数学建模最大流问题.docx
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数学建模最大流问题.docx
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数学建模最大流问题
建模作业之最大流问题
一、问题重述
在交通领域,不论是火车还是汽车甚至是飞机的起航与降落,都涉及到了流量问题。
顺利地解决最大流量问题,可以便利的解决交通方面日益突出的问题,更能让资源更充分更优化地得到利用。
所以,学者们对最大流量问题的各个方面进行了不同的研究并把所得结论运用到实践中,因此而极大地促进了经济文化的发展。
本题就是这样一个最基础的最大流量问题。
二、符号说明
X(i,j):
i流出到j的实际流量
C(i,j):
i流出到j的最大流量
三、模型假设
由于要计算0与1流到5、6、7的流量涉及到2个流出口与3个流进口,对计算十分不利,对模型的建立也增加了难度。
所以在本题1与2之前增加一个流出口S,在5、6、7之后增加一个流进口T,从而,本题的目标函数就变成从S流出到T的最大流量。
题中所涉及的变量一些是数字,一些是字母,对模型的建立十分不利。
所以,我们在建立模型前,将图中的S设定为1号,0到7号设定为2到9号,剩下的T则为10号。
本题所求的最大流量即为从1号的流出量或者10号的流进量。
目标函数:
max=X(1,2)+X(1,3)
约束条件:
X(i,j)<=C(i,j)i=(1…10),j=(1…10)
k=(1…10)
根据函数建模,由lingo得出结果,最大流量即为25.
四、附录
所建lingo模型如下;
sets:
a/1..10/;
do(a,a):
x,c;
endsets
max=x(1,2)+x(1,3);
@for(do:
x @for(a(k)|k#ne#1#and#k#ne#10: @sum(a(i): x(i,k))=@sum(a(j): x(k,j))); data: c=0,12,20,0,0,0,0,0,0,0 0,0,0,12,0,0,0,0,0,0 0,0,0,0,20,0,0,0,0,0 0,0,0,0,6,3,6,0,0,0 0,0,0,0,0,7,0,0,9,0 0,0,0,2,0,0,5,8,0,0 0,0,0,0,0,0,0,0,0,100 0,0,0,0,0,0,0,0,0,100 0,0,0,0,0,0,0,4,0,100 0,0,0,0,0,0,0,0,0,0; enddata end 经lingo求解得如下结果: Globaloptimalsolutionfoundatiteration: 0 Objectivevalue: 25.00000 VariableValueReducedCost X(1,1)0.0000000.000000 X(1,2)9.0000000.000000 X(1,3)16.000000.000000 X(1,4)0.0000001.000000 X(1,5)0.0000001.000000 X(1,6)0.0000000.000000 X(1,7)0.0000000.000000 X(1,8)0.0000000.000000 X(1,9)0.0000000.000000 X(1,10)0.0000000.000000 X(2,1)0.0000000.000000 X(2,2)0.0000000.000000 X(2,3)0.0000000.000000 X(2,4)9.0000000.000000 X(2,5)0.0000000.000000 X(2,6)0.0000000.000000 X(2,7)0.0000000.000000 X(2,8)0.0000000.000000 X(2,9)0.0000000.000000 X(2,10)0.0000000.000000 X(3,1)0.0000000.000000 X(3,2)0.0000000.000000 X(3,3)0.0000000.000000 X(3,4)0.0000000.000000 X(3,5)16.000000.000000 X(3,6)0.0000000.000000 X(3,7)0.0000000.000000 X(3,8)0.0000000.000000 X(3,9)0.0000000.000000 X(3,10)0.0000000.000000 X(4,1)0.0000000.000000 X(4,2)0.0000000.000000 X(4,3)0.0000000.000000 X(4,4)0.0000000.000000 X(4,5)0.0000000.000000 X(4,6)3.0000000.000000 X(4,7)6.0000000.000000 X(4,8)0.0000000.000000 X(4,9)0.0000000.000000 X(4,10)0.0000000.000000 X(5,1)0.0000000.000000 X(5,2)0.0000000.000000 X(5,3)0.0000000.000000 X(5,4)0.0000000.000000 X(5,5)0.0000000.000000 X(5,6)7.0000000.000000 X(5,7)0.0000000.000000 X(5,8)0.0000000.000000 X(5,9)9.0000000.000000 X(5,10)0.0000000.000000 X(6,1)0.0000000.000000 X(6,2)0.0000001.000000 X(6,3)0.0000001.000000 X(6,4)0.0000001.000000 X(6,5)0.0000001.000000 X(6,6)0.0000000.000000 X(6,7)2.0000000.000000 X(6,8)8.0000000.000000 X(6,9)0.0000000.000000 X(6,10)0.0000000.000000 X(7,1)0.0000000.000000 X(7,2)0.0000001.000000 X(7,3)0.0000001.000000 X(7,4)0.0000001.000000 