New methods for students' evaluation using fuzzy sets

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E L S E V I E R Fuzzy Sets and Systems 104 (1999) 209 218

F U | Z Y

sets and systems

New methods for students' evaluation using fuzzy sets

Shyi-Ming Chen a'*, Chia-Hoang Lee b

"Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC bDepartment of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan, ROC

Received April 1996; received in revised form May 1997

Abstract

In an earlier work, Biswas (1995) presented two methods for the application of fuzzy sets in students' answerscripts evaluation. In this paper, we extend his work to propose two new methods for evaluating students' answerscripts using fuzzy sets. The proposed methods can overcome the drawbacks in Biswas (1995) due to the fact that they do not need to perform the complicated matching operations and they can evaluate students' answerscripts in a more fair manner. ((-" 1999 Elsevier Science B.V. All rights reserved.

Keywords:

Extended fuzzy grade sheet; Fuzzy set; Generalized extended fuzzy grade sheet; Letter-grade; Mid-grade- point

1. Introduction

In 1965, Zadeh proposed the theory of fuzzy sets [15]. In recent years, some research on the application of fuzzy set theory in education has begun [2, 4, 10]. In [10], Chiang et al. presented a method for the application of fuzzy set theory to teaching assessment. In [4], Chang et al. presented a method for fuzzy assessment of learning performance of junior school students. In [2], Biswas pointed out that the chief aim of education institutions should be to provide the students with the evaluation re- ports regarding their test/examination as sufficient as possible and with unavoidable error as small as

* Corresponding author.

possible. He also presented a fuzzy evaluation method (fern) for the application of fuzzy sets in students' answerscripts evaluation. The fern presented in [2] is a computer based fuzzy approach, where a vector valued marking is used. Further- more, in F2], Biswas also generalized the fern to propose a generalized fuzzy evaluation method (gfem) in which a matrix-valued marking is ad- opted. However, the methods presented in [2] have the following drawbacks:

(1) Because they use a matching function S to measure the degrees of similarity between the standard fuzzy sets and the fuzzy marks of the questions, they will take a large amount of time to perform the matching operations.

(2) In Biswas's methods, two different fuzzy marks may be translated into the same awarded grade and this is unfair in students' evaluation. 0165-0114/99/$ see front matter © 1999 Elsevier Science B.V. All rights reserved.

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210 S . - ~ Chen, C-H. Lee / Fuzzy Sets and Systems 104 (1999) 209-218

Because Biswas's methods have the above two drawbacks in the task of students' answerscripts evaluation, it is necessary to develop new methods to overcome the above drawbacks.

In this paper, we present two new methods for the application of fuzzy sets in students' answerscripts evaluation. They can overcome the drawbacks of the ones presented in [2]. The proposed methods have the advantages of much faster execution and are more fair in the task of students' evaluation than the ones presented in [-2].

The rest of this paper is organized as follows. In Section 2, we briefly review the theory of fuzzy sets from [-5, 6, 12, 15, 16]. In Section 3, we briefly review Biswas's methods for students' answerscripts evaluation. In Section 4, we present a new method for students' answerscripts evaluation using fuzzy sets. In Section 5, we present a generalized fuzzy evaluation method for students' evaluation. The conclusions are discussed in Section 6.

2. Fuzzy set theory

The theory of fuzzy sets was proposed by Zadeh in 1965 [-15]. Roughly speaking, a fuzzy set is a class with fuzzy boundaries. Let X be the universe of discourse, X = {x1, x 2 . . . Xn} , and let A be a fuzzy set of X, then the fuzzy set A can be represented as

A = {(xI,fA(X1)), (x2, fA(X2)), ... ,(x,,fA(X,))}, (1) wherefa is the membership function of the fuzzy set

A, f a : X ~ [-0, 1], fA(xi) indicates the degree of membership of xi in A. If the universe of discourse X is an infinite set, then the fuzzy set A can be expressed as

A = ~ fa(xi)/xi, xi ~ X. (2)

, ) x

Example 2.1. Let X be the universe of discourse, X = {red, black, yellow, blue, white, brown, green}, and let "dark" be a fuzzy set of the universe of discourse X subjectively defined as follows: dark = {(red, 0.5), (black, 1.0), (yellow, 0.1),

(blue, 0.6), (white, 0.0), (brown, 0.8),

(green, 0.3)}, (3)

where "black" has the largest membership value (i.e., 1.0) in the fuzzy set "dark", and "white" has the smallest membership value (i.e., 0.0) in the fuzzy set "dark". Thus, "black" is most pertinent to the fuzzy set "dark", and "white" is impertinent to the fuzzy set "dark".

