美国大学生数学建模竞赛B题
美国大学生数学建模比赛2014年B题
Team # 26254
Page 2 oon ............................................................................................................................................................. 3 2. The AHP .................................................................................................................................................................. 3 2.1 The hierarchical structure establishment ....................................................................................................... 4 2.2 Constructing the AHP pair-wise comparison matrix...................................................................................... 4 2.3 Calculate the eigenvalues and eigenvectors and check consistency .............................................................. 5 2.4 Calculate the combination weights vector ..................................................................................................... 6 3. Choosing Best All Time Baseball College Coach via AHP and Fuzzy Comprehensive Evaluation ....................... 6 3.1 Factor analysis and hierarchy relation construction....................................................................................... 7 3.2 Fuzzy comprehensive evaluation ................................................................................................................... 8 3.3 calculating the eigenvectors and eigenvalues ................................................................................................ 9 3.3.1 Construct the pair-wise comparison matrix ........................................................................................ 9 3.3.2 Construct the comparison matrix of the alternatives to the criteria hierarchy .................................. 10 3.4 Ranking the coaches .....................................................................................................................................11 4. Evaluate the performance of other two sports coaches, basketball and football.................................................... 13 5. Discuss the generality of the proposed method for Choosing Best All Time College Coach ................................ 14 6. The strengths and weaknesses of the proposed method to solve the problem ....................................................... 14 7. Conclusions ........................................................................................................................................................... 15
2019年美国大学生数学建模竞赛B题
2019 MCMProblem B: Send in the Drones: Developing an Aerial Disaster Relief Response SystemBackground: In 2017, the worst hurricane to ever hit the United States territory of Puerto Rico (see Attachment 1) left the island with severe damage and caused over 2900 fatalities. The combined destructive power of the hurricane’s storm surge and wave action produced extensive damage to buildings, homes, and roads, particularly along the east and southeast coast of Puerto Rico. The storm, with its fierce winds and heavy rain, knocked down 80 percent of Puerto Rico's utility poles and all transmission lines, resulting in loss of power to essentially all of the island's 3.4 million residents. In addition, the storm damaged or destroyed the majority of t he island’s cellular communication networks. The electrical power and cell service outages lasted for months across much of the island, and longer in some locations. Widespread flooding blocked and damaged many highways and roads across the island, making it nearly impossible for emergency services ground vehicles to plan and navigate their routes. The full extent of the damage in Puerto Rico remained unclear for some time; dozens of areas were isolated and without communication. Demands for medical supplies, lifesaving equipment, and treatment strained health-care clinics, hospital emergency rooms, and non-governmental organizations’ (NGOs) relief operations. Demand for medical care continued to surge for some time as the chronically ill turned to hospitals and temporary shelters for care.Problem: Non-governmental organizations (NGOs) are often challenged to provide adequate and timely response during or after natural disasters, such as the hurricane that struck the United States territory of Puerto Rico in 2017. One NGO in particular – HELP, Inc. - is attempting to improve its response capabilities by designing a transportable disaster response system called “DroneGo.” DroneGo will use rotor wing drones to deliver pre-packaged medical supplies and provide high-resolution aerial video reconnaissance. Selected drones should be able to perform these two missions – medical supply delivery and video reconnaissance – simultaneously or separately, depending on relief conditions and scheduling. HELP, Inc. has identified various candidate rotor wing drones that it would like your team to consider for possible use in designing its DroneGo fleet (see Attachments 2, 3).DroneGo’s pre-packaged medical supplies, called medical packages, are meant to augment, not replace, the supplies provided by local medical assistance organizations on-site within the country affected by the disaster. HELP, Inc. is planning on three different medical packages referred to as MED1, MED2, and MED3. Drones will carry these medical packages within drone cargo bays for delivery to selected locations (see Attachments 4, 5). Depending on the specific drone being used to transport medical supplies, it may be possible that multiple medical packages can be transported in a single drone cargo bay. Note that drones must land on the ground to offload medical supplies from the drone cargo bays. The video capability of the drones will provide high-resolution video of damaged and serviceable transportation road networks to HELP, Inc.’s command and contro l center for ground-based route planning.HELP, Inc. will use International Standards Organization (ISO) standard dry cargo containers to quickly transport a complete DroneGo disaster response system to a particular disaster area. The individual shipping containers for all drones in the DroneGo fleet, along with all requiredmedical packages, must fit within a maximum of three of the ISO cargo containers to be delivered to a single location, or up to three different locations if three cargo containers are used in the disaster area. Each shipping container’s contents should be packed in order to minimize any need for buffer materials for unused space. Table 1 shows the dimensions of an ISO standard dry cargo container.Table 1. Standard ISO Container DimensionsExterior Interior Door OpeningLength Width Height Length Width Height Width Height 20’20’8’8’6”19’3”7’8”7’ 10”7’8”7’5”StandardDryContainerHELP, Inc. is asking your team to use the 2017 situation in Puerto Rico to design a DroneGo disaster response system that will fit within the containers noted while meeting the anticipated medical supply demands during a potential similar future disaster scenario. It is possible that the demand requirements of this scenario may exceed the capabilities of the drone fleet your team identifies. If this occurs, HELP, Inc. wants to clearly understand any tradeoffs that it must make for implementing solutions to address these shortcomings.Part 1. Develop a DroneGo disaster response system to support the Puerto Rico hurricane disaster scenario.Consider the background information, the requirements identified in the problem statement, and the information provided in the problem attachments to address the following.A.Recommend a drone fleet and set of medical packages for the HELP, Inc. DroneGodisaster response system that will meet the requirements of the Puerto Rico hurricanescenario. Design the associated packing configuration for each of up to three ISO cargocontainers to transport the system to Puerto Rico.B.Identify the best location or locations on Puerto Rico to position one, two, or three cargocontainers of the DroneGo disaster response system to be able to conduct both medicalsupply delivery and video reconnaissance of road networks.C.For each type of drone included in the DroneGo fleet:a.Provide the drone payload packing configurations (i.e. the medical packagespacked into the drone cargo bay), delivery routes and schedule to meet the identifiedemergency medical package requirements of the Puerto Rico hurricane scenario.b.Provide a drone flight plan that will enable the DroneGo fleet to use onboardvideo cameras to assess the major highways and roads in support of the Help, Inc.mission.Part 2. MemoWrite a 1–2 page memo to the Chief Operating Officer (CEO) of HELP, Inc. summarizing your modeling results, conclusions, and recommendations so that she can share with her Board of Directors.Your MCM team submission should consist of:∙One-page Summary Sheet,∙One- to Two-page memo to the HELP, Inc. CEO∙Your solution of no more than 20 pages, for a maximum of 23 pages with your summary and memo.∙Note: Reference list and any appendices do not count toward the 23-page limit and should appear after your completed solution.Attachments:1.Map of Puerto Rico2.Potential Candidate Drones for DroneGo Fleet Consideration (with Drone payloadcapability)3.Drone Cargo Bay Packing Configuration/Dimensions by Type4.Anticipated Medical Package Demand5.Emergency Medical Package Configuration/DimensionsAttachment 1: Map of Puerto RicoAttachment 2: Potential Candidate Drones for DroneGo Fleet Consideration (with Drone Payload Capability)Shipping ContainerDimensions PerformanceCharacteristics/CapabilitiesConfigurations CapabilitiesDroneLength (in.) Width (in.) Height (in.) Max PayloadCapability(lbs.) Speed(km/h)Flight Time No Cargo (min)Video Capable Medical Package Capable Drone Cargo Bay Type* A 45 45 25 3.5 40 35 Y Y 1 B 30 30 22 8 79 40 Y Y 1 C 60 50 30 14 64 35 Y Y 2 D 25 20 25 11 60 18 Y Y 1 E 25 20 27 15 60 15 Y Y 2 F 40 40 25 22 79 24 N Y 2 G 32 32 17 20 64 16 Y Y 2 H Tethered657541N/AN/AIndefiniteNNN/A*Note that cargo bays are affixed to the drone and that drone must be on the ground to offload cargo. SeeAttachment 3 for Drone Cargo Bay Type Configuration/Dimensions.Attachment 3: Drone Cargo Bay Packing Configuration/Dimensions by TypeDrone Cargo Bay Type Length (in) Width (in) Height (in)1 8 10 14 Top Loaded2 24 20 20 Top Loaded Attachment 4: Anticipated Medical Package DemandDelivery Location Emergency Medical Packages ** Location Name Latitude Longitude Requirement Quantity FrequencyCaribbean Medical Center 18.33 -65.65 MED 1 1 DailyJajardo MED 3 1 DailyHospital HIMA 18.22 -66.03 MED 1 2 DailySan Pablo MED 3 1 Daily Hospital Pavia Santurce 18.44 -66.07 MED 1 1 DailySan Juan MED 2 1 Daily Puerto Rico Children's Hospital 18.40 -66.16 MED 1 2 DailyBayamon MED 2 1 DailyMED 3 2 Daily Hospital Pavia Arecibo 18.47 -66.73 MED 1 1 DailyArecibo**See Attachment 5 for Emergency Medical Packages 1, 2, and 3 Configurations/Dimensions.A ttachment 5: Emergency Medical Package Configuration/DimensionsEmergency Medical Package ConfigurationPackage ID Weight(lbs.)Package Dimensions(in.)(L × W × H)MED 1 2 14 × 7 × 5 MED 2 2 5 × 8 × 5 MED 3 3 12 × 7 × 4Glossary:Cargo Container (Shipping Container): a large rectangular container with doors on the ends for loading and packing, and made of material suitable for shipping, storing, and handling in many weather and climate conditions.Drone (Unmanned Aerial Vehicle, UAV): a flying robot that can be remotely controlled or fly autonomously through software-controlled flight plans in their embedded systems that work in conjunction with onboard sensors and GPS.Drone Cargo Bay: For rotor wing drones, this is an externally carried “box” used to transport materials. For this problem, the drones under consideration have one of two types (sizes) of cargo bays. Note that each drone must land for the medical packages to be unloaded from the bay at its destination.Drone Fleet: a set of drones for a particular mission or purpose. For this problem, the total set of drones by type (A to H) and Payload Capability (Visual and Medical) needed to meet the requirements of HELP, Inc.Drone Payload Packing Configuration: how the drone payload bays are packed. For this problem, how the medical packages being transported by a drone are packed inside the drone cargo bay.Medical Package: a predetermined set of medical supplies packed in a single container. For this problem, there are three Medical Package Configurations (MED1, MED2, MED3) available for transport by a drone from a deployed cargo container location to the demand location.Non-governmental Organization (NGO): Usually non-profit and sometimes international organization independent of government and governmental organizations that is active in humanitarian, educational, healthcare, social, public policy, human rights, environmental and other areas in attempts to affect change.Payload Capability: the carrying capacity of an aircraft or launch vehicle, usually measured in terms of weight. For this problem, the capability/capacity of the drone to carry medical packages.。
美赛数学建模B题
老傅美赛建模B题思路
