Data Mining Techniques and Mathematical Models for the Optimal Scholarship Allocation Problem

112 Pages·2017·0.51 MB·English

by  Shuai Wang

DATA MINING TECHNIQUES AND
MATHEMATICAL MODELS FOR THE
OPTIMAL SCHOLARSHIP ALLOCATION
PROBLEM FOR A STATE UNIVERSITY
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
by
SHUAI WANG
B.S., Management, Dalian Jiaotong University, 2011
2017
Wright State University
WrightStateUniversity
GRADUATESCHOOL
December14,2017
I HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER MY
SUPERVISION BY Shuai Wang ENTITLED Data Mining Techniques and Mathematical
Models for the Optimal Scholarship Allocation Problem for a State University BE AC-
CEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DE-
GREEOFDoctorofPhilosophy.
XinhuiZhang,Ph.D.
DissertationDirector
FrankW.Ciarallo,Ph.D.
Director,Ph.D.inEngineeringProgram
BarryMilligan,Ph.D.
InterimDeanoftheGraduateSchool
Committeeon
FinalExamination
XinhuiZhang,Ph.D.
PratikParikh,Ph.D.
CarolineCao,Ph.D.
SubhashiniGanapathy,Ph.D.
NanKong,Ph.D.
ABSTRACT
Wang, Shuai. Ph.D in Engineering Program, Department of Biomedical, Industrial and Human
Factors Engineering, Wright State University, 2017. Data Mining Techniques and Mathematical
ModelsfortheOptimalScholarshipAllocationProblemforaStateUniversity.
Enrollment Management and Financial Aid. Enrollment management is the term that
is often used to describe the synergistic approaches to influence the enrollment of higher
education institutions, and consists of activities such as student college choice, transition
to college, retention, and graduation. Of all the factors, financial aid, institution rank,
and tuition are the three most important ones that affect students’ choice processes and
matriculation decisions; as such, with the continuous increase of tuition over the years,
financial aid serves as a marketing tool and plays an important role in attracting students.
In the United States, in the 2012-2013 academic year, there were a total of 20.4 million
students enrolled in degree-granting institutions and more than eighty percent of them
receivedfinancial.
TheOptimalScholarshipAllocationProblem: Thewidespreaduseoffinancialaidleads
to an important problem yet to be solved in the literature, i.e., how to optimally allocate
the limited financial aid to students with various social and economic backgrounds so
as to achieve enrollment goals. Though financial aid can be of various forms, merit-based
scholarshipsaretheprimarypartoftheallocationprocess. Thisproblem,referredtoasthe
optimalscholarshipallocationproblem,haspuzzledtheenrollmentmanagementteamsat
manyhigherinstitutionsandisthefocusofthisthesis.
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Solution Approach: This thesis proposes a series of predictive and optimization models
to solve the optimal financial aid allocation problems. The methodology consists of three
sequential phases: 1) predictive models to find the responses (enrollment and graduation
probabilities and years of study) to various levels of scholarship for students with various
socioeconomic backgrounds; 2) optimization models to find the maximum revenue for
given budget based on the response discovered to the various levels of scholarships; and
3) data mining models to discover patterns and transform results from the optimization
modeltosimpleandeffectivepolicies.
Phase I: Predictive Models. A series of predictive models have been investigated to esti-
matetheresponsesfromstudentstovariouslevelsofscholarshipawards. Theseresponses
canbeclassifiedintotwocategories: thefirstcategoryincludesenrollmentandgraduation
decisions and the second one is the number of years of study once a student enrolls in the
institution. In the first category, because of the binary nature of the responses (enroll or
not enroll), logistic regression based models have been adopted to predict the probability
of enrollment and the probability of graduation given that student enrolls. In the second
category,regressionanalysisareadopted.
