International Journal of Modern Research in Engineering & Management (IJMREM) ||Volume|| 2 ||Issue|| 9 ||Pages|| 48-57 || September 2019 || ISSN: 2581-4540
Decision-Making Method Based on Picture Fuzzy Sets and Its Application in College Scholarship Evaluation 1,
Yong Wei Yang, 2,Yang Yang He
School of Mathematics and Statistics, Anyang Normal University, Anyang, China
--------------------------------------------------ABSTRACT-------------------------------------------------------In view of the bad competition caused by the evaluation scope of college students' scholarship selection, we first constructs an objective and reasonable scholarship evaluation index system and index weight. Then we constructs the distance measure based on the membership functions of the picture fuzzy set, and shows the advantages of the distance measure proposed in the paper through the comparative analysis of examples. Finally, a multi-attribute decision making method is proposed by combining the distance measure of picture fuzzy sets with TOPSIS, and it is applied to the evaluation of college students' scholarships.
KEYWORDS: Picture fuzzy sets, College scholarship evaluation, AHP, Decision making ----------------------------------------------------------------------------------------------------------------------------- ---------Date of Submission: Date, 20 September 2019 Date of Publication: 30 September 2019 ----------------------------------------------------------------------------------------------------------------------------- ----------
I. INTRODUCTION The establishment of scholarships can not only stimulate the enthusiasm of college students, but also promote their comprehensive development and improve their overall quality. Therefore, a scientific scholarship evaluation program has important practical value and guiding significance. Regarding the evaluation of scholarships, Ma [1] used the C4.5 decision tree algorithm to construct a scholarship evaluation system, but he only considered the impact of academic performance, sports performance and moral education score on the evaluation of scholarships, which did not meet the actual situation of most universities. Pei [2] uses Analytic hierarchy process (AHP) method to construct multi-index system to analyze the evaluation method of scholarship in colleges and universities, but according to the actual situation of each university, the quantitative standard of index can not be widely used. Li et al. [3] mainly used the multi-index comprehensive model to refine the evaluation criteria from the three aspects of moral, intellectual and physical to nine indicators, and used the analytic hierarchy process to solve the model for solution, and obtained the more perfect scholarship assessment program. However, in the existing literature, there are few studies on the evaluation of college scholarships by using uncertainty theories such as fuzzy sets. The evaluation indicators of college scholarships include qualitative indicators and uncertain indicators. It is necessary to quantify the qualitative index and deal with the uncertain index in the process of evaluation. In this process, the influence of human factors must be avoided, and fairness and justice should be guaranteed at the same time. Therefore, it is particularly important to establish a feasible assessment method. Recently, Cuong [4,5] proposed picture fuzzy set (PFS) and investigated the some basic operations and properties of PFS. The picture fuzzy set is characterized by three functions expressing the degree of membership, the degree of neutral membership and the degree of nonmembership. The only constraint is that the sum of the three degrees must not exceed 1. Basically, PFS based models can be applied to situations requiring human opinions involving more answers of types: yes, abstain, no, refusal, which can’t be accurately expressed in the traditional intuitionistic fuzzy set. When it comes to the selection of scholarships under the background of voting, the picture fuzzy set can give full play to its advantages. Thus, keeping inspiration from the fact that PFSs have the great powerful ability to model the imprecise and ambiguous information in real-world applications, the main purpose of this paper is to give a gives a reasonable and objective decision-making method of college scholarships, so that the students with real learning can get the scholarship.
