Matlab Assignment 1

Lecture 4&5 - Linear Programming (LP) TTK4135 - Optimalisering og regulering

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We have the problem

$$\min J_{raw}+J_{prod},s.t.\{\begin{array}{l}3x_1+x_2+x_3\le 60\\ x_1+x_2+x_3+x_4=100\\ x_j\ge 0,j=1,\dots ,4\end{array}$$

Where

$$\begin{array}{rl}{J}_{raw}& =800(3x_1+x_2+x_3)+200(4x_2+8x_4)+100(12x_3)\\ J_{prod}& =10x_1+30x_2+50x_3+100x_4\end{array}$$

To use linear programming we need to put it on the form

Link to original

To use the linprog function in matlab, we need it on the following form

$$\underset{x\in {\mathbb{R}}^n}{\min }{c}^{T}x,s.t.\{\begin{array}{l}Ax\le b\\ A_{eq}x=b_{eq}\\ lb\le x\le ub\end{array}$$

Where

$$\begin{array}{rl}A& =\left[\begin{array}{cccc}3& 1& 1& 0\end{array}\right]\\ b& =60\\ A_{eq}& =\left[\begin{array}{cccc}1& 1& 1& 1\end{array}\right]\\ b_{eq}& =100\\ lb& =\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]\\ ub& =\left[\begin{array}{cccc}\infty & \infty & \infty & \infty \end{array}\right]\\ c& =\left[\begin{array}{cccc}2410& 1630& 2050& 1700\end{array}\right]\end{array}$$

The following matlab script solves this

% Define the matrices needed for linprog
c = [2410 1630 2050 1700];
A = [3 1 1 0];
b = [60];
Aeq = [1 1 1 1];
beq = [100];
lb = [0 0 0 0]'; % No negative production
ub = [inf inf inf inf]'; % No upper limits on variables
 
% Solve the optimization problem
[x_opt, J_opt] = linprog(c, A, b, Aeq, beq, lb, ub);