2016美国大学生数学建模特等奖论文集（ICM，含赛题）C47823.pdf资源-CSDN文库

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For$office$use$only

________________

Team$Control$Number$

47823

Problem$Chosen$

For$office$use$only

________________

2016

MCM/ICM

Summary Sheet

(Your team's summary should be included as the first page of your electronic submission.)

Type$a$summary$of$your$results$on$this$page.$Do$not$include$the$name$of$your$school,$advisor,$or$team$members$on$this$

page.$

$ We$develop$a$model$to$determine$an$optimal$investment$strategy$to$improve$

the$performance$of$undergraduate$students$in$the$US.$Our$model$has$three$parts:$

$ In$ the$ first$ part,$ we$ collect$ data$ about$ the$ focus$ of$ other$ foundations’$

investment$by$subjects$and$locations.$We$consider$the$charitable$identity$of$the$

Goodgrant$as$well.$Then$we$set$out$to$decide$our$focus,$which$is$to$invest$more$

on$ those$ schools$ with$ more$ minority$ races,$ lower$ educational$ performance,$

higher$debt$ratio$and$ so$on.$In$this$part,$we$also$classify$the$data$into$two$groups,$

one$for$school$selecting,$and$another$for$ROI$determining.$

$ In$ the$ second$ part,$ as$ a$ data$ extraction,$ we$ build$ a$ efficient$ and$ intuitive$

model$ to$ rank$ the$ candidate$ schools$ in$ accordance$ with$ the$ correlation$ of$ our$

focus,$using$the$PCA$ method.$After$ that,$the$ top$50$ schools$are$ selected$as$ our$

target$schools.$

$ In$the$third$part，we$make$a$key$assumption$that$the$social$utility$of$a$school$

has$ logarithmic$ relationship$ with$ the$ earnings$ of$ graduated$ students$ and$ the$

graduation$rate.$More$over,$we$create$a$parameter$!$to$denote$the$marginal$rate$

of$ substitution$ (MRS)$ between$ the$ two$ factors$ above.$ After$ that,$ we$ come$ to$

define$the$ROI$function$of$each$target$school$as$the$incremental$utility.$

$ We$further$discuss$how$to$devise$the$best$strategy$with$several$methods.$At$

last,$ we$ choose$ the$ improved$ PSO$ algorithm$ based$ on$ augmented$ Lagrange$

function.$ This$ algorithm$ is$ a$ typical$ method$ to$ solve$ the$ multivariable$

optimization$problem$with$constraint$conditions.$Then$we$offer$a$recommending$

list$by$the$cumulative$ROI$in$five$years.$What’s$more,$our$model$is$broad$enough$

to$accommodate$any$nonXlinear$constraint$optimization$problem.$

$ Finally,$ we$ change$ the$ numerical$ value$ of$ parameter$ k$ to$ examine$ the$

sensitivity$of$our$investment$strategy.$The$result$shows$that$our$model$is$robust.$ $ $

The$Optimal$Investment$Strategy$Based$on$the$LargeXscale$

NonXlinear$Constrain t$O pt im ization $M et ho ds$ $

Contents$

1$ Problem$Statement$.....................................................................................................................$3$

2$ Planned$Approach$......................................................................................................................$3$

3$ Assumptions$..................................................................................................................................$4$

4$ Data$Analysis$and$Focus$Decision$.......................................................................................$5$

4.1$ Data$Analysis$...................................................................................................................$5$

4.2$ Focus$Decision$................................................................................................................$6$

5$ School$Selecting$...........................................................................................................................$8$

5.1$ Manual$Selection$............................................................................................................$8$

5.2$ PCA$Selection$...................................................................................................................$8$

5.2.1$ Standardization$.................................................................................................$8$

5.2.2$ Calculation$..........................................................................................................$9$

5.2.3$ Principle$Components$....................................................................................$9$

5.2.4$ PCA$Results$.........................................................................................................$9$

6$ Strategy$Making$.........................................................................................................................$10$

6.1$ The$ROI$Function$.........................................................................................................$11$

6.2$ Optimizing$the$Total$ROI$..........................................................................................$12$

6.2.1$ KarushXKuhnXTucker$Conditions$............................................................$14$

6.2.2$ PSO$Algorithm$.................................................................................................$16$

6.2.3$ Improved$ PSO$ algorithm$ based$ on$ augmented$ Lagrange$

function$(LA_PSO_GT)$.................................................................................................$16$

7$ Result$.............................................................................................................................................$16$

7.1$ Optimal$Investment$Strategy$and$Recommending$List$..............................$17$

8$ Testing$our$Model$.....................................................................................................................$18$

8.1$ Sensitivity$Analysis$.....................................................................................................$18$

8.2$ Strengths$.........................................................................................................................$19$

8.3$ Weaknesses$....................................................................................................................$19$

9$ Conclusion$....................................................................................................................................$20$

10$ Letter$to$the$CFO$of$the$Goodgrant$Foundation$.......................................................$20$

11$ References$.................................................................................................................................$22$

12$ Appendix$1:$Recommending$List$.....................................................................................$23$

13$ Appendix$2$An$Introduction$to$the$Improved$PSO$ $

Algorithm$Based$on$Augmented$Lagrange$Function$........................................................$25$

