历年数学建模竞赛试题竞选.doc

历年数学建模竞赛试题竞选1997年全国大学生数学建模竞赛题目 A题 零件的参数设计一件产品由若干零件组装而成，标志产品性能的某个参数取决于这些零件的参数。零件参数包括标定值和容差两部分。进行成批生产 时，标定值表示一批零件该参数的平均值，容差则给出了参数偏离其标定值的容许范围。若将零件参数视为随机变量，则标定值代表期望 值，在生产部门无特殊要求时，容差通常规定为均方差的3 倍。 进行零件参数设计，就是要确定其标定值和容差。这时要考虑两 方面因素：一是当各零件组装成产品时，如果产品参数偏离预先设定的目标值，就会造成质量损失，偏离越大，损失越大；二是零件容差 的大小决定了其制造成本，容差设计得越小，成本越高。 试通过如下的具体问题给出一般的零件参数设计方法。 B题 截断切割某些工业部门（如贵重石材加工等）采用截断切割的加工方式。这 里“截断切割”是指将物体沿某个切割平面分成两部分。从一个长方体中加工出一个已知尺寸、位置预定的长方体（这两个长方体的对应表面是平行的），通常要经过6次截断切割。 设水平切割单位面积的费用是垂直切割单位面积费用的r 倍，且当先后两次垂直切割的平面(不管它们之间是否穿插水平切割)不平行时，因调整刀具需额外费用e。 试为这些部门设计一种安排各面加工次序（称“切割方式”）的方法，使加工费用最少。（由工艺要求，与水平工作台接触的长方体底面是事先指定的） 详细要求如下： 1）需考虑的不同切割方式的总数。2）给出上述问题的数学模型和求解方法。3）试对某部门用的如下准则作出评价：每次选择一个加工费用最少的待切割面进行切割。4）对于e = 0的情形有无简明的优化准则。5）用以下实例验证你的方法：待加工长方体和成品长方体的长、宽、高分别为10、14.5、 19和3、2、4，二者左侧面、正面、底面之间的距离分别为6、7、9（单位均为厘米）。垂直切割费用为每平方厘米1元，r和e的数据有以下4组： a. r =1, e = 0; b. r =1.5, e =0; c. r =8, e =0; d. r =1.5; 2 = e = 15. 对最后一组数据应给出所有最优解，并进行讨论。 1998年全国大学生数学建模竞赛题目A题 投资的收益和风险市场上有n种资产（如股票、债券、）Si ( i=1,n) 供投资者选择，某公司有数额为M的一笔相当大的资金可用作一个时期的投资。公司财务分析人员对这n种资产进行了评估，估算出在这一时期内购买Si的平均收益率为ri，并预测出购买Si的风险损失率为qi。考虑到投资越分散，总的风险越小，公司确定，当用这笔资金购买若干种资产时，总体风险可用所投资的Si中最大的一个风险来度量。购买Si要付交易费，费率为pi，并且当购买额不超过给定值ui时，交易费按购买ui计算（不买当然无须付费）。另外，假定同期银行存款利率是r0, 且既无交易费又无风险。（r0=5%）1. 已知n = 4时的相关数据如下： Siri(%)qi(%)pi(%)ui(元) S1282.51103S2211.52198S3235.54.552S4252.66.5402. 试给该公司设计一种投资组合方案，即用给定的资金M，有选择地购买若干种资产或存银行生息，使净收益尽可能大，而总体风险尽可能小。 3. 试就一般情况对以上问题进行讨论，并利用以下数据进行计算。 Siri(%)qi(%)pi(%)ui(元)S19.6 422.1181S218.5543.2407S349.4606.0428S423.9421.5549S270S614393.4397S740.7685.6178S8220S933.653.32.7475S1036.8402.9248S1111.8315.1195S1295.55.7320S1335462.7267S328S1515237.6131B题 灾情巡视路线下图为某县的乡（镇）、村公路网示意图，公路边的数字为该路段的公里数。今年夏天该县遭受水灾。为考察灾情、组织自救，县领导决定，带领有关部门负责人到全县各乡（镇）、村巡视。巡视路线指从县政府所在地出发，走遍各乡（镇）、村，又回到县政府所在地的路线。1. 若分三组（路）巡视，试设计总路程最短且各组尽可能均衡的巡视路线。 2. 假定巡视人员在各乡（镇）停留时间T=2小时，在各村停留时间t=1小时，汽车行驶速度V=35公里/小时。要在24小时内完成巡视，至少应分几组；给出这种分组下你认为最佳的巡视路线。 3. 在上述关于T , t和V的假定下，如果巡视人员足够多，完成巡视的最短时间是多少；给出在这种最短时间完成巡视的要求下，你认为最佳的巡视路线。 4. 若巡视组数已定(如三组），要求尽快完成巡视，讨论T，t和V改变对最佳巡视路线的影响。 99创维杯全国大学生数学建模竞赛题目 A题 自动化车床管理 一道工序用自动化车床连续加工某种零件，由于刀具损坏等原因该工序会出现故障，其中刀具损坏故障占95%, 其它故障仅占5%。工序出现故障是完全随机的, 假定在生产任一零件时出现故障的机会均相同。工作人员通过检查零件来确定工序是否出现故障。现积累有100次刀具故障记录，故障出现时该刀具完成的零件数如附表。现计划在刀具加工一定件数后定期更换新刀具。 已知生产工序的费用参数如下： 故障时产出的零件损失费用 f=200元/件； 进行检查的费用 t=10元/次； 发现故障进行调节使恢复正常的平均费用 d=3000元/次(包括刀具费)； 未发现故障时更换一把新刀具的费用 k=1000元/次。 1)假定工序故障时产出的零件均为不合格品，正常时产出的零件均为合格品, 试对该工序设计效益最好的检查间隔（生产多少零件检查一次）和刀具更换策略。 2)如果该工序正常时产出的零件不全是合格品，有2%为不合格品；而工序故障时产出的零件有40%为合格品，60%为不合格品。工序正常而误认有故障仃机产生的损失费用为1500元/次。对该工序设计效益最好的检查间隔和刀具更换策略。 3）在2)的情况, 可否改进检查方式获得更高的效益。 附:100次刀具故障记录(完成的零件数) 4593626245425095844337488155056124524349826407425657065936809266531644877346084281153593844527552513781474388824538862659775859755649697515628954771609402960885610292837473677358638699634555570844166061062484120447654564339280246687539790581621724531512577496468499544645764558378765666763217715310851 B题 钻井布局 勘探部门在某地区找矿。初步勘探时期已零散地在若干位置上钻井，取得了地质资料。进入系统勘探时期后，要在一个区域内按纵横等距的网格点来布置井位，进行“撒网式”全面钻探。由于钻一口井的费用很高，如果新设计的井位与原有井位重合（或相当接近），便可利用旧井的地质资料，不必打这口新井。因此，应该尽量利用旧井，少打新井，以节约钻探费用。比如钻一口新井的费用为500万元，利用旧井资料的 费用为10万元，则利用一口旧井就节约费用490万元。 设平面上有n个点Pi，其坐标为(ai,bi),i=1,2,n，表示已有的n个井位。新布置的井位是一个正方形网格N的所有结点（所谓“正方形网格”是指每个格子都是正方形的网格；结点是指纵线和横线的交叉点）。假定每个格子的边长（井位的纵横间距）都是1单位（比如100米）。整个网格是可以在平面上任意移动的。若一个已知点Pi与某个网格结点Xi的距离不超过给定误差（=0.05单位），则认为Pi处的旧井资料可以利用，不必在结点Xi处打新井。 为进行辅助决策，勘探部门要求我们研究如下问题： 1)假定网格的横向和纵向是固定的（比如东西向和南北向），并规定两点间的距离为其横向距离（横坐标之差绝对值）及纵向距离（纵坐标之差绝对值）的最大值。在平面上平行移动网格N，使可利用的旧井数尽可能大。试提供数值计算方法，并对下面的数值例子用计算机进行计算。 2)在欧氏距离的误差意义下，考虑网格的横向和纵向不固定（可以旋转）的情形，给出算法及计算结果。 