2016年美国大学生数学建模竞赛C题H奖论文.pdf

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1、For office use onlyT1 _T2 _T3 _T4 _Team Control Number 44952Problem Chosen CFor office use onlyF1 _F2 _F3 _F4 _One in a hundred-The Goodgrant Challenge Summary The Goodgrant Foundation wants to donate money to an appropriate group of schools per year,we were asked to develop a model to determine an

2、optimal investment strategy.Firstly,filtering data.Because the problem has a large number of additional data,we must preliminarily classify the data.We use SPSS to achieve above purpose,after that we get the valid data of 2936 potential candidate schools.Secondly,using Grey Relational Analysis (GRA)

3、to optimize the data roughly.By using MATLAB we calculate the grey relational coefficient and grey weight relation.And then we compare the size of every grey weight relation,taking the first 300 potential candidate schools as the evaluation objects.Thirdly,using Principal Component Analysis (PCA)to

4、build up the ROI (return on investment)Model.During the process of GRA,there are so many indexes influence the result,we can t get the accurate solution.So we use“PCA”method to simplify the problem and evaluate the ROI of every school.By using SPSS,we determine the main components and calculate the

5、comprehensive score.We rank the evaluation objects on the basis of score and choose the first 40 objects as the optimization result.And then,testing the reliability.We standardize the indexes of PCA,by comparing the optimization result with well-known universities which we don t choose(such as Harva

6、rd University),we can examine the rationality of our model and get the candidates list.Finally,using TOPSIS method to build up the Investment Prediction Model.We use the previous data(20092013)to make the Linear Regression Forecasting(LRF),after that we get the predicted data from 2016 to 2020.We us

7、e TOPSIS method to evaluate the candidate school list.According to the comprehensive score from 2016 to 2020,we find the first 20 schools which get the higher score are always higher than other school from 2016 to 2020.So we choose that 20 school which both have higher ROI and development potential

8、as the final candidate school list.According to the final candidate school list,we use the TOPSISmethod again to calculate the final comprehensive score,According to the score,we decide how much money The Goodgrant Foundation should invest for each school which is showed in Table 6.2.Key Words:GRA,P

9、CA,TOPSIS,ROI,LRFContents1.Introduction.11.1 Background.11.2 Foundation&ROI.12 Task.13 Fundamental assumptions .24 Definitions and Notations .25 Models.35.1 Filter data.35.2 Object Selection Model(Grey Relational Analysis).45.2.1 Model analysis.45.2.2 Model solution.45.3 ROI Model(Principal Componen

10、t Analysis).55.3.1 Model analysis.55.3.2 Model solution.65.4 Verify the possibility.95.4.1 Comparison.95.4.2 External factor.10 5.5 Investment Forecast Model.11 5.5.1 Linear Regression Forecasting Model.11 5.5.2 School potential Prediction(TOPSIS).12 5.5.3 Final investment(TOPSIS).13 6 Conclusions .

11、167 Strengths and Weaknesses.18 7.1 Strengths.19 7.2 Weaknesses.20 8 Letter to Mr.Alpha Chiang .219 References.22Team#44952 Page 1 of 22 1 Introduction1.1 Background The Goodgrant Foundation is a charitable organization that wants to help improve educational performance of undergraduates attending c

12、olleges and universities in the United States.To do this,the foundation intends to donate a total of$100,000,000(US100 million)to an appropriate group of schools per year,for five years,starting July 2016.In doing so,they do not want to duplicate the investments and focus of other large grant organi

13、zations such as the Gates Foundation and Lumina Foundation.Our team has been asked by the Goodgrant Foundation to develop a model to determine an optimal investment strategy that identifies the schools,the investment amount per school,the return on that investment,and the time duration that the orga

14、nization s money should be provided to have the highest likelihood of producing a strong positive effect on student performance.This strategy should contain a 1 to N optimized and prioritized candidate list of schools you are recommending for investment based on each candidate schools demonstrated p

15、otential for effective use of private funding,and an estimated return on investment(ROI)defined in a manner appropriate for a charitable organization such as the Goodgrant Foundation.1.2 Foundation&ROI Foundation(charitable foundation)refers to the nonprofit legal person who uses the property of the

16、 natural persons,legal persons or other organizations to engage in public welfare undertakings.In terms of its nature,foundation is a kind of folk non-profit organizations.ROI is a performance measure used to evaluate the efficiency of an investment or to compare the efficiency of a number of differ

17、ent investments.ROI measures the amount of return on an investment relative to the investments cost.To calculate ROI,the benefit(orreturn)of an investment is divided by the cost of the investment,and the result is expressed as a percentage or a ratio.2 Task One-page summary for our MCM submission Us