老傅美赛建模B题思路（喝多了做）先提醒，本次思路在QQ群首发，并欢迎各位转载！载转注明：fugn690621 的思路！！！！！！！！（7个感叹号哦，另外老傅喝多了）B题原题：多车道划分的封闭式收费公路使用“匝道收费”和“障碍收费”从司机那里收取通行费。
匝道收费是设置在入口或出口匝道的公路上一种收费机制，这是我们不用考虑的。
障碍收费是一排垂直于交通流的方向、横跨高速公路的收费亭。
通常（总有）收费亭的数量要多于进入收费场车道的数量（详见2005 MCM problem ）,因此当车辆离开障碍收费的收费亭时，车辆必须从大量的收费站出口车道“扇入”到数量较少的常规车道。
收费广场是在公路上的一个区域，以方便障碍收费，包括扇出区之前的障碍收费，收费障碍本身，和在收费障碍区后的扇入区域。
例如，一个三车道的公路（单方向）可以使用8个障碍收费亭。
交费后，车辆可在具有与进入收费广场相同数量的车道（在该示例中为三个）的高速公路上继续行驶。
考虑到一个收费高速公路在每个方向上具有L个行驶车道和B个障碍收费站（B>L）。
确定紧邻收费障碍的区域的形状，大小和合并模式，在此区域车辆从B 收费亭出口车道下行到L车道。
在您的模型中，包含重要要考虑的因素包括事故预防，吞吐量（每小时的车辆通过广场的一端加入L传出的交通车道），和成本（土地和道路建设是昂贵的）。
特别是,该问题并不仅仅要求针对任何可能已经实施的特定收费广场设计的性能分析。
重点是确定要提出是否有比任何常用的更好的解决方案（形状，大小和合并模式）。
确定在通畅和拥堵的交通状况中你的解决方案，随着更多的（自主）自驾车加入到交通组合中，你的方案如何改变？你的解决方案如何受到收费站比例的影响，这些收费站包括常规的（人员配置）收费站，精确找零（自动）收费站和电子收费站（例如通过车辆中的转发器进行电子收费）？您的MCM提交报告应该包括1页的摘要，1-2页给New Jersey Turnpike Authority（美国新泽西州收费高速公路当局）的信件，以及您的解决方案（不超过20页），最多23页，注意：附录和参考文献不计入23页的限制中。
14美国数学建模B题
For office use only T1________________ T2________________ T3________________ T4________________Team Control Number30213Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________The Keep-Right-Except-To-Pass RuleAbstractIn this paper, five mathematical models are proposed, respectively around the performance of the keep-right-except-to-pass rule on multi-lane freeways in light and heavy traffic.Three mathematical models are established to analyze tradeoffs between multi-line traffic flow and safety, traffic speed, and other factors under the keep-right-except-to-pass rule. At first, by taking overtaking and mixed vehicle types as viscous resistance, and according to the mass conservation law, we develop continuous and discrete fluid dynamic traffic flow model. Then by taking three variables (space and temporal distance between vehicles, vehicle speed) into account, we establish traffic safety control model. Finally, by considering synthetically vehicle operation efficiency, security, comfort, fuel economy and so on, we develop a multi-objective programming model to limit the highest speed.In this paper we give several different traffic rules, and present some improvement for the keep-right-except-to-pass rule, such as vehicles of different types on different lanes, and different lanes with different speed limits. In a segment of a multi-lane freeway, new measures can improve 15.11% of traffic flow.We can apply our mathematical models to the keep-left-except-to-pass rule with a simple change of orientation. For example, the drivers need drive at the right of the car for more safety.Two mathematical models are established to analyze the traffic flow of multi-lane freeways under the control of an intelligent system. At first, we present five security operating patterns basing on seven indices in security operation about person, vehicle and surroundings. We thus propose a microscopic safety control model basing on RBF neural network, and conduct a simulation test of recognition of security operating patterns. Finally, we study the vehicle’s speed change, total service flow of the freeway, the vehicle’s travel time. We thus propose three different macroscopic optimal control models for freeway sections with low traffic density, medium traffic density and high traffic density, respectively. The three models are dynamic nonlinear programming models. The result shows that the effect of the intelligent system promotes the traffic flow effectively.1 Introduction (1)2 Definitions and Variables (1)3 Assumptions (2)4 Establishment and Solution of Three Models for Problem 1 (2)4.1 Model Constraints (2)4.2 Fluid Dynamic Traffic Flow Model (3)4.2.1 Impact of mixed-type vehicles and overtaking (3)4.2.2 Continuous fluid dynamic traffic flow model (4)4.2.3 Discrete fluid dynamic traffic flow model (5)4.3 Traffic Flow Safety Control Model (6)4.3.1 Determination of the vehicle operating safety (6)4.3.2 Traffic flow safety control model [4] (7)4.4 Multi-objective Programming Model of the Maximum Speed Limit [9] (8)4.4.1 Performance indicators of multi-objective programming model [5] (8)4.4.2 Constraints about security and comfort property (8)4.4.3 Establishment of a multi-objective programming model to limit maximum speed (9)4.5 Improvement and Evaluation of the Traffic Rules (10)4.5.1 Improvement of the traffic rules (10)4.5.2 Evaluation of the traffic rules (10)5 Analysis of Problem 2 (11)6 Establishment and Solution of Two Models for Problem 3 (11)6.1 Microscopic Safety Control Model Basing on RBF Neural Network (12)6.1.1 Indices of security operation about person, vehicle and surroundings (12)6.1.2 Safe operation patterns (12)6.1.3 Microscopic security control model based on neural network (12)6.2 Macroscopic Optimal Control Model of the Traffic Flow (14)6.2.1 Control model with variable speed limitation (15)6.2.2 Constraints of the optimal control model (15)6.2.3 Performance index of the macroscopic optimal control model (15)6.2.4 Three macroscopic optimal control models of the traffic flow (16)7 Analysis and Evaluation of Models (17)7.1 Evaluation of fluid dynamic traffic flow models (17)7.2 Evaluation of the safety control model to limit speed (17)7.3 Evaluation of the microscopic safety control model with RBF neural network (18)7.4 Evaluation of the macroscopic optimal control model (18)7.5 Improvements of the Models (18)References (18)1 IntroductionSome countries, such as USA, China and so on, have the keep-right-except-to-pass rule on multi-lane freeways. This rule requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane. But there are also many countries providing the keep-left-except-to-pass rule. The traffic rule relies upon human judgment for compliance.To analyze the performance of the traffic rule in light andheavy traffic, we study some mathematical models to solve thefollowing problems.Problem 1: Examine tradeoffs between traffic flow and safety,the role of under- or over-posted speed limits, and other factorsfor the keep-right-except-to-pass rule. Judge the efficiency of therule in promoting traffic flow. Suggest alternatives that mightpromote greater traffic flow, safety, or other factors if the rule isnot effective.Problem 2: Argue whether or not above analysis can be carriedover to the keep-left-except-to-pass rule.Problem 3: Stuffy the performance of the traffic rule under thecontrol of an intelligent system-either part of the road networkor imbedded in the design of all vehicles using the roadway.Show the changing extent of the results of earlier analysis.In order to reach Problem 1, we first take overtaking and mixed vehicle types as viscous resistance, and develop continuous and discrete fluid dynamic traffic flow models according to the mass conservation law. Then we take space and temporal distance between vehicles and vehicle speed into account to establish traffic safety control model. Finally, we consider synthetically vehicle operation efficiency, security, comfort property and fuel economy and develop a two-objective programming model to limit the highest speed. We also look for other traffic rules and make comparison.For Problem 2, we give a simple change of orientation in order to apply our mathematical models to the keep-left-except-to-pass rule. For example, the drivers need drive at the right of the vehicles with different types on different lanes on different lanes with different speed limit for more safety and more traffic flow.In order to reach Problem 3, we study security operating patterns basing on indices in security operation about person, vehicle and surroundings. By applying RBF neural network, we propose a microscopic safety control model, and conduct a corresponding simulation test. Next, we research the vehicle’s speed change, total service flow of the freeway, the vehicle’s travel time and freeway sections with low traffic density, medium traffic density and high traffic density. Then according to the thinking of dynamic nonlinear programming, we propose three different macroscopic optimal control models. At last, we test the efficiency of the intelligent traffic system.2 Definitions and VariablesThe following definitions and variables will be used in our discussion of the traffic rule problem.·A rule is drivers drive in the right-most lane on multi-lane freeways.·u is the speed of the car in the freeways.·k is the density of the traffic flow .·q is the traffic flow on the freeways·w τis viscous drag ，mains micelles produced fluid occurs between the relative tangential slip resistance·μ is viscosity coefficient·η is the proportion of small cars·T is the driver's reaction time·A T is safe distance from the vehicle·A K is safety traffic density.·K is the density of the·L is safe distance between vehicles·()t k i is the average density of the i vehicle·g P is gasoline prices3 Assumptions(1) All factors that cause damage to the vehicle speed are defined as the viscous resistance w ι, including every factor that overtaking and mixed vehicle types and so on.(2) There are three types of vehicles in freeway: small car, midsize car, and super-huge type. Because the super-huge vehicle occupies a least proportion, we regard super-huge vehicle as midsize car.(3) Traffic flow is stable traffic flow which is continuous, uninterrupted, evenly distributed.(4)Traffic speed, flow, traffic density are continuous functions with respect to special and temporal variables.(5) There are five kinds of vehicle operation patterns: following pattern, accelerate pattern, retard pattern, overtake pattern, break pattern.(6) In low traffic density area, the mutual interference between vehicles is very small.(7) Under the intelligent control systems, pilots, vehicles, environment and other factors have some influence on the traffic safety.4 Establishment and Solution of Three Models for Problem 14.1 Model ConstraintsTo study the performance of the keep-right-except-to-pass rule, we consider the three elements of traffic flow: traffic flow, safety and speed limits. In China, we have a traffic speed limit rule on the roadway for safety, as follows:On the basis of security, we set up a model to analyze traffic flow, speed and vehicle density. First, speed u , density k and flow q satisfy thatku q = (1)For multi-lane freeway with small k , the driver will drive free at the speed of f u . According to Mass Conservation Law, we set up the following single-lane model without lane change0=∂∂+∂∂xq t k (2) For multi-lane with lane change, we should add a generation item of traffic flow to above equation. Denote the flow in and out of the lane line by r . We establish the following equationdxdr x q t k =∂∂+∂∂ (3) 4.2.1 Impact of mixed-type vehicles and overtaking In fluid mechanics, the fluid viscosity is defined as tangential resistance generating when fluid micelle occurs relative slippage. The size of resistance is in proportion to contact area and velocity gradient. There we assume that all factors that causes damage to the vehicle speed are defined as the viscous resistance w ι, including every factor that overtaking and mixed vehicle types and so on. The higher the degree of mixed vehicle types (Viscosity coefficient of performance μ) is, the more influence traffic stream has; the greater the mixed degree is, the greater the viscosity coefficient μ is; a single model of the traffic has the minimum mutual interference between them. There are three types of vehicles in freeway: small car, midsize car, and super-huge type. Because the super-huge vehicle occupies a least proportion, we regard super-huge vehicle as midsize car. According to [1], we assume that the ratio η of μand small cars to meetηημ++=312 (4) The viscous resistance which date from mixed models is also proportional to the difference between the free-flow speed f μμ-and density gradient between lanes k .drivers are inpursuit of the free-flow speed, and the vehicle speed, difference between the free stream velocity and the size of the density joint determines the possible of overtaking. The smaller the difference, the greater the density is, the more impossible of overtaking. Moreover, inter-lane overtaking has a great impact on the upstream traffic flows. The greater ratio of overtaking is, the greater impact on productivity of downstream, the greater the resistance. This is because the viscous resistance which date from mixed models is proportional to the difference between the free-flow speed and density gradient between lanes .Through the foregoing analysis, the viscous resistance w ιcan be represented by the follow formula:x k n u u f w ∂∂+-=])([μι (5) Corresponding model equations is:w e t x u k u Tdt du ι+-=)],()([1 (6) where T means that the reaction time of the driver and()⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧<<⎪⎪⎭⎫ ⎝⎛-≤≤----≤≤⎥⎥⎦⎤⎢⎢⎣⎡-+=n j n j q r j f n j f n j qr j f n f n j j q r f f e e k k e k e k k m u e k k k m m e k e k k m u e k m m e k k k e k k m m u u 21121141411411 (7) 4.2.2 Continuous fluid dynamic traffic flow modelBecause speed is a function with distance and time, hence we can getxu u t u dt dx x u t u dt t x du dt du a ∂∂+∂∂=⋅∂∂+∂∂===),( (8) We take the second equation and the fifth equation