3)如果有n口旧井，给出判定这些井均可利用的条件和算法（你可以任意选定一种距离）。 数值例子n=12个点的坐标如下表所示： i123456789101112ai0.501.413.003.373.404.724.725.437.578.388.989.50bi2.003.501.503.515.502.0014.503.410.80 99创维杯全国大学生数学建模竞赛题目（大专组） C题 煤矸石堆积 煤矿采煤时，会产出无用废料煤矸石。在平原地区，煤矿不得不征用土地堆放矸石。通常矸石的堆积方法是： 架设一段与地面角度约为 =25 的直线形上升轨道（角度过大，运矸车无法装满），用在轨道上行驶的运矸车将矸石运到轨道顶端后向两侧倾倒，待矸石堆高后，再借助矸石堆延长轨道，这样逐渐堆起如下图所示的一座矸石山来。 现给出下列数据： 矸石自然堆放安息角（矸石自然堆积稳定后，其坡面与地面形成的夹角）=55； 矸石容重（碎矸石单位体积的重量）约2吨/米3； 运矸车所需电费为 0.50元/度（不变）； 运矸车机械效率（只考虑堆积坡道上的运输）初始值（在地平面上）约30%，坡道每延长10米，效率在原有基础上约下降2%； 土地征用费现值为8万元/亩，预计地价年涨幅约10%； 银行存、贷款利率均为5%； 煤矿设计原煤产量为300万吨/年； 煤矿设计寿命为20年； 采矿出矸率（矸石占全部采出的百分比）一般为7%10%。 另外，为保护耕地，煤矿堆矸土地应比实际占地多征用10%。 现在煤矿设计中用于处理矸石的经费（只计征地费及堆积时运矸车用的电费）为100万元/年，这笔钱是否够用？试制订合理的年度征地计划，并对不同的出矸率预测处理矸石的最低费用。 D题 钻井布局（同 B 题） 勘探部门在某地区找矿。初步勘探时期已零散地在若干位置上钻井，取得了地质资料。进入系统勘探时期后，要在一个区域内按纵横等距的网格点来布置井位，进行“撒网式”全面钻探。由于钻一口井的费用很高，如果新设计的井位与原有井位重合（或相当接近），便可利用旧井的地质资料，不必打这口新井。因此，应该尽量利用旧井，少打新井，以节约钻探费用。比如钻一口新井的费用为500万元，利用旧井资料的费用为10万元，则利用一口旧井就节约费用490万元。 设平面上有n个点Pi，其坐标为(ai,bi),i=1,2,n，表示已有的n个井位。新布置的井位是一个正方形网格N的所有结点（所谓“正方形网格”是指每个格子都是正方形的网格；结点是指纵线和横线的交叉点）。假定每个格子的边长（井位的纵横间距）都是1单位（比如100米）。整个网格是可以在平面上任意移动的。若一个已知点Pi与某个网格结点Xi的距离不超过给定误差（=0.05单位），则认为Pi处的旧井资料可以利用，不必在结点Xi处打新井。 为进行辅助决策，勘探部门要求我们研究如下问题： 1)假定网格的横向和纵向是固定的（比如东西向和南北向），并规定两点间的距离为其横向距离（横坐标之差绝对值）及纵向距离（纵坐标之差绝对值）的最大值。在平面上平行移动网格N，使可利用的旧井数尽可能大。试提供数值计算方法，并对下面的数值例子用计算机进行计算。 2)在欧氏距离的误差意义下，考虑网格的横向和纵向不固定（可以旋转）的情形，给出算法及计算结果。 3)如果有n口旧井，给出判定这些井均可利用的条件和算法（你可以任意选定一种距离）。 数值例子n=12个点的坐标如下表所示： i123456789101112ai0.501.413.003.373.404.724.725.437.578.388.989.50bi2.003.501.503.515.502.0014.503.410.80 美国大学生数模赛题选 1996年赛题 1997年赛题 1998年赛题 1999年赛题 2002年赛题1996 美国大学生数模竞赛题Problem AThe worlds oceans contain an ambient noise field. Seismic disturbances,surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, developa method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed freqency and amplitude.Problem BWhen determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Lets say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest constains both the number of judges that can be obtained and amount of time that they can judge. For eample if P=100, then J=8 is typical.Ideally, each judge would read paper and rank-order them, but there are toomany papers for this. Instead, there will be a number of screening rounds in which each judge will read some number of papers and give them scores. Then some selection scheme is used to reduce the number of papers underconsideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from 1 to 100),then all papers below some cut-off level could be rejected.The new pool of papers is then passed back to the judges, and the process is repeated. A concern is then the total number of papers that judge reads must be substantially less than P. The process is stopped when there are only W papers left. There are the winners. Typically for P=100, W=3.Your task is to determine a selection scheme, using a combination ofrank-ordering, numerical scoring, and other methods, by which the final Wpapers will include only papers from among the best 2W papers. (By best,we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers. Among all such methods, the one that required each judge to read the least number of papers is desired.Note the possibility of systematic bias in a numerical scoring scheme. Forexample, for a specific collection of papers, one judge could average 70points, while another could average 80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)?1997 年美国大学生数模竞赛题 Problem A: The Velociraptor ProblemThe Velociraptor, Velociraptor mongoliensis, was a predatory dinosaur thatlived during the late Cretaceous period, approximately 75 million yearsago. Paleontologists think that it was a very tenacious hunter, and mayhave hunted in pairs or larger packs. Unfortunately, there is no way toobserve its hunting behavior in the wild as can be done with modernmammalian predators. A group of paleontologists has approached your teamand asked for help in modeling the hunting behavior of the velociraptor.They hope to compare your results with field data reported by biologistsstudying the behaviors of lions, tigers, and similar predatory animals.The average adult velociraptor was 3 meters long with a hip height of 0.5meters and an approximate mass of 45 Kg. It is estimated that the animalcould run extremely fast, at speeds of 60 km/hr., for about 15 seconds.After the initial burst of speed, the animal needed to stop and recoverfrom a buildup of lactic acid in its muscles.Suppose that Velociraptor prey on Thescelosaurus neglectus, a herbivorousbiped approximately the same size as the Velociraptor. A biomechanicalanalysis of a fossilized thescelosaurus indicates that if could run at aspeed of about 50km.hr. for long periods of time.Part1Assuming the velociraptor is a solitary hunter, design a mathematical modelthat describes a hunting strategy for a single velociraptor stalking andchasing a single thescelosaurus as well as the evasive strategy of theprey. Assume that the thecelosaurus can always detect the velociraptor whenin comes within 15 meters, but may detect the predator at even greaterranges (up to 50 meters) depending upon the habitat and weather conditions.Additionally, due to its physical structure and strength, the velociraptorhas a limited turning radius when running at full speed. This radius isestimated to be three times the animals hip height. On the other hand, thethescelosaurus is extremely agile and has a turning radius of 0.5 meters.Part 2Assuming more realistically that the velociraptor hunted in pairs, design anew model that describes a hunting strategy for two velociraptors stalkingand chasing a single thescelosaurus as well as the evasive strategy of theprey. Use the other assumptions and limitations given in Part 1.Problem B: Mix Well For Fruitful Discussions Small group meetings for the discussion of important issues, particularly long-rang planning, are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings the board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run risk of control by a dominant personality. In an attempt to reduce this danger it is common to schedule several sessions with a different mix of people in each group. A meeting of an Tostal Corporation will be attended by 29 Board Members of which nine are in-horse members(i.e., corporate employees). The meeting is to be an all-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each morning session will consist of six discussion group with each discussion group led by one of the corporations six senior officers. None of these of officers are board members. Thus each senior officer will lead threedifferent discussion groups. The sessions will consist of only fourdiscussion groups.The president of the corporation wants a list of board-member assignmentsto discussion group for each of seven sessions. The assignments shouldachieve as much of a mix of members as much as possible. The idealassignment would have each