关于美国高铁的数学建模(全英文)

28号交流的时间10:00----10:10 How should the national rail system be built?what the nation high speed rail will bring to American?SummaryFirst of all, we select the top 20 states through the rankings of the total GDP of the U.S. states divided into seven economic zones in each economic region between First select total GDP and GDP per capita, the top row and a total population of more citiesdesign of high-speed rail line. After we are through the high-speed rail line to connect each economic zone, and eventually form a complete high-speed rail network. In the design of high-speed rail line, we use the prim algorithm. By constructing a minimum spanning tree to determine the railway between which two points is the best.The construction of high-speed rail manufacturing relationship with the economy, tourism, Select economically developed regions reasons.First, according to the economic divide block, we get seven economic blocks through the analysis of GDP of 20 states, and then consolidated the top of the urban population and 50 selected priority to the construction of high-speed rail city, and finally we decided the 31 cities priority the construction of high-speed rail.The second year the main purpose is to connect some of the blocks, integrated urban population in 100 cities elected before connecting the city and some of the more populous city, and finally selected 26 new high-speed rail city.The third and fourth years of the connection of the eastern and western regions of the central region.The first five years the construction of the difficulty of some larger and some minor high-speed rail construction.The impact of the aircraft on the high-speed railway, we assume that its effect is linear.800 miles ago, the choice of high-speed rail and aircraft are fixed, the number of aircraft in the 800 miles increased linearly select number of high-speed rail will decrease linearly.Key wordsThe prim algorithm, Economic zones Linear effectsContents1. Introduction (3)1.1 Why does toll way collects toll? (3)1.2 Toll modes (3)1.3 Toll collection methods (3)1.4 Annoyance in toll plazas (3)1.5 The origin of the toll way problem (3)1.6 Queuing theory (4)2. The Description of Problem (5)2.1 How do we approximate the whole course of paying toll? (5)2.2 How do we define the optimal configuration? (5)2.2.1 From the perspective of motorist (5)2.2.2 From the perspective of the toll plaza (6)2.2.3 Compromise (6)2.3 Overall optimization and local optimization (6)2.4 The differences in weights and sizes of vehicles (7)2.5 What if there is no data available? (7)3. Models (7)3.1 Basic Model (7)3.1.1 Symbols and Definitions (7)3.1.2 Assumptions (8)3.1.3 The Foundation of Model (9)3.1.4 Solution and Result (11)3.1.5 Analysis of the Result (11)3.1.6 Strength and Weakness (13)3.2 Improved Model (14)3.2.1 Extra Symbols (14)3.2.2 Additional Assumptions (14)3.2.3 The Foundation of Model (14)3.2.4 Solution and Result (15)3.2.5 Analysis of the Result (18)3.2.6 Strength and Weakness (19)4. Conclusions (19)4.1 Conclusions of the problem (19)4.2 Methods used in our models (19)4.3 Application of our models (19)5. Future Work (19)5.1 Another model (19)5.2 Another layout of toll plaza (23)5.3 The newly- adopted charging methods (23)6.References (23)7.Appendix (23)Programs and codes (24)I. IntroductionAmerican has it’s well developed highway system already,the reason why we are on track to creating a modern ,high-speed and interconnected rail system is that the rail system will support sustained economic growth,move people and goods with greater speed and efficiency, relieve growing congestion,and provide another choice for the public’s mobility needs.1.1 More people, more fright, more challengeThe United States and world economies are experiencing an increased demand for rail.Expanding U.S.passenger and freight mobility will require a networked railroad system that is able to modernize and increase capacity.With an estimated U.S.population growth of 70 million people over the next 25 years,mostly centered in metropolitan regions,a