《数学建模》美赛优秀论文：2001b 评论s Commentary The Outstanding Hurricane Evacuation Papers.pdf_大学文库

Judges commentary 337 Judge's Commentary The Outstanding Hurricane Evacuation Papers Mark parker Department of Mathematics, Engineering, and Computer Science Carroll college Helena, mr 59625 marker@carroll. edu http://web.carroll.edu/mparker Introduction Once again, Problem B proved to be quite challenging-both for the stu- dent teams and for the judges! The students were challenged by a multifaceted problem with several difficult questions posed, and the judges were challenged to sort through the wide range of approaches to find a small collection of the best papers. It is worth reminding participants and advisors 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 same way. The approaches taken by this year's teams can be divided into two genera categories: Macroscopic: Traffic on a particular highway or segment of highway was con- sidered to be a stream and a flow rate for the stream was characterized Among the successful approaches in this category were fluid dynamics and network flow algorithms Microscopic: These can be considered car-following models, where the spacing between and the speeds of individual vehicles are used to determine the flow. Among the successful approaches were discrete event simulations (including cellular automata)and queuing systems The UIMAP Journal 22(3)(2000)337-343. @ Copyright 2001 by COMAP, Inc. All rights reserve 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 profit 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

Judgeís Commentary 337 Judgeís Commentary: The Outstanding Hurricane Evacuation Papers Mark Parker Department of Mathematics, Engineering, and Computer Science Carroll College Helena, MT 59625 mparker@carroll.edu http://web.carroll.edu/mparker Introduction Once again, Problem B proved to be quite challengingóboth for the student teams and for the judges! The students were challenged by a multifaceted problem with several difficult questions posed, and the judges were challenged to sort through the wide range of approaches to find a small collection of the best papers. It is worth reminding participants and advisors that Outstanding 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 same way. The approaches taken by this yearís teams can be divided into two general categories: Macroscopic: Traffic onaparticular highway or segment of highway was considered to be a stream, and a flow rate for the stream was characterized. Among the successful approaches in this category were fluid dynamics and network flow algorithms. Microscopic: These can be considered car-following models, where the spacing between and the speeds of individual vehicles are used to determine the flow. Among the successful approaches were discrete event simulations (including cellular automata) and queuing systems. The UMAP Journal 22 (3) (2000) 337ñ343. 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 use is granted without fee provided that copies are not made or distributed for profit 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

338 The UMAP Journal 22.3 (2001) By far, the most common approach was to determine that the flow q, or flux, is a function of the density ρ of cars on a highway and the average speed v of those cars: q = ρv. Successful approaches identified the following characteristics of the basic traffic flow problem: • When the vehicle density on the highway is 0, the flow is also 0. • As density increases, the flow also increases (up to a point). • When the density reaches its maximum, orjam density ρ0, the flow must be 0. • Therefore, the flow initially increases, as density does, until it reaches some maximum value. Further increase in the density, up to the jam density, results in a reduction of the flow. At this point, many teams either derived from first principles or used one of the many resources available on traffic modeling (such as Garber and Hoel [1999]) to find a relationship between the density and the average speed. Three of the common macroscopic models were: • a linear model developed by Greenshield: v = v0 1 − ρ ρ0 , so q = ρv0 1 − ρ ρ0 ; • a fluid-flow model developed by Greenberg: v = v0 log ρ ρ0 , so q = ρv0 log ρ ρ0 ; or • a higher-order model developed by Jayakrishnan: v = v0 1 − ρ ρ0 a , so q = ρv0 1 − ρ ρ0 a , where v0 represents the speed that a vehicle would travel in the absence of other traffic (the speed limit). By taking the derivative of the flow equation with respect to speed (or density), teams then found the optimal speed (or density) to maximize flow. Many teams took the optimal flow from one of the macroscopic approaches 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 had to find a way realistically to regulate traffic density to meet these optimality conditions. Many teams did this by stipulating that ramp metering systems (long term) or staggered evacuations (short term) could be used to control traffic density

Judge's Commentary 339 here were a number of mathematically rigorous papers that started with a partial differential equation, derived one of the macroscopic formulas, de termined appropriate values for the constants, calculated the density giving the optimal flow, and incorporated this flow value into an algorithm for de- termining evacuation time. In spite of the impressive mathematics, if no plan was given to regulate traffic density, 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 equations from first 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 judging No paper completely analyzed all 6 questions, so the judges were intrigued by what aspects of the problem that a team found 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 address 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 treat 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. Two examples that come to mind are a unique 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 the problem, but deficiencies as minor as missing parame ter descriptions or model implementation details prevented them from being considered Outstanding

Judgeís Commentary 339 There were a number of mathematically rigorous papers that started with a partial differential equation, derived one of the macroscopic formulas, determined appropriate values for the constants, calculated the density giving the optimal flow, and incorporated this flow value into an algorithm for determining evacuation time. In spite of the impressive mathematics, if no plan was given to regulate traffic density, 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 deriving these equations from first principles and researching them from a book. Whether derived or researched, it is imperative to demonstrate an understanding of the model you are using. The Judging No paper completely analyzed all 6 questions, so the judges were intrigued by what aspects of the problem that a team found 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 address 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 interchange and the congestion problem in Columbia. Many teams chose to treat 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. Two examples that come to mind are a unique 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 the problem, but deficiencies as minor as missing parameter descriptions or model implementation details prevented them from being considered Outstanding

340 The UMaP Journal 22.3 (2001) The Outstanding Papers The six papers selected as outstanding were recognized as the best of the submissions because they e 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, 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 flow, for individual road segments. They then formulated a maximum flow problem, with intersections and 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 defined to be sinks. each city was then assigned an optimal evacuation route, and total evacuation times under the different scenarios were examined The Duke team also used a basic car-following model from the traffic modeling literature This model provided the foundation of a one-dimensional cellular automata simulation. They did a particularly good job of defining evac- uation performance measures--maximum traffic flow and minimum transit time, and analyzing traffic mergers and bottlenecks-aspects of the problem gored 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, this team did the best job of considering advanced parameters--including road grade, non-ideal drivers, and heavy-vehicle modification. 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 the 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 model against. They researched the literature to find traffic-flow 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 heir model to changes in parameter values The INFORMS prizewinner, from Lawrence Technical University, combined Greenshields model with a discrete event simulation. The judges saw thisentry as a solid paper with logical explanations and a good analysis. The teams

340 The UMAP Journal 22.3 (2001) The Outstanding Papers The six papers selected as Outstanding were recognized as the best of the submissions because 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, 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 flow, for individual road segments. They then formulated a maximum flow problem, with intersections and 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 defined to be sinks. Each city was then assigned an optimal evacuation route, and total evacuation times under the different scenarios were examined. The Duke team also used a basic car-following model from the trafficmodeling literature. This model provided the foundation of a one-dimensional cellular automata simulation. They did a particularly good job of defining evacuation performance measuresómaximum traffic flow and minimum transit time, and analyzing traffic mergers and bottlenecksóaspects of the problem ignored by many other teams. What discussion of Outstanding papers would be complete without a Harvey Mudd team? Of the teams that utilized literature-based models, this team did the best job of considering advanced parametersóincluding road grade, non-ideal drivers, and heavy-vehicle modification. 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 the 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 Governorís School team began their analysis by reviewing the current South Carolina evacuation plan, a baseline to compare their model against. They researched the literature to find traffic-flow 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 prizewinner, from Lawrence Technical University, combined Greenshieldís model with a discrete event simulation. The judges saw this entry as a solid paper with logical explanations and a good analysis. The teamís