2000年美国大学生数学建模竞赛试题 (2000年2月4日2月7日举行) 2000 Mathematical Contest in Modeling Dedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation Agency Problem a: Air traffic Control To improve safety and reduce air traffic controller workload, the Federal Aviation Agency(FAA)is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAa r traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the faa has posed the following problems Requirement A: Given two airplanes flying in space, when should the air traffic controller ld the air traffic controllerconsider the objects to be too close and to require intervention? Requirement B: An airspace sector is the section of three-dimensional airspace that one air Requirement b An airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector(1)at any one instant? (2)during any given interval of time? (3)durina particular time of day? How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity In addition to the guidelines for your report, write a summary (no more than two pages) that the In addition to the guidelines for your report, write a summary(no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions 问题A空间交通管制 为加强安全并减少空中交通指挥员的工作量,联邦航空局(FAA)考虑对空中交通管制系统添加软件 以便自动探测飞行器飞行路线可能的冲突,并提醒指挥员。为完成此项工作,FAA的分析员提出了下列问 题 要求A:对于给定的两架空中飞行的飞机,空中交通指挥员应在什么时候把该目标视为太靠近,并予 以干预 要求B:空间扇形是指某个空中交通指挥员所控制的三维空间部分。给定任意一个空间扇形,我们怎 样从空中交通工作量的方位来估量它是否复杂?当几个飞行器同时通过该扇形时,在下面情形所确定的复 杂性会达到什么程度:(1)在任一时刻?(2)在任意给定的时间范围内?(3)在一天的特别时间内? 在此期间可能出现的冲突总数是怎样影响着复杂性来的? 提出所添加的软件工具对于自动预告冲突并提醒指挥员,这是否会减少或增加此种复杂性? 在作出你的报告方案的同时,写出概述(不多于二页)使FAA分析员能提交给FAA当局 Jane Garvey, 并对你的结论进行答辩 唐云译
2000 年美国大学生数学建模竞赛试题 （2000 年 2 月 4 日-2 月 7 日举行） 2000 Mathematical Contest in Modeling Dedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation Agency Problem A: Air traffic Control To improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA r traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA has posed the following problems Requirement A: Given two airplanes flying in space, when should the air traffic controller ld the air traffic controllerconsider the objects to be too close and to require intervention? Requirement B: An airspace sector is the section of three-dimensional airspace that one air Requirement B: An airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector (1) at any one instant? (2) during any given interval of time?(3) durina particular time of day? How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions 问题 A 空间交通管制 为加强安全并减少空中交通指挥员的工作量，联邦航空局(FAA)考虑对空中交通管制系统添加软件， 以便自动探测飞行器飞行路线可能的冲突，并提醒指挥员。为完成此项工作，FAA 的分析员提出了下列问 题。 要求 A：对于给定的两架空中飞行的飞机，空中交通指挥员应在什么时候把该目标视为太靠近，并予 以干预。 要求 B：空间扇形是指某个空中交通指挥员所控制的三维空间部分。给定任意一个空间扇形，我们怎 样从空中交通工作量的方位来估量它是否复杂？当几个飞行器同时通过该扇形时，在下面情形所确定的复 杂性会达到什么程度：（1）在任一时刻？（2）在任意给定的时间范围内？（3）在一天的特别时间内？ 在此期间可能出现的冲突总数是怎样影响着复杂性来的？ 提出所添加的软件工具对于自动预告冲突并提醒指挥员，这是否会减少或增加此种复杂性？ 在作出你的报告方案的同时，写出概述（不多于二页）使 FAA 分析员能提交给 FAA 当局 Jane Garvey ， 并对你的结论进行答辩。 唐云译
