Journal of Software Engineering and Applications, 2013, 6, 4-7
doi:10.4236/jsea.2013.63b002 Published Online March 2013 (http://www.scirp.org/journal/jsea)
Application of Mathematical Model in Evaluating
Undergraduate’s Degree Paper*
Pin-fei Wang, Wei-ling Peng
1College of Business Administration, Tonghua Normal University, Tonghua, China; 2College of Mathematics, Tonghua Normal
University, Tonghua, China.
Received 2013
ABSTRACT
The level of undergraduate's degree paper is one of the important indicators of teaching quality. In this paper, mathe-
matical modeling of FAHP (Fuzzy Analytic Hierarchy Process) is given, and then undergraduate’s degree paper of the
college with an example is elaborated a comprehen sive evaluation of quan titative science, in order to fully mobilize the
enthusiasm of teachers and students, and constantly promote the improvement of the quality of college teaching.
Keywords: Factor Set; Fuzzy Analytic Hierarchy Process; Weight Coefficient
1. Introduction
The undergraduate paper is academic thesis which is
written independently by undergraduate student with the
requirement of the teaching programs under the guidance
of the experienced teachers before student graduate. It is
an important part of the training scheme and comprehen-
sive examination of knowledge ability and quality. With
the development of the social economy and scientific
technology, it put forward more higher requirement of
quality for colleges and universities, which improve cor-
respondingly the quality and requirements of the paper.
But it is a complicated work to evaluate it, because the
process and results are restricted in many aspects, for
example, the analysis and judgment have fuzziness and
uncertainty. It is very difficult to guarantee the thinking
coherence in dealing many indexes for traditional Ana-
lytical Hierarchy Process(AHP). In this circumstances, it
can resolve the problem to combine Fuzzy Mathematics
with AHP(AHP). By this means, many factors reflected
paper level can be calculated according to relativity and
subjection relation from top to b ottom, changing qualita-
tive problems to quantitative problems, which make the
result more correct and scientifical mathematical model.
2. Mathematical Model
Here are main steps of the Fuzzy Analytical Hierarchy
Process(FAHP). First, hierarchical structure of the sys-
tems is establish. Second, weights of the every factor are
calculated. Third, the degree of membership is decided
by fuzzy comprehensive evaluation. Finally, its final
value is computed.
2.1. Factor Set
It is crucial to establish the evalua ting indexes system for
undergraduate paper. The evaluation indexes system is
concretization of evaluation standard and core of evalua-
tion scheme. There are many factors that influence the
scientificity and reliability, and every factor contains a
number elements, so that the ratio nal indexes system can
outstand the characteristic and innovation.
The factor set is divided in to many layers, the first
one is
1
,(, ),
n
ii k
i
UUUU ik 



the second one is
1,(, ),
i
n
iijij ik
j
UUUU jk 




the last one is
1,(, ).
ij
n
ij ijk ijkijl
k
UUUUlk 




2.2. Weight Coefficient
The important degree of every index in the index system
is different, the difference can be represented by different
weight coefficient, which is equal to a mapping
)1,0(: 
i
Uw , i.e.
*Supported by JInlin education scie nce a nd pl annin g issues (GH12426)

Application of Mathematical Model in Evaluating Undergraduate’s Degree Paper
5
(),(1,2,,).
iii
UwUai n
Let the first weight coefficient be
12
(, ,, )
n
A
aa a
the second one b e
i12
A(,,,)
i
ii in
aa a
and the last one be
12
{, ,,}
ij
ijij ijijn
Aaa a.
There are many method to establish weight coefficient,
for example binomial coefficient, neighboring compara-
tive gathering statistics iterative, analytical hierarchy
process and so on.
2.3. Evaluation Set
The evaluation set is divided into many indexes. It is
represented by membership degree, and it can be re-
flected correctly the result. Let evaluation set be
12
{, ,,}
m
Vvv v,
which is applicable for every layer and factor.
2.4. FAHP Model
The ordinary model is principal-factor-decision (,)M

