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2004美国大学生数学建模竞赛试题

2004美国大学生数学建模竞赛试题
参考译文:


题一:指纹是独一无二的吗?
人们普遍认为世界上每一个活人的指纹都是不一样的,请设计一个模型,并且用该模型分析以上说法正确的可能性,比较一下因为指纹相同导致确认身份时产生错误的可能性和因为DNA相同导致产生错误的可能性。


题二:更快的快通系统
现在的快通系统在收费站、娱乐公园和其他的地方,正在被越来越频繁的使用,来减少人们排队等候的时间,现在我们考虑为一个娱乐公园所设计的快通系统,在一次测试当中,这个公园在几个游客比较多的景点旁边都设置了快通系统,这个系统的设计创意是对于那些比较热门的景点,可以到旁边的一个机器,将门票插入后出来一张纸条,上面写着在具体的时间段你可以回来,比如说你把你的门票在1:15查到机子里,系统就告诉你你可以在3:30-4:30回来,这个时候队伍就比较短,你可以凭你的纸条加入这个队伍,很快就可以进入景点,为了防止游客同时在几个景点使用这个系统。系统的机器只允许你一次在一个景点排队等待。
现在你是几个被公园雇佣的相互竞争的一个,你的职责是改善快通系统的运行。很多游客都在抱怨测试期间系统的异常现象,比如说有一次系统提供的回到景的时间是4小时以后,但是才过一小会,在相同的景点系统提供的时间只有1小时。在另外一些时候根据快通系统组织起来的游客的等候队伍,就和普通的队伍一样长一样慢。
现在的问题是要提出并且测试一个模型,这个模型能让快通系统的等候纸条的发放能增加人们在公园的乐趣的目的。问题的一部分就是首先要决定衡量不同模型的标准,在你提交的报告里还要附带一份非技术性的总结,以便公园的领导,在不同的顾问所提的模型当中选择。


原文:
MCM: The Mathematical Contest in Modeling

2004 MCM Problems

PROBLEM A: Are Fingerprints Unique?

It is a commonplace belief that the thumbprint of every human who has ever lived is different. Develop and analyze a model that will allow you to assess the probability that this is true. Compare the odds (that you found in this problem) of misidentification by fingerprint evidence against the odds of misidentification by DNA evidence.

PROBLEM B: A Faster QuickPass System


"QuickPass" systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time.

You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines.

The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park executives who must choose between alternatives from competing consultants.
MCM: The Mathematical Contest in Modeling

2004 MCM Problems

PROBLEM A: Are Fingerprints Unique?


It is a commonplace belief that the thumbprint of every human who has ever lived is different. Develop and analyze a model that will allow you to assess the probability that this is true. Compare the odds (that you found in this problem) of misidentification by fingerprint evidence against the odds of misidentification by DNA evidence.

PROBLEM B: A Faster QuickPass System


"QuickPass" systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time.

You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines.

The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park executives who must choose between alternatives from competing consultants.


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