X(7,5)0.0000001.000000 X(7,6)0.0000000.000000 X(7,7)0.0000000.000000 X(7,8)0.0000000.000000 X(7,9)0.0000000.000000 X(7,10)8.0000000.000000 X(8,1)0.0000000.000000 X(8,2)0.0000001.000000 X(8,3)0.0000001.000000 X(8,4)0.0000001.000000 X(8,5)0.0000001.000000 X(8,6)0.0000000.000000 X(8,7)0.0000000.000000 X(8,8)0.0000000.000000 X(8,9)0.0000000.000000 X(8,10)8.0000000.000000 X(9,1)0.0000000.000000 X(9,2)0.0000001.000000 X(9,3)0.0000001.000000 X(9,4)0.0000001.000000 X(9,5)0.0000001.000000 X(9,6)0.0000000.000000 X(9,7)0.0000000.000000 X(9,8)0.0000000.000000 X(9,9)0.0000000.000000 X(9,10)9.0000000.000000 X(10,1)0.0000000.000000 X(10,2)0.0000001.000000 X(10,3)0.0000001.000000 X(10,4)0.0000001.000000 X(10,5)0.0000001.000000 X(10,6)0.0000000.000000 X(10,7)0.0000000.000000 X(10,8)0.0000000.000000 X(10,9)0.0000000.000000 X(10,10)0.0000000.000000 C(1,1)0.0000000.000000 C(1,2)12.000000.000000 C(1,3)20.000000.000000 C(1,4)0.0000000.000000 C(1,5)0.0000000.000000 C(1,6)0.0000000.000000 C(1,7)0.0000000.000000 C(1,8)0.0000000.000000 C(1,9)0.0000000.000000 C(1,10)0.0000000.000000 C(2,1)0.0000000.000000 C(2,2)0.0000000.000000 C(2,3)0.0000000.000000 C(2,4)12.000000.000000 C(2,5)0.0000000.000000 C(2,6)0.0000000.000000 C(2,7)0.0000000.000000 C(2,8)0.0000000.000000 C(2,9)0.0000000.000000 C(2,10)0.0000000.000000 C(3,1)0.0000000.000000 C(3,2)0.0000000.000000 C(3,3)0.0000000.000000 C(3,4)0.0000000.000000 C(3,5)20.000000.000000 C(3,6)0.0000000.000000 C(3,7)0.0000000.000000 C(3,8)0.0000000.000000 C(3,9)0.0000000.000000 C(3,10)0.0000000.000000 C(4,1)0.0000000.000000 C(4,2)0.0000000.000000 C(4,3)0.0000000.000000 C(4,4)0.0000000.000000 C(4,5)6.0000000.000000 C(4,6)3.0000000.000000 C(4,7)6.0000000.000000 C(4,8)0.0000000.000000 C(4,9)0.0000000.000000 C(4,10)0.0000000.000000 C(5,1)0.0000000.000000 C(5,2)0.0000000.000000 C(5,3)0.0000000.000000 C(5,4)0.0000000.000000 C(5,5)0.0000000.000000 C(5,6)7.0000000.000000 C(5,7)0.0000000.000000 C(5,8)0.0000000.000000 C(5,9)9.0000000.000000 C(5,10)0.0000000.000000 C(6,1)0.0000000.000000 C(6,2)0.0000000.000000 C(6,3)0.0000000.000000 C(6,4)2.0000000.000000 C(6,5)0.0000000.000000 C(6,6)0.0000000.000000 C(6,7)5.0000000.000000 C(6,8)8.0000000.000000 C(6,9)0.0000000.000000 C(6,10)0.0000000.000000 C(7,1)0.0000000.000000 C(7,2)0.0000000.000000 C(7,3)0.0000000.000000 C(7,4)0.0000000.000000 C(7,5)0.0000000.000000 C(7,6)0.0000000.000000 C(7,7)0.0000000.000000 C(7,8)0.0000000.000000 C(7,9)0.0000000.000000 C(7,10)100.00000.000000 C(8,1)0.0000000.000000 C(8,2)0.0000000.000000 C(8,3)0.0000000.000000 C(8,4)0.0000000.000000 C(8,5)0.0000000.000000 C(8,6)0.0000000.000000 C(8,7)0.0000000.000000 C(8,8)0.0000000.000000 C(8,9)0.0000000.000000 