A paper mill manufactures its paper in standard reels 6 metres wide and cuts these to the widths required by its customers. This may result in some waste, which must be discarded as there is no room to store it.
It has received the following orders:
- 2.7m wide, 30 reels ordered
- 2.1m wide, 150 reels ordered
- 1.65m wide, 65 reels ordered
My working out;
To begin this problem I wanted to find out how many different options there were to cut the reels in order to minimise wastage.
Ways denoted by pi, Width (in metres), Wastage
p1 3*1.65=4.95, wastage = 1.05
p2 2*1.65 + 2.1=5.4 wastage= 0.6
p3 1.65 + 2*2.1=5.85 wastage= 0.15
p4 2.1 + 2.7=4.8 wastage= 1.2
p5 2*2.7=5.4 wastage= 0.6
p6 2.7 + 2*1.65=6 wastage= 0
Decision variables:
Let x1 be the number of times we use p1, x2 be the number of times we use p2, x3 be the number of times we use p3 and so on.
Constraints:
I don't think my constraints are right? please can some one help me
$3x_1 + 2x_2+ x_3 + 2x_6 = 65$ (to satisfy order of 1.65m)
$x_2 + 2x_3 + x_4 = 150$ (to satisfy order of 2.1m)
$x_4 + 2x_5 + x_6 = 30$ (to satisfy order of 2.7m)
$x_i \geq 0 \ \ \forall \ i=1,..,6$
Objective function:
We want to minimise waste so;
$$Min \ \ Z= 1.05x_1 + 0.6x_2 + 0.15x_3 + 1.2x_4 + 0.6 x_5$$
To solve this problem I used the Solver function on Excel.
Best Answer
Use the variables of the options you´ve introduced: $p_1,p_2,p_3,p_4,p_6$
Then we want to minimize the waste:
$$\texttt{Min} \ = 1.05\cdot p_1 + 0.6\cdot p_2 + 0.15\cdot p_3 + 1.2\cdot p_4 + 0.6\cdot p_5$$
I just changed the notation from $x_i$ to $p_i$.
I summarize your options in a table:
$$\begin{array}{|c|c|c|c|} \hline & 2.7 & 2.1 & 1.65 & \texttt{waste} \\ \hline p_1 &0&0&3&1.05 \\ \hline p_2&0&1&2&0.6 \\ \hline p_3 &0 &2&1 &0.15 \\ \hline p_4 &1&1&0&1.2 \\ \hline p_5&2&0&0&0.6 \\ \hline p_6& 1&0&2& 0 \\ \hline \end{array}$$
Now we have the constraint that we need at least $30$ reels with a width of $2.7 \ m$. Just read off the coefficient of the column which is lables as $2.7$
$1p_4+2p_5+p_6\geq 30$
Similar for the constraints of reels with the widths $2.1$ and $1.65$
$p_2+2p_3+1p_4\geq 150$
$ 3p_1+2p_2+1p_3+2p_6\geq 65 $
Finally the domain of the variables: $p_i\in \mathbb N_0 \ \forall i=1,2,...,6$