18、ing our models to achieve the candidate list of schools Calculate the time durati on that the organizations money should be provided to have the highest likelihood of producing a strong positive effect on student performance Calculate the investment amount Goodgrant Foundation would pay for each sch

19、ool Calculate the ROI of the Goodgrant Foundation Forecast the development of this kind of investment mode Write a letter to the CFO of the Goodgrant Foundation,Mr.Alpha Chiang,that describes the optimal investment strategy Team#44952 Page 2 of 22 3 Fundamental assumptions1)The indexes of GRA(such a

20、s ACT、SAT、Pell Grant、Graduation Rate、Retention Rate、Graduates income)are the most influential factor that affect the use potential of school funds,what s more,the indexes have the same weight 2)For four-year universities,their C200_L4_POOLED_SUPP、RET_FTL4、RET_PTL4 are zero;For two-year colleges,thei

21、r C150_4_POOLED_SUPP、RET_FT4、RET_PT4 are zero 3)For public institutions，their NPT4_PRIV、NPT41_PRIV、NPT42_PRIV、NPT43_PRIV、NPT44_PRIV、NPT45_PRIV are zero;For private for-profit and nonprofit institutions,their NPT4_PUB、NPT41_PUB、NPT42_PUB、NPT43_PUB、NPT44_PUB、NPT45_PUB are zero 4)We define“NULL”appears

22、 in the data except appears in 3)as the average of that series5)Ignore the influence of degree-conferring situation,race,religion,region6)Schools data of SAT and ACT develop in a linear trend 4 Definitions and Notations Table A:The Excel which contain the IPEDS UID for Potential Candidate Schools Ta

23、ble B:The Excel which contain the Most Recent Cohorts Data(Scorecard Elements):The weight of the first k index)(ki:Grey relational coefficient ri:Grey weight relation:Standardized index value:Index value:Sample average:Sample standard deviation:Standardized index vector:The correlation coefficient y

24、i:Main components:Rate of contribution:The cumulative contribution rate:The comprehensive score kwaijaijjjsxjrijbjpZTeam#44952 Page 3 of 22 5 Models 5.1 Filter data Because the topic has a large number of additional data,we should classify the data based on the College Scorecard Data Dictionary whic

25、h we can find from the official website.And then,according to the flow diagram5.1.1 as follow,we can filtering data.By using that flow diagram,we set up limits to filter the data circularly.We use SPSS to achieve above purpose,after that we get the valid data of 2936 potential candidate schools.Yes

26、No Figure 5.1.1 Flow Diagram Input Table A,Table B Based on the ID in Table A,merge Table A and B Table A has the ID which Table B don t Delete that ID in Table A Output Table A Output Table A Team#44952 Page 4 of 22 5.2 Object Selection Model(Grey Relational Analysis)5.2.1 Model analysis Not only t

27、here is a large amounts of missing in the original data,but also even after a preliminary screening,the data volume is still very large.We find there are more than 50 factors which affect us optimizing the school,what s more,the link between each factor we cant find accurately.So the normal model ha

28、s no use to predict and evaluate the data.But Grey correlation analysis method is both suitable for irregular data and normal data.The quantitative results are consistent with qualitative analysis perfectly.Therefore,we choose that method to further narrowing the scope of the data.Grey correlation a

29、nalysis is based on the similarity degree of various factors changing curve,it can judge the correlation degree of each index.Through the quantitative analysis of the dynamic process,we can get the geometrical relationship of statistical data in the system and the grey correlation degree between the

30、 reference sequence and compare sequence.The greater the comparative sequence s correlation degree is,the closer the relationship between the reference sequence and compare sequence will be.The basic idea is to standardized the original observation and calculate the correlation coefficient,correlati

31、on degree.And then rank the indexes according to the size of the correlation.The application of GRA involves many fields,especially in the field of social economy,such as the ROI of the national economy departments,analyzing regional economic advantages,industrial structure adjustment,GRA has a good

32、 application effect.5.2.2 Model solution1)We have 2936 objects（potential candidate schools）and 7 evaluation indexes（ACT、SAT、Pell Grant、Graduation Rate、Retention Rate、Graduates income），the reference sequence is 7,.,2,1|)(00kkxx,the compare sequence is 2936,.,2,1,7,.,2,1|)(ikkxiix；2)The weight of ever

33、y index is,.,71www，)7,.,2,1(kwkmeans the first k index s weight,in that model we assume every index s weight is equal.3)|)()(|maxmax|)()(|)()(|maxmax|)()(|minmin)(0000txtxtxkxtxtxtxtxkstsistsstsi4)(71kikiiwr5)By using MATLAB,we can finish above data calculation process,and then we can make the figur

34、e of each evaluation objects grey weight relation as follow:Team#44952 Page 5 of 22 Figure 5.2.1 Distribution Diagram of Grey Weight Relation According to Figure 5.2.1,we find that 90%objectsgrey weight relation is under 0.60.We use MATLAB to rank the evaluation objects,because the greater the grey