For convenience, if an element xi has zero membership value in a fuzzy set A (i.e.,fa(X3 = 0), then the ordered pair (X,fA(Xi)) can be discarded from the representation of the fuzzy set A. Thus, in the above example, the fuzzy set "dark" also can be written as follows:

dark = {(red, 0.5), (black, 1.0), (yellow, 0.1),

(blue, 0.6), (brown, 0.8), (green, 0.3)}. (4)

Example 2.2. Let X be the universe of discourse, X = [,0, 100]. Then, the fuzzy sets "young" and "old" may subjectively be defined as follows: fyoung(X) 1, 0 < x ~ 2 0 , (l + ((x -- 20)/15)2) - 1 , 20 < x ~ 100, (5) fo~d(x) = { 0 , (1 + ((x -- 40)/10)-2) -1, 0 < x ~<40, 40 < x ~< 100, (6)

where fyoung and fold are the membership functions of the fuzzy sets "young" and "old", respectively.

3. Biswas's methods for students' answerscripts evaluation

In [2], Biswas used a matching function S to measure the degree of similarity between two fuzzy sets. Let A and B be two fuzzy sets of the universe of discourse X, where

A : {(xI, fA(X1)), (x2, fA(X2)), -.. ,(Xn,fA(Xn))},

B = {(xx, f,~(x~)), (x~, fB(x~)) . . . . ,(x., f,,(x,,))}, x = {x l, ~ . . . ~.}.

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S.-M. Chen, C.-H. Lee /Fuz~ Sets and Systems 104 (1999) 209 218 21l By using the vector representation method, the

fuzzy sets A and B can be represented by the vectors /1 and/~, respectively, where

A = (fA{Xl),fA{X2) . . . . ,fa(X,)}, B = ( f u ( x l ) , f R ( x 2 ) . . . fB(X,)}.

Then, the degree of similarity S(A, B) between the fuzzy sets A and B can be defined by

A . B

S(A, B) = Max(A. A, B. B)' (7) where S ( A , B ) e [0, 1]. The larger the value of

S(A, B), the more the similarity between the fuzzy sets A and B.

Based on the matching function S, Biswas introduced a "fuzzy evaluation method" (fern) for evaluating students' answerscripts. In the following, we briefly review Biswas's methods for students' answerscripts evaluation. In [2], Biswas used five fuzzy linguistic hedges (called Standard Fuzzy Sets (SFS)) for students' answerscripts evaluation, i.e., E (excellent), V (very good), G (good), S (satisfactory), and U (unsatisfactory), where

X = {0%, 20%, 40%, 60%, 80%, 100%}, E = {(0%,0), (20%,0), (40%,0.8), (60%,0.9),

(80%, 1), (100%, 1)},

V = {(0%, 0), (20%, 0), (40%, 0.8), (60%, 0.9), (80%, 0.9), (100%, 0.8)}, G = {(0%, 0), (20%, 0.1), (40%, 0.8), (60%, 0.9), (80%, 0.4), (100% , 0.2)}, S = {(0%, 0.4), (20%, 0.4), (40%, 0.9), (60%, 0.6), (80%, 0.2), (100%, 0)}, U = {(0%, 1), (20%, 1), (40%, 0.4), (60%, 0.2),

(80%, o), (lOO%, o)}.

Based on the vector representation method, the fuzzy sets E, V, G, S, and U can be represented by the vectors E, V, G, S, and /.7, respectively, where E = (0, O, 0.8, 0.9, 1, 1}, 17 = (0, 0, 0.8, 0.9, 0.9, 0.85, 6 = (0, 0.1, 0.8, 0.9, 0.4, 0.2}, = (0.4, 0.4, 0.9, 0.6, 0.2, 05, U = (1, 1, 0.4, 0.2, 0, 05.

