please I am good in linear algebra, but I dont know model this problem:
A company requires large quantities of iron and coal to produce steel. Three HC, FC, IL suppliers will be purchased over the next three months. The following shows the quantities of tons of iron and coal that will be purchased and the costs in dollars per ton of each supplier. What is the most profitable provider?
Thanks.
Each supplier has different costs for iron and coal, which can be represented by this matrix: $$ \begin{bmatrix} 450&550&650 \\ 700&650&600 \end{bmatrix} $$ For each month, there is a certain number of tons needed of iron and coal. This is represented by: $$ \begin{bmatrix} 9&8 \\ 5&7 \\ 6&4\end{bmatrix} $$ By multiplying these matrices together, we obtain a matrix showing the different costs for each month for each supplier. $$ \begin{bmatrix} 9&8 \\ 5&7 \\ 6&4\end{bmatrix}\begin{bmatrix} 450&550&650 \\ 700&650&600 \end{bmatrix}=\begin{bmatrix} 450\times9+700\times8&550\times9+650\times8&650\times9+600\times8 \\ 450\times5+700\times7&550\times5+650\times7&650\times5+600\times7 \\ 450\times6+700\times4&550\times6+650\times4&650\times6+600\times4\end{bmatrix} $$ Evaluating this gives: $$ \begin{bmatrix} 9650&10150&10650\\7150&7300&7450\\5500&5900&6300\end{bmatrix} $$ By taking the sum of each column, we find the total cost for each supplier. $$ 9650+7150+5500=22300\\ 10150+7300+5900=23350\\ 10650+7450+6300=24400 $$ This shows that supplier HC is the cheapest, followed by FC, and IL is the most expensive supplier.
The matrix multiplication takes the values for each supplier, multiplies them by the amount of materials needed, and shows the cost for each month for each supplier. By adding together the costs for each month, we find the total cost for each supplier.