into the third equation, gaining another mixed equation.xk n u u u k u T x u u t u f e ∂∂+-+-=∂∂+∂∂])([])([1μ (9) Therefore, we combine the third equation with the eighth equation for obtaining a new kinetic model as follows:()[]()[]x k n u u t t u k u Tx u u t u dxdr x q t k f w w e ∂∂+-=+-=∂∂+∂∂=∂∂+∂∂μ1 (10) This is the hydrodynamic model which corresponding with the mixed traffic flow that has the phenomenon of overtaking. The model is continuous model . Dispersing the density andvelocity by using numerical differential methods to ninth equation then we can create Discrete Models.4.2.3 Discrete fluid dynamic traffic flow modelt r u k u k xt k k n i n i n i n i n i n i n i ∆+-∆∆+=--+)(111 (11) ()()()()]()⎩⎨⎧⎭⎬⎫-+-+-+-⨯-+∆∆+=---+n i n i n i f n i n i e n i n i e n i n i n i n i k k n u u u u T u u u u x t u u 1111[1μμ (12) where ,3,2,1=i …, ,3,2,1,0=n …. The finite difference numerical format only keeps stability when it meets the following V on Neumann conditions, that ismuu k k x t f j ∆∆≤∆ (13) We use MA TLAB software to simulate the above established mathematical model and then get three-dimensional map which includes changes in the density and the corresponding density and in the speed and corresponding speed when other factors remain unchanged. Drawn diagram below:Figuer1 Density - time curve Figuer2 Three-dimensional map of the changes of densityFiguer3 Velocity- time curve Figuer4 Three-dimensional map of the changes of velocityThrough simulating the change map of speed and density, we can clear observe the relationship between the velocity, density and traffic flow.Utilizing the shifty relation diagram with speed, density and flow, we work out the vehicle flow ’s shifty diagram under low load and high load with the rule. As follow:Figure 5 Low load and high load ’s shifty vehicle flowBy observing the shifty diagram of vehicle under low load and high load, we can study clearly the affect of the right rule. On the low load condition, changes in volatility of vehicle are small. There is a rush hour under high load. Letter on, it would tend to smooth and steady.4.3 Traffic Flow Safety Control ModelFluid dynamic traffic flow model study the relationship between the flow, speed and density and the driver driving safety [4]. In the traffic, we should also consider security problems. Then we improve the fluid dynamic traffic flow model, establishing a new model on flow, speed and security.4.3.1 Determination of the vehicle operating safetyThere are many factors affecting the safety of vehicle running. We study spacing, distance and speed.A T is the shortest length time of don’t crash when the ahead vehicle stop suddenly. Then:321t t t T A ++= (14)In above equation, 1t is the driver perception reaction time, it depends on the driver’s visual function space, motion perception sensitive, other physiological condition and weather condition. 2t denotes the time from life to the breaking, it depends on the driver’s reaction capacity, operational experience, vehicle performance. 3t is the time from breaking start to the end of the break, it depends vehicle performance, road conditions, meteorological condition.Safety clearance A L is the shortest distance between cars of don’t crash when the ahead vehicle stop suddenly. Suppose V is the speed when the vehicle break, safety distance A L can be expressed when vehicle operating.()()212254A K V L L t t V h g =+++- (15)In above equation, K L is a constant,h is adhesion coefficient on tire and road, g is corrected parameter for h with special meteorological conditions such as rain, snow, ice and so on4.3.2 Traffic flow safety control model [4]Suppose f V is pass impeded speed,j K is jam density, X is the length on a road, traffic flow is continuous, uninterrupted, evenly distributed, consisting of small cars. m L is the length of the car, q is a traffic flow into the road, references[4], Select the security control model for the road traffic flowVK V K K qV V V V VKq j f j f f f -=⎥⎥⎦⎤⎢⎢⎣⎡-±==4212 ()[]TL V X L L X K m =+= (16) ()()AAK A A V V K K i g h V V t t L L L t t t T T ≤≤±-+++=≥++=≥254221321In this formula, K is vehicle density,L is vehicle spacing,T is vehicle interval,A Vis safety speed, A K is safety traffic density.Figure 6 Three-dimensional map of speed-density-flowUnder the condition of abiding by the rules of right of traffic, When other factors unchanged,different cars have a higher requirement for the driver.4.4 Multi-objective Programming Model of the Maximum Speed Limit [9]Application of multi-objective optimization ideas proposed maximum speed limit method based on the operational efficiency, safety, economy and comfort of the highway. The time and fuel consumption costs of highway speed to a minimum as the goal , accident mortality and comfort to satisfy the corresponding allowable range as constraints, and we have established a multi-objective programming model of a maximum speed of about costs, fuel costs and other factors limit.4.4.1 Performance indicators of multi-objective programming model [5]4.4.1.1 Time cost function for operating efficiencyPassengers’ time in transit can not to reach its ideal vehicle speed which resulting in the decreased value created that due to the increased time in transit as the cost of highway travelers. The fee of highway passenger time in transit expressed as follows⎪⎪⎭⎫ ⎝⎛-⨯⨯⨯⨯=i t v L v L q E G C 8365 (17) where t C represents the cost of highway travelers’ time in transit:, G represents the GDP per capita, E represents the average load factor, q represents the number of traffic ，L represents the length of the highway ，v represents the running speed ，i v represents the ideal vehicle speed of drivers.4.4.1.2 Fuel cost function about fuel consumptionWhen the vehicle is traveling on the highway, the increasing of speed causes the fuel consumption. Therefore, we have established a highway fuel cost function as follows:())(100497.222518.00013.02e d g g v v Q P L v v C >⨯⨯⨯+-= where g C represents d riving on the highway fuel consumption costs ，g P represents market price of gasoline ，d Q represents the number of daily traffic ，e v represents economic speed .4.4.2 Constraints about security and comfort property4.4.2.1 Constraints about securityDue to maximum speed limit on the highway accident mortality should ensure that the value is less than the acceptable tolerance, and We constructed the constraints that based on security for highest highway speed limits as follows:()[]Death h h h h Death I v v E v v E v v E v v E I <--+-----+--=)(13314)(176)(076234 (18)where h v represents the maximum speed limit ，v represents average operating speed ，[]Death I represents the tolerability of traffic accident mortality law.4.4.2.2 Constraints about comfort propertyIn the course of traveling, everyone has a comfortable speed, here we choose t he highest threshold with more comfortable as constraints of in highway speed limits maximum comfort as follows:h km v h /132≤. The following table shows the relationship between speed and comfortable of the driver's.4.4.3 4.4.3.1 Single goal programming model of the maximum operating efficiencyWe take the maximum operating efficiency (the minimum cost)as the goal that can limit the maximum speed of the highway. we establish efficient single target programming model of the highest speed limits, as follows ：⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-⨯⨯⨯⨯=i t v L v L q E G C 8365min )min( (19) 4.4.3.2 Single goal programming model of the maximum economic limitWe take the minimum fuel consumption (the minimum cost of the fuel consumption)as the goal that can limit the maximum speed of the highway. we establish economic target programming model of the highest speed limits, as follows ：())(100497.222518.00013.0min )min(2e d g g v v Q P L v v C >⎥⎦⎤⎢⎣⎡⨯⨯⨯+-= (20) 4.4.3.3 Multi-objective programming model to limit the maximum speedConsidering both operating efficiency and fuel consumption performance indicators, we establish a multi-objective programming model of the maximum speed limit, as follows:()[]hb g hb t v C v C +)(min ⎥⎥⎦⎤⎪⎪⎭⎫ ⎝⎛⨯⨯-⨯⎢⎢⎢⎢⎣⎡+⨯⨯+⨯-⨯=t g g g v GE GL P v E G QL v P QL v P QL 183********.2283652518.01000013.0100min 2 (21) []⎪⎩⎪⎨⎧≤≤<<132.hbDeath Death i hb e v I I v v v t s (22)where hb v represents small cars on the highway the maximum speed limit reference value.4.4.3.4 Solution of the multi-objective programming modelAfter calculation, we obtain that the multi-objective programming model has only one real solution. Represented by the following formula: ()()()()()2920,)0351.2(,2056.262736546273654613132322313232E G f P E e P E d de ef d f d e f d d e e f d f d e f d d vg g ⨯=⨯-=⨯-=++++++++=- (23) According to the solution of our model, we can get the maximum speed limit under the different design in the following table.and heavy traffic.4.5 Improvement and Evaluation of the Traffic Rules4.5.1 Improvement of the traffic rulesThere are many kinds of safe driving of the traffic rules, for example:(1)Driving on the right, allowing the overtaking(2)Driving on the right, and each lane no overhead(3)The lane with the same speed limit(4)Each lane has different speed limitIn these traffic rules, (1) and (4) combined is most efficient. Besides, in order to improve the traffic flow, we put forward some improvement measures. i.e.:(5)constant driving rules, increase the number of lanes(6)In a case of a safe, flexible choice about overtaking lane(7)Large and medium-sized car in the right lane, small cars can be arbitrary choice(8)Each lane have different speed limit. For example, in the case of four lanes, left lane speed limit for 90km/h-120km/h, right lane speed limit for 60km/h-90km/h .4.5.2 Evaluation of the traffic rulesWe evaluate above the traffic rules, in the case of guarantee the safe operation of the vehicle, we study traffic flow on the combination of (1),（7）and (8),then we compare the results with previous model.Under the condition of without passing phenomenon, we use MA TLAB software to solve the traffic flow, speed and density. Results of different speed limits are in the following tables.Table 5 Results of different speed limitMaxspeed(km/h)The initial density The final density The original speed The final speed The initial flow The final flow 12052.1 12.42 44.9 93.35 2339.3 1160.4 90 52.1 23.75 44.9 83.28 2339.3 1978.260 52.1 30.15 44.9 88.51 2339.3 2682.7Table6 The traffic condition with or without overtakingTypes Maxspeed(km/h)TheinitialdensityTheFinaldensityTheOriginalspeedThefinalspeedTheinitialflowThefinalflowDon’t overtake 120 52.1 12.42 44.9 93.35 2339.3 1160.4 90 52.1 23.75 44.9 83.28 2339.3 1978.2 60 52.1 30.15 44.9 88.51 2339.3 2682.7overtake 120 52.1 44.83 44.9 87.66 2339.3 3930.3 According to table 5 and table 6, we work out the traffic flow, the traffic flow is 4660.9 when we not improve the traffic rules. The traffic flow is 5365.4 when we improve the traffic rules. Increased by 15.11% between the two traffic rules. We obtain the most of the traffic follow the combination of (1),（7）and(8),and it have high degree of safety.5 Analysis of Problem 2In the countries with Left-hand drive and right lines，drivers require driving on the left lane on the highway. When they want to overtake other cars, they first travel to the right lane, then return to the original lane. This is right in principle to the rule that driving on the right . Nowadays, the number of left-hand drive and right line is third more than the number of the right-hand and left line. However, it not mean the rule, left-hand drive and right line, is superior the rule, right-hand drive and left line. This lies on the driving habits of the drive’s own, geographic position, environmental factors, vehicle driving position and so on. For instants, in countries in the northern hemisphere, vehicles produce a slight shift to the right in the direction of running because of the earth rotation. At the same time, driving on the right is more beneficial to traffic safety. Vehicle driving position of the right of vehicle is on the left side. In this way, drivers can obtain a broader vision for traffic safety.To apply the model fluid dynamic traffic flow model built in the rule of to left-hand drive and right line to the rule of right-hand drive and left line, we should modify the modified model based on relative requirements, for instants:(1) Converting the driving position of vehicles of left-hand drive and right line into right side;(2) Driving on the left line and allowing overtaking;(3) Large and medium-sized cars drive on the left line, cars run on any freeway;(4) Limit speed in every freeway is different.6 Establishment and Solution of Two Models for Problem 3We studied the traffic rules, and the improved depends on the people follow the traffic rules and other factors on the road，under the supervision of intelligent system，The car will be increased relative sensitivity，conduct to the vehicle safe and efficient operation，At the same time greatly enhance the utilization of freeway lanes. We have established hydrodynamic traffic flow model can be modified, create