In [2], Biswas pointed out that "A", "B", "C", "D", and "E" are called letter grades, where 0 ~< E < 30, 30 ~< D < 50, 50 ~< C < 70, 7 0 ~ < B < 9 0 , 90~<A ~<100.

Furthermore, he also introduced the concept of mid-grade-point, where the mid-grade-point of A = 95 is denoted by P(A), B = 80 by P(B), C = 60 by P(C), D = 40 by P(D), E = 15 by P(E). Assume that an evaluator is to evaluate the ith question (i.e., Q.i) of an answerscript of a student using a fuzzy grade sheet shown in Table 1.

In the first row of Table 1, the fuzzy mark (fum) to the answer of question Q.1 shows the degrees of the evaluator's satisfication for that answer in 0%, 20%, 40%, 60%, 80%, and 100% are 0, 0.1, 0.2, 0.4, 0.4, and 0.6, respectively. Let the fuzzy mark of the answer of question Q.1 be denoted by F1. Then, we can see that F1 is a fuzzy set of the universe of discourse X, where

X = {0% , 2 0 % , 4 0 % , 60% , 80%, 100% ], F1 = {(0%, 0), (20%, 0.1), (40%, 0.2),

{60%, 0.4}, (80%, 0.4), (100%, 0.6}}.

Biswas's algorithm [2] for students' answerscript evaluation is summarized as follows.

Step 1: For each attempted question in the answerscript repeatedly perform the following steps:

Table !

A fuzzy grade sheet Question No. Q.I Q.2 Q.3 Fuzzy mark 0 0.1 0.2 0.4 0.4 Grade 0.6 Total mark =

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212 S.-M. Chen, C.-H. L e e / F u z z y Sets and Systems 104 (1999) 209-218

(1) The evaluator awards a fuzzy mark Fi to the question Q.i by his best possible judgement and fills up the cells of the ith row for the first seven columns. Let F~ be the vector representation of F~.

(2) Calculate the following degrees of similarit-

ies:

S(E, Fi), S(V, El), S(G, f ~), S(~g, Fi),

and

S(U, f ~),

where E, V, G, S, and U are the vector representa- tions of the standard fuzzy sets E (excellent), V (very good), G (good), S (satisfactory), and U (unsatisfactory), respectively.

(3) Find the maximum among the five values

S(E, F~), S(V, F~), S(G, F,), S(S, Fi),

and

S(U, F~).

Assume that

S(V, Fi)

is the maximum value among

the values of

S(E, Fi), S(V, Fi), S(G, Fi), S(S, Fi),

and

S(U, F~),

then award grade "B" to the question Q.i due to the fact that grade "B" corresponds to V (very good) of the standard fuzzy set.

Step

2: Calculate the total score using the following formula:

1

Total score = T~X,, ~ [T(Q.i) x

P(gi)],

I U U - -

(8)

where T(Q.i) is the mark alloted to Q.i in the

question paper, and

gi

is the grade awarded to Q.i

Table 2

A generalized fuzzy grade sheet

Question No. gfum Grade

Q1 Q2 F l l 01l F12 912 F13 013 F14 gi4 F21 gzl F22 922 F23 gz3 F24 ,q24 M a r k ml m2 Total m a r k =

by Step 1 of the algorithm. Put this total score in the appropriate box at the bottom of the fuzzy grade sheet.

Furthermore, in [2], Biswas also presented a generalized fuzzy evaluation method (gfem), where a generalized fuzzy grade sheet shown in Table 2 is used to evaluate the students' answerscripts. In the grade sheet of Table 2, for all

j = 1, 2, 3, 4, and for all

i, gi~

is the calculated grade

by fem for the awarded rum Fu, and rn~ is the calculated mark to be awarded to the attempted question Q.i using the formula:

1 4-

mi = 40-6" T

(Q.i)"

~ P(gij)

(9)

j = l

and Total mark = 52 mi.

