1、Judge?s Commentary337Judge?s Commentary:The Outstanding HurricaneEvacuation PapersMark ParkerDepartment of Mathematics, Engineering, and Computer Science Carroll CollegeHelena, MT 59625 /mparkerIntroductionOnce again, Problem B proved to be quite challenging?bo

2、th for thestu-dent teams and for the judges! The students were challenged by a multifacetedproblem with several dif?cult questions posed, and the judges were challenged to sort through the wide range of approaches to ?nd a small collection of the best papers. It is worth reminding participants and a

3、dvisors that Outstand- ing papers are not without weaknesses and even mathematical or modeling errors. It is the nature of judging such a competition that we must trade off the strengths, both technical and expository, of a given paper with its weaknesses, and make comparisons between papers the sam

4、e way.The approaches taken by this year?s teams can be divided into two general categories:Macroscopic: Traf?c ona particular highway or segment of highway was con-sidered to be a stream, and a ?ow rate for the stream was characterized. Among the successful approaches in this category were ?uid dyna

5、mics and network ?ow algorithms.Microscopic: These can be considered car-following models, where the spacingbetween and the speeds of individual vehicles are used to determine the?ow. Among the successful approaches were discrete event simulations (including cellular automata) and queuing systems.Th

6、e UMAP Journal 22 (3) (2000) 337?343. c Copyright 2001 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for pro?t or commercial advantage and

7、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.更多數(shù)學(xué)建模資料請(qǐng)關(guān)注微店店鋪“數(shù)學(xué)建模學(xué)習(xí)交流”ht

8、tps://RHO6PSpA338The UMAP Journal22.3 (2001)By far, the most common approach was to determine that the ?ow q, or ?ux,is a function of the density of cars on a highway and the average speed v ofthose cars: q = v.Successful approaches identi?ed the following characteris-tics of the basic

9、traf?c ?ow problem: When the vehicle density on the highway is 0, the ?ow is also 0. As density increases, the ?ow also increases (up to a point). When the density reaches its maximum, or jam density 0 , the ?ow must be 0. Therefore, the ?ow initially increases, as density does, until it reaches som

10、e maximum value. Further increase in the density, up to the jam density, results in a reduction of the ?ow.At this point, many teams either derived from ?rst principles or used one of themany resources available on traf?c modeling (such as Garber and Hoel 1999) to ?nd a relationship between the dens

11、ity and the average speed. Three of the common macroscopic models were: a linear model developed by Greenshield: 0 0v = v1 ;soq = v1 ,00 a ?uid-?ow model developed by Greenberg:0v = v0 log,soq = v0 log;0or a higher-order model developed by Jayakrishnan:aa 0 0v = v01 soq = v01 ,where v0 represents th

12、e speed that a vehicle would travel in the absence ofother traf?c (the speed limit). By taking the derivative of the ?ow equation with respect to speed (or density), teams then found the optimal speed (or density) to maximize ?ow.Many teams took the optimal ?ow from one of the macroscopic approaches

13、 and used it as the basis for a larger model. One of the more common models was simulation, to determine evacuation times under a variety of scenarios.In order to make it beyond the Successful Participant category, teams hadto ?nd a way realistically to regulate traf?cdensity to meet these optimalit

14、yconditions. Many teams did this by stipulating that ramp metering systems(long term) or staggered evacuations (short term) could be used to control traf?c density.Judge?s Commentary339There were a number of mathematically rigorous papers that started witha partial differential equation, derived one

15、 of the macroscopic formulas, de- termined appropriate values for the constants, calculated the density giving the optimal ?ow, and incorporated this ?ow value into an algorithm for de- termining evacuation time. In spite of the impressive mathematics, if no plan was given to regulate traf?c density

16、, the team missed an important concept of the MCM: the realistic application of a mathematical solution to a real-world problem.One key to successful model building is to adapt existing theory or models properly to the problem at hand, so judges see little difference between deriv- ing these equatio

17、ns from ?rst principles and researching them from a book. Whether derived or researched, it is imperative to demonstrate an understand- ing of the model you are using.The JudgingNo paper completely analyzed all 6 questions, so the judges were intrigued by what aspects of the problem that a team foun

18、d most important and/or interesting. We were similarly interested in determining what aspects of the problem a team found least relevant and how they divided their effort among the remaining questions. To be considered Outstanding, a paper had to meet several minimum requirements: the paper must add

19、ress all 6 questions, all required elements (e.g., the newspaper article) must be included, and some sort of validation of the model must be included.We were also particularly interested in how teams modeled the I-26/I-95 inter- change and the congestion problem in Columbia. Many teams chose to trea

20、t Columbia as the terminal point of their model and assumed that all cars arriving there would be absorbed without creating backups.To survive the cut between Honorable Mention and Meritorious, a paper had to have a unique aspect on some portion of the problem. Twoexamples that come to mind are a un