35、weight relation is,the better the evaluation result is.We choose the first 300 objects as the new potential candidate schools.But during the process of GRA,the weight of evaluation indexes has deviation,we cant get the accurate solution,so we need build another model to analyze in detail.5.3 ROI Mod

36、el(Principal Component Analysis)5.3.1 Model analysis According to the topics requirements,we should choose the potential candidate schools by their money using ability,but there are so many indexes influence the result,we cant get reliable evaluation only by one factor.In the study of that practical

37、 problem,in order to comprehensively and systematically analyze problems,we must consider many factors.These involved factors generally referred to the index,also known as a variable in the multivariate statistical analysis.Because each variable reflects some information of the research question and

38、 all of them has a certain correlation,the information one index reflects may overlap another.PCA is to use less variables to explain most variables of the original data,it transforms many high correlation variables into uncorrelated variables.Usually the number of new variables is smaller than the

39、original variables,we call it principal component and use it to explain the comprehensive index.This method simplify the problem,at the same time make the result more scientific and effective Therefore,it seems easier to use PCA solving the Optimization problem.By reducing dimension,we transform man

40、y indexes(such as Net Price,Repayment of Debt,Repayment Rate,Graduates Income)into a few principal components.Team#44952 Page 6 of 22 5.3.2 Model solution1)Standardized the original data,15,.,2,1,300,.,2,1,jisjjijijaa;15,.,2,1,)(13001,3001300123001jaaijijjiijjs15,.,2,1,jsxxjjjj2)Calculate the correl

41、ation coefficient matrix,15*15)(rijR，15,.,2,1,1300*3001jiaarkjkkiij3)Calculate the eigenvalue and the feature vectors,By using SPSS,we calculate the eigenvalue of R,0.1521，And the standardized feature vector,.,1521by using the feature vector,we get 15 new indexes，,.,.,.151515221511151515152222112215

42、1512211111xxxxxxxxxyyy4)Choose P（15P）principal component，calculate the comprehensive score Calculate the contribution rate of)15,.,2,1(jjand cumulative contribution rate 151151,jjpkkjjbbBy using SPSS,we get the contribution rate of 15 eigenvalue as follow:Team#44952 Page 7 of 22 Table 5.3.1 Contribu

43、tion Rate Analyzing Table 5.3.1，we find the cumulative contribution rate of the first three eigenvalue is more than 90%,the model has a well result.Choose the first three principal component for a comprehensive evaluation,ybjpjjZ1By using SPSS,we get the feature vectors of the first three characteri

44、stic root,shows in Table 5.3.2 Table 5.3.2 The Feature Vectors Team#44952 Page 8 of 22 We make the first three principal components contribution rate as weight,build up the principal component comprehensive evaluation model,yyyZ3211031.01153.06892.0Put every optimized schools three principal compone

45、nts into above equation，we get the comprehensive evaluation result of 300 new potential candidate schools,shows in figure 5.3.3，Fac_1 on behalf of y1，Fac_2 on behalf of y2,Fac_3 on behalf of y3，Total on behalf of Z，Figure 5.3.3 Comprehensive evaluation result of 300 schools We choose the first 40 sc

46、hools as the final potential candidates list，the comprehensive evaluation result shows in figure 5.3.4，Figure 5.3.4 Comprehensive evaluation result of 40 final potential schools Team#44952 Page 9 of 22 Because the indexes(such as Net Price,Repayment of Debt,Repayment Rate,Graduates Income)of this mo

47、del are closely related to ROI,we use the comprehensive score to evaluate schools ROI.We ranked it in Table 5.3.3,Table 5.3.3 Rank 5.4 Verify the possibility 5.4.1 Comparison Through comparison we found that many famous universities such as Harvard University does not appear in our preferred list,th

48、is can t help but let us create confusion,what is the reason causes this kind of phenomenon?IDNAMESCORERANKRhode Island School of Design1.154 1MCPHS University1.113 2School of the Art Institute of Chicago1.106 3Worcester Polytechnic Institute1.006 4Rose-Hulman Institute of Technology1.005 5New York

49、University0.983 6Quinnipiac University0.938 7Emerson College0.925 8Bryant University0.918 9Rensselaer Polytechnic Institute0.917 10Seattle University0.912 11Kettering University0.905 12Drexel University0.900 13Albany College of Pharmacy and Health Sciences0.892 14Boston University0.855 15Carnegie Me

50、llon University0.854 16Providence College0.846 17Loyola University Maryland0.845 18Wentworth Institute of Technology0.842 19Baylor University0.826 20Loyola Marymount University0.816 21University of St Thomas0.815 22Fordham University0.812 23University of Puget Sound0.809 24Saint Louis University0.80