2012年美国数学建模优秀论文(3)

Results of the2012MCM193 MCM Modeling Forum

Results of the2012

Mathematical Contest in Modeling William P.Fox,MCM Director

Dept.of Defense Analysis

Naval Postgraduate School

1University Circle

Monterey,CA93943–5000

wpfox@067aad14c281e53a5802ff30

Introduction

A total of3,697teams of undergraduates from hundreds of institutions and departments in16countries spent a weekend in February working on applied mathematics problems in the28th Mathematical Contest in Modeling(MCM)R .

The2012MCM began at8:00P.M.EST on Thursday,February9,and ended at8:00P.M.EST on Monday,February13.During that time,teams of up to three undergraduates researched,modeled,and submitted a solution to one of two open-ended modeling problems.Students registered,obtained contest materials,downloaded the problem and data,and entered completion data through COMAP’s MCM Website.After a weekend of hard work,solution papers were sent to COMAP on Monday.Two of the top papers appear in this issue of The UMAP Journal,together with commentaries.

In addition to this special issue of The UMAP Journal,COMAP offers a supplementary2012MCM-ICM CD-ROM containing the press releases for the two contests,the results,the problems,unabridged versions of the Outstanding papers,and judges’067aad14c281e53a5802ff30rmation about ordering is at http: //067aad14c281e53a5802ff30/product/cdrom/index or at(800)772–6627.

Results and winning papers from the?rst27contests were published in special issues of Mathematical Modeling(1985–1987)and The UMAP Journal (1985–2011).The1994volume of Tools for Teaching,commemorating the tenth anniversary of the contest,contains the20problems used in the?rst10years of the contest and a winning paper for each year.That volume and the special The UMAP Journal33(3)(2012)193–204.c Copyright2012by COMAP,Inc.All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro?t or commercial advantage and that copies bear this notice.Abstracting with credit is permitted,but copyrights for components of this work owned by others than COMAP must be honored.To copy otherwise, to republish,to post on servers,or to redistribute to lists requires prior permission from COMAP.

194The UMAP Journal33.3(2012)

MCM issues of the Journal for the last few years are available from COMAP.The 1994volume is also available on COMAP’s special Modeling Resource CD-ROM. Also available is The MCM at21CD-ROM,which contains the20problems from the second10years of the contest,a winning paper from each year,and advice from advisors of Outstanding teams.These CD-ROMs can be ordered from COMAP at 067aad14c281e53a5802ff30/product/cdrom/index.

This year,the two MCM problems represented signi?cant challenges:?Problem A,“The Leaves of a Tree,”asked teams to model the leaves on a tree,classifying their shapes,investigating the effect of leaf shape on tree pro?le and branching,and estimating the leaf mass of a tree.?Problem B,“Camping Along the Big Long River,”asked teams to design a management plan for scheduling recreational multi-day rafting tours down

a long stretch of a river.The goal was to maximize the number of trips,

optimize campsite usage,and offer an optimal mix of trip lengths.

COMAP also sponsors:

?The MCM/ICM Media Contest(see p.202).

?The Interdisciplinary Contest in Modeling(ICM)R ,which runs concurrently with MCM and next year will offer a modeling problem involving network science.Results of this year’s ICM are on the COMAP Website at http: //067aad14c281e53a5802ff30/undergraduate/contests.The contest report,an Outstanding paper,and commentaries appear in this issue.

?The High School Mathematical Contest in Modeling(HiMCM)R ,which offers high school students a modeling opportunity similar to the MCM. Further details are at 067aad14c281e53a5802ff30/highschool/contests.

2012MCM Statistics

?3,697teams participated(with1,329more in the ICM)

?8high school teams(<0.5%)

?341U.S.teams(9%)

?2,428foreign teams(91%),from Canada,China,Finland,Germany,India, Indonesia,Ireland,Malaysia,Mexico,Palestine,Singapore,South Africa, South Korea,Spain,Turkey,and the United Kingdom

?10Outstanding Winners(<0.5%)

?17Finalist Winners(<0.5%)

?405Meritorious Winners(11%)

?1,048Honorable Mentions(28%)

?2,211Successful Participants(60%)

Results of the2012MCM195

Problem A:The Leaves of a Tree

“How much do the leaves on a tree weigh?”How might one estimate the actual weight of the leaves(or for that matter any other parts of the tree)?How might one classify leaves?Build a mathematical model to describe and classify leaves.Consider and answer the following:

?Why do leaves have the various shapes that they have?

