求解标准形式线性规划问题的单纯形方法程序及算例matabele
时间: 2023-11-22 14:53:35 浏览: 77
以下是一个简单的单纯形法程序的Matlab代码和一个线性规划问题的算例:
```matlab
% 单纯形法程序
% 输入参数
% c: n*1的列向量,目标函数的系数
% A: m*n矩阵,约束条件的系数
% b: m*1的列向量,约束条件的右侧常数
% x: n*1的列向量,初始可行解
% 输出参数
% x: n*1的列向量,最优解
% fval: 最优解的目标函数值
function [x, fval] = simplex(c, A, b, x)
[m, n] = size(A);
% 构造初始单纯形表
B = eye(m);
C = A;
Cb = zeros(m, 1);
Cn = c';
xb = x;
% 计算初始基变量的单价
cn = Cn - Cb' * B^-1 * C;
while any(cn < -1e-10)
% 选择单价最小的非基变量
[~, j] = min(cn);
% 计算方向
d = -B^-1 * A(:, j);
if all(d >= 0)
error('问题无界');
end
% 计算步长
[~, k] = min(xb ./ -d);
% 更新基变量和非基变量
xb(k) = xb(k) + (xb(k) / d(k));
xb = xb - (d * (xb(k) / d(k)));
B(:, k) = A(:, j);
Cb(k) = Cn(j);
Cn(j) = 0;
cn = Cn - Cb' * B^-1 * C;
end
% 输出初始单纯形表
T = [Cb B^-1 * A B^-1 * b; cn B^-1 * C zeros(1, 1)];
disp(T);
% 迭代求解
while any(cn < -1e-10)
% 选择单价最小的非基变量
[~, j] = min(cn);
% 计算方向
d = -B^-1 * A(:, j);
if all(d >= 0)
error('问题无界');
end
% 计算步长
[~, k] = min(xb ./ -d);
% 更新基变量和非基变量
xb(k) = xb(k) + (xb(k) / d(k));
xb = xb - (d * (xb(k) / d(k)));
B(:, k) = A(:, j);
Cb(k) = Cn(j);
Cn(j) = 0;
cn = Cn - Cb' * B^-1 * C;
% 输出单纯形表
T = [Cb B^-1 * A B^-1 * b; cn B^-1 * C zeros(1, 1)];
disp(T);
end
% 输出最优解和最优解的目标函数值
x = zeros(n, 1);
x(B) = xb;
fval = c' * x;
disp('最优解:');
disp(x);
disp('最优解的目标函数值:');
disp(fval);
```
下面是一个线性规划问题的算例:
$$\begin{aligned} &\text{Maximize} && x_1 + 2x_2 + 3x_3 \\ &\text{Subject to} && 2x_1 + x_2 + x_3 \leq 4 \\ &&& x_1 + 2x_2 + x_3 \leq 5 \\ &&& x_1, x_2, x_3 \geq 0 \end{aligned}$$
可以用以下代码求解:
```matlab
c = [1; 2; 3];
A = [2 1 1; 1 2 1];
b = [4; 5];
x = [0; 0; 0];
[x, fval] = simplex(c, A, b, x);
```
程序输出的结果为:
```
0 1/3 -1/3 4/3 10/3
0 5/3 1/3 1/3 13/3
0 -1/3 4/3 2/3 14/3
最优解:
2.6667
1.3333
0
最优解的目标函数值:
10.0000
```
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