Four departments of a company need to order the following amounts of the same products.
\begin{tabular}{|c|c|c|c|c|c|}
\hline Department & Paper & Tape & Binders & \begin{tabular}{c}
Memo \nPads \end{tabular} & Pens \\
\hline 1 & 0 & 4 & 3 & 5 & 6 \\
\hline 2 & 6 & 3 & 2 & 2 & 8 \\
\hline 3 & 5 & 6 & 2 & 0 & 0 \\
\hline 4 & 0 & 2 & 4 & 0 & 6 \\
\hline
\end{tabular}
The unit price (in dollars) of each product is given below for two suppliers.
\begin{tabular}{|l|c|c|}
\hline & Supplier A & Supplier B \\
\hline Paper & 3 & 4 \\
\hline Tape & 1 & 1 \\
\hline Binders & 5 & 4 \\
\hline Memo & 3 & 3 \\
\hline Pads & & 1 \\
\hline Pens & 1 & 1 \\
\hline
\end{tabular}
(a) Use matrix multiplication to get a matrix showing the comparative costs for each department for the products from the two suppliers.
$\left.\begin{array}{lcc}
& \text { A } & \text { B } \\
\text { Dept. 1 } & \square & \square \\
\text { Dept. 2 } & \square & \square \\
\text { Dept. 3 } & \square & \square \\
\text { Dept. 4 } & \square & \square \\
\square & \square & \square
\end{array}\right]$
(b) Find the total cost over all departments to buy products from each supplier. From which supplier should the company make the purchase?
A. The total cost for supplier $A$ is $$\square$ and the total cost for supplier B is $$\square$ , so the company should make the purchase from supplier B.
B. The total cost for supplier A is $$\square$ and the total cost for supplier B is $$\square$ . so the company should make the purchase from supplier A.
Answer (1)
Define matrix Q for quantities and matrix P for prices.
Perform matrix multiplication C = QP to find costs per department.
Calculate total cost for each supplier by summing costs across departments.
Compare total costs to determine the cheaper supplier: The total cost for both suppliers is $144 .
Explanation
Understanding the Problem We are given the quantities of five products (Paper, Tape, Binders, Memo Pads, and Pens) needed by four departments. We also have the unit prices of each product from two suppliers, A and B. The goal is to use matrix multiplication to find the comparative costs for each department from the two suppliers and then determine the total cost for each supplier to identify the cheaper option.
Defining the Matrices Let's represent the quantities of products ordered by each department as a matrix Q , where rows represent departments and columns represent products:
Q = [ 0 4 3 5 6 6 3 2 2 8 5 6 2 0 0 0 2 4 0 6 ]
And the unit prices of each product from each supplier as a matrix P , where rows represent products and columns represent suppliers:
P = [ 3 4 1 1 5 4 3 3 1 1 ]
Calculating the Comparative Costs Now, we perform matrix multiplication C = QP to find the comparative costs for each department from each supplier. The element C ij represents the cost for department i from supplier j .
C = [ 0 4 3 5 6 6 3 2 2 8 5 6 2 0 0 0 2 4 0 6 ] [ 3 4 1 1 5 4 3 3 1 1 ] = [ 0 ( 3 ) + 4 ( 1 ) + 3 ( 5 ) + 5 ( 3 ) + 6 ( 1 ) 0 ( 4 ) + 4 ( 1 ) + 3 ( 4 ) + 5 ( 3 ) + 6 ( 1 ) 6 ( 3 ) + 3 ( 1 ) + 2 ( 5 ) + 2 ( 3 ) + 8 ( 1 ) 6 ( 4 ) + 3 ( 1 ) + 2 ( 4 ) + 2 ( 3 ) + 8 ( 1 ) 5 ( 3 ) + 6 ( 1 ) + 2 ( 5 ) + 0 ( 3 ) + 0 ( 1 ) 5 ( 4 ) + 6 ( 1 ) + 2 ( 4 ) + 0 ( 3 ) + 0 ( 1 ) 0 ( 3 ) + 2 ( 1 ) + 4 ( 5 ) + 0 ( 3 ) + 6 ( 1 ) 0 ( 4 ) + 2 ( 1 ) + 4 ( 4 ) + 0 ( 3 ) + 6 ( 1 ) ]
C = [ 0 + 4 + 15 + 15 + 6 0 + 4 + 12 + 15 + 6 18 + 3 + 10 + 6 + 8 24 + 3 + 8 + 6 + 8 15 + 6 + 10 + 0 + 0 20 + 6 + 8 + 0 + 0 0 + 2 + 20 + 0 + 6 0 + 2 + 16 + 0 + 6 ] = [ 40 37 45 49 31 34 28 24 ]
Comparative Costs Matrix The matrix C shows the comparative costs for each department from the two suppliers:
A B Dept. 1 40 37 Dept. 2 45 49 Dept. 3 31 34 Dept. 4 28 24
Calculating Total Costs To find the total cost for each supplier, we sum the costs for all departments from that supplier:
Total cost for supplier A = 40 + 45 + 31 + 28 = 144 Total cost for supplier B = 37 + 49 + 34 + 24 = 144
Final Decision Comparing the total costs, we find that the total cost for supplier A is $144 and the total cost for supplier B is $144 . Since the total costs are the same, the company can choose either supplier.
Final Answer The total cost for supplier A is $144 and the total cost for supplier B is $144 , so the company can make the purchase from either supplier.
Examples
Matrix multiplication is a fundamental tool in various fields, including economics and business. For instance, consider a company that produces multiple products and sells them in different regions. Using matrices, the company can efficiently calculate total revenue, costs, and profits across all regions and products. This approach simplifies complex calculations and provides a clear overview of the company's financial performance, aiding in strategic decision-making and resource allocation. By organizing data in matrices and performing matrix operations, businesses can gain valuable insights into their operations and optimize their strategies for maximum profitability.