national rail plan is needed to ensure a coordinated and intelligent system that provides safe,reliable,and efficient passenger and freight rail service.1.2 T ask and ApproachesThe United States now faces new challenges spurred by unprecedented population growth,economic transformations,and technological innovations. High-speed rail, now established in marry developed nations, is positioned to benefit the United States as the States’plans mature and projects are implemented. With FY2009 kick starting U.S. High-speed rail investment and planning,follow-on funding and strategic investment will bring us closer to realizing the vision outlined by Congress and the Administration.The ext American transformation will require an interconnected and balanced transportation network that maximizes the benefits of every mode. A key to integrating these systems is higher-performing rail, including the full spectrum of high-speed and intercity passenger rail, commuter rail,and freight rail.These interconnected rail systems will relieve congestion,promote livable communities, facilitate economic expansion, respect environmental sustainability and provide choices for the American public. This investment will set set the stage for job creation, sustainable economic competitiveness, a more resilient infrastructure and a lasting prosperity.In order to construct a reasonable rail system ,we plan to accomplish goals as follows:⑴To find the economically developed region ,and assign this area as the strongly construct area.⑵To find the metropolitan areas as the strongly construct area;⑶To consider the developed and intensive area as the megaregions, there will have the densely railway network.⑷To design the railway in the megaregions.⑸To connect the different megaregions.II. Basic Assumptions⑴In the future, the developed regions remain developed.⑵The inhabitants will live in these metropolitan areas and most of the population growth will occur there.⑶The data we used is accurete and reliable.⑷Assuming the railway between cities is straight.⑸The state with less people do not construct.III. The different megaregions3.1 To certain the metropolitan①According to the relevant references,we get the Figture 1②According to the relevant references,we get Figure 2③integrate economic and population ,we can get the megaregionsPart I: Washington State; (northwest)Part II: California (western)Part III: Minnesota, Wisconsin, Michigan; (North)Part IV: New Y ork, Pennsylvania, Massachusetts, Massachusetts, Pennsylvania, New Jersey, Maryland (Eastern)Part V: Illinois, Tennessee, Missouri, West V irginia, V irginia, North Carolina, South Laizhou (Central)Part VI: Florida, Georgia (southeast)Part VII: Texas, Mississippi, Louisiana (South)District of Columbia as the fourth, fifth part of the point of contact.In every part of the city economy, population ranked each part of the center of the city:Part I: SeattlePart II: Los AngelesPart III: MinneapolisPart IV: New Y orkPart V: ChicagoPart VI: AtlantaPart VII: DallasIV. The best route modelThere are six megaregions in the map,and every magaregion has different circumstance,if there are less than three cities in one magaregion, then we can connect them into a triangle. If the number bigger than three, we can use prim algorithm to find the shortest way .What we have construct was our five years plan, next we will base on the per capita GDP to certain the timing.4.1 Certain the route in every part4.1.1 Part oneIn part one, there are only three bigger cities,as we wanted we connect them as a trianglePart IStarting point Ending point The distance(in miles)Seattle Portland 144.990Portland Kennewick 170.753Seattle Kennewick 172.4814.1.2 Part twoIn part two, the densely cities are on a line, maybe it ’s the best answer that we want,we connect them like a line.4.1.3 Part sixIn part six, there are only four cities in there ,it ’s also easy to find the best route.Part VIStarting point Ending point The distant(in miles) Jacksonville Orlando 125.600 Orlando Tampa 77.272 OrlandoMiami203.8804.1.4 The rest partThe