However, the methods presented in [-2] have the following drawbacks:

(1) Because they use a matching function S to measure the degrees of similarity between the standard fuzzy sets and the fuzzy marks of the questions, it will take a large amount of time to perform the matching operations. Especially, when the number of questions in the test/examination is very big.

(2) In Biswas's method, two different fuzzy marks may be translated into the same awarded grade and this is unfair in students' evaluation. For

example, let Fi and

Fj

be two different fuzzy marks

represented by fuzzy sets of the universe of discourse X, respectively, and let E (excellent), V (very good), G (good), S (satisfactory), and U (unsatisfactory) be standard fuzzy sets of the universe of discourse X, where X = {0%, 20%, 40%, 60%, 80%, 100%} and the corresponding awarded grade of the standard fuzzy sets "E', "V", "G", "S", and "U" are "A', "B', "C", "D', and "E', respectively. Then, based on [2], we can calculate the following degrees of similarities:

Case

1: If

S(V, Fi)

the values of

S(E, F~), S(V, Fi), S(G, F~), S(S, Fi),

S(U, F~),

then the fuzzy mark Fi is translated to the awarded grade "B" due to the fact that the grade "B" corresponds to V (very good).

Case

2: If

S(V, F~)

the values of

S(E, Fj), S(V, Fj), S(G, Fj), S(S, Fj),

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S.-M. Chen, (2-H. Lee / Fuzzy Sets and Systems 104 (1999) 209 218 213

awarded grade "B" due to the fact that the grade "B" corresponds to V (very good).

F r o m Cases 1 and 2, we can see that two different fuzzy marks Fi and Fj are translated to the same awarded grade "B", and this is unfair in the task of students' answerscripts evaluation.

Because Biswas's methods have the above two drawbacks in the task of students' answerscripts evaluation, a new method for students' answerscripts evaluation is required to overcome the above drawbacks.

4. A new method for student's evaluation using fuzzy sets

Table 3

Satisfication levels and satisfication

their corresponding degrees

Satisfication levels Degrees of satisfication Extremely good (EG)

Very very good (VVG) Very good (VG) Good (G)

More or less good (MG) Fair (F)

More or less bad (MB) Bad (B)

Very bad (VB) Very very bad (VVB) Extremely bad (EB)

100% (i.e., 1.00) 91% 99% i.e., 0.91 0.99) 81%-90% i.e., 0.81 0.90t 71% 80% i.e.,0.71 0.80) 61% 70% i.e.,0.61 0.70) 51%--60% i.e., 0.51-0.60) 41'7o 50% i.e., 0.41 0.50) 25%~40% i.e., 0.25-0.40) 10%--24% i.e., 0.10-0.24) 1% 9% (i.e., 0.01 0.09) 0% (i.e., 0) of

In this section, we present a new method for students' answerscripts evaluation. Assume that there are eleven satisfication levels to evaluate the students' answerscripts regarding a question of a test/examination, i.e., extremely good (EG), very very good (VVG), very good (VG), good (G), more or less good (MG), fair (F), more or less bad (MB), bad (B), very bad (VB), very very bad (VVB), and extremely bad (EB), where the degrees of satisfication of the eleven satisfication levels are shown in Table 3.

Let X be a set of satisfication levels, X = {extremely good (EG), very very good (VVG), very good (VG), good (G), more or less good (MG), fair (F), more or less bad (MB), bad (B), very bad (VB), very very bad (VVB), extremely bad (EB)}, and let T be a mapping function which maps a satisfication level to the maximum degree of satisfication of the corresponding satisfication level, where T : X - - * [0, 1]. F r o m Table 3, we can see that

T (extremely good) = 1.00 (i.e., T (EG) = 1.00), T (very very good) = 0.99

(i.e., T ( V V G ) = 0.99),

T (very good) = 0.90 (i.e., T (VG) = 0.90), T (good) = 0.80 (i.e., T (G) = 0.80), T (more or less good) = 0.70

(i.e., T ( M G ) = 0.70),

T (fair) = 0.60 (i.e., T (F) = 0.60),

T ( m o r e or less bad) = 0.50 (i.e., T ( M B ) = 0.50),

T (bad) = 0.40 (i.e., T (B) = 0.40), T (very bad) = 0.24 (i.e., T (VB) = 0.24), T (very very bad) = 0.09 (i.e., T (VVB) = 0.09), T(extremely bad) = 0 (i.e., T(EB) = 0). (10)

Assume that an evaluator can evaluate the students' answerscripts using extended fuzzy grade sheets. The definition of the extended fuzzy grade sheets is presented as follows.

