78] A4 0.1107 0.2836 0.3285 0 0 0.2772 [0.59,0.81]

[image:6.595.89.506.660.736.2]

If the second dimension language is not taken into account, the utility of the four projects is:

1 2 3

( ) 0.82, ( ) ( )

u A  u A u A = (u A₄) 0.72, cannot distinguish the advantages and disadvantages of the three projects of A2, A3 and A4. However, the traditional two-dimensional semantic approach cannot integrate the semantic information on different frames.

Concluding

Two-dimensionality semantic information has unique advantages in representing different quality decision information. This paper extends the classic two-dimensionality semantic information and proposes a two-dimensionality semantic decision method based on evidential reasoning. This paper mainly does the following work:

(1) The methods of identifying the equivalent of different identification frames and the transformation of evidence body on the recognition frames and the transformation of semantic consistency of two dimensions of different granularity are given.

(2) By using the fuzzy membership degree of dimension II language, a semantic representation method of two dimensions is presented, and the semantic representation of two dimensions is represented as evidence body with weight, which objectively reflects the connotation of the semantic semantics of two dimensions.

Acknowledgement

Thanks for the guidance and Suggestions from the evaluation experts, and thanks for the support from the natural science foundation of zhejiang province (LQ15G010006) and the talent project of zhejiang a&f university (2015FR001, 2015FR011).

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