board member with each other board member in adiscussion group the same number of times while minimizing commonmembership of groups for the different sessions.The assignments should also satisfy the following criteria:1.For the morning sessions, no board member should be in the same seniorofficers discussion group twice.2.No discussion group should contain a disproportionate number of in-housemembers.Give a list of assignments for members 1-9 and 10-29 and officers 1-6.Indicate how well the criteria in the precious paragraphs are met. Since itis possible that some board members will cancel at the last minute or thatsome not scheduled will show up, an algorithm that the secretary could useto adjust the assignments with an user to make assignments for futuremeetings involving different levels of participation for each type ofattendee.1998 年美国大学生数模竞赛题Problem A: Introduction: Industrial and medical diagnostic machines known as Magnetic Resonance Imagers (MRI) scan a three-dimensional object such as a brain, and deliver their results in the form of a three-dimensional array of pixels. Each pixel consists of one number indicating a color or a shade of gray that encodes a measure of water concentration in a small region of the scanned object at the location of the pixel. For instance, 0 can picture high water concentration in black (ventricles, blood vessels), 128 can picture a medium water concentration in gray (brain nuclei and gray matter), and 255 can picture a low water density in white (lipid-rich white matter consisting of myelinated axons). Such MRI scanners also include facilities to picture on a screen any horizontal or vertical slice through the three-dimensional array (slices are parallel to any of the three Cartesian coordinate axes). Algorithms for picturing slices through oblique planes, however, are proprietary. Current algorithms are limited in terms of the angles and parameter options available; are implemented only on heavily used dedicated workstations; lack input capabilities for marking points in the picture before slicing; and tend to blur and feather out sharp boundaries between the original pixels. A more faithful, flexible algorithm implemented on a personal computer would be useful (1) for planning minimally invasive treatments, (2) for calibrating the MRI machines, (3) for investigating structures oriented obliquely in space, such as postmortem tissue sections in animal research, (4) for enabling cross-sections at any angle through a brain atlas consisting of black-and-white line drawings. To design such an algorithm, one can access the values and locations of the pixels, but not the initial data gathered by the scanner. Problem: Design and test an algorithm that produces sections of three-dimensional arrays by planes in any orientation in space, preserving the original gray-scale values as closely as possible. Data Sets: The typical data set consists of a three-dimensional array A of numbers A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)_ijk . Typically, A(i,j,k) can range from 0 through 255. In most applications; the data set is quite large. Teams should design data sets to test and demonstrate their algorithms. The data sets should reflect conditions likely to be of diagnostic interest. Teams should also characterize data sets that limit the effectiveness of their algorithms. Summary: The algorithm must produce a picture of the slice of the three-dimensional array by a plane in s