Definition 4.1. Extended fuzzy grade sheet: An ex- tended fuzzy grade sheet is a matrix type structure containing thirteen columns and n rows, where n is the total number of questions in a test/examination. An example of an extended fuzzy grade sheet is shown in Table 4. At the bottom of the sheet there is a box which tells the total score. The first column reveals the serial numbers of the questions; in any row, the columns from the second to the twelfth shows the fuzzy mark awarded to the answer to the corresponding question in the first column, where the fuzzy mark is represented as a fuzzy set in the universe of discourse X, X = {extremely good (EG), very very good (VVG), very good (VG), good (G), more or less good (MG), fair (F), more or less bad (MB), bad (B), very bad (VB), very very bad (VVB), extremely bad (EB)}, The last (i.e., the

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214 S.-M. Chen, C.-H. Lee / Fuzzy Sets and Systems 104 (1999) 209-218 Table 4

An extended fuzzy grade sheet Question No. Q.1 Q.2 Q.n Satisfication levels EG VVG VG G MG F MB B VB VVB EB Table 5

An example of an extended fuzzy grade sheet

Degree of satisfication Total mark = Question No. Q.1 Satisfication levels EG VVG VG G MG F MB B VB VVB EB 0 0.9 0.8 0.5 0 0 0 0 0 0 0 Degree of satisfication Total mark =

thirteenth) column shows the degree of satisfication evaluated by the p r o p o s e d method awarded to each question. The box at the b o t t o m shows the total m a r k awarded to the student.

F o r example, assume that an evaluator is using an extended fuzzy grade sheet to evaluate the fuzzy m a r k of the first question (i.e., Q. 1) of a test/examination of a student as shown in Table 5. F r o m Table 5, we can see that the satisfication level regarding the first question of the student's answerscript is represented by a fuzzy set F ( Q . 1) of the universe of discourse X, where X = {EG, VVG, VG, G, M G , F, MB, B, VB, VVB, EB}, and F(Q. 1) = {(EG, 0), (VVG, 0.9), (VG, 0.8),

(G, 0.5), (MG, 0), (V, 0), (MB, 0), (B, 0), (VB, 0), (VVB, 0), (EB, 0)}. (11) F o r convenience, the fuzzy set F(Q.1) can also be abbreviated into

F(Q. 1) = {(VVG, 0.9), (VG, 0.8), (G, 0.5)}. (12)

It indicates that the satisfication level of the student's answerscript with respect to the first question is described as 90% very very good, 80% very good, and 50% good.

The method for students' answerscripts evaluation is now presented as follows:

Step 1: Assume that the fuzzy m a r k of the question Q. i of a student's answerscript evaluated by an evaluator is shown in Table 6, where Yi ~ [0, 1] and 1 ~<i~< 11. F r o m formula (10), we can see that T(EG) = 1, r (VVG) -- 0.99, T (VG) -- 0.90, T (G) = 0.80, T ( M G ) = 0.70, T ( F ) = 0.60, T ( M B ) = 0.50, T(B) = 0.40, T(VB) = 0.24, T(VVB) = 0.09, and T(EB) = 0. In this case, the degree of satisfication D(Q.i) of the question Q . i of the student's answerscript can be evaluated by the function D, D(Q.i)

Yx * T ( E G ) + Y2* T ( V V G ) + ... + Yal * T(EB) Yt + Y 2 + "" + Y l l

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S.-M. Chen, C.-H. Lee / Fuzzy Sets and Systems 104 (1999) 209 218