a new mathematical model for the intelligent system control，to follow traffic rules traffic situation. Under the intelligent system control, there is the impact of macro and micro effects of the impact on the vehicle runs. Macroeconomic impact of the shift, highway traffic services, vehicle travel time and other factors that together determine the vehicle, the microscopic effects are determined by personal factors, vehicle performance, weather and other factors. In this regard, we have established asecure micro-and macro-control network model optimal control model.6.1 Microscopic Safety Control Model Basing on RBF Neural Network On the highway, there are 5 types of security operating patterns on the multi-lane traffic, including following pattern, accelerating pattern, decelerating pattern, overtaking pattern and breaking pattern. For simplicity, we study the security operating patterns on two-way four-lane freeways.6.1.1 Indices of security operation about person, vehicle and surroundingsIn the driving process, considering the impact of various factors about the drivers, vehicle and traffic environment, security operating pattern is determined by the following seven indices:（1）absolute speed of vehicle 1v ;（2）the proportion of the vehicle speed 1v and speed 2v of the vehicle in front ；（3）the distance of the car and the vehicle in front ；（4）the distance of the car and the vehicle in latter on the overtaking freeway ；（5）overtaking signal of the car in latter （0 means no overtaking ，1 means overtaking ）；（6）driver's fatigue condition （0 means relaxation ，1 means fatigue ）；（7）the weather condition （0 means sunny ，1 means snow ）.6.1.2 Safe operation patternsThe above seven characteristics determine the following five indicators in the safe operation mode highway vehicles [3]：（1）accelerating pattern ：Measured in run mode is under the distance the car and the vehicle in front ，and no vehicle enters the ultra-lane carriageway circumstances taken a run mode.（2）overtaking pattern ：Measured in run mode is under the vehicle speed is up to speed with the vehicle in front ，and distance before the car reaches a range of overtaking, nor behind the vehicle taken by a passing mode of operation.（3）following pattern ：Measured in run mode is ready to overtake ， but after overtaking car overtaking signal taken a run mode ；（4）decelerating pattern: there is a world run mode is the speed of the vehicle and the vehicle in front of a large vehicle, and the close proximity of the vehicle in front or behind the vehicle overtaking under taken by one mode of operation;（5）breaking pattern ：This model is in front of unexpected events occur, impassable taken the safe operation mode.6.1.3 Microscopic security control model based on neural networkCars run on the highway, accidents often occur in a relatively short period of time. On the background of highway operation decision-making research, real time is the most critical point. The BP neural network is a kind of approximation network, training quickly and meeting the real-time requirements. The BP network simulation diagram is figure 5. Input layer choose 7 neurons, i.e. the input vector is 17 feature index vector of the network. The radial base (implicit strata) choose 5 neurons, the target output layer choose a neuron, the target vector T =[1 2 3 4 5] represent the five kinds of safe operation of the model。
美国大学生数学建模竞赛试题AB题中文
A 题热水澡一个人进入浴缸洗澡放松。
浴缸的热水由一个水龙头放出。
然而浴缸不是一个可以水疗泡澡的缸，没有辅助加热系统和循环喷头，仅仅就是一个简单的盛水容器。
过一会，水温就会显著下降。
因此必须从热水龙头里面反复放水以加热水温。
浴缸的设计就是当水达到浴缸的最大容量，多余的水就会通过一个溢流口流出。
做一个有关浴缸水温的模型，从时间和地点两个方面来确定在浴缸中泡澡的人能采用的最佳策略，从而泡澡过程中能保持水温并在不浪费太多水的情况下使水温尽量接近最初的水温。
用你的模型来确定你的策略多大程度上依赖于浴缸的形状和容量，浴缸中的人的体型/体重/体温，以及这个人在浴缸中做出的动作。
如果这个人在最开始放水的时候加入了泡泡浴添加剂，这将会对你的模型结果有什么影响？要求提交一页MCM的总结，此外你的报告必须包括一页给浴缸用户看的非技术性的解释，其中描述了你的策略并解释了在泡澡过程中为什么保持平均的水温会非常困难。
B题太空垃圾地球轨道周围的小碎片的数量受到越来越多的关注。
据估计，目前大约有超过50万片太空碎片被视为是宇宙飞行器的潜在威胁并受到跟踪，这些碎片也叫轨道碎片。
2009年2月10号俄罗斯卫星科斯莫斯-2251与美国卫星iridium-33相撞的时候，这个问题在新闻媒体上就愈发受到广泛讨论。
已经提出了一些方法来清除这些碎片。
这些方法包括小型太空水流喷射器和高能量激光来瞄准具体的碎片，还有大型卫星来清扫碎片等等。
这些碎片数量和大小不一，有油漆脱离的碎片，也有废弃的卫星。
碎片高速转动使得定位清除变得困难。
建一个随时间变化的模型来确定一个最佳选择或组合的选择提供给一家私人公司让它以此为商业机遇来解决太空碎片问题。
你的模型应该包括对成本、风险、收益的定量和/或定性分析以及其他重要因素的分析。
你的模型应该既能够评估单个的选择也能够评估组合的选择，且能够探讨一些重要的”what if ”情景。
用你的模型来确定是否存在这样的机会，在经济上很有吸引力；或是根本不可能有这样的机会。
2019年美国大学生数学建模竞赛B题
2019 MCMProblem B: Send in the Drones: Developing an Aerial Disaster Relief Response SystemBackground: In 2017, the worst hurricane to ever hit the United States territory of Puerto Rico (see Attachment 1) left the island with severe damage and caused over 2900 fatalities. The combined destructive power of the hurricane’s storm surge and wave action produced extensive damage to buildings, homes, and roads, particularly along the east and southeast coast of Puerto Rico. The storm, with its fierce winds and heavy rain, knocked down 80 percent of Puerto Rico's utility poles and all transmission lines, resulting in loss of power to essentially all of the island's 3.4 million residents. In addition, the storm damaged or destroyed the majority of t he island’s cellular communication networks. The electrical power and cell service outages lasted for months across much of the island, and longer in some locations. Widespread flooding blocked and damaged many highways and roads across the island, making it nearly impossible for emergency services ground vehicles to plan and navigate their routes. The full extent of the damage in Puerto Rico remained unclear for some time; dozens of areas were isolated and without communication. Demands for medical supplies, lifesaving equipment, and treatment strained health-care clinics, hospital emergency rooms, and non-governmental organizations’ (NGOs) relief operations. Demand for medical care continued to surge for some time as the chronically ill turned to hospitals and temporary shelters for care.Problem: Non-governmental organizations (NGOs) are often challenged to provide adequate and timely response during or after natural disasters, such as the hurricane that struck the United States territory of Puerto Rico in 2017. One NGO in particular – HELP, Inc. - is attempting to improve its response capabilities by designing a transportable disaster response system called “DroneGo.” DroneGo will use rotor wing drones to deliver pre-packaged medical supplies and provide high-resolution aerial video reconnaissance. Selected drones should be able to perform these two missions – medical supply delivery and video reconnaissance – simultaneously or separately, depending on relief conditions and scheduling. HELP, Inc. has identified various candidate rotor wing drones that it would like your team to consider for possible use in designing its DroneGo fleet (see Attachments 2, 3).DroneGo’s pre-packaged medical supplies, called medical packages, are meant to augment, not replace, the supplies provided by local medical assistance organizations on-site within the country affected by the disaster. HELP, Inc. is planning on three different medical packages referred to as MED1, MED2, and MED3. Drones will carry these medical packages within drone cargo bays for delivery to selected locations (see Attachments 4, 5). Depending on the specific drone being used to transport medical supplies, it may be possible that multiple medical packages can be transported in a single drone cargo bay. Note that drones must land on the ground to offload medical supplies from the drone cargo bays. The video capability of the drones will provide high-resolution video of damaged and serviceable transportation road networks to HELP, Inc.’s command and contro l center for ground-based route planning.HELP, Inc. will use International Standards Organization (ISO) standard dry cargo containers to quickly transport a complete DroneGo disaster response system to a particular disaster area. The individual shipping containers for all drones in the DroneGo fleet, along with all requiredmedical packages, must fit within a maximum of three of the ISO cargo containers to be delivered to a single location, or up to three different locations if three cargo containers are used in the disaster area. Each shipping container’s contents should be packed in order to minimize any need for buffer materials for unused space. Table 1 shows the dimensions of an ISO standard dry cargo container.Table 1. Standard ISO Container DimensionsExterior Interior Door OpeningLength Width Height Length Width Height Width Height 20’20’8’8’6”19’3”7’8”7’ 10”7’8”7’5”StandardDryContainerHELP, Inc. is asking your team to use the 2017 situation in Puerto Rico to design a DroneGo disaster response system that will fit within the containers noted while meeting the anticipated medical supply demands during a potential similar future disaster scenario. It is possible that the demand requirements of this scenario may exceed the capabilities of the drone fleet your team identifies. If this occurs, HELP, Inc. wants to clearly understand any tradeoffs that it must make for implementing solutions to address these shortcomings.Part 1. Develop a DroneGo disaster response system to support the Puerto Rico hurricane disaster scenario.Consider the background information, the requirements identified in the problem statement, and the information provided in the problem attachments to address the following.A.Recommend a drone fleet and set of medical packages for the HELP, Inc. DroneGodisaster response system that will meet the requirements of the Puerto Rico hurricanescenario. Design the associated packing configuration for each of up to three ISO cargocontainers to transport the system to Puerto Rico.B.Identify the best location or locations on Puerto Rico to position one, two, or three cargocontainers of the DroneGo disaster response system to be able to conduct both medicalsupply delivery and video reconnaissance of road networks.C.For each type of drone included in the DroneGo fleet:a.Provide the drone payload packing configurations (i.e. the medical packagespacked into the drone cargo bay), delivery routes and schedule to meet the identifiedemergency medical package requirements of the Puerto Rico hurricane scenario.b.Provide a drone flight plan that will enable the DroneGo fleet to use onboardvideo cameras to assess the major highways and roads in support of the Help, Inc.mission.Part 2. MemoWrite a 1–2 page memo to the Chief Operating Officer (CEO) of HELP, Inc. summarizing your modeling results, conclusions, and recommendations so that she can share with her Board of Directors.Your MCM team submission should consist of:∙One-page Summary Sheet,∙One- to Two-page memo to the HELP, Inc. CEO∙Your solution of no more than 20 pages, for a maximum of 23 pages with your summary and memo.∙Note: Reference list and any appendices do not count toward the 23-page limit and should appear after your completed solution.Attachments:1.Map of Puerto Rico2.Potential Candidate Drones for DroneGo Fleet Consideration (with Drone payloadcapability)3.Drone Cargo Bay Packing Configuration/Dimensions by Type4.Anticipated Medical Package Demand5.Emergency Medical Package Configuration/DimensionsAttachment 1: Map of Puerto RicoAttachment 2: Potential Candidate Drones for DroneGo Fleet Consideration (with Drone Payload Capability)Shipping ContainerDimensions PerformanceCharacteristics/CapabilitiesConfigurations CapabilitiesDroneLength (in.) Width (in.) Height (in.) Max PayloadCapability(lbs.) Speed(km/h)Flight Time No Cargo (min)Video Capable Medical Package Capable Drone Cargo Bay Type* A 45 45 25 3.5 40 35 Y Y 1 B 30 30 22 8 79 40 Y Y 1 C 60 50 30 14 64 35 Y Y 2 D 25 20 25 11 60 18 Y Y 1 E 25 20 27 15 60 15 Y Y 2 F 40 40 25 22 79 24 N Y 2 G 32 32 17 20 64 16 Y Y 2 H Tethered657541N/AN/AIndefiniteNNN/A*Note that cargo bays are affixed to the drone and that drone must be on the ground to offload cargo. SeeAttachment 3 for Drone Cargo Bay Type Configuration/Dimensions.Attachment 3: Drone Cargo Bay Packing Configuration/Dimensions by TypeDrone Cargo Bay Type Length (in) Width (in) Height (in)1 8 10 14 Top Loaded2 24 20 20 Top Loaded Attachment 4: Anticipated Medical Package DemandDelivery Location Emergency Medical Packages ** Location Name Latitude Longitude Requirement Quantity FrequencyCaribbean Medical Center 18.33 -65.65 MED 1 1 DailyJajardo MED 3 1 DailyHospital HIMA 18.22 -66.03 MED 1 2 DailySan Pablo MED 3 1 Daily Hospital Pavia Santurce 18.44 -66.07 MED 1 1 DailySan Juan MED 2 1 Daily Puerto Rico Children's Hospital 18.40 -66.16 MED 1 2 DailyBayamon MED 2 1 DailyMED 3 2 Daily Hospital Pavia Arecibo 18.47 -66.73 MED 1 1 DailyArecibo**See Attachment 5 for Emergency Medical Packages 1, 2, and 3 Configurations/Dimensions.A ttachment 5: Emergency Medical Package Configuration/DimensionsEmergency Medical Package ConfigurationPackage ID Weight(lbs.)Package Dimensions(in.)(L × W × H)MED 1 2 14 × 7 × 5 MED 2 2 5 × 8 × 5 MED 3 3 12 × 7 × 4Glossary:Cargo Container (Shipping Container): a large rectangular container with doors on the ends for loading and packing, and made of material suitable for shipping, storing, and handling in many weather and climate conditions.Drone (Unmanned Aerial Vehicle, UAV): a flying robot that can be remotely controlled or fly autonomously through software-controlled flight plans in their embedded systems that work in conjunction with onboard sensors and GPS.Drone Cargo Bay: For rotor wing drones, this is an externally carried “box” used to transport materials. For this problem, the drones under consideration have one of two types (sizes) of cargo bays. Note that each drone must land for the medical packages to be unloaded from the bay at its destination.Drone Fleet: a set of drones for a particular mission or purpose. For this problem, the total set of drones by type (A to H) and Payload Capability (Visual and Medical) needed to meet the requirements of HELP, Inc.Drone Payload Packing Configuration: how the drone payload bays are packed. For this problem, how the medical packages being transported by a drone are packed inside the drone cargo bay.Medical Package: a predetermined set of medical supplies packed in a single container. For this problem, there are three Medical Package Configurations (MED1, MED2, MED3) available for transport by a drone from a deployed cargo container location to the demand location.Non-governmental Organization (NGO): Usually non-profit and sometimes international organization independent of government and governmental organizations that is active in humanitarian, educational, healthcare, social, public policy, human rights, environmental and other areas in attempts to affect change.Payload Capability: the carrying capacity of an aircraft or launch vehicle, usually measured in terms of weight. For this problem, the capability/capacity of the drone to carry medical packages.。
