Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

A NEW FUZZY SYNTHETIC EVALUATION MODEL FOR TEACHING QUALITY

HUI-ZHU TIAN!, CHEN-XIA JIN2

lHebei College of Industry and Technology, Shijiazhuang 050091, China, 2School of Economy and Management, Hebei University of Science and Technology, Shijiazhuang 050018, China

E- MAIL: tianhuizhu88@163.com.jinchenxia2005@126.com

Abstract: In this paper, based on the concept of core samples set, we

give an attribute correlation metric method. Further, combining

with Choquet fuzzy integrals, we establish a new fuzzy synthetic

evaluation model (denoted as BFM-FSEM) to deal with correlation indexes. And finally we analyze the effectiveness of

BFM-FSEM and give its implementation steps through a teaching

quality evaluation case. Theoretical analysis and computation results show that BFM-FSEM has good operability and

interpretability. So, it can be applied in many fields such as information fusion, synthetic evaluation and fuzzy decision­

making.

Keywords: Fuzzy measures; Fuzzy integrals; Synthetic evaluation

1. Introduction

Synthetic evaluation is a multi-factor decision-making method of giving a comprehensive evaluation to object. It is the core problem of resource management, complex system optimization and so on. Constructing a fuzzy synthetic evaluation model which can deal with the interaction is considered widely in academic fields. Many scholars gave many discussions under different background and also obtained many important research results. Mi et al. [1] did an assessment of environment lodging stress for maize using fuzzy synthetic evaluation and provide a scientific basis for maize variety extension and recommendation and comprehensive management to reduce maize planting risk and loss. Xu et al.[2] proposed a fuzzy synthetic evaluation model aiming at risk assessment of PPP (P ublic-private partnerships) projects in China, which provided a deeper understanding of managing different types of PPP projects. Wang et al. [3] did a fuzzy synthetic evaluation of wetland soil quality degradation of northeast of China with a case study on the Sanjiang P lain. It is of scientific and practical significance for protection and management of soils and for sustainable development of agriculture. Ren et al. [4] provided a method of fuzzy synthetic evaluation of location plan of city distribution

978-1-4673-1487 -9/12/$31.00 ©2012 IEEE

center. It successfully eliminated the fuzziness and uncertainties in determining the location of city distribution center.

At present, the commonly used fuzzy synthetic evaluation methods are dependent on some weight systems. They are only suitable for some special evaluation problems and all have some shortcomings which are difficult to overcome. With the development of the fuzzy measure and fuzzy integrals theory, many scholars describe the interaction between the evaluation indexes by using the non-additivity of the fuzzy measure, and they put forward many fuzzy synthetic evaluations based on different fuzzy integrals (such as Sugeno fuzzy integrals, Choquet fuzzy integrals and (N) fuzzy integrals). These methods can solve information fusion with the interaction in theory, but how to determine an appropriate fuzzy measure is very difficult. For the determining methods, except for some methods given by some domain experts, the used common ones are obtained according to study from some known decision data. Keller et al. [5] studied a method based on confusion matrix. Grabisch [6] presented a method based on quadratic programming to identify fuzzy measures. Hu [7] put forward a genetic algorithm based method to determine fuzzy measures, in which operation process is simple and easy to understand, while the algorithm design depended on the type of training data.

In this paper, according to the influence on decision system by condition attributes, based on the concept of core sample set, we first give an attribute correlation metric method, by using the change of the core samples set as a strategy of measuring the importance of condition attributes sets. Second, we establish a fuzzy synthetic evaluation model to deal with correlation indexes (BF M-FSE M), combining with Choquet fuzzy integrals. Finally, we analyze the characteristics and effectiveness of this method and FSE M through a case-based example.

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

2. Preliminaries

Fuzzy measures essentially widen the classic metric, it is

made up for the deficiency in dealing with the non-additive

phenomenon.