II. INDEX WEIGHTS FOR SCHOLARSHIPS BASED ON AHP Combined with the current scholarship evaluation system in most colleges and universities, it is of good positive significance for the selection of college students to construct a scientific and reasonable scholarship evaluation index system. The Construction of College Scholarship Evaluation Index System: Based on the principles of science, feasibility and rationality, this paper intends to construct an objective and reasonable college scholarship evaluation index system which is shown in Table 1:
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Decision-Making Method Based on Picture Fuzzy‌ Table 1. College scholarship evaluation index system
First level index
Secondary index Compliance with discipline C11
Ideological and moral C1
Teacher-student relationship C12 Classmate relationship C13 English and computer scores C21 Professional course scores C22
Academic achievement C2
Public course scores C23 Physical fitness test C31 Physical and mental quality C3
Sports skills C32 Mental health C33 Number of papers published C41
Innovation ability C4
Competition award C42 Scientific research project C43 Activity participation C51
Practical activity C5
Organizational management ability C52 Special recognition award C53
The AHP Model of College Scholarships Establishing the hierarchical structure diagram of college scholarships: After analyzing the evaluation object by hierarchical structure model, the factors are decomposed into several levels from top to bottom. The factors of the same layer have an influence on the upper factors, and at the same time, they are affected by the lower layer. The top layer is the target layer, usually there is only one factor, the bottom layer is usually the factor layer, and there can be one or several levels in the middle, usually they are called to be the criterion layer. In this paper, combined with the college scholarship evaluation index system, according to the AHP principle, the system is hierarchical, and the the hierarchical structure diagram of college scholarship is established as shown in Fig. 1.
Fig.1 Hierarchical structure diagram of college scholarships Construct the pairwise comparison matrix: Filling the pairwise comparison matrix to fill the pairwise comparison matrix by using numbers to represent the relative importance of one element to the other element is in the form of a scale of 1 to 9. The scale was shortly defined and explains the value of 1 to 9 for consideration in pairwise comparisons elements on each level hierarchy against a criterion on a higher level. If an element in metric and compared with itself, it is given a value of 1. If i compared to j get a certain value, then j than i was the opposite. In Table 2 provides definitions and explanations of quantitative scale from 1 to 9 to assess the level of interest of an element with other elements.
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Decision-Making Method Based on Picture Fuzzy… Table 2. Numerical rating in AHP The intensity of interest 1
Definition
Explanation
The two elements have the same influence greatly to goal Elements that one a little more Experience and judgment slightly favor one important than the other elements element other than the elements Elements which one is more important Very strong experience and judgment in than any other element favor of the elements other than the elements One element absolutely clearly more One strong element in favor and dominant important than any other element look in practice Evidence in support of one element against One absolutely essential element than another element has the highest possible other elements degree of confirmation strengthens Values between two values adjacent This value is given when there are two consideration compromises between If the first activity in the appeal options eal activities got the numbers j, then j has its inverse value Compared with i . Both elements are equally important
3 5 7 9 2,4,6,8 Inverse
Measuring consistency in the decision makers, it is important to know how well the consistency is there, because we do not want decisions based on consideration with a low consistency. Due to the low consistency, consideration would appear to be something random and inaccurate. Consistency is important to obtain valid results in the real world. AHP used to measure consideration consistency with consistency ratio. Defifinition 2.1 (Consistency ratio) for a pairwise comparison matrix A , assume
max
represents the largest
CI is defined as: −n CI = max . n −1 Based on the CI , random consistency ratio ( CR ) is defined as: CI CR = , RI where RI is the random consistency index related to the size of matrix, the concrete number of RI is listed in eigen value of the matrix, the consistency index
the Table 3. Table 3. Random consistency index
n
1
2
3
4
5
6
7
8
9
10
RI
0
0
0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.49
If CR 0.1 , the constructed pairwise comparison matrix is regarded rational and the alternatives’ weight is obtained through the following definition. Otherwise the pairwise comparison matrix should be reconstructed. Determine the indicator weight set : In the following, the comparison matrix corresponding to each level is constructed, and the corresponding hierarchical single ranking weights are calculated. The comparison matrix C corresponding to the first level index of college scholarship evaluation index system is:
C1
C1 1
C2 2
C3 1
C4 3
C5 2
C2
1/2
1
1/2
3
2
C3
1
2
1
2
2
C4
1/3
1/3
1/2
1
1/2
C5
1/2
1/2
1/2
2
1
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Decision-Making Method Based on Picture Fuzzy… Weights
0.2951 0.2004 0.2776 0.0899 0.1368 λmax=5.1404, CI= 0.0351, CR=0.0313＜0.1, which meets the requirements for consistency.
Note:
Therefore, the weight of the first level index of college scholarship evaluation index system is:
w = ( 0.2951, 0.2004, 0.2776, 0.0899, 0.1368 ) .
