2016年O奖论文C42939Tsinghua University China

For office use only T1 T2 T3 T4 Team Control Number 42939 ProblemChosen C For office use only F1 F2 F3 F4 2016 Mathematical Contest in Modeling (MCM) Summary Sheet (Attach a copy of this page to each copy of your solution paper.) Summary In order to determine the optimal donation strategy, this paper proposes a data- motivated model based on an original definition of return on investment (ROI) appropriate for charitable organizations. First, after addressing missing data, we develop a composite index, called the performance index, to quantify students educational performance. The perfor- mance index is a linear composition of several commonly used performance indi- cators, like graduation rate and graduates earnings. And their weights are deter- mined by principal component analysis. Next, to deal with problems caused by high-dimensional data, we employ a lin- ear model and a selection method called post-LASSO to select variablesthat statis- tically significantly affect the performance index and determine their effects (coef- ficients). Wecall them performance contributing variables. In this case, 5 variables are selected. Among them, tuition PPTUG_EF is share of students who are part- time;Carnegie_HighResearchActivity is Carnegie classifica- tion basic: High Research Activity The results presented in Table 3 are consistent with common sense. For instance, the pos- itive coefficient of High Research Activity Carnegie classification implies that active research activity helps students educational performance; and the negative coefficient of Student-to- Faculty ratio suggests that decrease in faculty quantity undermines students educational per- formance. Along with the large R square value and small p-value for each coefficient, the post-LASSO procedure proves to select a valid set of performance contributing variables and describe well their contribution to the performance index. 5Determining Investment Strategy based on ROI Weve identified 5 performance contributing variables via post-LASSO. Among them, tu- ition fROIiis fittedROIs ofperformance contribut- ing variables; Piis a tuning parameter to adjust the fitted ROIs of performance contributing variables. As we can see, ROI is institution-specific and dependent on increase in donation amount (X). And because increase in donation amount is discrete,we geta finite set of pos- sible ROI forevery institution. Team # 42939Page18of24 5.3School Selection and Scenario 2 is where we predict the trends and determine 5-year investment strategy only based on the original dataset. In Scenario 1, we adopt a dynamic calibration method based on the outcome of institutions we have patronized. To be more specific, we first determine the investment strategy for the first year. And in year 2, we would have had access to new data of year 1. Then we apply the baseline model again with the new information or merged dataset. This would generate new results after the model learning from the latest behaviors of each institution and making the calibrations on the predicting patterns. Similarly, this updating method can be used in the donation years afterwards. In Scenario 2, we adopt a partial calibration method based on the coefficients we calculate with the original dataset. According to assumption A1, we can simply treat status of institu- tions we have not patronized as fixed. And the only values changed are donation amounts and values of performance contributing variables of patronized institutions. This barely af- fects the post-LASSO procedure considering the largequantityof variablesused in the process. However, this would lead to movement of points of patronized institutions in the scatterplots of performance contributing variables and donation amount. So we recommend to recalcu- late the fitted curve and ROI and determine the investment strategy for year 2. As for how to determine the movement of points, below is the procedure: Find the institutions you have patronized in year 1; Add the amount you have patronized to their original donation amount; XI= X + X Calculate the values of performance contributing variables as follows: Fi= Fi+ ROIi X The results for Scenario 2 are presented below in Table 5 to Table 8: Team # 42939Page20of24 Table5: InvestmentStrategyofYear2 UNITIDnamesROIdonation 197027United States Merchant Marine Academy21.82%2500000 102711AVTEC-Alaskas Institute of Technology21.25%7000000 187745Institute of American Indian and Alaska Native Culture20.93%2000000 262129NewCollegeofFlorida20.73%6500000 216296ThaddeusStevensCollegeofTechnology20.62%3000000 229832Western Texas College20.23%10000000 196158SUNY at Fredonia20.23%5500000 234155VirginiaStateUniversity20.02%9000000 196200SUNYCollege atPotsdam19.74%5000000 178615TrumanStateUniversity19.64%3000000 199120University of North Carolina at Chapel Hill19.52%3000000 101648Marion Military Institute19.48%2500000 187912New Mexico Military Institute19.31%1500000 227386Panola College19.22%9500000 434584Ilisagvik College19.19%4500000 199184University of North Carolina School of the Arts19.18%2500000 413802EastSan GabrielValleyRegional OccupationalProgram19.13%6000000 174251University ofMinnesota-Morris19.07%8000000 159391Louisiana State University and Agricultural & Mechanical College19.07%8500000 403487WabashValleyCollege19.00%500000 Table6: InvestmentStrategyofYear3 UNITIDnamesROIdonation 197027United States Merchant Marine Academy21.65%2000000 102711AVTEC-Alaskas Institute of Technology21.29%7000000 187745Institute of American Indian and Alaska Native Culture21.01%2000000 216296ThaddeusStevensCollegeofTechnology20.99%3500000 262129NewCollegeofFlorida20.87%6000000 229832Western Texas College20.42%9500000 196158SUNY at Fredonia20.42%5500000 234155VirginiaStateUniversity20.32%9000000 196200SUNYCollegeatPotsdam20.27%5000000 178615TrumanStateUniversity20.06%3000000 187912New Mexico Military Institute20.00%2000000 101648Marion Military Institute19.96%2500000 199120University of North Carolina at Chapel Hill19.93%2500000 227386Panola College19.87%9500000 434584Ilisagvik