Cutting bars problem with integer programming.

Question

How to model the Linear Programming for the problem below with the most complete + reasonable constraints.
A production facility has 2 types of reinforcement bars 6m, 8m long (unlimited quantity). Need to process 100 2.4m and 150 2.8m sections. Ask how to cut rebar to save the most?
This is a practice homework in my school. It bother me for a while. I don't know how to set variables. I set number of 2.4m bar, 2.8m bar in 6m bar is x, y; 2.4m bar, 2.8m bar in 8m bar is a, b; then number of 6m, 8m bar is n1, n2. Then one of constraints is $n_1x+n_2a\leq100$. That is non-linear so I can't continue solving with integer programming. Can someone help? Thank you.

Answer 1

How can we cut the bars to 2.4 and 2.8 pieces? Let $x_i$ and $y_i$ be the different cuts as follows

$$\begin{aligned} x_1 &: 6 = 2.4+2.4+\color{red}{1.2} \\ x_2 &: 6 = 2.4+2.8+\color{red}{0.8} \\ x_3 &: 6 = 2.8+2.8+\color{red}{0.4} \\ y_1 &: 8 = 2.4+2.4+2.4+\color{red}{0.8} \\ y_2 &: 8 = 2.4+2.4+2.8+\color{red}{0.4} \\ y_3 &: 8 = 2.4+2.8+2.8 \\ \end{aligned}$$

So we should minimize the number of bars used.

\begin{align} \min \quad& z= x_1+x_2+x_3+y_1+y_2+y_3 \label{z1} \\ \text{s.t.} \quad& 2x_1+x_2+3y_1+2y_2+y_2 \ge 100 \tag{1}\label{c1} \\ & x_2+2x_3+y_2+2y_3 \ge 150 \tag{2}\label{c2} \\ & x_i, y_j \geq 0 \space \text{(int)}. \tag{3}\label{c3} \end{align}