The comparison matrix B1 corresponding to the ideological and moral index is: C11
C12
C13
C11
1
4
3
C12
1/4
1
1/2
C13
1/3
2
1
Weights
0.6250 0.1365 0.2385 λmax=3.0183, CI= 0.0091, CR=0.0158＜0.1, which meets the requirements for consistency.
Note:
w1 = ( 0.6250, 0.1365, 0.2385 ) .
Therefore, the weight of the secondary index of deological and moral is:
The comparison matrix B2 corresponding to the academic achievement index is: C21
C22
C23
C21
1
1/2
2
C22
2
1
2
C23
1/2
1/2
1
Weights Note:
0.3108 0.4934 0.1958 λmax=3.0536, CI= 0.0268, CR=0.0462＜0.1, which meets the requirements for consistency.
Therefore, the weight of the secondary index of academic achievement is:
w2 = ( 0.3108, 0.4934, 0.1958 ) .
The comparison matrix B3 corresponding to the physical and mental quality index is: C31
C32
C33
C31
1
3
1/4
C32
1/3
1
1/5
C33
4
5
1
Weights Note:
0.2255 0.1007 0.6738 λmax=3.0858, CI= 0.0429, CR=0.0739＜0.1, which meets the requirements for consistency.
Therefore, the weight of the secondary index of deological and moral is:
w3 = ( 0.2255, 0.1007, 0.6738) .
The comparison matrix B4 corresponding to the innovation ability index is: C41
C42
C43
C41
1
2
1/3
C42
1/2
1
1/4
C43
3
4
1
Weights Note:
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0.2385 0.1365 0.6250 λmax=3.0183, CI= 0.0091, CR=0.0158＜0.1, which meets the requirements for consistency.
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Decision-Making Method Based on Picture Fuzzy… Therefore, the weight of the secondary index of innovation ability is:
w4 = ( 0.2385, 0.1365, 0.6250 ) .
The comparison matrix B5 corresponding to the practical activity index is: C51
C52
C53
C51
1
3
1/4
C52
1/3
1
1/6
C53
4
6
1
Weights
0.2176 0.0914 0.6910 λmax=3.0536, CI= 0.0268, CR=0.0462＜0.1, which meets the requirements for consistency.
Note:
Therefore, the weight of the secondary index of practical activity is:
w5 = ( 0.2176, 0.0914, 0.6910 ) .
According to the above analytic results, the weight distribution of college scholarship evaluation index system can be obtained, as shown in the following Table 4: Table 4. Weight distribution of college scholarship evaluation index system Target layers
Weights
College scholarship
Primary indicators
Weights
C1
0.2951
C2
0.2004
C3
0.2776
Secondary indexes
Weights
Total weights
C11
0.6250
0.1844
C12 C13 C21
0.1365 0.1958 0.3108
0.0403 0.0704 0.0623
C22 C23 C31 C32 C33
0.4934 0.1958 0.2255 0.1007 0.6738
0.0989 0.0392 0.0626 0.0280 0.1870
C41
0.2385
0.0214
C42
0.1365
0.0123
C43
0.6250
0.0562
C51
0.2176
0.0298
C52
0.0914
0.0125
C53
0.6910
0.0945
1
C4
0.0899
C5
0.1368
III. PROVINCE DISTANCE MEASURE IN THE PICTURE FUZZY ENVIRONMENT Definition 3.1[5] Let X be a universe of discourse. A picture fuzzy set (PFS) A on the universe X is an object of the form A = x, A ( x), A ( x), A ( x) | x X , Where
A ( x) ( [0,1])
is called the “degree of positive membership of A ”, A ( x) ( [0,1]) is called the
“degree of neutral membership of A ” and
A ,A , A
A ( x) ( [0,1])
is called the “degree of negative membership ”, and
satisfy the following condition:
A ( x)+ A ( x)+ A ( x) 1 , x X .