rest is harder to railroad, because the city is intensive, and there are densely population. Now we try to use prim algorithm to find the shortest route. According to the relevant reference,we obtain T able 3.We can see from T able 3, we find the cities ’s position, then we get the distant between different cities, now we base the thoughts of prim algorithm , to find the shortest way. （i ） { P = v1} ，Q = Φ；（ii ）while P ~=V ;find the shortest edge pv ， p ∈ P ,v ∈V − PStarting point Ending point The distance(in miles) San Francisco Sacramento 76.2114 San Francisco Los Angeles 350.793 Los AngelesSan Diego113.002P = P +{v}Q = Q +{pv}endNow we start the MA TLAB simulation .Results and discussions of the simulation1 2 3 17 16 15 5 142 3 17 16 15 5 14 121.19282.5708 1.5621 0.2791 1.1207 1.4416 1.5997 0.8004 12 3 11 13 10 8 7 710 11 13 4 8 7 9 61.581 1.6658 1.6878 0.8344 3.33892.1544 0.7281 1.36784From the conclusion, we can get that the path is that 1→2→3→17→16→15→5→14→12→10; 3→11→13→4; 10→8→7→9; 7→6;The final answer is shown as the following figure4.1.5 Optimize the networkWe take the first year construct city as the center ,year by year ,we can connect the vice city, eventually we will get a whole network.4.2 The whole networkAfter we get the separate route, we can use the next few years to finish the rest route.The red line indicates the first year of laying railway.The blue line represents the second year of laying railway.The yellow line represents the third year of laying railway.The green line represents the fourth year of laying railway.The brown line represents the fifth year of laying railway.4.3 Evaluation of the model4.3.1 Strengths①prim algorithm can precisely find the shortest way. It build up an equation to calculate the shortest path in the magaregions.② prime algorithm has a strong flexibility. It can be applied to other region .4.3.2 weaknesses① prim algorithm get the routine too ideally and it can not consider the other factors.②V. The influence nation rail system will bring toOn one hand, The high-speed and interconnected rail system will support sustained economic growth, move people and goods with greater speed and efficiency, relieve growing congestion,ang provide another choice for the public’s mobility needs. On the other hand,it’s unpleasant for the airline, because someone will take train nor plane in the future. Obviously, it will do harm to their benefit.VI. Prediction of the airline passenger lossHaving analyzed the two sides influence, we can sure that it’s bad news for the airline.But why do not the passenger to take the high-speed rail for the rail more punctual and comfortable, Y es,that’s many of the passenger’s thought. But what proportion will the passenger to take high-speed rail instead of plane. And how many losses the airline will have?6.1 liner Equation Model6.1.1 Basic AssumptionSuppose there are N passengers to travel with high-speed rail or plane throughout a year, and if the distance is less than 800km, then eighty percent of the passenger will take the high-speed rail, if the distance is greater than 800km, then the number of passenger decreased by 20% per 200km,the same time ,the number of passenger by plane will increase 20%. And suppose the travel distance X~U[600,1600].6.1.2 Finding the distance that harm the airline mostAccording to the basic assumption,we can get the following figure ,the high-speed rail curve and the plane curve .we can also get it’s equations:⑴ 1y =0.8N (X<=800)1y =0.8N-N*(X-800)/1000 (X>=800)⑵ 2y =0.2N (X<=800)2y =0.2N+N*(X-800)/1000 (X>=800) For X~U[600,1600], ix P =1/1000 ;P(600<=X<=800)=200*1/1000=0.2; P(800<=X<=1600)=800*1/1000=0.8;So the passenger loss L=0.2*0.8N+0.8*(0.8N-N*(X-800)/1000)=0.8N-0.8*(N*(X-800)/1000); The conclusion:When the travel distance is 800km, the passenger loss is biggest.Then from the map ,we can find several route, in these routes 80% of the passenger was lost.6.2 Evaluations of our module6.2.1 strengths ① Linear model has enormous edibility. For instance, not only it can be applied to the transportation, but it can also be used to simulate the other situations. ② Linear model is easy to calculate. It ’s the feature of linear equation. 