AMC美国数学竞赛AMC.B 试题及答案解析
2003A M C10 B 1、Which of the following is the same asSolution2、Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs more than a pink pill, and Al’s pills cost a total of for the two weeks. How much does one green pill costSolution3、The sum of 5 consecutive even integers is less than the sum of the rst consecutive odd counting numbers. What is the smallest of the even integersSolution4、Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the gure. She plants one flower per square foot in each region. Asters cost 1 each, begonias 1.50 each, cannas 2 each, dahlias 2.50 each, and Easter lilies 3 each. What is the least possible cost, in dollars, for her gardenSolution5、Moe uses a mower to cut his rectangular -foot by -foot lawn. The swath he cuts is inches wide, but he overlaps each cut by inches tomake sure that no grass is missed. He walks at the rate of feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawnSolution.6、Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is . The horizontal length of a “-inch” television screen is closest, in inches, to which of the followingSolution7、The symbolism denotes the largest integer not exceeding . For example. , and . ComputeSolution.8、The second and fourth terms of a geometric sequence are and . Which of the following is a possible first termSolution9、Find the value of that satisfies the equationSolution10、Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increasedSolution11、A line with slope intersects a line with slope at the point . What is the distance between the -intercepts of these two linesSolution12、Al, Betty, and Clare split among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of . Betty and Clare have both doubled their money, whereas Al has managed to lose . What was Al’s original portionSolution.13、Let denote the sum of the digits of the positive integer . For example, and . For how many two-digit values of isSolution14、Given that , where both and are positive integers, find the smallest possible value for .Solution15、There are players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest players are given a bye, and the remaining players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played isSolution16、A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the yearSolution.17、An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly ll the cone. Assume that the melted ice cream occupies of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radiusSolution18、What is the largest integer that is a divisor offor all positive even integersSolution19、Three semicircles of radius are constructed on diameter of a semicircle of radius . The centers of the small semicircles divide into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicirclesSolution20、In rectangle , and . Points and are on so that and . Lines and intersect at . Find the area of .Solution21、A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacementsSolution22、A clock chimes once at minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February , , on what date will the chime occurSolution23、A regular octagon has an area of one square unit. What is the area of the rectangleSolution24、The rst four terms in an arithmetic sequence are , , , and, in that order. What is the fth termSolution25、How many distinct four-digit numbers are divisible by and have as their last two digitsSolution。
北美数学建模选题B水资源
目录摘要 3 一、导论错误!未定义书签。
二、假定 3三、基于多项式曲线拟合法对2025年水需求的预测 3四、基于主成分分析建立的水资源短缺省份的可持续用水计划 4 （1）主成分分析 5 （2）主成分分析的过程 5 （3）对所选主成分做经济解释 8五、基于聚类分析对各地区水资源短缺状况做进一步探讨 10六、结语13 附录:表a.中国各省份16因素情况摘要近年来，水资源问题日益严重，并引起人全世界的重视。
我国是一个水资源贫乏的国家，人均水资源量只有全世界人均占有量的四分之一，而且时空分布极不均匀。
本文旨在构建一种水资源合理利用的综合计划。
首先，我们用模型一做一个预测，通过使用matlab工具对2025年水资源的需求作出预测。
然后，在我们的模型二中，用主成分分析方法和数据软件spss，我们最终得出结论：影响水资源的主要因素是用水情况、平均气温、造林森林面积、人口自然增长率和GDP增长率。
最后，在模型三中，用聚类分析法将全国省份化为五类，根据具体情况分析水资源问题，并给出具体的解决措施。
水，水，哪里都有——中国水资源策略模型一、导论水是基础性的自然资源，战略性的经济资源。
水环境是人类生存和发展的基础。
随着社会的进步和经济的发展，中国的水资源紧张问题变得日益严重。
本文从中国水资源现状和特点出发，分析了影响中国水资源问题的因素，指出了水资源存在的危机，并根据实际情况提出转变观念，改善生态环境，提高生产技术，加强管理水平和基础设施建设，发展污水处理技术，提高水资源循环利用率及跨流域调水等相应的解决对策。
二、假定①不考虑气候变化导致的冰川融化和全球变暖对淡水利用和储存的影响。
②在水资源利用，工业，经济和环境保护方面不会进行彻底改革。
③模型忽略自然水资源净化。
三、基于多项式曲线拟合法对2025年水需求的预测用指数模型进行拟合为了预测2025年水需求总量，我们搜集了2000年到2010年的水需求量，并将数据代入matlab进行拟合，程序如下：x=1:11y=[5497.59 5567.43 5497.28 5320.4 5547.8 5632.98 5794.969019 ...5818.668804 5909.95 5965.154577 6021.994773];plot(x,y,'+')xlabel('时间')ylabel('用水量')p4=polyfit(x,log(y),2)Y4=exp(polyval(p4,x));semilogy(x,y,'+',x,Y4)x=12:26Y5=exp(polyval(p4,x))运行程序可得如下结果：x =1 2 3 4 5 6 7 8 9 10 11p4 =0.0011 -0.0022 8.6094x =12 13 14 15 16 17 18 19 20 21 22 23 24 25 26Y5 =1.0e+04 *Columns 1 through 120.6230 0.6385 0.6558 0.6749 0.6961 0.7196 0.7453 0.7737 0.8049 0.8391 0.8766 0.9178Columns 13 through 150.9630 1.0125 1.0669根据预测结果写出拟合指数曲线-=xxyln2+00116094.80022.0.0各column的值即为10年之后历年的预测数据：2011 2012 2013 2014 2015 2016 2017 2018 6230 6385 6558 6749 6961 7196 7453 7737 2019 2020 2021 2022 2023 2024 20258049 8391 8766 9178 9630 10125 10669我们取2025年的数据，即10669，就是我们所预测的2025年的水需求量。
连环犯罪预测-2010美国大学生数学建模B题(MCM)
For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number7370Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________2010 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of your solution paper.)Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.SummaryIn our paper, we define two types of serial murder, the single-hunting area type and the multi-hunting area type, and we mainly do research on serial murder of these two types.We work out a method to determine if a given serial murder belongs to the single-hunting area type or the multi-hunting area type, or neither of them.To do research on the cases of the single-hunting area type, we developed two sub-models, the Concentration Driven Model (CDM) and the Centro-based Model (CM). We use the “center of mass” and the search-radius calculated by an algorithm to figure out the scope of hunting area.We discretize the hunting area by divided it into small action spaces and all of our models depend on the discretized one.The CDM is based a truth that the distribution of preys will affect the criminal’s moving to a certain extent. We develop a discrete algorithm to calculate variation of the prey’s distribution, and an algorithm to search for the next possible crime locations.The CM is based on the fact that the criminal tends to move to some places far away from those latest crime locations to avoid be arrested, and CM also includes the random factors. We develop a partially-random algorithm to determine the next possible directions the criminal may choose based on the time and locations of the past crimes. We determine the next possible distance the criminal will go basing on a reference.We then develop an algorithm to solve cases of multi-hunting area based on the sub-models in our model development.For the sub-Models have take into consideration some direct driven factors, in our Combined Model, we consider some indirect factors that may affect the moving of criminals. We develop a method to figure out the main common properties among the locations of past crimes, and then use an matching-algorithm to figure out the superiority for each result that figured out by the sub-models, basing on the assumption that if a place have much common property with the past crime locations, there’s more chances that it will be the next crime location. And the police can select some places with high superiority to investigate.In our paper, we study three cases of single-hunting area type and one case of multi-hunting area type to determine the best value of the key coefficients; we also use them to do the Sensitivity and Reliability Analysis, and we find that after the sample size approximately increases to 7, our model will give out relatively accurate results.We write a report to the police whether our model can be used to solve a given case, and if yes, when the model will give accurate results in terms of the sample size. We also give some suggestions that should be taken care of to the executive officer.Key words: serial-crime; prey-concentration; single-hunting area; multi-hunting areaTo Seize the Devils!Content1. Introduction (2)2. Terminology (2)3. Assumptions and Symbol (3)4. Sub-Models (3)4.1 Preparation (3)4.1.1 Determine the Search Radius and the Center of Mass (4)4.1.2 Hunting Area Division (5)4.1.3 Normalized Distance (ND) (6)4.2 Concentration Driven Model (CDM) (6)4.2.1 Overview (6)4.2.2 Frame of CDM (6)4.2.3 Model Definition (7)4.2.3.1 Concentration Calculator (7)4.2.3.2 Points Generator (8)4.3 Centro-based Model (CM) (10)4.3.1 Overview (10)4.3.2 Model Definition (10)4.3.2.1 Random Direction Algorithm (10)4.3.2.2 Random Distance Algorithm (12)4.4 Residence of the criminal (13)4.5 Parameter Determination (13)5. Model Comparison (13)6. Sub-Model Development (14)6.1 Problem Addressing (14)6.2 Identify the type of a case (14)6.3 Use the Basic Model (16)7. Model Combining (16)7.1 Overview (16)7.2 The Property Vector (16)7.3 Model Definition (17)8. Strengths and Weaknesses of the Model (18)8.1 Strengths (18)8.2 Weaknesses (18)9. Case Study (19)10. Sensitivity Analysis (20)11. Model Reliability (21)12. Executive Summary (22)13. References (24)Appendix (25)Appendix I (25)Appendix II (25)Appendix III (26)Appendix IV (26)1.I ntroductionSerial crime is generally defined as crimes in a serial or repetitive nature. [1]Serial murder, serial rape and serial arson are the crimes regarded as serial crimes. Serial crime is a frightening and perplexing phenomenon that has proven to be a difficult puzzle for both criminal investigators and criminological researchers. Despite being a rare event, this crime has a broad-based impact on the larger community (Jenkins, 1992a; Silverman & Kennedy, 1993). [2] In our paper, we try to develop a method to aid in a local police agency’s investigations of serial criminals.The objectives in our study are:z Make use of at least two different schemes to generate a geographical profile of serial criminals; z Develop a technique to combine the results of the different schemes and generate a useful prediction;z Provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings;z Provide an additional two-page executive summary, which provides a broad overview of the potential issues and an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool;Firstly, we determine what the meaning of the geographical profiling is. Geographic profiling is a criminal investigative methodology that analyzes the locations of a connected series of crimes to determine the most probable area of the murderer’s residence or his next crime location. [3] To generate a geographical profile of serial criminal, we should provide the murderer’s sphere of activities; predict the murderer’s next crime location.In our paper, firstly we work out a Concentration Driven Model (CDM) to make a prediction providing some kind of guidance about possible locations of the next crime from the microscopic point of view. Secondly, we add the concept of the center of hunting area and build a Centro-based Model (CM) to macroscopically predict the possible locations of the next crime. Thirdly, we combine both models and come up with a combined model aiming at generate a more reliable prediction of the possible locations of the next crime. Finally, we analyze the sensitivity and the reliability of our models. At the end of this paper, we provide an executive summary for a chief of police on the strategy for investigations of serial criminals.2.T erminologyz Crime location: a place where has just experienced a crime actionz Hunting area: a concentrated area that has experience a series of crimes. An area can be assumed to be hunting area only when the distance between two successive crime locations doesn’t vary much as the number of the crimes increases. In our model, the hunting area is a square.z Action space: we divided the hunting area into several small parts, and each part is an action space. The action space represents a district, a city, a block, or someplace else. It depends on the scale of a certain serial crime.z Single-hunting area and multi-hunting area: in our paper, we just investigate two types of serial crime. One is single-hunting area and the other is multi-hunting area.