Definition 2.1 [8] Let X be a nonempty set, � be a nonempty class of subsets of X, J1: B �[o, 00] and satisfies: 1) When ° E B , J1(0) =0; 2) For A, BE B , and A c B imply p(A) �p(B) ; 3) For {4.}��1 cB, 4c4c",c4,c···, and

U:�1 An E B, limp(An) = P (U:�1 An) ; (4) For {An}��1 c B , n-->oo 4:::J�:::J"':::JA,,:::J"" and n:�IA"EB,limP(A,,)= P(n:�IA,,) . n-->oo Then J1 is called a fuzzy measure on (X, B ), and (X, B , J1) is called a fuzzy measure space. Especially, when p(X) = 1 , J1 is

said a normalized fuzzy measure.

Definition 2.2 [8] Let (X, .ow, p) be a fuzzy measure

space, f(x) be a non-negative measurable function on

(X, .ow, p) (that is {x I x E X, f(x) > a} E.OW for any

a E [0, + 00)), BE.OW. Then

is called the Choquet fuzzy integral off{x) on B about fuzzy measure space (X, S9, p)

For convenience, in the following, we will introduce the concept of core sample set, let (U, C, d, F) to be a decision information system, where U is a universe of samples; C={ C1, C2, " ', Cs} is the condition attribute set, V( C:;) = {£1h £12, "', �}

is the range of Ci; d is the decision attribute and V(d)={dJ, 0 0 . '

dn} is the range of d; F=ifj, h, " ', Is, Jd} is the information function. Rei is an equivalence relation by the values of

attribute Ci, and Rdis also an equivalence relation by attribute d. For the equivalence relation R on U, let [uJR be the R-equivalence class of u, u/R is the sets of R-equivalence class; we call (U, B, d, FB) the subsystem of (U, C, d, F) for B c C , FB = {fd , fk I Ck E B} .

Definition 2.3[9] For the decision information system (U, C, d, F), if (C1 = Clil' C2 = C2i" "' , Cs = Csi,) *-0 , and

then (C1 = Clil' C2 = C2i,, ' '', Cs = csi,; d = dj) is called an

elementary knowledge factor of (U, C, d, F), (c:; = Clil , Cz = �i, ,

"', Cs = CSi,) is called the support samples set of (C1 = Clil ,

C2 = C2i,, ' 0 0, Cs = CSi,; d = dj) , and

Ue= u

is called the core samples set of (u, C, d, F). Where Ik is the family of all elementary knowledge factors of (U, C, d, F).

Theorem 2.1 Let (U, C, d, FB) be the subsystem of (U, C, d, F), then UB cUe. Here, UB and Uc are the core samples sets of (U, C, d, FB) and (U, C, d, F) respectively.

The core samples set is the supporting body of knowledge in information system, it reflects the influence on information system of each attributes. So the core samples set can be used as the metric basis of attribute importance. Let (U, C, d, F) be a decision information system, P( C) be the power set of C,

BEP(C) , {I UB 1/ I Ue I,

p(B) = 1,

0,

IUe 1*-0,

IUel= O and B*-0, (4)

B = 0.

Here, IAI is the number of elements of A, UB and Uc is the core samples set of (U, C, d, FB) and (U, C, d, F) separately, B E p( C) .

It is easy to see, (4) is a metric describing the support degree to (U, C, d, F) by condition attributes set B, and it is an attribute correlation metric method based on knowledge, reflecting the influence on the generated knowledge by the attributes value in B. It can be used in many issues such as synthetic evaluation, information fusion, pattern recognition and multiple attribute decision making.

3. A fuzzy synthetic evaluation model based on Choquet fuzzy integrals

Synthetic evaluation is the key focus in many fields such as resource management, complex system optimization and so on. And its implementation steps are listed as follows:

Step 1 Build the factors set X = {Ul' U2, .. " us} ;

Step 2 Build the evaluation set V = {vp V2,00, vm};

Step 3 Determine the single factor evaluation matrix by some methods such as fuzzy statistics, expert scoring method. Here, rij E [0,1] means the degree of the evaluation objects

to each Vi on each Ui, i = 1, 2,00', s; j = 1, 2,00', m;

R = [} ;� ••• ): ] = (RPR2,00. ,Rm)

81 rs2 8m

Step 4 Choose a suitable fuzzy synthetic function S(xp X2,00 ., xJ satisfying the following conditions: 1)

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

S(Xp x2, .··, xJ: [0, 11' � [0, 1]; 2) S(x, X, .··, x) = X ; 3) It

is monotone nondecreasing on each variable Xi; 4) It is continuous on each variable Xi.