6.2.2 weakness① Linear model can not to predict precisely, because the passenger ‘s change with the distance can not be linear ,it can be complex, so the outcome we finally get was roughly② the parameter of the model is subjective ,it is not through the actual survey . ③If the distance is shorter than 400km ,there will not be plane , but in our model,we suppose there plane still be there.VII Contribution to the American economicAs we have analyses before ,the construction of high-speed rail will play a great role to theVIII ConclusionIX future workReferences[1]Albert.R & Barabasi, A.-L.(2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74,47-97.[2]American Radio Relay League (2008). The Arrl Operating Manual for Radio Amateurs. ARRL OPERA TING MANUAL. Amer Radio Relay League.[3]Barclay,L, & Institution of Electrical Engineer(2003). Propagation of radiowaves. Electromagnetic Waves. Institution of Electrical Engineers.[4]Federal Communications Commission (2011). About the fcc.URL /aboutus.html[5]/[6]/wiki/%E7%BE%8E%E5%9B%BD%E5%90%84%E5%B7%9E%E5%9 B%BD%E5%86%85%E7%94%9F%E4%BA%A7%E6%80%BB%E5%80%BC%E5%88%97%E8%A1%A8[7]Straw, R, Cebik, L. Hallidy, D, & Jansson, D. (2007). The ARRL Antenna Book .ARRL ANTENNA BOOK. ARRL.[8]ukrepeaters, The UK Amateur Radio Repeater Resource (2011). 2 metres band repeaters(coverage)Appendix 1American GDP per stat ranking （2010）GDPstate GDPpercentage ofnationalGDPpopulationGDP percapitaGDP percapitarankingsranking(milliordollor)(millior) (dollor) United States14,657,80100 308.7 47,4821 California1,936,400 13.34 37.3 51,914 122 Texas 1,207,432 7.95 25.1 45,940 243 New York 1,156,500 7.68 19.4 57,423 74 Florida 754,000 5.2 18.8 40,106 405 Illinois 644,200 4.44 12.8 50,328 156 Pennsylvania 575,600 3.97 12.7 45,323 257 New Jersey 497,000 3.42 8.8 56,477 88 Ohio 483,400 3.33 11.5 42,035 339 Virginia 427,700 2.95 8 53,463 910 North Carolina 407,400 2.81 9.5 42,884 3111 Georgia 404,600 2.79 9.7 41,711 3512 Massachusetts 377,700 2.6 6.5 58,108 613 Michigan 372,400 2.57 9.9 37,616 4214 Washington 351,100 2.42 6.7 52,403 1015 Maryland 300,000 2.07 5.8 51,724 1316 Indiana 267,600 1.84 6.5 41,169 3617 Minnesota 267,100 1.84 5.3 50,396 1418 Arizona 261,300 1.8 6.4 40,828 3919 Colorado 259,700 1.79 5 51,940 1120 Wisconsin 251,400 1.73 5.7 44,105 2921 Tennessee 250,300 1.72 6.3 39,730 41Table 1. the American per stat ranking(source:U.S. Census Bureau)Appendix 2The first yearStarting point Ending point The distant(in miles) Seattle Portland 144.990San Francisco Sacramento 76.2114San Francisco Los Angeles 350.793Los Angeles San Diego 113.002Phoenix Coctus 109.391Houston Dallas 230.225Houston San Antonio 186.698Dallas Austin 182.682Austin San Antonio 76.4823Jacksonville Orlando 125.600Orlando Tampa 77.272Orlando West Palm Beach 149.751West Palm Beach Miami 67.1118Atlanta Charlotte 224.081Charlotte Raleigh 130.743Nashville Atlanta 215.622Minneapolis Milwaukee 302.516Milwaukee Chicago 83.601Chicago Indianapolis 161.628Chicago Detroit 259.963Chicago Springfield 177.601Springfield St Louis 86.056Indianapolis Columbus 168.986New Y ork Philadelphia 80.326Philadelphia Washington 123.266The first year of the total length of the laying of high-speed rail:3904.5985milesThe second yearStarting point Ending point The distant(in miles) Portland San Francisco 532.850Los Angeles Phoenix 356.266Denver Pueblo 108.682Pueblo Santa Fe 188.330 Albuquerque Santa Fe 55.2531 Oklahoma City Tulsa 94.297Toledo Fort Wayne 92.8754Fort Wayne Indianapolis 107.213 Indianapolis Cincinnati 97.127 Indianapolis Louisville 104.933Louisville Lexington 70.166Cincinnati Lexington 73.706Cleveland Akron 31.818Akron Pittsburgh 89.693Pittsburgh Washington 190.470 Philadelphia Allentown 51.050Atlanta Macon 77.543Atlanta Birmingham 137.838Memphis Nashville 194.249Nashville Louisville 156.843Louisville Lexington 71.592 Montgomery Birmingham 83.9005 Montgomery Atlanta 146.289 Birmingham Atlanta 140.149 