¾The type of single-hunting area contains only one hunting area.¾The type of multi-hunting area contains multiple hunting areas, and the distance of each pair of hunting areas is much larger than the distance between two successive crime locations within one hunting area.z The search-radius and the core-radius:¾The search-radius limits the search scale of the criminal, and the criminal cannot reachbeyond it. In our paper, the search-radius is half the side-length of the hunting area.¾The core-radius is an abstract valuable; it varies as the number of crimes increases. We will define it later in our paper.Here is a graph to illustrate the terms above:1 Graph 1z Latest crime location: the location of the latest crimez Prey: certain kind of people that the murderer tends to kill3.A ssumptions and SymbolAssumptions:z We treat the action space as a unit. A murderer’s basic step is to stay in a action space, or to move to the very neighboring action space.z The prey concentration of each action space is the natural distribution through the hunting area before the very first crime of the serial crime happened. Also, we consume that the natural distribution remains constant.Symbol:z K: the number of parts we divided the length of the hunting area intoz , … , … :the action space located in i row and the j column in the hunting areaz , :the concentration or the number of prey in Sq ,z … … :the array of concentration through the hunting areaz : the longitude of the location of i crimez : the latitude of the location of i crimez : the longitude of the “mass of center”z : the latitude of the “mass of center”4.S ub-Models4.1PreparationWe develop two sub-models in this part, both of which are based on a single-hunting area. In this part, we introduce several important concepts, such as the search radius, the center of mass, the ND, etc. Now, we are going to definite the scope of the hunting area. And the search radius and thecenter of mass are both used to describe the scope of the hunting area. One thing that we must bear in mind is that the center of mass is used to determine the scope of hunting area, rather than a place that the murderer will come back to after a crime action.4.1.1 Determine the Search Radius and the Center of MassThe search radiusThe search radius of the murderer’s hunting area is measured as of the maximum distance from “center of mass” to crime location. The following regression equation reflects the maximum distance from “center of mass” to crime site against maximum distance between crime sites y a x 0.61(Equation 4.1.1)Where:y is the maximum distance in miles from the “center of mass” to crime site; and x is the maximum distance in miles between crime sites.The gradient of a in Equation 4.1.1 indicates an eccentric placement of the “center of mass” vis-à-vis the crime sites (a perfect centric placement would yield a gradient of 0.5). Regressions for U.S. and British serial murder crime location data yield values of 0.81 and 0.79, respectively, and an Australian study found gradients of 0.77 for rape, 0.60 for arson, and 0.65 for burglary. [2] The center of massCalculate the “center of the mass” by using Centroid-Formula: Longitude∑ ,Latitude ∑ VerificationWe review the serial murder of Peter Sutcliffe. Graph 2 shows Yorkshire Ripper locations within West Yorkshire (Victims 6 & 9 are off this map to the south west). In Graph 2 the labels of 1-13 were the victims’ locations where the bodies were found. The labels of 14-17 were the survivors and the label of 18 was the place where he was arrested.2 Graph 2Figure 1shows the quantified original image of the crimes, and Figure 2 shows the image without singularities.3 Figure 14 Figure 2Take the order of occurrence of the murder into consideration; we do the following works as the serial number n increases:1) Calculate the “center of the mass” by using Centroid-Formula: Longitude∑ ,Latitude ∑ ; 2) Calculate the maximum distance from the “center of mass” to crime site by using Equation4.1.1.Figure 3 shows all the positions of the “center of mass”, from which we can find that the “center of mass” tend to be lumped as we take more crimes into consideration.Figure 4 shows the variation of y as the serial number n increases. As is shown in 4.2, y is the maximum distance from the “center of mass” to crime location. Y tends to be stable as n increases. It means that the murder’s hunting area is limited to an area that centered with the “center of mass” and that has a radius of y.5 Figure 36 Figure 4All the above proves that it is reasonable to describe the scope of a hunting area by the center of mass and the search radius.4.1.2 Hunting Area DivisionIn order to apply our model to different scales of serial crime, we should normalize the hunting area, and treat it as a square 1 on a side. The procedure is as follows: We define the normalizationparameter as 2 times the length of search radius, and then get the scope of the hunting area normalized by dividing the distance by the normalization parameter. We divide the hunting area into k k action spaces. Each of the action spaces is a square on a side.4.1.3 Normalized Distance (ND)It is difficult to deal with problems of different order of magnitude. As the scope of the murderer’s hunting area differs at a wide range, we introduce the concept of Normalized Distance (ND) to make our model more tractable.We get the ND of a crime location by dividing the actual distance by 2 times the length of search radius.4.2 Concentration Driven Model (CDM)4.2.1 OverviewThe Concentration Driven Model (CDM) is a discrete model. In this model, we suppose that the murderer’s moving from one action space to another depends on preys’ distribution through the hunting area. Meanwhile, as the crimes being committed sequentially, the distribution will change.We develop an algorithm (we call it concentration calculator) to calculate the current preys’ distribution using the happening time and the location of the past crimes scenes. Then, we develop an algorithm (we call it points-generator) to search the next possible crime locations. The points-generator is a ‘discrete ‘steepest descent method.4.2.2 Frame of CDM7 Frame of CDM4.2.3Model Definition4.2.3.1 Concentration Calculatorz Change of the Prey’s DistributionAccording to our assumption above, each action space own a property of prey concentration. So we investigate the transfer of preys among the many action spaces to estimate the change of the prey distribution.z Causes for Transfer¾ A crime has just been committedAccording to our assumptions, when a crime has just been committed in an action space, all the preys in the action space will soon run away to other action spaces. This assumption is reasonable. In fact, people who will emerge in a certain place always for the reason that he will pass through the place on the way to work or some places else in his daily life, and he will go round if an crime has just happened in the place. This is always true when people choose the way to work or people choose a place for purchase, etc.Suppose the current crime location is represented by C , . We let the preys transfer to the eight neighboring action spaces according to a certain principle. We limit the transfer just among the crime location .We divided the eight spaces into two groups of A and B, Figure 5 can illustrate it:8 Figure 5We assume that each action space of group A will receives a% of the preys in the crime location, and each action space of group B receives b%, where a ,a b 25.This is for the reason that action space of group A connects with the crime location basing on edge, while group B basing on angle, and we think that the larger the contact area is, the more preys will transfer.Therefore, after a crime is committed, the instant transfer follows the principle below:1)When the coordinates of an action space i′,j′ is one of i 1,j , i 1,j , i,j 1 ,i,j 1 , C ′, ′ a% C , ;2)When the coordinates of an action space i′,j′ is one of （i 1,j 1）, i 1,j 1 ,i 1,j 1 , i 1,j 1 , C ′, ′ b% C , ;3)C ′, ′ 0.¾As time goes by, preys that run away to other action spaces will come back After a crime being committed, the number of the preys in the crime location will reduce, but as time goes by, the influence will gradually reduce, preys will gradually come back because the way contain the crime location or the supermarket they choose in their daily life before the crime’s happening is their usual choice and it must be the best one. Thus the distribution will recover. We put forward assumptions below:In every time unit, there are a preys come back from each action space of group A, whileb preys from each action space of group B. We assume that the preys don’t transfer on largescale any more until the distribution recover to the normal level.Apart from the ending situation above, there is another ending situation, that is, when the concentration or number of preys in an action space of A or B group being affected by another crime later, that means it also exchange preys with other action spaces later, it may not return all the preys who come from the crime location .Under this circumstances, the return or the transfer between the certain action space and the crime location will terminate.¾Exception on the borderFor two kinds of points on the border shown in Figure 6 and Figure 7, we also divided the action spaces around the crime location into group of A and B, and then figure deal with it in the similar way:9Figure 6 10Figure 7z Current Distribution of the PreysAccording to our assumptions, the distribution of the preys through the hunting area is already known, and represented by a two-dimensional array C . The distribution after the ith crime action represented by C (i=1…n), and we will figure out C during the simulation, the process as below:1)After the i th crime action, we can calculate the C based on the C2)According to the algorithm above, and repeat n times, we could figure out C .4.2.3.2 Points GeneratorWe develop an algorithm to search for the next possible action spaces that would welcome a crime action; we called the algorithm points-generator. The algorithm derives from Gradient descent, but we improve it so that it could be used in a discrete problem.z Gradient DescentGradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.z Gradient descent in discrete useWe replace the Gradient in the Gradient descent with the difference of concentration.Though we use a method developed from an optimization algorithm, we don’t mean to search for the local optimization result, we just use the method to simulate the moving of the criminal.Why not to search for the optimization result will be explained below. And we will obtain a series of action space that will welcome a crime action.Why not search for the optimization result?There two reasons to explain it:1)The first reason is that the total iteration number to find a local maximum result would bevery large, and a large iteration number may not be realistic because the criminal’s ability, physical strength, or his financial condition is limited and always cannot afford to go so far.2)The second reason is that even though a criminal is deeply affected by the preyconcentration, his destination may not be the optimal solution but a place that is above his standard. This is because the criminal is somewhat shortsighted and he may satisfy with the places above his standard.z Figure out the next action spaceWe replace the gradient in the Gradient descent with the difference of prey numbers between the adjacent action spaces, as follows:We still use the local area as is shown in Figure 8 (the action space on the border of the hunting area can be proposed in the similar way)11 Figure 8For that in every iterative round, the algorithm will generate a series of points, for each point/current iterative point, we do things as follows:For each action space marked from 1 to 8 around current iterative point in the figure above, we calculate the difference of prey concentration between the marked action space and the current iterative point, choose those action spaces with a corresponding difference that is larger than a certain value of h to be a part of the iterative