?Do the shapes“minimize”overlapping inpidual shadows that are cast, so as to maximize exposure?Does the distribution of leaves within the “volume”of the tree and its branches effect the shape?

?Speaking of pro?les,is leaf shape(general characteristics)related to tree pro?le/branching structure?

?How would you estimate the leaf mass of a tree?Is there a correlation between the leaf mass and the size characteristics of the tree(height,mass, volume de?ned by the pro?le)?

In addition to your one-page summary sheet,prepare a one-page letter to an editor of a scienti?c journal outlining your key?ndings.

Problem B:Camping along the Big Long River

Visitors to the Big Long River(225miles)can enjoy scenic views and exciting white water rapids.The river is inaccessible to hikers,so the only way to enjoy it is to take a river trip that requires several days of camping.

River trips all start at First Launch and exit the river at Final Exit,225miles downstream.Passengers take either oar-powered rubber rafts,which travel on average4mph or motorized boats,which travel on average8mph.The trips range from6to18nights of camping on the river,start to?nish.

The government agency responsible for managing this river wants every trip to enjoy a wilderness experience,with minimal contact with other groups of boats on the river.

Currently,X trips travel down the Big Long River each year during a six-month period(the rest of the year it is too cold for river trips).There are Y camp sites on the Big Long River,distributed fairly uniformly throughout the river corridor.Given the rise in popularity of river rafting,the park managers have been asked to allow more trips to travel down the river.They want to determine how they might schedule an optimal mix of trips,of varying duration (measured in nights on the river)and propulsion(motor or oar)that will utilize the campsites in the best way possible.In other words,how many more boat trips could be added to the Big Long River’s rafting season?

196The UMAP Journal33.3(2012)

The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river,remembering that no two sets of campers can occupy the same site at the same time.

In addition to your one-page summary sheet,prepare a one-page memo to the managers of the river describing your key?ndings.

The Results

The solution papers were coded at COMAP headquarters so that names and af?liations of the authors would be unknown to the judges.Each paper was then read preliminarily by two“triage”judges at either Appalachian State University(Leaf Problem)or at the National Security Agency(River Problem) or at Carroll College(River Problem).At the triage stage,the summary and overall organization are the basis for judging a paper.If the judges’scores perged for a paper,the judges conferred;if they still did not agree,a third judge evaluated the paper.

Additional Regional Judging sites were created at the 067aad14c281e53a5802ff30itary Academy, the Naval Postgraduate School,and Carroll College to support the growing number of contest submissions.

Final judging took place at the Naval Postgraduate School,Monterey,CA. The judges classi?ed the papers as follows:

Honorable Successful

Outstanding Finalist Meritorious Mention Participation Total Leaf Problem482264828621,582

River Problem691795661,3492,109

10174051,0482,2113,691 We list here the10teams that the judges designated as Outstanding;the list of all participating schools,advisors,and results is at the COMAP Website.

Outstanding Teams

Institution and Advisor Team Members

Leaf Problem

“How to Measure the Weight of Leaves on a Tree”

Hong Kong Baptist University Kowloon,Hong Kong

Alex Wing Kee Mok Xiaotian Wu Qingran Li Jin Liang

Results of the2012MCM197“A Close Look at Leaves”

Shanghai Foreign Language School Shanghai,China

YiJung Wang Bo Zhang Yi Zhang TianKun Lu

“Geometrical Tree”

National University of Singapore Singapore

Weizhu Bao Wenji Xu Jing Zhang Jingyi Lu

“The Secrets of Leaves”

Zhejiang University Hangzhou,China Zhiyi Tan

Cheng Fu

Danting Zhu

Hangqi Zhao River Problem

“Best Schedule to Utilize the Big Long River”

Peking University Beijing,China

Liu Xu Feng Nan Bi Chenwei Tian Yuan Liu

“Computing Along the Big Long River”

Western Washington University Bellingham,WA

Edoh Y.Amiran Chip Jackson Lucas Bourne Travis Peters

“Optimization of Seasonal Throughput and Campsite Utilization on the Big Long River”

University of Colorado Boulder,CO

Anne M.Dougherty Stephen M.Kissler Christopher Corey Sean Wiese

“Getting Our Priorities Straight”

Bethel University Arden Hills,MN Nathan Gossett Michael D.Tetzlaff Blaine Goscha Jacob Smith

198The UMAP Journal33.3(2012)“Optimal Scheduling for the Big Long River”

University of Colorado Boulder,CO

Anne M.Dougherty Tracy Babb Christopher V.Aicher Daniel J.Sutton

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