Birmingham Savannah 225.946 Savannah Jacksonville 130.365 Jacksonville Montgomery 311.175Raleigh Washington 233.375Boston New Y ork 191.973The second year of the total length of the laying of high-speed rail:4345.967milesThe third yearStarting point Ending point The distant(in miles) Denver Salt Lake City 368.745 Albuquerque Tucson 316.996 Albuquerque Dallas 586.689Dallas Oklahoma City 188.002Wichita Kansas City 181.989 Louisiana New Orleans 80.628New Orleans Mobile 128.703Mobile Tallahassee 224.573 Tallahassee Jacksonville 161.590Mobile Montgomery 155.973Birmingham Jackson 213.116Jackson Louisiana 145.713Columbus Charleston 133.827Cleveland Buffalo 173.795Buffalo Boston 399.072The thirty year of the total length of the laying of high-speed rail:3459.411miles The forth yearStarting point Ending point The distant(in miles) Sacramento Reno 109.336Reno Salk Lake City 427.061Los Angeles Flagstaff 380.012Flagstaff Phoenix 127.505Kansas City Hays 385.565Hays Denver 309.963Kansas City Lowa 212.521Lowa Minneapolis 192.885Dallas Little Rock 293.249Little Rock St Louis 289.804Lexington Parkersburg 178.301Parkersburg Washington 244.502Richmond Norfolk 76.779Raleigh Charlotte 126.143Charlotte Charleston 176.901New Y ork Watertown 242.678The fourth year of the total length of the laying of high-speed rail:3773.205milesThe fifth yearStarting point Ending point The distant(in miles) Seattle Kennewick 172.481Kennewick Boise 229.268Boise Salt Lake City 299.174San Diego Las V egas 253.996Memphis Little Rock 129.070Memphis Birmingham 216.322St Louis Nashville 253.171Roanoke Greensboro 87.004Raleigh Norfolk 153.169Boston Augusta 151.061Boston Montpelier 152.706The fifth year of the total length of the laying of high-speed rail:2097.422milesAppendix 3clc;clear;a=zeros(17);a(1,2)=1.1928;a(1,3)=3.7065;a(1,4)=4.9103;a(1,5)=5.7873;a(1,6)=14.0866;a(1,7)=13.1131;a(1,8)=11.6331;a(1,9)=12.6520;a(1,10)=8.3259;a(1,11)=3.3859;a(1,12)=6.9160;a(1,13)=4.5627;a(1,14) =6.3636;a(1,15)=5.4361;a(1,16)=6.0495;a(1,17)=5.2392;a(2,3)=2.5708;a(2,4)=4.6055;a(2,5)=5.0081;a(2,6)=13.6692;a(2,7)=12.6168;a(2,8)=11.0061;a(2,9) =12.2062;a(2,10)=7.7681;a(2,11)=2.6129;a(2,12)=6.4720;a(2,13)=4.0805;a(2,14)=5.8336;a(2,15)=4.4466;a(2,16)=4.9533;a(2,17)=4.0736;a(3,4)=4.0330;a(3,5)=3.1663;a(3,6)=12.1817;a(3,7)=11.0016;a(3,8)=9.1665;a(3,9)=10.7029;a(3,1 0)=6.2005;a(3,11)=1.6658;a(3,12)=5.2830;a(3,13)=3.2225;a(3,14)=4.5046;a(3,15)=2.1130;a(3,16) =2.3976;a(3,17)=1.5621;a(4,5)=2.3645;a(4,6)=9.1800;a(4,7)=8.2365;a(4,8)=6.9241;a(4,9)=7.7514;a(4,10)=3.5900;a(4,11) =2.4363;a(4,12)=2.0540;a(4,13)=0.8344;a(4,14)=1.7307;a(4,15)=3.4125;a(4,16)=4.5332;a(4,17)= 4.8947;a(5,6)=9.0212;a(5,7)=7.8400;a(5,8)=6.0604;a(5,9)=7.5371;a(5,10)=3.0409;a(5,11)=2.4140;a(5,12) =2.3874;a(5,13)=1.7855;a(5,14)=1.5997;a(5,15)=1.4416;a(5,16)=2.4444;a(5,17)=3.2468;a(6,7)=1.3784;a(6,8)=3.5308;a(6,9)=1.4842;a(6,10)=5.9925;a(6,11)=11.1360;a(6,12)=7.2016;a(6, 13)=9.5919;a(6,14)=7.8494;a(6,15)=10.2716;a(6,16)=10.8269;a(6,17)=12.0084;a(7,8)=2.1544;a(7,9)=0.7281;a(7,10)=4.8600;a(7,11)=10.0434;a(7,12)=6.2037;a(7,13)=8.5668;a(7 ,14)=6.7846;a(7,15)=9.0345;a(7,16)=9.5353;a(7,17)=10.7355;a(8,9)=2.3046;a(8,10)=3.3389;a(8,11)=8.3951;a(8,12)=4.8852;a(8,13)=7.0832;a(8,14)=5.2881;a(8 ,15)=7.1220;a(8,16)=7.5143;a(8,17)=8.7484;a(9,10)=4.5091;a(9,11)=9.6681;a(9,12)=5.7488;a(9,13)=8.1341;a(9,14)=6.3786;a(9,15)=8.7934;a( 9,16)=9.3691;a(9,17)=10.5352;a(10,11)=5.1843;a(10,12)=1.5810;a(10,13)=3.7639;a(10,14)=1.9626;a(10,15)=4.3674;a(10,16)=5. 1045;a(10,17)=6.1632;a(11,12)=3.9926;a(11,13)=1.6878;a(11,14)=3.2890;a(11,15)=2.2128;a(11,16)=3.1135;a(11,17)=2. 8942;a(12,13)=2.3917;a(12,14)=0.8004;a(12,15)=3.8277;a(12,16)=4.8043;a(12,17)=5.6203;a(13,14)=1.8013;a(13,15)=2.6303;a(13,16)=3.7481;a(13,17)=4.0583;a(14,15)=3.0352;a(14,16)=4.0345;a(14,17)=4.8218;a(15,16)=1.1207;a(15,17)=1.8152;a(16,17)=.2791;a=a+a';a(find(a==0))=inf;result=[];p=1;tb=2:length(a);while length(result)~=length(a)-1temp=a(p,tb);temp=temp(:);d=min(temp);[jb,kb]=find(a(p,tb)==d);j=p(jb(1));k=tb(kb(1));result=[result,[j;k;d]];p=[p,k];tb(find(tb==k))=[];endresult。