points of the next round.In fact, the threshold h can describe the criminal’s sensibility degree to prey concentration, and if threshold h is large, indicating that the criminal is sensitive to the difference of prey concentration, otherwise the opposite.z When to Terminate the IterationAccording to the analysis above, we terminate the iteration when the number of iteration steps has reached a constant value of d, the value of d is constant as the scale of the crime varies, and in a given serial murder, it is also constant.Why it is constant as scale varies?In fact, no matter it is crime in a large scale with automobiles as its transportation or a crime in small scale with transportation just base on foot, the d is constant among all kinds of cases because we have normalized the scope of the case, thus for a large scale case one iterative step (move from current action space to another action space) represents a longer distance than in a small scale case.Evidence for a constant d in a given serial murderWe also have found that in one case, the distance between two crime-sites that happened in succession don’t vary largely, that means that to assume that the value of d is constant in a given serial murder is reasonable.We studied the case of Peter Sutcliff, and his moving track during the case is in Figure 9.12Figure 9In this case, there are 15 distances, we calculate the mean m and the standard deviation σ of them, and =0.2903, the value is minor and that means the distance don’t vary very much. On the other hand, the conclusion above can be explained as that the criminal’s physicalstrength, or his financial condition or something else would limit him.4.3 Centro-based Model (CM)4.3.1 OverviewThis is a macro-dynamic mode. We divide the hunting area into k action spaces, in which we find the possible locations of the next crime based on the time and locations of the past crime scenes. We determine the next possible crime locations by confirming the murderer’s choice of the next crime location’s direction and distance.In CM, the murderer’s choice of the direction of the next possible crime location is based on the time and locations of the past crime scenes. The direction is a stochastic variable, following a certain distribution. The distribution is influenced by the time and locations of the past crime scenes. We design the probability density function of the murderer’s next direction choice on the basis of his past crime scenes’ time and locations. And then we generate a crime location’s direction through a simulation on the basis of the distribution.We got a buffered distance-decay function [2], and construct a probability density function of the ND of the murderer’s next crime location in accordance with the curve’s variation law. Then we run a simulation to get the murderer’s next crime location’s distance.4.3.2 Model Definition4.3.2.1 Random Direction Algorithm1. Calculate the weights f t of the murderer’s past crime scene at the time of t . We opt for thefunction defined as follows:f t a lg b t 1 ,a 0, 0(4.3.2.1-1) Wheret is the time of the murderer’s past crime scenes. The crime happened earlier as the valueof t getting larger. a and b are determined by concrete serial crime case.13Figure 10Figure 10 shows the function curve of this function.We can see that f 0 0, which means the murdererwon’t choose the same direction as that of his latest crimelocation. This may be on account of the large possibility ofbeing arrested at the very direction. We can also find thatf t tends to get larger as t is increasing. This is becausethat as the crime happened earlier the impact of itsdirection on the murderer’s next direction choice is gettingsmaller and smaller. 2. Using the weights f t direction choice as follows:g θ w f t 1 cos θ θ,θ π,π(4.3.2.1-2) Whereθ is the argument that can uniquely determine an action space;θ is the argument of the murderer’s past crime location at the time of t ; and w ∑.Figure 11 shows the definition of the argument.14 Figure 113. Calculate the function of possibility distribution as follows:G θ g θ d θθ π,θ π,π(4.3.2.1-3) 4. Divide the range of θ into h equal non-overlapping intervals. Accordingly, we divide the rangeof G θ into h non-overlapping intervals that are not always of equal width.5. We run a simulation to get a random sampling on an interval 0,1 . If this sampling fall into thei th interval of G θ , we calculate the average of the two endpoints of the i th interval of θ. And it is the direction of the murderer’s next crime location.4.3.2.2 Random Distance AlgorithmBuffered distance-decay function:It is a dynamic process of a murderer’s target selection. Crimes occur in those areas where suitable targets overlap the offender’s awareness space. Offenders then search outward from these areas, the search behavior following some form of distance-decay function. There is usually a “buffer zone,” however, centered the criminal’s residence. Within this zone, targets are viewed as less desirable because of the perceived level of risk associated with operating too close to home. For the offender, this area represents an optimized balance between the maximization of opportunity and the minimization of risk. Figure 12 shows an example, derived from a serial rape case, of a typical buffered distance-decay function .Algorithm:1. Confirm the type of the serial crime, and determine the gradient of a in Equation 4.1.1;2. Calculate the maximum distance from residence to crime site y on the basis of the maximumdistance between the past crime sites x, using the Equation 4.1.1;3. Calculate the ND of the actual distance d by using the equation ND; 4. Construct a probability density function d ND of the ND of the murderer’s next crime locationin accordance with the buffered distance-decay function curve’s variation law. We study Figure 12, and design the probability density function in form of the formula below: d ND q 1 q 2 10 ND q 3 2,0 ND 0.2k x b,0.2 1 (4.3.2.2-1) Figure 13 shows the probability density function curve. We can find that it preserves the approximate characters of Figure 12, and we can use this probability density function to simulate the ND of the murderer’s next crime location.15 Figure 1216Figure 135. Calculate the function of possibility distribution as follows:D ND D ND d ND D,ND 0,1 (4.3.2.2-2)6. Divide the range of ND into h equal non-overlapping intervals. Accordingly, we divide the rangeof D ND into h non-overlapping intervals that are not always of equal width;7. We run a simulation to get a random sampling on an interval 0,1 . If this sampling fall into thei th interval of D ND , we calculate the average of the two endpoints of the i th interval of ND ;8. Calculate the result of ND y , and it is the distance of the murderer’s next crime location.。
数学建模入门基本知识
数学建模知识——之新手上路一、数学模型的定义现在数学模型还没有一个统一的准确的定义，因为站在不同的角度可以有不同的定义。
不过我们可以给出如下定义：“数学模型是关于部分现实世界和为一种特殊目的而作的一个抽象的、简化的结构。
”具体来说，数学模型就是为了某种目的，用字母、数学及其它数学符号建立起来的等式或不等式以及图表、图像、框图等描述客观事物的特征及其内在联系的数学结构表达式。
一般来说数学建模过程可用如下框图来表明：数学是在实际应用的需求中产生的，要解决实际问题就必需建立数学模型，从此意义上讲数学建模和数学一样有古老历史。
例如，欧几里德几何就是一个古老的数学模型，牛顿万有引力定律也是数学建模的一个光辉典范。
今天，数学以空前的广度和深度向其它科学技术领域渗透，过去很少应用数学的领域现在迅速走向定量化，数量化，需建立大量的数学模型。
特别是新技术、新工艺蓬勃兴起，计算机的普及和广泛应用，数学在许多高新技术上起着十分关键的作用。
因此数学建模被时代赋予更为重要的意义。
二、建立数学模型的方法和步骤1. 模型准备要了解问题的实际背景，明确建模目的，搜集必需的各种信息，尽量弄清对象的特征。
2. 模型假设根据对象的特征和建模目的，对问题进行必要的、合理的简化，用精确的语言作出假设，是建模至关重要的一步。
如果对问题的所有因素一概考虑，无疑是一种有勇气但方法欠佳的行为，所以高超的建模者能充分发挥想象力、洞察力和判断力，善于辨别主次，而且为了使处理方法简单，应尽量使问题线性化、均匀化。
3. 模型构成根据所作的假设分析对象的因果关系，利用对象的内在规律和适当的数学工具，构造各个量间的等式关系或其它数学结构。
这时，我们便会进入一个广阔的应用数学天地，这里在高数、概率老人的膝下，有许多可爱的孩子们，他们是图论、排队论、线性规划、对策论等许多许多，真是泱泱大国，别有洞天。
不过我们应当牢记，建立数学模型是为了让更多的人明了并能加以应用，因此工具愈简单愈有价值。
MCM B题 个人思路
【MCM-2017】2017年数学建模美赛题目原文及翻译-B （个人思路）Problem B: Merge After Toll2017 MCM :???Multi-lane divided limited-access toll highways use “ramp tolls” and “barrier tolls” to collect tolls from motorists. A ramp toll is a collection mechanism at an entrance or exit ramp to the highway and these do not concern us here. A barrier toll is a row of tollbooths placed across the highway, perpendicular to the direction of traffic flow. There are usually (always) more tollbooths than there are incoming lanes of traffic (see former 2005 MCM Problem B). So when exiting the tollbooths in a barrier toll, vehicles must “fan in” from the larger number of tollbooth egress lanes to the smaller number of regular travel lanes. A toll plaza is the area of the highway needed to facilitate the barrier toll, consisting of the fan-out area before the barrier toll, the toll barrier itself, and the fan-in area after the toll barrier. For example, a three-lane highway (one direction) may use 8 tollbooths in a barrier toll. After paying toll, the vehicles continue on their journey on a highway having the same number of lanes as had entered the toll plaza (three, in this example).?Consider a toll highway having L lanes of travel in each direction and a barrier toll containing B tollbooths (B > L) in each direction. Determine the shape, size, and merging pattern of the area following the toll barrier in which vehicles fan in from B tollbooth egress lanes down to L lanes of traffic. Important considerations to incorporate in your model include accident prevention, throughput (number of vehicles per hour passing the point where the end of the plaza joins the L outgoing traffic lanes), and cost (land and road construction are expensive). In particular, this problem does not ask for merely a performance analysis of any particular toll plaza design that may already be implemented. The point is to determine if there are better solutions (shape, size, and merging pattern) than any in common use.?Determine the performance of your solution in light and heavy traffic. How does your solution change as more autonomous (self-driving) vehicles are added to the traffic mix? How is your solution affected by the proportions of conventional (human-staffed) tollbooths, exact-change (automated) tollbooths, and electronic toll collection booths (such as electronic toll collection via a transponder in the vehicle)?Link 2005 MCM Problem B （Not Problem-B 2017）PROBLEM B: TollboothsHeavily-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, theflow 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.问题B：收费后合并收取过路费（多车道划分限制访问的收费公路）使用“坡道收费”和“障碍收费”来收取驾驶员费用。
2014年美国数学建模比赛B题(最佳教练)
B
Summary
2014 Mathematical Contest in Modeling (MCM) Summary Sheet
This paper mainly sets a model of selecting five “best all time college coach” in basketball, football and hockey respectively. The model is separated to six sub-problems and its solution in our paper:1)Selecting the best coach in one NCAA basketball season; 2)Screening the college coach at top-20 to top-30 in the past 100 years; 3)Further analysis to these selected coaches, and ranking the top-5; 4)Applying this model to football and hockey; 5)Gender impact analysis and time impact analysis; 6)Analyzing the reasons that some famous coach selected by some magazines and media are not on our ranking list. For sub-problem1:Identify four indicators, and confirm the weight of each indicator by Analytic Hierarchy Process, then transformer the indicator data into scores and add these scores multiplied by the weight of itself to get the final scores. The best coach in one season is the one with highest score. For sub-problem2: Use the method above to score for all of the coaches in the past 100 years , then select the top-20.For purpose of reducing the influence of subjective factors, we use Principal Component Analysis to get another ranking list, also, we select the top-20.And union these two top-20. For sub-problem3:Surveying these coaches in the union to learn how many NBA players they've respectively brought up. And regarding this as an indicator, then dividing all indicators into Experience,Leading ability and Ability of player cultivation three aspects, and using Multilevel Hierarchical Analysis to weight them. Next, score for each person by the Fuzzy Criterion of Composite and rank according to the scores. With the purpose of reducing the influence of subjective factors, using Grey Correlation Analysis to calculate the grey correlation degree between “perfect coach” and these coaches. After that, using t-test to judge whether the two results with significant differences. The conclusion obtained: There are no significant difference. Namely, this method is generality. For sub-problem4:Fine-tuning part of indicators and weights, then apply this model in football and hockey. For sub-problem5:As One-way Analysis of Variance used, and the conclusion obtained: gender doesn’t affect the overall score or the ranking of coaches, but female coach has unique advantages in women's basketball league. And the time has no significant effect on the comprehensive score ,but the most excellent coach were in the 70s. Combined with the history of NCAA development, we think that it is necessary to improve the weight of the indicator---champions that gotten recently, making this model better. .For sub-problem6: Comparing the data of their career with our standards to find the reasons.