,
principal-factor-outstanding (, )M , (or (, )M ), and
the weighted mean (, )M . The weighted mean is
adopted because it is suitable for the factor with weight.
By experiment the algorithm is effective and simple.
Step one considering the third layer comprehensive
evaluation 1, 2,
{,}
ij
ijij ijijn
Uuu u, let fuzzy mapping
:()
ij ij
f
UFV
, i.e.
12
()(, ,, )()
ij ijkijkijkijkm
f
urr rFV
,
then the third fuzzy relation matrix is built
11 121
21 222
12
ij ijij
ijijijm
ijijij m
ij
ijnijnijn m
rr r
rr r
R
rr r











,
where
01,1,2, ,,1,2, ,,1,2, , ,
ijkqi ij
ri nj nk n 
1, 2,q,m
ijkq
r



,

stands for the number of the expert that ijk u be-
long to q
V, and

stands for the number of the all ex-
pert.
The third layer fuzzy comprehensive evaluation is
calculated according to weight distribution
1, 2,
(,)
ij
ijij ijijn
Aaa a,
11 121
21 222
1, 2,
12
12 ,
(,)
(, ,,)
ij
ij ijij
ij
ijijij m
ijijijm
ijijij ijijn
ijnijnijn m
ij ijijm
B
rr r
rr r
AR aaa
rr r
bb b



 










where
1
ij
n
ijqijp ijpq
p
bar



,
(1,2,,,1,2, ,,1,2, ,)
i
injnqm  
Step two considering the second layer comprehensive
evaluation 1, 2, {,}
i
iii in
UUU U，the second fuzzy relation
matrix is obtained, i.e.
11112 1
22122 2
12
m
m
nnnnm
Bbb b
Bbbb
R
Bbb b

 
 
 
 
 
 




If weight distribution is
12
(, ,, )
n
A
aa a
then the first fuzzy comprehensive evaluation is obtained
12
(,,,) ()
m
B
AR bbbFV 
where
1
,(1,2,, )
n
qppq
p
babq m

 
.
3. Application Example
The evaluation index system is different for various uni-
versity. Here is an example of undergraduate’s degree
thesis in Tonghua Normal university.
3.1. Evaluation Index System
There are three indexes in the first layer of evaluation
index system, thirteen indexes in the second layer, and
sixteen indexes in the third layer, which is in the paper
quality because of its importance.
3.2. Weight Coefficient
A number of experienced experts who is invited score
according to their importance, and then calculate their
weight coefficient by superi ority chart.

Application of Mathematical Model in Evaluating Undergraduate’s Degree Paper
6
3.3. Evaluation Result
Ten experienced examinants give the grade according to
evaluation index system.
Table 1. Weight codfficient.
The First
Index The Second Index The Third Index
Discipline(0.15)
Activeness(0.15)
Consulting(0.15)
Investigation(0.40)
Attitude
(0.15)
Writing (0.15)
Innovation level(0.3)
Academic value(0.36)
Application value(0.18)
Ability for
selecting topic
(0.25) Difficult degree (0.16)
Richness(0.27)
Reality(0.27)
Timeliness(0.27)
Choice of data
(0.15) Correlation(0.19)
Argument establishment(0.34)
Verification method (0.20)
Material structure(0.17)
Verifying
ability
(0.35) Logic structure(0.29)
Title(0.16)
Abstract , keywords(0.36)
Language, punctuation, signal(0.32)
Quality
(0.70)
Expression
ability
(0.25) Notes(0.16)
Content introduce
(0.40)
Answering
problem (0.35)
Language (0.15)
Answering
statement
(0.15) Politeness,
apparatus (0.10)
Table 2. Evaluation grade.
Evaluation Set
No. Index A B C D
1 Discipline 8 2 0 0
2 Activeness 9 1 0 0
3 Consulting 8 1 1 0
4 Investigation 9 1 0 0
5 Writing 9 1 0 0
6 Innovation level 7 2 1 0
7 Academic value 8 1 1 0
8 Application value 7 1 1 1
9 Difficult degree 8 1 1 0
10 Richness( 7 2 1 0
11 Reality 8 1 1 0
12 Timeliness 7 1 1 0
13 Correlation 8 0 1 1
14 Argument establishment 8 2 0 0
15 Verification method 8 1 1 0
16 Material structure 7 2 1 0
17 Logic structure 9 1 0 0
18 Title 9 1 0 0
19 Abstract , keywords 9 0 1 0
20 Language, punctuation, signal 8 2 0 0
21 Notes 7 1 1 1
22 Content introduce 9 1 0 0
23 Answering problem 8 1 1 0
24 Language 9 1 0 0
25 Politeness, apprance 10 0 0 0
3.4. Fuzzy Comprehensive Evaluation
Fuzzy comprehensive evaluation of the third layer fac-
tor set is 21211 212213 214
{, , , }UUUUU
21 21 21
,
0.70.2 0.10
0.80.1 0.10
(0.70.10.1 0.1
0.80.1 0.10
()
0.3,0.36,0.18,0.16
0.752,0.13,0.10,0.018
)BAR