College19.87%4500000 199184University of North Carolina School of the Arts19.73%2500000 413802East SanGabriel ValleyRegional Occupational Program19.69%6000000 174251University of Minnesota-Morris19.25%8000000 142179EasternIdahoTechnicalCollege19.15%1000000 159391Louisiana State University and Agricultural & Mechanical College19.04%7000000 403487WabashValleyCollege3.16%2000000 Team # 42939Page21of24 Table7: InvestmentStrategyofYear4 UNITIDnamesROIdonation 197027United States Merchant Marine Academy21.72%2000000 102711AVTEC-Alaskas Institute of Technology21.34%6500000 187745Institute of American Indian and Alaska Native Culture21.25%2000000 216296ThaddeusStevensCollegeofTechnology21.15%4000000 262129NewCollegeofFlorida21.13%6500000 196158SUNY at Fredonia20.98%2500000 229832Western Texas College20.92%10000000 234155VirginiaStateUniversity20.80%9000000 187912New Mexico Military Institute20.73%3000000 101648Marion Military Institute20.68%2500000 196200SUNYCollege atPotsdam20.66%1500000 178615TrumanStateUniversity20.37%4000000 199184University of North Carolina School of the Arts20.15%2000000 227386Panola College19.66%10000000 434584Ilisagvik College19.52%4500000 199120University of North Carolina at Chapel Hill19.45%2000000 413802EastSan GabrielValleyRegional OccupationalProgram19.45%6000000 174251University ofMinnesota-Morris19.36%8000000 142179EasternIdahoTechnicalCollege19.35%1500000 159391Louisiana State University and Agricultural & Mechanical College19.13%7000000 111188CaliforniaMaritimeAcademy18.98%5500000 Table8: InvestmentStrategyofYear5 UNITIDnamesROIdonation 197027United States Merchant Marine Academy21.65%2000000 102711AVTEC-Alaskas Institute of Technology21.20%6500000 187745Institute of American Indian and Alaska Native Culture21.20%2500000 216296ThaddeusStevensCollegeofTechnology21.05%4000000 262129NewCollegeofFlorida20.83%6500000 196158SUNY at Fredonia20.78%2500000 229832Western Texas College20.56%10000000 234155VirginiaStateUniversity20.34%10000000 187912New Mexico Military Institute20.25%3000000 101648Marion Military Institute20.21%2500000 196200SUNYCollege atPotsdam20.03%1500000 178615TrumanStateUniversity20.00%4000000 199184University of North Carolina School of the Arts19.76%2000000 227386Panola College19.58%10000000 434584Ilisagvik College19.48%4500000 199120University of North Carolina at Chapel Hill19.46%2000000 413802EastSan GabrielValleyRegional OccupationalProgram19.61%6000000 174251University ofMinnesota-Morris19.48%8000000 142179EasternIdahoTechnicalCollege19.35%500000 159391Louisiana State University and Agricultural & Mechanical College18.95%7000000 111188CaliforniaMaritimeAcademy19.04%5000000 By following above mentioned methods, it is viable to determine the time duration or the investment strategy across time dimension. This approach can effectively avoid waste on do- nations by calibrating or partially calibrating the baseline model. Team # 42939Page22of24 6.2Geographical Distribution Another element that can be incorporated into the baseline model is the geographical distri- bution of donations. It matters in two ways. First, regional equality often raises heated debate among citizens and demands appropri- ate treatment. Consequently, charitable organizations are supposed to avoid displaying clear pattern of regional disparity on donations. Second, according to decreasing marginal utility theory, its reasonable to diversify investment with respect to regions. Assuming graduates are more likely to get employed where their colleges are, the marginal utility of corresponding social benefits, such as more taxpayers, smaller crime rate, and increase in GDP, would decline asdonationsarecentralized. To deal with this factor, we can simply divide institutions to different geographical groups, split the donations and then apply the model to every group. Another proposed approach is to apply the baseline model with discounted ROI rather than the original definition. If the baseline model selects more than a predetermined number of institutions from a single region, we discount the ROI of these investments. For example, we could halve the ROI of excessive investments (Specific discounting method might need further exploration). Then we deter- mine the strategy to maximize the sum of discounted ROI. In this manner, we could effectively alleviate regional related problems. 7Conclusions and Discussion Conclusions: This paper manages to develop a fully data-based model to produce a provi- dent investment strategy that maximizing ROI. Our model exhibits a great potential in drawing theconclusionsbelow: We formulate a performance index for each institution with principal component analysis and develop an appropriate concept for return of investment (ROI) for the charitable foundations like Goodgrant Foundation. We identify three main performance contributing variables that generate a strong impact on the performance index with post-LASSO procedure: percentage of students who re- ceive a Pell grant amount, the students that are part time and the student-to-faculty ratio. We derive the relation between the performance contributing variables and donation amount from a GAM fitting model to predict ROI of performance contributing variables. The final list of selected institutions and appropriate amount of donation is determined by a two-step selection algorithm. Limitation andextensions: Though our model successfullyproduced an investment strategy, it can be improved in several ways: Since only cross-section data is available, our results of time duration of donation can be improvedif we haveaccesstotime-series. The post-LASSO selection only allows for a relatively simple linear model. A more gen- eral selection method is needed when applying to a complicated model. Team # 42939Page23of24 References 1 Schifrin, M., Chen L. (2015). Top50ROIColleges:2015GratefulGradsIn- dex. gradsindex/#285011fc15cc/ 2 Burke,J.C.(2002).Fundingpubliccollegesanduniversitiesforperformance.SUNYPress. 3 Cave, M. (1997). The use of performance indicators in higher education: The challenge of thequalitymovement.JessicaKingsleyPubli