Team$#47823$

Page$3$of$27$ $ $

1( Problem(Statement(

$ Private$foundations$are$created$by$an$individual,$family,$or$business$to$fulfill$ $

specific$charitable$missions.$Those$like$Gates$foundation$and$Lumina$foundation$

make$ great$ efforts$ to$ improve$ the$ quality$ of$ health$ and$ education$ in$ relatively$

poor$ areas.$ We$ must$ set$ big$ goals$ and$ spare$ no$ effort$ on$ the$ way$ because$ the$

world$won’t$get$better$by$itself.$The$Goodgrant,$one$of$the$foundations,$intends$

to$ help$ improving$ educa tional$ performance$ of$ undergraduates$ attending$

colleges$and$universities$in$the$United$States.$Given$its$potential$donation$of$100$

million$dollars$per$year$in$five$years,$what$is$the$best$investment$strategy?$We$

are$tasked$with$creating$models$that$can$be$applied$in$the$universities$ across$the$

nation.$The$solution$ proposed$within$this$paper$ will$ offer$an$insight$to$ use$the$

big$data$and$will$objectively$devise$the$investment$strategy$including$the$target$

schools,$investment$amount$and$duration.$

2( Planned(Approach(

$ Our$objective$is$to$set$out$the$best$strategy$including$three$components:(1)$

target$ schools;(2)$ the$ investment$ amount$ per$ school;$ (3)$ the$ investment$

duration.$ And$ also$ we$ will$ offer$ an$ optimized$ and$ prioritized$ recommendation$

list$of$candidate$schools$based$on$each$school’s$return$on$investment$(ROI).$

$ Faced$ with$ the$ big$data$ problem,$ we$ can’t$ use$the$ data$directly$ because$ of$

the$ limitation$ of$ our$ personal$ computers$ and$ the$ length$ of$ the$ contest.$ If$ the$ $ $

data$are$directly$applied,$the$computing$system$will$run$several$days$or$weeks.$

As$a$result,$the$data$selection$is$extremely$important,$which$will$also$reflect$the$

focus$of$the$foundation.$To$determine$the$most$effective$computing$system,$we$

divide$the$problem$into$three$parts$together$with$the$procedures$as$follows:$

$ Part(one:$Data$Analysis$and$Focus$Decision$

1. We$ will$give$ an$ analysis$ of$ the$ big$ data$of$ the$ problem,$which$ includes$

information$of$near$3000$schools.$ $

2. Based$ on$ the$ data$ given$ and$ the$ statistics$ of$ the$ focus$ of$ foundations$

collected$ from$ the$ Internet,$ we$ will$ decide$ the$ focus$ of$ the$ Goodgrant,$

avoiding$ duplicating$ the$ investment$ and$ focus$ of$ other$ large$ grant$

organizations.$

$ Part(two:$School$Selecting$

Team$#47823$

Page$4$of$27$ $ $

1. Manual$ selection.$ We$ have$ taken$ some$ schools$ out$ of$ consideration$ for$

certain$ reasons$ (the$ reason$ will$ be$ explained$ below).$ For$ example,$ we$

exclude$ the$ schools$ located$ at$ NY,$ CA,$ WA$ and$ MA$ due$ to$ the$ large$

amount$of$existing$grant$foundations.$

2. PCA$ (principle$ component$ analysis)$ selection.$ According$ to$ part$ of$ the$

data,$ the$ PCA$ method$ can$ rank$ the$ candidate$ schools$ by$ the$ degree$ of$

correlation$of$our$focus.$The$top$50$schools$will$be$selected$out.$

$ Part(three:$Strategy$Making$

1. Derive$ a$ ROI$ function$ that,$ given$ the$ year$ and$ a$ specific$ investment$

amount$ of$ a$ candidate$ school,$ can$ output$ the$ utility$ in$ an$ appropriate$

manner.$ The$ function$ is$ based$ on$ the$ graduation$ rate,$ earnings$ of$

graduated$students$and$so$on.$

2. Utilize$ an$ optimization$ algorithm$ to$ maximize$ the$ total$ utility$ of$ the$

target$50$schools$(in$part$two$(2)),$return$the$amount$of$investment$and$

the$time$duration$per$school.$

3( Assumption s(

$ Due$ to$ limited$ data$ about$ the$ educational$ performance$ of$ the$ candidate$

schools,$ the$ performance$ of$ the$ undergraduate$ students$ and$ the$ specific$

distribution$of$other$grants$by$subjects,$ ra ces$and$locations,$we$use$the$following$

assumptions$to$complete$our$model.$These$simplified$assumptions$will$be$used$

through$our$paper$and$can$be$improved$with$more$reliable$data.$

! The$ stat ist ics$ of$ the$ candidate$ schools$ can$ be$ regarded$ as$ constant$

within$ five$ years.$ This$ assumption$ is$ reasonable$ to$ a$ large$ extent$

because$ the$ identities$ of$ a$ specific$ college$ won’t$ change$ a$ lot$ in$ five$

years.$

! The$ school$ will$ devote$ all$ the$ funding$ received$ this$ year$ to$improving$

the$students’$performance.$ $

! The$appropriate$manner$to$measure$the$return$on$the$investment$is$the$

school’s$ incremental$ utility.$ The$ utility$ function$ must$ be$ concave$

(𝜕

𝑦/𝜕𝑥

<0).$If$not,$we$should$give$the$whole$100$million$to$one$college$

to$maximize$the$total$incremental$utility,$which$is$opposite$against$the$

common$sense.$And$it’s$ reasonable$ in$economic$consideration,$as$with$

the$capital$growth,$the$marginal$production$will$be$less$and$less.$So$we$

assume$ the$ utility$ function$ has$ this$ typical$ formulation$ 𝑈

= log!(𝑥),$

where$ 𝑈

$ is$the$utility,$ 𝑥 $ are$the$independent$variables.$

! Neglect$the$discount$rate$of$the$capital.$In$this$paper,$we$do$not $tak e$the$

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