 




likewise, 22 (),0.746,0.108,0.10,0.046B
23 (),0. 812,0.151,0.037,0B
24 ()0.83 6,0.096,0. 052,0. 016B
Fuzzy comprehensive evaluation of the second layer
factor set is 111121314
{, , , }UUUUU
111
(
0.80.200
0.9 0.100
0.15,0.15,0.15 0.40,0.15)0.80.10.1 0
0.9 0.100
0.90.1 00
(0.87 ,0.115,0.015,0).
,
BAR











333
(
0.9 0.100
0.80.1 0.1 0
0.40,0.35,0.15 0.10)0.90.100
1.000 0
,
(0.875,0.09,0.035,0 ).
ARB










222
()
0.752 0.130.10.018
0.746 0.1080.10.046
0.8120.151 0.0370
0.836 0.096 0.052 0.016
0.25,0. 1 5,0.35,0.25
(0.7931,0.1256,0.0659,0.01 54 ).
BAR









Fuzzy comprehensive evaluation of the first layer
factor set is 123
{,, }UUUU,then

Application of Mathematical Model in Evaluating Undergraduate’s Degree Paper
7
Table 3. Grades and score segments.
Number Grades Score segments
1 Excellent 90------100
2 Good 80------90
3 Pass 60------80
4 Fail 60 Below
()
0.870.115 0.0150
0.79310.12560.06590.0154
0.875 0.0900.0350
0. 15,0.70,0.15
( 0. 8169,0.1186,0.0537,0.0108).
BAR








The results indicate that 81.69 % of experts think it is
excellect, that 11.86% of experts think it is good,
5.37% of experts think it is pass, and anothers think it is
fail. According to maximum subordination principle, the
result of above evaluation is excellent.
The grade theory domain is used for the sake of ob-
serving visual image. The evaluation set is described by
different score segments, see Table 3.
Let 95,85,75,35 represent the score of the different
grades, then the grad e matrix is (95,85,75,35)T C. The
final score is
95
85
(0.8169,0.1186,0.0537,0.0108) 92.09.
75
35
T
FBC 







It is well know that the grade of the undergraduate’s
degree paper is excellect .
4. Conclusion
The undergraduate paper is one of the important indica-
tor that show practical teaching of university. It is very
crucial to evaluate undergraduate paper. FAHP is not
only considering the internal relationship between the
various indicator fuzziness of the system, but also pos-
sessing the basis of Fuzzy Mathematics, Matrix Theory,
and AHP, therefore FAHP can accurately reflect the level
of the undergraduate‘s degree thesis, this method is sci-
entific.
The steps is clear, the judgment is simple in entire
model, and it can be calculated by using mathematical
software Mathematica and Mathlab when calculation
amount is very complicated, so FAHP can avoid confu-
sion that caused by inaccurate scoring and reduce the
workload of teacher.
It is rigorous in theory, and convenient in application,
especially, the programming effective of the FAHP is
verified through some examples, there is extensive ap-
plication space and wonderful development prospects in
the field of life and production.
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[3] Jing Yang, “FAHP-Basic Comprehensive Economic As-
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