2012年美国大学生数学建模竞赛B题特等奖文章翻译要点
2012年美赛B题题目翻译：到Big Long River（225英里）游玩的游客可以享受那里的风景和振奋人心的急流。
远足者没法到达这条河，唯一去的办法是漂流过去。
这需要几天的露营。
河流旅行始于First Launch，在Final Exit结束，共225英里的顺流。
旅客可以选择依靠船桨来前进的橡皮筏，它的速度是4英里每小时，或者选择8英里每小时的摩托船。
旅行从开始到结束包括大约6到18个晚上的河中的露营。
负责管理这条河的政府部门希望让每次旅行都能尽情享受野外经历，同时能尽量少的与河中其他的船只相遇。
当前，每年经过Big Long河的游客有X组，这些漂流都在一个为期6个月时期内进行，一年中的其他月份非常冷，不会有漂流。
在Big Long上有Y处露营地点，平均分布于河廊。
随着漂流人数的增加，管理者被要求应该允许让更多的船只漂流。
他们要决定如何来安排最优的方案：包括旅行时间（以在河上的夜晚数计算）、选择哪种船（摩托还是桨船），从而能够最好地利用河中的露营地。
换句话说，Big Long River在漂流季节还能增加多少漂流旅行数？管理者希望你能给他们最好的建议，告诉他们如何决定河流的容纳量，记住任两组旅行队都不能同时占据河中的露营地。
此外，在你的摘要表一页，准备一页给管理者的备忘录，用来描述你的关键发现。
沿着大朗河露营摘要我们开发了一个模型来安排沿大河的行程。
我们的目标是为了优化乘船旅行的时间，从而使6个月的旅游旺季出游人数最大化。
我们模拟团体从营地到营地旅行的过程。
根据给定的约束条件，我们的算法输出了每组沿河旅行最佳的日程安排。
通过研究算法的长期反应，我们可以计算出旅行的最大数量，我们定义为河流的承载能力。
我们的算法适应于科罗多拉大峡谷的个案分析，该问题的性质与大长河问题有许多共同之处。
最后，我们考察当改变推进方法，旅程时间分布，河上的露营地数量时承载能力的变化的敏感性。
我们解决了使沿大朗河出游人数最大化的休闲旅行计划。
2021 himcm b题解析
2021年HIMCM B题解析1. 简介2021年HIMCM（美国数学建模竞赛）B题涉及了一个复杂的实际问题，要求参赛者结合数学建模方法对美国新冠疫情期间的电子商务发展进行分析与预测。
该题目涉及到宏观经济、市场趋势、数据分析等多个领域，需要综合运用数学、统计学和经济学等知识，因此具有一定的难度和挑战性。
2. 数据收集和处理在解析这个题目时，首先需要对所涉及到的数据进行收集与处理。
参赛者可以利用美国政府发布的新冠疫情数据、电子商务交易额数据等进行分析。
另外，还需要获取相关的市场调研报告和企业财报等资料，对电子商务行业的发展趋势进行深入了解。
3. 建模方法针对这个题目，可以采用多种建模方法进行分析。
可以使用时间序列分析方法对疫情期间的电子商务交易额进行预测，也可以利用回归分析方法研究疫情对消费者购买行为的影响。
还可以运用数据挖掘和机器学习技术，对大量的电子商务数据进行挖掘和分析，挖掘出潜在的商业机会和趋势。
4. 主要结论在完成建模分析后，需要对所得到的结果进行总结和归纳。
可以得出一些有价值的结论，比如疫情期间电子商务的市场规模扩大、消费者购买行为发生变化等。
这些结论对于行业从业者和决策者具有一定的参考意义。
5. 个人观点作为文章写手，我个人对这个题目的理解是，它涉及到了疫情期间的新兴产业发展和市场变化，对于我们理解经济学和市场营销学的相关知识具有一定的帮助。
通过对这个题目的深入研究与分析，可以更好地理解现代经济发展趋势与商业模式的变迁，对于未来的职业规划和发展具有一定的启发作用。
总结解析2021年HIMCM B题需要从多个方面进行深入研究和分析，涉及到的知识面较广，对于参赛者的综合素质有一定的考验。
通过对这个题目的深入思考和分析，可以更好地理解数学建模在实际问题中的应用，提高自身的综合分析和解决问题的能力。
希望我的文章能够对你有所帮助，如有需要，欢迎继续沟通。
在解析2021年HIMCM B题时，我们需要从多个方面进行深入研究和分析。
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2019 MCMProblem B: Send in the Drones: Developing an Aerial Disaster Relief Response SystemBackground: In 2017, the worst hurricane to ever hit the United States territory of Puerto Rico (see Attachment 1) left the island with severe damage and caused over 2900 fatalities. The combined destructive power of the hurricane’s storm surge and wave action produced extensive damage to buildings, homes, and roads, particularly along the east and southeast coast of Puerto Rico. The storm, with its fierce winds and heavy rain, knocked down 80 percent of Puerto Rico's utility poles and all transmission lines, resulting in loss of power to essentially all of the island's 3.4 million residents. In addition, the storm damaged or destroyed the majority of t he island’s cellular communication networks. The electrical power and cell service outages lasted for months across much of the island, and longer in some locations. Widespread flooding blocked and damaged many highways and roads across the island, making it nearly impossible for emergency services ground vehicles to plan and navigate their routes. The full extent of the damage in Puerto Rico remained unclear for some time; dozens of areas were isolated and without communication. Demands for medical supplies, lifesaving equipment, and treatment strained health-care clinics, hospital emergency rooms, and non-governmental organizations’ (NGOs) relief operations. Demand for medical care continued to surge for some time as the chronically ill turned to hospitals and temporary shelters for care.Problem: Non-governmental organizations (NGOs) are often challenged to provide adequate and timely response during or after natural disasters, such as the hurricane that struck the United States territory of Puerto Rico in 2017. One NGO in particular – HELP, Inc. - is attempting to improve its response capabilities by designing a transportable disaster response system called “DroneGo.” DroneGo will use rotor wing drones to deliver pre-packaged medical supplies and provide high-resolution aerial video reconnaissance. Selected drones should be able to perform these two missions – medical supply delivery and video reconnaissance – simultaneously or separately, depending on relief conditions and scheduling. HELP, Inc. has identified various candidate rotor wing drones that it would like your team to consider for possible use in designing its DroneGo fleet (see Attachments 2, 3).DroneGo’s pre-packaged medical supplies, called medical packages, are meant to augment, not replace, the supplies provided by local medical assistance organizations on-site within the country affected by the disaster. HELP, Inc. is planning on three different medical packages referred to as MED1, MED2, and MED3. Drones will carry these medical packages within drone cargo bays for delivery to selected locations (see Attachments 4, 5). Depending on the specific drone being used to transport medical supplies, it may be possible that multiple medical packages can be transported in a single drone cargo bay. Note that drones must land on the ground to offload medical supplies from the drone cargo bays. The video capability of the drones will provide high-resolution video of damaged and serviceable transportation road networks to HELP, Inc.’s command and contro l center for ground-based route planning.HELP, Inc. will use International Standards Organization (ISO) standard dry cargo containers to quickly transport a complete DroneGo disaster response system to a particular disaster area. The individual shipping containers for all drones in the DroneGo fleet, along with all requiredmedical packages, must fit within a maximum of three of the ISO cargo containers to be delivered to a single location, or up to three different locations if three cargo containers are used in the disaster area. Each shipping container’s contents should be packed in order to minimize any need for buffer materials for unused space. Table 1 shows the dimensions of an ISO standard dry cargo container.Table 1. Standard ISO Container DimensionsExterior Interior Door OpeningLength Width Height Length Width Height Width Height 20’20’8’8’6”19’3”7’8”7’ 10”7’8”7’5”StandardDryContainerHELP, Inc. is asking your team to use the 2017 situation in Puerto Rico to design a DroneGo disaster response system that will fit within the containers noted while meeting the anticipated medical supply demands during a potential similar future disaster scenario. It is possible that the demand requirements of this scenario may exceed the capabilities of the drone fleet your team identifies. If this occurs, HELP, Inc. wants to clearly understand any tradeoffs that it must make for implementing solutions to address these shortcomings.Part 1. Develop a DroneGo disaster response system to support the Puerto Rico hurricane disaster scenario.Consider the background information, the requirements identified in the problem statement, and the information provided in the problem attachments to address the following.A.Recommend a drone fleet and set of medical packages for the HELP, Inc. DroneGodisaster response system that will meet the requirements of the Puerto Rico hurricanescenario. Design the associated packing configuration for each of up to three ISO cargocontainers to transport the system to Puerto Rico.B.Identify the best location or locations on Puerto Rico to position one, two, or three cargocontainers of the DroneGo disaster response system to be able to conduct both medicalsupply delivery and video reconnaissance of road networks.C.For each type of drone included in the DroneGo fleet:a.Provide the drone payload packing configurations (i.e. the medical packagespacked into the drone cargo bay), delivery routes and schedule to meet the identifiedemergency medical package requirements of the Puerto Rico hurricane scenario.b.Provide a drone flight plan that will enable the DroneGo fleet to use onboardvideo cameras to assess the major highways and roads in support of the Help, Inc.mission.Part 2. MemoWrite a 1–2 page memo to the Chief Operating Officer (CEO) of HELP, Inc. summarizing your modeling results, conclusions, and recommendations so that she can share with her Board of Directors.Your MCM team submission should consist of:∙One-page Summary Sheet,∙One- to Two-page memo to the HELP, Inc. CEO∙Your solution of no more than 20 pages, for a maximum of 23 pages with your summary and memo.∙Note: Reference list and any appendices do not count toward the 23-page limit and should appear after your completed solution.Attachments:1.Map of Puerto Rico2.Potential Candidate Drones for DroneGo Fleet Consideration (with Drone payloadcapability)3.Drone Cargo Bay Packing Configuration/Dimensions by Type4.Anticipated Medical Package Demand5.Emergency Medical Package Configuration/DimensionsAttachment 1: Map of Puerto RicoAttachment 2: Potential Candidate Drones for DroneGo Fleet Consideration (with Drone Payload Capability)Shipping ContainerDimensions PerformanceCharacteristics/CapabilitiesConfigurations CapabilitiesDroneLength (in.) Width (in.) Height (in.) Max PayloadCapability(lbs.) Speed(km/h)Flight Time No Cargo (min)Video Capable Medical Package Capable Drone Cargo Bay Type* A 45 45 25 3.5 40 35 Y Y 1 B 30 30 22 8 79 40 Y Y 1 C 60 50 30 14 64 35 Y Y 2 D 25 20 25 11 60 18 Y Y 1 E 25 20 27 15 60 15 Y Y 2 F 40 40 25 22 79 24 N Y 2 G 32 32 17 20 64 16 Y Y 2 H Tethered657541N/AN/AIndefiniteNNN/A*Note that cargo bays are affixed to the drone and that drone must be on the ground to offload cargo. SeeAttachment 3 for Drone Cargo Bay Type Configuration/Dimensions.Attachment 3: Drone Cargo Bay Packing Configuration/Dimensions by TypeDrone Cargo Bay Type Length (in) Width (in) Height (in)1 8 10 14 Top Loaded2 24 20 20 Top Loaded Attachment 4: Anticipated Medical Package DemandDelivery Location Emergency Medical Packages ** Location Name Latitude Longitude Requirement Quantity FrequencyCaribbean Medical Center 18.33 -65.65 MED 1 1 DailyJajardo MED 3 1 DailyHospital HIMA 18.22 -66.03 MED 1 2 DailySan Pablo MED 3 1 Daily Hospital Pavia Santurce 18.44 -66.07 MED 1 1 DailySan Juan MED 2 1 Daily Puerto Rico Children's Hospital 18.40 -66.16 MED 1 2 DailyBayamon MED 2 1 DailyMED 3 2 Daily Hospital Pavia Arecibo 18.47 -66.73 MED 1 1 DailyArecibo**See Attachment 5 for Emergency Medical Packages 1, 2, and 3 Configurations/Dimensions.A ttachment 5: Emergency Medical Package Configuration/DimensionsEmergency Medical Package ConfigurationPackage ID Weight(lbs.)Package Dimensions(in.)(L × W × H)MED 1 2 14 × 7 × 5 MED 2 2 5 × 8 × 5 MED 3 3 12 × 7 × 4Glossary:Cargo Container (Shipping Container): a large rectangular container with doors on the ends for loading and packing, and made of material suitable for shipping, storing, and handling in many weather and climate conditions.Drone (Unmanned Aerial Vehicle, UAV): a flying robot that can be remotely controlled or fly autonomously through software-controlled flight plans in their embedded systems that work in conjunction with onboard sensors and GPS.Drone Cargo Bay: For rotor wing drones, this is an externally carried “box” used to transport materials. For this problem, the drones under consideration have one of two types (sizes) of cargo bays. Note that each drone must land for the medical packages to be unloaded from the bay at its destination.Drone Fleet: a set of drones for a particular mission or purpose. For this problem, the total set of drones by type (A to H) and Payload Capability (Visual and Medical) needed to meet the requirements of HELP, Inc.Drone Payload Packing Configuration: how the drone payload bays are packed. For this problem, how the medical packages being transported by a drone are packed inside the drone cargo bay.Medical Package: a predetermined set of medical supplies packed in a single container. For this problem, there are three Medical Package Configurations (MED1, MED2, MED3) available for transport by a drone from a deployed cargo container location to the demand location.Non-governmental Organization (NGO): Usually non-profit and sometimes international organization independent of government and governmental organizations that is active in humanitarian, educational, healthcare, social, public policy, human rights, environmental and other areas in attempts to affect change.Payload Capability: the carrying capacity of an aircraft or launch vehicle, usually measured in terms of weight. For this problem, the capability/capacity of the drone to carry medical packages.。