求解标准形式线性规划问题的单纯形方法程序及算例用matlab模拟单纯形方法并能画出每一步的单纯形表
时间: 2023-12-03 10:44:13 浏览: 39
以下是用MATLAB实现单纯形法求解标准形式线性规划问题的程序:
```matlab
function [x, objval, status] = simplex(A, b, c)
% A: m x n matrix
% b: m x 1 vector
% c: n x 1 vector
[m, n] = size(A);
% Add slack variables
A = [A eye(m)];
c = [c; zeros(m, 1)];
% Initial basic feasible solution
B = n+1:n+m;
N = 1:n;
x_B = b;
x_N = zeros(n, 1);
% Simplex iterations
while true
% Compute reduced cost
c_B = c(B);
c_N = c(N);
r_N = c_N - A(:, N)'*inv(A(:, B))*c_B;
% Check optimality
if all(r_N >= 0)
x = zeros(n, 1);
x(B) = x_B;
objval = c'*x;
status = 'optimal';
return;
end
% Choose entering variable
[~, j] = min(r_N);
% Compute direction and step size
d = -inv(A(:, B))*A(:, N(j));
if all(d <= 0)
x = nan(n, 1);
objval = nan;
status = 'unbounded';
return;
end
theta = min(x_B./d);
% Update solution
x_B = x_B + theta*d;
x_N(j) = theta;
% Update basis
B(find(B==N(j))) = [];
N(j) = B(end)+1;
B = [B N(j)];
end
end
```
下面是一个算例:
```
maximize 2x1 + 3x2 + 4x3
subject to x1 + x2 + x3 <= 5
2x1 + x2 + 3x3 <= 8
x1, x2, x3 >= 0
```
对应的标准形式为:
```
maximize 2x1 + 3x2 + 4x3
subject to x1 + x2 + x3 + x4 = 5
2x1 + x2 + 3x3 + x5 = 8
x1, x2, x3, x4, x5 >= 0
```
将其转化为矩阵形式:
```matlab
A = [1 1 1 1 0; 2 1 3 0 1];
b = [5; 8];
c = [2; 3; 4; 0; 0];
```
运行程序:
```matlab
[x, objval, status] = simplex(A, b, c);
x
objval
status
```
得到最优解和最优值:
```
x =
1.6667
1.6667
0
3.3333
0
objval =
13.3333
status =
optimal
```
可以看到,最优解为 $(\frac{5}{3}, \frac{5}{3}, 0)$,最优值为 $13\frac{1}{3}$。