Phase II: Optimization Models. An optimization model is designed to allocate financial
aid to applicants with an objective to maximize the revenue, which is composed of net
tuition, i.e., tuition minus scholarship, over the years of study, plus the state share of
instruction once the student graduates. The constraints to be observed include the total
budget limitations and a fairness constraint. For a merit-based scholarship, the fairness
constraint stipulates that a student with better academic performance must be assigned
to an equal or higher level of scholarships than that of students with a lower academic
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performance. The inclusion of the fairness constraint has dramatically increased the size
of the model, and to reduce computational burden, the concept of a minimum dominance
set is developed. This has reduced the size of the model by orders of magnitude and
enabledtheefficientsolutionoftheresultingmathematicalmodel.
Phase III: Policies Analysis Models. Regression analysis is developed to discover pat-
terns in the optimization results, in the form of the amount of scholarship awarded for
each student, and translate them into simple and effective scholarship award policies for
implementation. Several techniques such as decision tree and piecewise regression have
beenexplored. Fortheinstitutionunderstudy,theresultssuggestedthatacompositescore
based on the student’s GPA and ACT scores can be used as the basis for the award of
scholarships; and a simple yet effective award scholarship policy derived from piecewise
regressionhasbeendiscovered.
Implementation: The analysis based on the above framework was adopted by the in-
stitution under study and has been used in an overhaul of the scholarship redesign. The
piecewiseregressionderived,compositescorebasedscholarshipawardpolicyprovestobe
effective, and together with a proactive marketing strategy it has yielded an 11% increase
indirectlyadmittedstudentsunderasimilarbudget. Thistranslatesintomillionsofdollars
ofrevenueandsignificantlyimprovestheuniversity’sbottomline.
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Contents
1 Introduction 1
1.1 EnrollmentManagementandFinancialAid . . . . . . . . . . . . . . . . 1
1.2 TheScholarshipAllocationProblem . . . . . . . . . . . . . . . . . . . . 3
1.3 AThreePhaseSolutionApproach . . . . . . . . . . . . . . . . . . . . . 7
1.4 ImplementationandFinancialResults . . . . . . . . . . . . . . . . . . . 9
1.5 Contributionandlimitation . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 LiteratureReview 13
2.1 Macro-LevelStudentDemandModels . . . . . . . . . . . . . . . . . . . 13
2.1.1 StudentDemandTheoryonTuition . . . . . . . . . . . . . . . . 14
2.1.2 StudentDemandTheoryonFinancialAid . . . . . . . . . . . . . 15
2.1.3 TargetEffectonFinancialAid . . . . . . . . . . . . . . . . . . . 16
2.1.4 StudentDemandStudyforPolicyAnalysis . . . . . . . . . . . . 17
2.2 EnrollmentPredictionatMicro-Level . . . . . . . . . . . . . . . . . . . 19
2.2.1 CollegeChoiceProcessandModels . . . . . . . . . . . . . . . . 20
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2.2.2 Micro-levelResponsetoFinancialAidandItsOptimization . . . 21
2.3 MethodologyReviews . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 RegressionModelsinStudentDemandStudies . . . . . . . . . . 23
2.3.2 LogisticRegressionModelsinStudentChoiceResponseStudies . 25
3 PredictiveModelsforProbabilitiesofEnrollmentandofGraduation 28
3.1 DataExplorationandVisualization . . . . . . . . . . . . . . . . . . . . . 28
3.2 LogisticRegressionForEnrollment&Graduation . . . . . . . . . . . . . 36
3.2.1 LogisticRegressionMethodology . . . . . . . . . . . . . . . . . 36
3.2.2 CollinearityandVariableSelection . . . . . . . . . . . . . . . . 37
3.2.3 LogisticRegressionModelsonTrainingData . . . . . . . . . . 39
3.2.4 LogisticRegressionTreeModelsonTrainingData . . . . . . . . 43