Problem b: Radio Channel assignments We seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3,.. Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span Let s be the length of a side of one of the hexagons We concentrate on the case that there are let s be the length of a side of one of the hexagons We concentrate on the case that there are two levels of interference Requirement A: There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2 Under these constraints, what can we say about the span in Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in al directions Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance differ by at least some given integer k, while those at distance at most must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k? Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider? Requirement E: Write an article(no more than 2 pages) for the local newspaper explaining your findings 问题B无线电信道分配 我们寻找无线电信道配置模型,在一个大的平面区域上设置一个传送站的均衡網絡,以避免干扰。 个基本的方法是将此区域分成正六边形的格子(蜂窝狀)。如图1。传送站安置在每个正六边形的中心点 容许频率波谱的一个区间作为各传送站的频率。将這一区间规则地分割成一些空间信道,用整数1,2, 3,…来表示。每一个传送站将被配置一正整数信道。同一信道可以在许多局部地区使用,前提是相邻 近的传送站不相互干扰。根据某些限制设定的信道需要一定的频率波谱,我们的目标是极小化频率波谱的 這个区间宽度。這可以用跨度這一概念。跨度是某一个局部区域上使用的最大信道在一切滿足限制的配置 中的最小值。在一个获得一定跨度的配置中不要求小於跨度的每一信道都被使用。 令s为一个正六边形的一側的长度我们集中考虑存在两种干扰水平的一种情况。 要求A:频率配置有几个限制,第一,相互靠近的两个传送站不能配给同一信道。第二,由於波谱的 传播,相互距离在2s內的传送站必須不配给相同或相邻的信道,它们至少差2。在這些限制下,关于跨度能 说些什么。 要求B:假定前述图1中的格子在各方向延伸到任意远,回答要求A
Problem B: Radio Channel Assignments We seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon. An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, ... . Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span. Let s be the length of a side of one of the hexagons. We concentrate on the case that there are Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference Requirement A: There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in, Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions. Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance differ by at least some given integer k, while those at distance at most must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k? Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider? Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings. 问题 B 无线电信道分配 我们寻找无线电信道配置模型，在一个大的平面区域上设置一个传送站的均衡網絡，以避免干扰。一 个基本的方法是将此区域分成正六边形的格子（蜂窝狀）。如图 1。传送站安置在每个正六边形的中心点。 容许频率波谱的一个区间作为各传送站的频率。将這一区间规则地分割成一些空间信道，用整数 1，2， 3，……来表示。每一个传送站将被配置一正整数信道。同一信道可以在许多局部地区使用，前提是相邻 近的传送站不相互干扰。根据某些限制设定的信道需要一定的频率波谱，我们的目标是极小化频率波谱的 這个区间宽度。這可以用跨度這一概念。跨度是某一个局部区域上使用的最大信道在一切滿足限制的配置 中的最小值。在一个获得一定跨度的配置中不要求小於跨度的每一信道都被使用。 令 s 为一个正六边形的一側的长度.我们集中考虑存在两种干扰水平的一种情况。 要求 A：频率配置有几个限制，第一，相互靠近的两个传送站不能配给同一信道。第二，由於波谱的 传播,相互距离在 2s 內的传送站必須不配给相同或相邻的信道,它们至少差 2。在這些限制下，关于跨度能 说些什么。 要求 B： 假定前述图 1 中的格子在各方向延伸到任意远，回答要求 A
要求C:在下述假定下,重复要求A和B。更一般地假定相互靠近的传送站的信道至少差一个给定的 整数k,同时那些隔开一点的保持至少差1。关于跨度和关于设计配置的有效策略作为k的一个函数能说 点什么 要求D:考虑问题的一般化,比如各种干扰水平,或不规则的传送站布局。其他什么因素在考虑中是 重要的。 要求E:写一篇短文(不超过两页)给地方报纸,阐述你的发现。 孙山泽译 Figure-1(图-1)
要求 C：在下述假定下，重复要求 A 和 B。更一般地假定相互靠近的传送站的信道至少差一个给定的 整数 k，同时那些隔开一点的保持至少差 1。关于跨度和关于设计配置的有效策略作为 k 的一个函数能说 点什么。 要求 D：考虑问题的一般化，比如各种干扰水平，或不规则的传送站布局。其他什么因素在考虑中是 重要的。 要求 E：写一篇短文（不超过两页）给地方报纸，阐述你的发现。 孙山泽译 Figure－1（图－1）