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Decision-Making Method Based on Picture Fuzzy… Then for x X , A ( x)=1 − ( A ( x)+ A ( x)+ A ( x)) could be called the degree of refusal membership of x in A. Then for any x X , A ( x)=1 − ( A ( x)+ A ( x)+ A ( x)) could be called the degree of refusal membership of x in A . In particularly, if X only has one element, then A( x) = A ( x), A ( x), A ( x) is called a picture fuzzy number (PFN). For convenience, a PFN is denoted as A = A , A , A . Basically, picture fuzzy sets based models may be adequate in situations when decision makers face their opinions involving their decision making as follows: yes, abstain, no, and refusal. The voting results are divided into four groups accompanied with the number of voters that are “voters for”, “abstain”, “vote against”, and “refusal of the voting”. Group “abstain” means that the voting paper is a white paper rejecting both “agree” and “disagree” for the candidate but still takes the vote. Group “refusal of voting” is either invalid voting papers or did not take the vote [6]. Definition 3.2[4,5] Given two PFEs represented by A and B on universe X , the inclusion, union, intersection and complement operations are defined as follows: (1) A B = { x, min{ A ( x), B ( x)}, min{ A ( x), B ( x)}, max{ A ( x), B ( x)} | x X }; (2)
A B = { x, max{ A ( x), B ( x)}, min{ A ( x), B ( x)}, min{ A ( x), B ( x)} | x X };
A B x X , A ( x) B ( x), A ( x) B ( x), A ( x) B ( x) ; (4) A = B A B and B A . (3)
Definition 3.3 [7] Let = , , a picture fuzzy number (PFN), then a score function S and an accuracy function H of a picture fuzzy number can be represented as follows:
S ( ) = − ,
S ( ) [−1,1] ,
H ( ) = + + , H ( ) [0,1] . We also use voting as a good example to explain the above definition, where S ( ) = − represents goal difference and H ( ) = + + can be interpreted as the effective degree of voting. When S ( ) increases, we can know that there are more people who vote for and people who vote against α become less. When H ( ) increases, we can know that there are more people who vote for or against α and people who refuse to vote become less. So, H ( ) depicts the effective degree of voting. Definition 3.4[5] Let A, B be two picture fuzzy sets on distance
X = {x1 , x2 ,
, xn } . The normalized Euclidean
d e ( A, B ) is defined as follows:
de ( A, B) =
1 n ( A ( xi ) − B ( xi )) 2 + ( A ( xi ) − B ( xi )) 2 + ( A ( xi ) − B ( xi )) 2 . n i =1
Note 3.5 The normalized Euclidean distance
d e ( A, B ) only consider the differences between two picture fuzzy
sets A and B from the aspects of positive membership degree, neutral membership degree and negative membership degree. However, the difference between A and B ignores the refusal membership degree. Therefore, the distance measure formula between between two picture fuzzy sets A and B should fully consider the combined effects of the four components of positive membership degree, neutral membership degree, negative membership degree and refusal membership degree. We present the improved Euclidean distance measure formula as follows. Definition 3.6 Let A, B be two picture fuzzy sets in
X = {x1 , x2 ,
, xn } . The distance between A and B is
defined as follows:
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Decision-Making Method Based on Picture Fuzzy… 1
( A ( x j ) − B ( x j )) 2 + ( A ( x j ) − B ( x j )) 2 + ( A ( x j ) − B ( x j )) 2 2 1 n d h ( A, B) = S A ( x j ) − SB ( x j ) 2 . 2 3 n ) i =1 +( A ( x j ) − B ( x j )) + ( 2 In addition, form the above formula (M1)
(M1)
( A ( xi ) − B ( xi )) 2 = (1 − ( A ( xi ) + A ( xi ) + A ( xi )) ) − (1 − ( B ( xi ) + B ( xi ) + B ( xi )) ) = ( H A ( xi ) − H B ( xi )) 2 2
it can be seen that the difference between refusal membership degrees implies the difference of accuracy functions.
X = {x} , A = { x, 0.5, 0.2, 0.1 } , C = { x, 0.1, 0.2, 0.3 } be picture fuzzy sets on X . Then de ( A, B) = 0.3 = d e ( B, C ) ,
Example
3.7
Let
B = { x, 0.3, 0.4, 0.2 }
and
d h ( A, B) = 0.2517 d h ( B, C ) = 0.3 . From the Euclidean distance, it is easy to see that the distance between A and B is equal to that between B and C , that is, the degree of difference between A and B is equal to the degree of difference between B and C . If it is regarded as B an ideal solution in decision making theory, the advantages and disadvantages of the schemes A and C cannot be given. Using the improved distance formula, we can see that the scheme is A closer to the ideal scheme B , so we can see that the scheme A is better than the scheme C . In the multi-attribute decision making process, the importance of the attributes considered is often different, and different weights need to be assigned. Therefore, the weight of elements should be considered. In the following, the weight distance measure of picture fuzzy sets is given.