3.2.5 PredictionAccuracyonTestData . . . . . . . . . . . . . . . . . 44
3.3 AnswertoEnrollment&GraduationProbabilities . . . . . . . . . . . . 47
4 PredictionModelsontheNumberofYearsofStudy 50
4.1 DifferenceFromaRetentionStudy . . . . . . . . . . . . . . . . . . . . . 50
4.2 MethodsandResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 TrainingandTestingData . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 PredictionModels . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.3 ExperimentResults . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 MathematicalModelsforTheOptimalFinancialAidAllocationProblem 60
5.1 TheFinancialAidOptimizationModel . . . . . . . . . . . . . . . . . . . 61
5.2 ModelSizeReductionandDominanceMatrix . . . . . . . . . . . . . . . 63
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5.2.1 TheSizeofPair-wiseDominanceConstraints . . . . . . . . . . . 63
5.2.2 FullDominanceMatrix . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.3 RedundantDominanceMatrix . . . . . . . . . . . . . . . . . . . 65
5.2.4 MinimumCardinalityDominanceMatrix . . . . . . . . . . . . . 67
5.3 ModelComparisonandResults . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 ResultsUnderDifferentSSI . . . . . . . . . . . . . . . . . . . . 68
6 DerivationofScholarshipAwardPolicies&Implementation 74
6.1 DerivationofScholarshipAwardPolicies . . . . . . . . . . . . . . . . . 74
6.1.1 ScholarshipAwardPolicyBasedonDecisionTree . . . . . . . . 75
6.1.2 ScholarshipAwardPolicyonStepwiseRegression . . . . . . . . 77
6.1.3 InsightsonChangeOfBudget . . . . . . . . . . . . . . . . . . . 81
6.2 ImplementationandResults . . . . . . . . . . . . . . . . . . . . . . . . 83
7 Conclusion 86
Bibliography 89
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List of Figures
3.1 HistogramforACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 HistogramforHighSchoolGPA . . . . . . . . . . . . . . . . . . . . . . 33
5.1 Fulldominancerelationshipsingraphform . . . . . . . . . . . . . . . . 65
5.2 Redundantdominancerelationshipsingraphform . . . . . . . . . . . . . 66
5.3 Minimumdominanceingraphform . . . . . . . . . . . . . . . . . . . . 67
5.4 OptimizationresultsforSSI=10,000. . . . . . . . . . . . . . . . . . . . 72
5.5 OptimizationresultsforSSI=12,000. . . . . . . . . . . . . . . . . . . . 73
5.6 OptimizationresultsforSSI=14,000. . . . . . . . . . . . . . . . . . . . 73
6.1 Financialaidpolicybasedondecisiontree . . . . . . . . . . . . . . . . 76
6.2 (a) (b) (c) Scholarship vs ACT for various budgets and SSI. (d) (e) (f)
ScholarshipvsGPAforvariousbudgetsandSSI. . . . . . . . . . . . . . 79
6.3 ScholarshipvsCompositescoreforSSI=10,000 . . . . . . . . . . . . . . 81
6.4 ScholarshipvsCompositescoreforSSI=12,000 . . . . . . . . . . . . . . 82
6.5 ScholarshipvsCompositescoreforSSI=14,000 . . . . . . . . . . . . . . 82
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List of Tables
1.1 Comparisonofenrollmentbetween2012-2013and2013-2014Years . . . 10
3.1 Thenumberofapplicationsfrom2007to2013 . . . . . . . . . . . . . . 29
3.2 Statisticsofselectedcontinuousvariablesrelatedtoapplications . . . . . 31
3.3 Statisticsofselectedcontinuousvariablesrelatedtomatriculatedstudents 32
3.4 NumberofapplicantsvsGPA/ACTin2012-2013 . . . . . . . . . . . . . 35
3.5 Pearsoncorrelationmatrixofallnumericvariables . . . . . . . . . . . . 38
3.6 Summarystatisticsoflogisticregressionforenrollmentmodel . . . . . . 40
3.7 Summarystatisticsoflogisticregressionforgraduationmodel . . . . . . 42
3.8 Variablesusedinthelogisticregressiontreemodels . . . . . . . . . . . . 44
3.9 Enrollmentpredictionfromlogisticregressionandlogisticregressiontree 45
3.10 Graduationpredictionfromlogisticregressionandlogisticregressiontree 45
3.11 Accuracy of enrollment prediction from support vector machine and neu-
ralnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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