Table 6

Fuzzy mark of question Q.i in an extended fuzzy grade sheet

215 Question No. Q.i Satisfication levels EG VVG VG G MG Yl Y2 Y3 Y4 Y5 F MB B Y6 Y7 Y8 VB VVB EB Y9 Yl0 Yll Degree of satisfication Total mark =

where D ( Q . i ) e [0,1]. The larger the value of D(Q. i), the more the degree of satisfication that the question Q.i of the student's answerscript satisfies the evaluator's opinion•

F o r example, let us consider the example shown in Table 5. F r o m formula (10), we can see that T (VVG) = 0.99, T (VG) = 0.90, and T (G) = 0.80. By applying formula (13), the degree of satisfication D(Q.1) of the student's answerscript regarding question Q.1 can be evaluated as follows:

D(Q. 1) = 0.9 • 0.99 + 0.8 • 0.90 + 0.5 • 0.80 0.9 + 0.8 + 0.5

= 0.9141. (14)

It indicates that the degree of satisfication of the question Q. 1 of the student's answerscript evaluated by the evaluator is 0.9141 (i.e., 91.41%).

Step 2: Consider a candidate's answerscript to a paper of 100 marks. Assume that in total there were n questions to be answered:

T O T A L M A R K S = 100 Q. 1 carries Sl marks Q.2 carries s2 marks

Q.n carries s, marks,

where Z'i'=lSi = 100, 0 -%< sl ~< 100, and 1 ~< i ~< n. Assume that the evaluated degree of satisfication of the question Q.1,Q.2 . . . and Q.n are D(Q.1),

D(Q. 2), ..., and D(Q. n), respectively, then the total score of the student can be evaluated as follows:

s t * D ( Q . 1 ) + s 2 * D ( Q . 2 ) + ... + sn*D(Q.n). (15) Put this total score in the appropriate box at the bottom of the extended fuzzy grade sheet•

In the following, we use an example to illustrate the students' answerscript evaluation process.

Example 4.1. Consider a candidate's answerscript to a paper of 100 marks. Assume that in total there were four questions to be answered:

T O T A L M A R K S = 100 Q. 1 carries 20 marks Q.2 carries 30 marks Q.3 carries 25 marks Q.4 carries 25 marks.

Assume that an evaluator awards the students' answerscript by an extended fuzzy grade sheet as shown in Table 7.

[Step 1] Based on formula (10) and by applying formula (13), we can see that

D(Q. 1) = 0.8. T(VVG) + 0 . 9 . T(VG) 0.8 + 0.9

0.8 • 0.99 + 0.9 • 0.90

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216 S.-M. Chen, C-H. Lee / Fuzzy Sets and Systems 104 (1999) 209 218 Table 7

Extended fuzzy grade sheet of Example 4.1 Question No. Q.1 Q.2 Q.3 Q.4 Satisfication levels i EG VVG VG G MG F MB B VB VVB EB 0 0.8 0.9 0 0 0 0 0 0 0 0 0 0 0 0.6 0.9 0.5 0 0 0 0 0 0 0 0.8 0.7 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 0.9 0.2 0 Degree of satisfication 0.9424 0.7050 0.8150 0.2713 Total mark = 67 0 . 6 , T(G) + 0.9* T ( M G ) + 0 . 5 , T ( F ) D(Q.2) = 0.6 + 0.9 + 0.5 0.6 • 0.80 + 0.9 * 0.70 + 0.5 * 0.60 0.6 + 0.9 + 0.5 = 0.7050. (17) 0 . 8 , T(VG) + 0 . 7 , T(G) + 0 . 5 , T ( M G ) D ( Q . 3 ) = 0.8 + 0.7 + 0.5 0.8 • 0.90 + 0.7 • 0.80 + 0.5 • 0.70 0.8 + 0.7 + 0.5 = 0.8150. (18) 0 . 5 , T(B) + 0 . 9 , T(VB) + 0 . 2 , T(VVB) D ( Q . 4 ) = 0.5 + 0.9 + 0.2 0.5 * 0.40 + 0.9 * 0.24 + 0.2 • 0.09 0.5 + 0.9 + 0.2 = 0.2713. (19)