21、ique modeling approach or some aspect of the problem analyzed particularly well. Thus, papers that failed to address all questions or had a fatal weakness that prevented their model from being extended could still be considered Meritorious. The Meritorious papers typically had very good insight into

22、 the problem, but de?ciencies as minor as missing parame- ter descriptions or model implementation details prevented them from being considered Outstanding.340The UMAP Journal22.3 (2001)The Outstanding PapersThe six papers selected as Outstanding were recognized as the best of the submissions becaus

23、e they: developed a solid model which allowed them to address all six questions, and analyze at least one very thoroughly; made a set of clear recommendations; analyzed their recommendations within the context of the problem; and wrote a clear and coherent paper describing the problem, their model,

24、and their recommendations.Here is a brief summary of the highlights of the Outstanding papers.The Bethel College team used a basic car-following model to determine an optimal density, which maximized ?ow, for individual road segments. They then formulated a maximum ?ow problem, with intersections an

25、d cities as vertices and road segments as arcs. The optimal densities were used as arc capacities, the numbers of vehicles to be evacuated from each city were used as the sources, and cities at least 50 miles inland were de?ned to be sinks. Each city was then assigned an optimal evacuation route, an

26、d total evacuation times under the different scenarios were examined.The Duke team also used a basic car-following model from the traf?c- modeling literature. This model provided the foundation of a one-dimensional cellular automata simulation. They did a particularly good job of de?ning evac- uatio

27、n performance measures?maximum traf?c ?ow and minimum transit time, and analyzing traf?c mergers and bottlenecks?aspects of the problem ignored by many other teams.What discussion of Outstanding papers would be complete without a Har- vey Mudd team? Of the teams that utilized literature-based models

28、, this team did the best job of considering advanced parameters?including road grade, non-ideal drivers, and heavy-vehicle modi?cation. They also did a very good job of comparing their model with the new South Carolina evacuation plan, recognizing the bottleneck problem in Columbia, and analyzing th

29、e impact of extra drivers from Florida and Georgia on I-95. Their entry was a nice example of a simple model that was well analyzed and thoroughly explained.The Virginia Governor?s School team began their analysis by reviewing the current South Carolina evacuation plan, a baseline to compare their m

30、odel against. They researched the literature to ?nd traf?c-?ow equations and then used a genetic algorithm to assign road orientation and evacuation start times for cities. They did an exceptionally good job of analyzing the sensitivity of their model to changes in parameter values.The INFORMS prize

31、winner, from Lawrence Technical University, combined Greenshield?s model witha discrete event simulation. The judges sawthisentry as a solid paper with logical explanations and a good analysis. The team?sJudge?s Commentary341model handled bottlenecks, and the team used a simulation of the actual 199

32、9evacuation to validate their model.The MAA and SIAM winner, from Wake Forest University, derived a car- following model from ?rst principles, which was then incorporated in a cellular automata type model. Like many of the best approaches, the parameters for their model came from the 1999 evacuation

33、. They provided a thoughtful, not necessarily mathematical, analysis of intersections and I-95.AdviceAt the conclusion of our judging weekend, the judges as a whole offered the following comments:Follow the instructions Answer all required parts. Make a precise recommendation. Don?t just copy the or

34、iginal problem statement, but provide us with your interpretation.Readability Make it clear in the paper where the answers are. Many judges ?nd it helpful to include a table of contents. Pictures and graphs can help demonstrate ideas, results, and conclu- sions. Use discretion: If your paper is exce

35、ssively long (we had a paper this year that was over 80 pp, not including computer program listing!), you should probably reconsider the relevance of all factors that you are discussing. Depending on what round of judging your paper is being read, judges typically have between 5 and 30 minutes to re

36、ad it.Computer Programs Make sure that all parameters are clearly de?ned and explained. When using simulation, you must run enough times to have statistically signi?cant output. A single run isn?t enough! Always include pseudocode and/or a clear verbal description.Reality Check Why do you think your

37、 model is good? Against what baseline can you compare/validate it? How sensitive is your model to slight changes in the parameters you have chosen? (sensitivity analysis)342The UMAP Journal22.3 (2001) Complete the analysis circle: Are your recommendations practical in the problem context?Before the

38、?nal judging of the MCM papers, a ?rst (or triage) round ofjudging is held. During triage judging, each paper is skimmed by two or three judges, who spend between 5 and 10 minutes each reading the paper. Typically, when you send your paper off to COMAP, you have about a 43% chance of being ranked hi

39、gher than Successful Participant. If, however, you survive the triage round, you have about an 80% chance of being ranked higher than Successful Participant. Head triage judge Paul Boisen offers the following advice to help you survive triage.Triage Judge Tips Your summary is a key component of the

40、paper; it needs to be clear and contain results. A long list of techniques can obscure your results; it is better to provide only a quick overview of your approach. The Lawrence Technical University paper is a good example of a clear and concise summary. Your paper needs to be well organized?can a triage judge understand the signi?cance of your paper in 6 to 10 minut