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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) Copyright © 2013 SciRes. 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. Email: thwpf@126.com 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 Copyright © 2013 SciRes. JSEA 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 Copyright © 2013 SciRes. JSEA 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 Copyright © 2013 SciRes. JSEA 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. REFERENCES [1] F. Chiclana, F. Herrera, E. Herrea-Vidm , “Integrating three representation models in fuzzy multipurpose deci- sion making based on fuzzy preference relations, ” Fuzzy Set and Systems. vol. 97,1998,pp.33-48. [2] Yong –yue Zhu, “FAHP-based evaluation of the service quality of private-owned express companies,” Proceed- ings of 2010 International Colloquium on Computing, Communication, Control, and Management (CCCM2010) Volume 2[C], 2010. [3] Jing Yang, “FAHP-Basic Comprehensive Economic As- sessment of Wind Power Project.” Proceedings of 2009 Asia-Pacific Conference on Information Processing (Volume 2) [C], 2009. [4] Deming Fan, “Research on the teaching effect evaluation of teaching website of Quality courses in universities.” Science and Technology Information , 2007(11): pp 110 - 111. |