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