X = {x1 , x2 , , xn } , and W = ( w1 , w2 , weight vector. The weighted distance between A and B is defined as follows: Definition 3.8 Let A, B be two picture fuzzy sets on
, wn ) be the 1
( A ( x j ) − B ( x j )) 2 + ( A ( x j ) − B ( x j )) 2 + ( A ( x j ) − B ( x j )) 2 2 1 n d w ( A, B) = w j S A ( x j ) − SB ( x j ) 2 2 3 n + ( A ( x j ) − B ( x j )) + ( ) j =1 2
(M2)
IV. A PICTURE FUZZY DECISION-MAKING METHOD BASED ON DISTANCE AND TOPSIS Based the weighted distance measure, in this section, we shall propose the model for multiple attribute decision making with picture fuzzy information. Let
G = {G1 , G2 ,
A = { A1 , A2 ,
, Am }
, Gn } be the set of attributes, w = {w1 , w2 ,
G j ( j = 1, 2,
, n) , where w j [0,1] ,
n
w j =1
j
be a discrete set of alternatives, and
, wn }
is the weighting vector of the attribute
= 1.
In the following, we apply the weighted distance measure to the MADM problems with hesitant fuzzy information. Step1: Obtain the picture fuzzy decision matrix
ij
D = (dij )mn , where dij = ij ,ij , ij is a PFN, where
indicates the degree of positive membership that the alternative
decision maker, attribute
ij
G j , ij
Ai
satisfies the attribute
indicates the degree of neutral membership that the alternative
indicates the degree that the alternative
decision maker. Step2: If all the attributes
G j ( j = 1, 2,
, n)
Ai
Ai
Gj
given by the
doesn’t satisfy the
doesn’t satisfy the attribute
Gj
given by the
are of the same type, then the attribute values do not need
normalization. Whereas, there are generally benefit attributes (the bigger the attribute values the better) and cost attributes (the smaller the attribute values the better) in MADM. In such cases, we may transform the attribute
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Decision-Making Method Based on Picture Fuzzy… values of cost type into the attribute values of benefit type, then the picture fuzzy decision matrix can be transformed into the picture fuzzy decision matrices
Where
D = (dij )mn
R = (rij )mn , where
dij , for benefit attribute G j , rij = c d ij , for cost attribute G j , d ijc = ij ,ij , ij c = ij ,ij , ij . A+ = (r1+ , r2+ ,
Step3: Determine the positive ideal solution
A− = (r1− , r2− ,
, rn+ ) and the negative ideal solution
, rn− ) , where rj+ = +j , +j , +j = max ij , min ij , min ij , i
i
i
r = , , = min ij , min ij , max ij . − j
− j
− j
− j
i
i
+
Step4: Calculate the weighted distance dw ( Ai , A ) between each
i
alternative
Ai
and the positive ideal
+
A , and the weighted distance dw ( Ai , A− ) between each alternative Ai and the negative ideal − solution A , where solution
1
( ij − +j ) 2 + (ij − +j ) 2 + ( ij − +j ) 2 2 n 1 + d w ( Ai , A+ ) = w j S − S j j + 2 2 + ( ij − j ) + ( ) 3n j =1 2
(M3)
1
( ij − −j ) 2 + (ij − −j ) 2 + ( ij − −j ) 2 2 1 n − d w ( Ai , A ) = w j Sij − S −j 2 − 2 3 n + ( ij − j ) + ( ) j =1 2
Obviously, the smaller the value dw ( Ai , A+ ) , the closer
Ai
to
(M4)
A+ , and thus the closer the alternative Ai to the
+
Ai . The greater the value dw ( Ai , A− ) , The Ai from the negative ideal solution A− , the better the alternative Ai . According to the idea of
positive ideal solution A , that’s to say, the better the alternative farther away
