【全集】2005 MCM B 公路收费亭的设置特等奖论文教程

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PROBLEM B: Tollbooths
Heavily-traveled toll roads such as the Garden State Parkway, Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.
Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.

MCM2005 B 题: 公路收费亭的设置
诸如美国新泽西州的风景区干道，95 号州际公路等交通繁忙的收费公路都是多车道的交通干线，每隔一定距离设有过路费收费区。由于收取过路费一般是不得人心的，因此通过限制由于过路费收费区造成的交通混乱把驾车人的烦恼减到极小是很值得做的。通常，收费区内收费亭的数目远多于进入过路费收费区的车道数。进入过路费收费区时，车流扇形散开分流，分别在各个收费亭交费；离开收费区时，车流又会汇合到和进入收费区时一样多的车道离开。因此，在交通繁忙时，通行的车辆会在离开收费区时出现拥塞；更严重的时候，收费站的入口也会出现拥堵。

试构建一个模型，用来决定拦路过路费收费区内收费亭的最优数目的配置。明确考虑如下情景，即在进来的每个车道恰好只有一个收费亭。在什么情况下你制订的方案要比现有的方案效率多少要高一点？注意：“最优”的定义要由你自己来决定。

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