C(8,10)100.00000.000000 C(9,1)0.0000000.000000 C(9,2)0.0000000.000000 C(9,3)0.0000000.000000 C(9,4)0.0000000.000000 C(9,5)0.0000000.000000 C(9,6)0.0000000.000000 C(9,7)0.0000000.000000 C(9,8)4.0000000.000000 C(9,9)0.0000000.000000 C(9,10)100.00000.000000 C(10,1)0.0000000.000000 C(10,2)0.0000000.000000 C(10,3)0.0000000.000000 C(10,4)0.0000000.000000 C(10,5)0.0000000.000000 C(10,6)0.0000000.000000 C(10,7)0.0000000.000000 C(10,8)0.0000000.000000 C(10,9)0.0000000.000000 C(10,10)0.0000000.000000 RowSlackorSurplusDualPrice 125.000001.000000 20.0000000.000000 33.0000000.000000 44.0000000.000000 50.0000000.000000 60.0000000.000000 70.0000000.000000 80.0000000.000000 90.0000000.000000 100.0000000.000000 110.0000000.000000 120.0000001.000000 130.0000000.000000 140.0000000.000000 153.0000000.000000 160.0000000.000000 170.0000001.000000 180.0000001.000000 190.0000001.000000 200.0000001.000000 210.0000001.000000 220.0000001.000000 230.0000000.000000 240.0000000.000000 250.0000000.000000 264.0000000.000000 270.0000001.000000 280.0000001.000000 290.0000001.000000 300.0000001.000000 310.0000001.000000 320.0000001.000000 330.0000000.000000 340.0000000.000000 350.0000000.000000 366.0000000.000000 370.0000001.000000 380.0000001.000000 390.0000001.000000 400.0000001.000000 410.0000001.000000 420.0000001.000000 430.0000000.000000 440.0000000.000000 450.0000000.000000 460.0000000.000000 470.0000001.000000 480.0000001.000000 490.0000001.000000 500.0000001.000000 510.0000001.000000 520.0000000.000000 530.0000000.000000 540.0000000.000000 552.0000000.000000 560.0000000.000000 570.0000000.000000 583.0000000.000000 590.0000000.000000 600.0000000.000000 610.0000000.000000 620.0000000.000000 630.0000000.000000 640.0000000.000000 650.0000000.000000 660.0000000.000000 670.0000000.000000 680.0000000.000000 690.0000000.000000 700.0000000.000000 7192.000000.000000 720.0000000.000000 730.0000000.000000 740.0000000.000000 750.0000000.000000 760.0000000.000000 770.0000000.000000 780.0000000.000000 790.0000000.000000 800.0000000.000000 8192.000000.000000 820.0000000.000000 830.0000000.000000 840.0000000.000000 850.0000000.000000 860.0000000.000000 870.0000000.000000 880.0000000.000000 894.0000000.000000 900.0000000.000000 9191.000000.000000 920.0000000.000000 930.0000000.000000 940.0000000.000000 950.0000000.000000 960.0000000.000000 970.0000000.000000 980.0000000.000000 990.0000000.000000 1000.0000000.000000 1010.0000000.000000 1020.0000001.000000 1030.0000001.000000 1040.0000001.000000 1050.0000001.000000 1060.0000000.000000 1070.0000000.000000 1080.0000000.000000 1090.0000000.000000 根据求解结果得出: 最大流量为25
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