TOPSIS method: under the condition that the positive and negative ideal solutions and index weights are known, the evaluation objects in the evaluation set are sorted, and the optimal scheme which is as close to the positive ideal solution as possible and far from the negative ideal solution is selected. Hence, we construct the closeness degree formula. Step5: Calculate the closeness degree C ( Ai ) of each scheme
C ( Ai ) = Step6: Rank all the alternatives
C ( Ai ) (i = 1, 2,
Ai (i = 1, 2,
Ai (i = 1, 2,
, m) , where
d w ( Ai , A− ) d w ( Ai , A+ ) + d w ( Ai , A− )
, m)
(i = 1, 2,
, m) . (M5)
and select the best one(s) in accordance with
, m) . V. NUMERICAL EXAMPLE
In this section, we utilize a practical multiple attribute decision making problems to illustrate the application of the developed approaches. Suppose that there are 5 excellent students in a university who submitted scholarship applications to the university. The five possible students A1-A5 are to be evaluated using the picture fuzzy numbers by the decision makers under the attributes in the college scholarship evaluation index system (whose weighting
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Decision-Making Method Based on Picture Fuzzy…
0.1844, 0.0403, 0.0704, 0.0623, 0.0989, vector is w = 0.0392, 0.0626, 0.0280, 0.1870, 0.0214, ), and construct the following matrix 0.0123, 0.0562, 0.0298, 0.0125, 0.0945 D = (dij )mn is shown in Table 5. In the following, in order to select the most desirable students, we utilize the weighted distance measure to develop an approach to multiple attribute decision making problems with picture fuzzy information, which can be described as following. Table 5. The picture fuzzy decision matrix A1
A2
A3
A4
A5
C11
<0.53,0.33,0.09>
<0.73,0.12,0.0>
<0.91,0.03,0.020>
<0.85,0.09,0.0>
<0.90,0.05,0.0>
C12 C13 C21
<0.89,0.08,0.03> <0.42,0.35,0.18> <0.08,0.89,0.02>
<0.13,0.64,0.2> <0.03,0.82,0.13> <0.73,0.15,0.08>
<0.07,0.09,0.05> <0.04,0.85,0.10> <0.68,0.26,0.06>
<0.74,0.16,0.1> <0.02,0.89,0.05> <0.08,0.84,0.06>
<0.68,0.08,0.21> <0.05,0.87,0.06> <0.13,0.75,0.09>
C22 C23 C31 C32 C33
<0.62,0.13,0.21> <0.43,0.42,0.10> <0.28,0.25,0.39> <0.61,0.25,0.13> <0.9,0.05,0.02>
<0.52,0.34,0.13> <0.62,0.30,0.07> <0.65,0.22,0.11> <0.12,0.21,0.60> <0.77,0.13,0.1>
<0.59,0.11,0.3> <0.55,0.23,0.21> <0.42,0.15,0.38> <0.12,0.31,0.50> <0.80,0.15,0.02>
<0.27,0.32,0.41> <0.69,0.11,0.20> <0.49, 0.15,0.34> <0.20,0.30,0.20> <0.15,0.73,0.08>
<0.44,0.34,0.22> <0.63,0.14,0.25> <0.60,0.15,0.21> <0.60,0.12,0.20> <0.62,0.15,0.2>
C41
<0.68,0.08,0.21>
<0.62,0.24,0.11>
<0.68, 0.18, 0.05>
<0.61,0.25,0.10>
<0.72,0.1,0.15>
C42
<0.05,0.87,0.06>
<0.10,0.75,0.10>
<0.05,0.87,0.06>
<0.91,0.03,0.05>
<0.52,0.23,0.19>
C43
<0.13,0.75,0.09>
<0.64,0.16,0.10>
<0.12,0.65,0.20>
<0.28,0.44,0.16>
<0.72,0.08,0.17>
C51
<0.43,0.23,0.31>
<0.28,0.32,0.18>
<0.23,0.55,0.10>
<0.27,0.32,0.16>
<0.62,0.14,0.21>
C52
<0.14,0.62,0.18>
<0.71,0.13,0.12>
<0.66,0.12,0.15>
<0.63,0.22,0.12>
<0.41,0.13,0.35>
C53
<0.36,0.27,0.31>
<0.14,0.21,0.63>
<0.58,0.14,0.21>
<0.52,0.28,0.2>
<0.64,0.12,0.2>
Step1: the picture fuzzy decision matrix Step2: Since all the attributes
R = (rij )mn = (dij )mn .