Step 5 Synthesize each column for a value bj = S(Rj)

= S(rjJ' rj2,···, rjs) by synthetic function R , then we get the

fuzzy synthetic evaluation results B= (b., , �,. .. , bm). Where bj

means the degree of the evaluation objects to each evaluation set Vj in synthetic sense, j = 1, 2, .··, m . That is

(5)

It is easy to see, the selection of a fuzzy synthetic function is very important for fuzzy synthetic evaluation.

In the following, by combining with attribute correlation metric in section 2 and Choquet fuzzy integrals, we give a fuzzy synthetic function to deal with the interaction relation.

Theorem 3.1 Let (C, p(C), Jl) be the attribute

correlation metric based on the knowledge of (U, C, d, F),

and C = {CI ' C2, · .. , Cs} • For any given Xi E [0, 1] ,

i = 1, 2, . .. , S , let x = (Xl, X2, . . . , xs) , Ix (CJ = Xi , then

S(XJ, X2, .··, xs) = r Jl(Na(fi))da

is a fuzzy synthetic function.

(6) is essentially a Choquet fuzzy integral on fuzzy measure space (C, P(C), Jl) , where Xl, X2, .··, Xs is

taken as the function of the corresponding value of the Cp C2, . .. , Cs. Suppose Xl � X2 � ... � Xs , then,

Where Xo = 0, Bk = {Ck, Ck+I"··, C,}, k = 1, 2, .·· , s .

For convenience, we call (6) a fuzzy synthetic evaluation model based on fuzzy measure (B CFI-FSE M), and denoted as

{B = SoR,

S(xJ, X2, .··, xs) = r' Jl(Na(fi))da,

(C, P(C), Jl) E (U, C, F, d).

(6)

(7)

(8)

Where, (U, C, F, d) is a decision information system

relevant to the synthetic evaluation.

4. Example analysis

In order to guarantee the teaching quality, one school considers the teachers' teaching situation synthetically from the aspects of professional level (Cj), teaching attitude (C2), teaching method (C3) and teaching effect (C4) , so as to find and solve the problems in the teaching process in time. In view of the evaluation of the teaching quality is related to many subjective and objective factors such as the students' basic situation, the teaching facilities and session, it is not suitable to describe it by a precise numerical value. Therefore, some schools decide to use fuzzy synthetic evaluation to examine the teachers' teaching situation, the corresponding evaluation is {very good, good, common, poor}.

It is easy to see that there are some correlations between the four evaluation factors above, so it is not suitable to use the weighted average synthetic function. How to determine correlation metric between the evaluation indexes is the key of constructing the evaluation. Because the teaching quality is a long-term concerned problem by education department and school, there are much accumulated and related information. Table 3 shows 20 information extracted at random from the database of the teaching quality evaluation of the school, d is the synthetic evaluation result. Therefore, we can use the fuzzy synthetic evaluation model (11) based on the knowledge system given in the section 4 as the evaluation pattern of the teaching quality. Its implementation steps are stated as follows:

Step 1 Determine attribute correlation metric of the evaluation indexes according to (4) (Table 2 is the results of Table 1);

Step 2 Let (6) be a synthetic function, combined with a concrete single factor evaluation matrix R. Then, we can get the fuzzy synthetic evaluation results (5).

