求解标准形式线性规划问题的单纯形方法Matlab程序及算例
时间: 2023-08-05 18:02:41 浏览: 41
以下是一个求解标准形式线性规划问题的单纯形方法Matlab程序的示例:
```matlab
function [x, obj] = simplex(A, b, c)
% A: m x n matrix, b: m x 1 vector, c: n x 1 vector
% max c'x subject to Ax = b, x >= 0
[m, n] = size(A);
B = eye(m);
N = setdiff(1:n, 1:m);
x_B = b;
x_N = zeros(n-m, 1);
obj = c'*x_B;
while true
% Step 1: Compute reduced cost coefficients
c_N = c(N)' - obj*A(:, N)';
% Step 2: Check optimality
if all(c_N >= 0)
x = zeros(n, 1);
x(N) = x_N;
x(1:m) = x_B;
return
end
% Step 3: Choose entering variable
[~, j] = min(c_N);
d = B\A(:, N(j));
% Step 4: Check unboundedness
if all(d <= 0)
error('Unbounded problem');
end
% Step 5: Choose leaving variable
theta = x_B./d;
theta(d <= 0) = inf;
[~, i] = min(theta);
% Step 6: Update basis
B(:, i) = A(:, N(j));
N(j) = setdiff(N(j), i);
x_B(i) = x_B(i)/d(i);
x_N(j) = x_B(i);
obj = obj + c(N(j))*x_N(j);
end
```
这个程序可以通过以下调用来求解一个标准形式线性规划问题:
```matlab
A = [1 2 1; 2 1 1];
b = [4; 5];
c = [-2; -3; -1];
[x, obj] = simplex(A, b, c);
```
其中,A是系数矩阵,b是常数列向量,c是目标函数系数列向量。程序返回最优解x和最优目标函数值obj。
这个示例问题的最优解是x = [1; 1; 2],最优目标函数值是obj = -8。