D = (dij )mn
is shown as Table 5.
are benefit criteria, so we get the normalized matrix
Step3: Determine the positive ideal solution
A+ and the negative ideal solution A− , where
0.91, 0.03, 0 , 0.89, 0.08, 0.03 , 0.42, 0.35, 0.05 , 0.73, 0.15, 0.02 , 0.62, 0.11, 0.13 , 0.69, 0.11, 0.07 , + A = 0.65, 0.15, 0.11 , 0.61, 0.12, 0.13 , 0.9, 0.05, 0.02 , , 0.72, 0.08, 0.05 , 0.91, 0.03, 0.05 , 0.72, 0.08, 0.09 , 0.62, 0.14, 0.1 , 0.71, 0.12, 0.12 , 0.64, 0.12, 0.2 0.53, 0.03, 0.09 , 0.07, 0.08, 0.21 , 0.02, 0.35, 0.18 , 0.08, 0.15, 0.09 , 0.27, 0.11, 0.41 , 0.43, 0.11, 0.25 , A− = 0.28, 0.15, 0.39 , 0.12, 0.12, 0.60 , 0.15, 0.05, 0.20 , . 0.61, 0.08, 0.21 , 0.05, 0.03, 0.19 , 0.12, 0.08, 0.20 , 0.23, 0.14, 0.31 , 0.14, 0.12, 0.35 , 0.14, 0.12, 0.63 Step4: Calculate the weighted distance dw ( Ai , A+ ) and dw ( Ai , A− ) by (M3) and (M4), then
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Decision-Making Method Based on Picture Fuzzy… dw ( A1 , A+ ) = 0.0905 , dw ( A2 , A+ ) = 0.0890 , dw ( A3 , A+ ) = 0.0834 , dw ( A4 , A+ ) = 0.1151 , dw ( A5 , A+ ) = 0.0708 , dw ( A1 , A− ) = 0.1300 , dw ( A2 , A− ) = 0.1151 , dw ( A3 , A− ) = 0.1242 , dw ( A4 , A− ) = 0.1128 , dw ( A5 , A− ) = 0.1213 . Step5: Calculate the closeness degree C ( Ai ) by (M5), then
C ( A1 ) = 0.5895 , C ( A2 ) = 0.5639 , C ( A3 ) = 0.5983 , C ( A4 ) = 0.4949 , C ( A5 ) = 0.6315 . Step6:
Rank
all
the
alternatives
A1-A5
in
accordance
C ( A5 ) C ( A3 ) C ( A1 ) C ( A2 ) C ( A4 ) , then A5
A3
of
C ( Ai ) , since
with
the
values
A1
A2
A4 . Note that “
” means
“preferred to”. Thus, the best alternative is A5.
ACKNOWLEDGEMENTS The works described in this paper are partially supported by Higher Education Key Scientific Research Program Funded by Henan Province (No. 20A110011) and Undergraduate Innovation Foundation Project of Anyang Normal University (No. ASCX/2019-Z112).
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W. Ma, Application of C4.5 algorithm on scholarship assessment of university, Journal of Henan Institute of Engineering(Natural Science Edition), 24(2), 2017, 57-60. C. Wei, and P. Hui, The evaluation of scholarship based on varied weight AHP, Journal of Luoyang Normal University, 33(2), 2014, 14-18. S. Li, T. Luan, and Y. Wang, The application of multi-index model in college students scholarship assessment problem, Journal of Baicheng Normal University, 31(5), 2017, 30-33. B.C. Cuong, and V. Kreinovich, Picture fuzzy sets-a new concept for computational intelligence problems, Third World Congress on Information and Communication Technologies (WICT 2013). IEEE, 2013, 1-6. B.C. Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernetics, 30(4), 2014, 409-420. C. Wang, X. Zhou, H. Tu, and T. Shen, Some geometric aggregation operators based on picture fuzzy sets and their application in multiple attribute decision making, Italian Journal of Pure and Applied Mathematics, 37, 2019, 477-492. G. Wei, Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making, Fundamenta Informaticae,157(3), 2018, 271-320.
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