I am required to send a request to a vendor for some pipes. They come in either 3m or 4m of length. I have a list of sections of length which are not really related to 3 or 6. However, I am required to find the number of 3m and 4m pipes each to minimize wastage.
Dimensions in mm. 1068.67, 630.8, 311.84, 400, 711.4, 440, 777.27, 502.49, 400, 528.9, 203.09, 121.85, 284.34, 700, 400
How do I solve this problem?
Let $L$ be the set of $k$ required lengths $L = \{ 1068.67, 630.8, \dots, \}$. We denote every required length in $L$ as $l_1 = 1068.67$ , $l_2 = 630.8$, etc. $\dots$$l_k$.
Now we define a binary variable $x_{l_i}, 1 \le i \le k$ to be 1 if length $l_i$ is assigned to the 3m unit, 0 otherwise. Similarly, we define $y_{l_i}, 1 \le i \le k$ to be 1 if length $l_i$ is assigned to the 4m unit, 0 otherwise.
Now, we define a wastage variable $z_{3}$ and $z_{4}$ as the wastage variable for each of the 3m and 4m lengths. We then write the constraint $z_{3} \ge 3000 - 1068.67 x_{l_1} - 630.8 x_{l_2} - \dots$. Similarly, we write $z_{4} \ge 4000 - 1068.67 y_{l_1} - 630.8 y_{l_2} - \dots$.
So, we write our optimization model as
$\min z_{3} + z_{4}$
subject to
$z_{3} \ge 3000 - 1068.67 x_{l_1} - 630.8 x_{l_2} - \dots$
$z_{4} \ge 4000 - 1068.67 y_{l_1} - 630.8 y_{l_2} - \dots$
$x_{l_{i}} + y_{l_{i}} = 1,\ 1 \le i \le k$
$x_{l_i}, y_{l_i} \in { 0,1 }.