TABLE 1. EVALUATION INFORMATION SYSTEM OF 20 TEACHERS' TEACHING QUALITY

Samples Professional level Cl Teaching attitude C2 Teaching method C3 Teaching effect C4 d

1 very good very good very good very good very good

2 good very good very good very good very good

3 common very good common good common

4 common Common poor poor poor

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

5 poor Good good common common

6 very good Common poor common common

7 good common good good good

8 good poor good good good

9 very good good common good good

10 good poor poor poor poor

11 good good common common common

12 poor very good very good good common

13 common very good very good good good

14 good common good common common

15 good good good good good

16 poor good very good common common

17 poor very good good common common

18 good common common good common

19 common good good good good

20 very good common poor poor poor

TABLE 2. ATTRIBUTE CORRELATION METRIC OF THE TEACHING QUALITY

knowledge system Core examples set UB IUBI fl(B)

(U, {Cd, F, d) {5, 12, 16, 17} 4 0.2

(U, {C2}, F, d) 0 0 0

(U, {C3}, F, d) 0 0 0

(U, {C4}, F, d) {I, 2, 4, 5, 6, 10, 11, 14, 16, 17, 20} 11 0.55

(U, {Cj, C2}, F, d) {I, 2, 4, 5, 9, 12, 16, 17, 19} 9 0.45

(U, {Cj, C3}, F, d) {1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 16, 17, 1� 19} 14 0.7

(U, {Cj, C4}, F, d) {I, 2, 4, 5, 6, 9, 10, 11, 12, 14, 16, 17, 20} 13 0.65

(U, {C2, C3}, F, d) {3, � 10, 1� 1� 18} 6 0.3

(U, {C2, C4}, F, d) {1, 2, 4, 5, 6, 8, 9, 10, 11, 14, 15, 16, 17, 19, 20} 15 0.75

(U, {C3, C4}, F, d) {I, 2, 4, 5, 6, 7, 8, 10, 11, 14, 15, 16, 17, 19, 20} 15 0.75

(U, {Cj, C2, C3}, F, d) {I, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19} 16 0.8

(U, {Cj, C2, C4}, F, d) {1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 1� 17, 19, 20} 16 0.8

(U, {Cj, C3, C4}, F, d) U 20 1

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

(U, {C2, C3, C4}, F, d) {I, 2, 3, 4, 5, 6,7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20} 18 0.9

(U, C, F, d)

According to the dlscusslOn above, If one teacher's single factor evaluation matrix is

lo.70 0.20 0.10 0.00] 0.35 0.55 0.10 0.00

R = = (R R R R) (9) 0.15 0.65 0.10 0.10 P 2' 3' 4

0040 0040 0.10 0.10

S(R,) = S(0.70, 0.35, 0.15, 0040)

U 20 1

then by usmg (11) m the sectlOn 4 we can get the following:

= (0.15 -O.OO),u(C) + (0.35 -0.15),u( {C" C, C.}) + (0040 -0.35),u( {C" C.} )+(0. 70 - OAO),u( {C,})

= 0.15 xl + 0.20 xO.8 + 0.05 xO.65 + 0.30x 0.2

= 004025;

S(R2) = S(0.20, 0.55, 0.65, 0040)

= (0.20 -O.OO),u(C) + (0040 -0.20),u( {C" C" C.}) + (0.55 -OAO),u( {C" C3} ))+ (0.65 -0.55),u( {C3})

= 0.20x 1 + 0.20x 0.9 + 0.15 x 0.3 + O.lOx 0

= 00425;

S(R3) = S(O.lO, 0.10, 0.10, 0.10) = (0.10 -O.OO),u(C) = 0.10 xl = 0.10 ;

S(R.) = S(O.OO, 0.00, 0.10, 0.10) = (0.10 -0.00),u( {C" C.}) = O.lOx 0.75 = 0.075 .

that is, the teacher's teaching quality evaluation is B = (S(RJ), S(R2), S(R3), S(R4)) = (004025, 00425, 0.1

0.075) , i.e. the degree of the teacher's teaching quality

corresponds to very good, good, common and poor are 004025, 00425, 0.10 and 0.075.

For the performance analysis of model (8), we further give fuzzy synthetic evaluation for several

different evaluation object, by using weighted average fuzzy synthetic function (6) with WJ=(0.25. 0.25, 0.2, 0.3), single determining fuzzy synthetic function (8) with W2=(0.8. 0.8, 0.6, 1), and fuzzy synthetic function (10) separately. The results are shown in Table 5.

