例 9   某公司在六個城市 c1,c2, …c6 中有分公司，從 i

ci 到 cj的直接航程票價記在下述矩陣的 (I,j) 位置上。（∞表示無直接航路），請幫助該公司設計一張城市c1 到其它城市間的票價最便宜的路線圖。



clc,clear 

a=zeros(6); 

a(1,2)=50;a(1,4)=40;a(1,5)=25;a(1,6)=10; 

a(2,3)=15;a(2,4)=20;a(2,6)=25; 

a(3,4)=10;a(3,5)=20; 

a(4,5)=10;a(4,6)=25; 

a(5,6)=55; 

a=a+a'; 

a(find(a==0))=inf; 

pb(1:length(a))=0;pb(1)=1;index1=1;index2=ones(1,length(a));

d(1:length(a))=inf;d(1)=0;temp=1; 

while sum(pb)<length(a) 

  tb=find(pb==0); 

  d(tb)=min(d(tb),d(temp)+a(temp,tb)); 

  tmpb=find(d(tb)==min(d(tb))); 

  temp=tb(tmpb(1)); 

  pb(temp)=1; 

  index1=[index1,temp]; 

  temp2=find(d(index1)==d(temp)-a(temp,index1)); 

  index2(temp)=index1(temp2(1)); 

end 

d, index1, index2
 

 編寫 LINGO 程式如下：

model: 

sets: 

cities/A,B1,B2,C1,C2,C3,D/; 

roads(cities,cities)/A B1,A B2,B1 C1,B1C2,B1 C3,B2 C1, 

B2 C2,B2 C3,C1 D,C2 D,C3 D/:w,x; 

endsets 

data: 

w=2 4 3 3 1 2 3 1 1 3 4; 

enddata 

[email protected](cities); !城市的個數; 

[email protected](roads:w*x); 

@for(cities(i)|i #ne#1 #and# i #ne#n: 

@sum(roads(i,j):x(i,j))
 
[email protected](roads(j,i):x(j,i)));

@sum(roads(i,j)|i #eq#1:x(i,j))=1; 

@sum(roads(i,j)|j #eq#n:x(i,j))=1; 

end

model: 

sets: 

cities/1..11/; 

roads(cities,cities):w,x; 

endsets 

data: 

w=0; 

enddata 

calc: 

w(1,2)=2;w(1,3)=8;w(1,4)=1; 

w(2,3)=6;w(2,5)=1; 

w(3,4)=7;w(3,5)=5;w(3,6)=1;w(3,7)=2; 

w(4,7)=9; 

w(5,6)=3;w(5,8)=2;w(5,9)=9; 

w(6,7)=4;w(6,9)=6; 

w(7,9)=3;w(7,10)=1; 

w(8,9)=7;w(8,11)=9; 

w(9,10)=1;w(9,11)=2;w(10,11)=4; 

@for(roads(i,j):w(i,j)=w(i,j)+w(j,i)); 

@for(roads(i,j):w(i,j)
 
[email protected](w(i,j) #eq# 0,1000,w(i,j))); 

endcalc 

[email protected](cities);  !城市的個數; 

[email protected](roads:w*x); 

@for(cities(i)|i #ne#1 #and# i #ne# 

n:@sum(cities(j):x(i,j))[email protected](cities(j):x(j,i)));

@sum(cities(j):x(1,j))=1; 

@sum(cities(j):x(j,1))=0; !不能回到頂點1; 

@sum(cities(j):x(j,n))=1; 

@for(roads:@bin(x)); 

end 

 

例12  用Floyd演算法求解例9。

矩陣path用來存放每對頂點之間最短路徑上所經過的頂點的序號。Floyd演算法的

Matlab程式如下：

clear;clc; 

n=6; a=zeros(n); 

a(1,2)=50;a(1,4)=40;a(1,5)=25;a(1,6)=10; 

a(2,3)=15;a(2,4)=20;a(2,6)=25;a(3,4)=10;a(3,5)=20; 

a(4,5)=10;a(4,6)=25; a(5,6)=55; 

a=a+a'; M=max(max(a))*n^2; %M為充分大的正實數 

a=a+((a==0)-eye(n))*M; 

path=zeros(n); 

for k=1:n 

  for i=1:n 

     for j=1:n 

        if a(i,j)>a(i,k)+a(k,j) 

           a(i,j)=a(i,k)+a(k,j); 

           path(i,j)=k; 

        end 

     end 

  end 

end 

a, path

    我們使用LINGO9.0編寫的FLOYD演算法如下：

model: 

sets: nodes/c1..c6/; 

link(nodes,nodes):w,path;  !path標誌最短路徑上走過的頂點; 

endsets 

data: 

path=0; 

w=0; 

@text(mydata1.txt)[email protected](nodes(i):@writefor(nodes(j):

@format(w(i,j),' 10.0f')),@newline(1)); 

@text(mydata1.txt)[email protected](@newline(1)); 

@text(mydata1.txt)[email protected](nodes(i):@writefor(nodes(j):

@format(path(i,j),' 10.0f')),@newline(1)); 

enddata 

calc: 

w(1,2)=50;w(1,4)=40;w(1,5)=25;w(1,6)=10; 

w(2,3)=15;w(2,4)=20;w(2,6)=25; 

w(3,4)=10;w(3,5)=20; 

w(4,5)=10;w(4,6)=25;w(5,6)=55; 

@for(link(i,j):w(i,j)=w(i,j)+w(j,i)); 

@for(link(i,j) |i#ne#j:w(i,j)[email protected](w(i,j)#eq#0,10000,w(i,j)));

@for(nodes(k):@for(nodes(i):@for(nodes(j): 

[email protected](w(i,j),w(i,k)+w(k,j)); 

path(i,j)[email protected](w(i,j)#gt#tm,k,path(i,j));w(i,j)=tm)));  

endcalc 

end