[Step

2] By applying formula (15), the total mark of the student can be evaluated as follows: 2 0 , D(Q.1) + 3 0 , D(Q.2) + 2 5 , D(Q.3) + 25 • D(Q.4) = 20*0.9424 + 30*0.7050 + 2 5 , 0 . 8 1 5 0 + 25 * 0.2713 = 18.848 + 21.15 + 20.375 + 6.7825 = 67.155

- 67 (assuming that no half mark is given in the total score). (20)

5. A generalized fuzzy evaluation method

In this section, we generalize the method presented in Section 4 to propose a weighted method for students' answerscripts evaluation using fuzzy sets. Consider a candidate's answerscript to a paper of 100 marks.

Step

1: Assume that in total there are n questions to be answered:

T O T A L M A R K S = 100 Q. 1 carries sl marks Q.2 carries s2 marks

Q.n carries s. marks.

Assume that an evaluator evaluates the questions of students' answerscripts using the following four criteria [2]:

CI: Accuracy of information, C2: Adequate converage, C3: Conciseness,

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S.-M. Chen, C-H. Lee / Fuzzy Sets and Systems 104 (1999) 209-218

Table 8

A generalized extended fuzzy grade sheet

217

Question Criteria Satisfication levels

No. E G VVG VG G M G F MB B VB VVB Q.1 C1 C2 C3 C4 Q.2 Ct C2 C3 C4 Q.n C1 C2 C3 C4 Degree of satisfication EB for criteria D ( C l l ) D(C12) D(CI3) D(C14) D(C21) D(C22) D(C23) D(C24) D(Cnl) D(Cn2) D(Cn3) D(Cn4) Degree of satisfication for questions P(Q. 1 ) P(Q.2) P(Q.n) Total mark = s~ * P(Q.1) + s2* P(Q.2) + .-. + s.*P(Q.n)

and assume that the weights of the criteria C1, C2, C3, and C4 are wl, w2, w3, and w4, respectively, where wi~ [0, 1] and 1 ~< i ~< 4. Furthermore, as- sume that the evaluator can evaluate each question of the students' answerscripts using the above four criteria based on the method described in Section 4. In this case, an evaluator can evaluate the student's answerscripts using a generalized ex- tended fuzzy grade sheet as shown in Table 8, where the degrees of satisfication of the question Q.i of a students' answerscript regarding to the criteria C1, C2, C3, and C4 evaluated by the method presented in ection 4 are

D(Cil), D(Ci2),

D(Ci3), and

D(Ci4), respectively, where 0 ~< D(Cil)

<~ 1, 0 <~ D(Ci2) <<. 1, 0 <~ D(Ci3) <~ 1, 0 <~ D(Ci4)

~< 1, and 1

<~i<~n.

Step 2: The degree of satisfication P(Q.i) of the

question Q.i of the student's answerscript can by evaluated as follows:

P(Q.i)

w l * D ( C i l ) + w2 * D(Ci2) + w 3 * D(Ci3) + w4 * D(Ci4)

W 1 -l- W 2 -}- /4' 3 -}- Wd.

(21) where P(Q.i)e [0, 1] and 1 ~< i ~< n. The total score of the student can be evaluated and is equal to s l * P ( Q . 1 ) + s 2 * P ( Q . 2 ) + -.. + s . * P ( Q . n ) . (22) Put this total score in the appropriate box at the bottom of the extended fuzzy grade sheet.

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218 S.-M. Chen, C.-H. Lee / Fuzzy Sets and Systems 104 (1999) 209-218

6. Conclusions

In [2], Biswas has presented two fuzzy evaluation methods for students' answerscripts evaluation. In this paper, we extend the work of [-2] to present two new methods for students' answerscripts evaluation. The proposed methods can be executed much faster than the ones presented in [-2] due to the fact that they do not need to perform the complicated matching operations. Furthermore, they can make a more fair evaluation of students' answerscripts. The proposed methods can overcome the drawbacks of the ones presented in [2].

Acknowledgements

The authors would like to thank the referees for providing very helpful comments and suggestions. This work was supported in part by the National Science Council, Republic of China, under Grant NSC 86-2213-E-009-018.

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