TABLE 3. RESULTS OF DIFFERENT FUZZY SYNTHETIC EVALUATION

results of evaluation

No. the single factor

the main factors determining evaluation matrix the weighted average type Model (8)

type

[0W 0." o.w 0 00] 1 0.35 0.55 0.10 0.00

(0.4125,0.4375,0.1000,0.0500) (0.7000,0.6000,0.1000,0.1000) (0.4025, 0.4250, 0.1000, 0.0750) 0.15 0.65 0.10 0.10 0.40 0.40 0.10 0.10

[OW OM 0." 0 00] 2

0.15 0.70 0.10 0.05 (0.1125,0.6275,0.1975,0.0625) (0.1500,0.7000,0.2500,0.1000) (0.1000,0.6125,0.2350,0.0825) 0.10 0.55 0.25 0.10

0.10 0.60 0.20 0.10

[0." 0." 0." 0 00] 3

0.55 0.35 0.10 0.00 (0.5600,0.3425,0.0975,0.0000) (0.7500,0.6000,0.1500,0.0000) (0.4850,0.2750,0.0950,0.0000) 0.20 0.65 0.15 0.00

0.65 0.25 0.10 0.00

[0.00 0." 0." OW] 4

0.75 0.15 0.10 0.00 (0.3175,0.165,0.4425,0.0850) (0.7500,0.3000,0.7500,0.2000) (0.1950,0.1400,0.5000,0.0200) 0.65 0.30 0.10 0.00

0.00 0.10 0.70 0.20

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

[0.00 0.00 0.30 0'" 1 5

0.75 0.25 0.00 0.00 (0.2775, 0.2025, 0.1350, 0.3850) 0.45 0.55 0.00 0.00

0.00 0.10 0.20 0.70 [0 00 0.00 O.W O�l

6 0.50 0.35 0.15 0.00

(0.1250,0.1525,0.3775,0.3450) 0.00 0.25 0.65 0.10 0.00 0.05 0.20 0.75

From Table 3, we can see the results of three synthetic evaluation have obvious differences. They are presented in the following: 1) Results for the main factors determining type obviously depend on the value of individual indexes such as No. 3, No. 4 and No.5 . 2) Results for the weighted average type and model (8) have similarity such as No. 1, No. 2 and No. 3, and also have differences such as No. 4, No. 5 and No. 6. But these results closely depend on weight system. 3) Model (8) just depends on the past data information, and it has a certain generality and objectivity.

5. Conclusions

In this paper, based on the concept of the core samples set, we establish an attribute correlation metric method. Further, combined with Choquet fuzzy integrals, we establish a new fuzzy synthetic evaluation model (B CFI-FSE M) to deal with correlation indexes. Then we analyze the effectiveness of this model through a cased-based example. Theoretical analysis and computation results show that the attribute correlation metric method has strong interpretability. So it can be applied in many problems such as information fusion, synthetic evaluation, fuzzy decision.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (71071049, 71210107002) and the Natural Science Foundation of Hebei Province (F2011208056), and Scientific Research Fund of Hebei College of Industry and Technology.

(0.7500, 0.5500, 0.3000, 0.7000) (0.1350,0.1350,0.1500,0.4550)

(0.5000, 0.3500, 0.6000, 0.7500) (0.0000,0.1050,0.4800,0.4875)

References

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[5] James M. Keller, P aul Gader, Hossein Tahani, Jung-Hsien Chiang, Magdi Mohamed. "Advances in fuzzy integration for pattern recognition", Fuzzy sets and systems, Vol. 65, pp. 273-283, 1994.

[6] M. Grabisch and J. M. Nicolas. " Classification by fuzzy integral: P erformance and tests", Fuzzy sets and systems, Vol. 65, No.2-3, pp. 255-271, 1994.

[7] yc. Hu. "Nonadditive grey single-layer perceptron with Choquet integral for pattern classification problems using genetic algorithms", Neurocomputing, Vol. 72, No. I-3, pp.331-340, 2008.

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