\[P_1, \: P_2\]
and three service departments \[S_1, \: S_2, |; S_3\]
. The monthly costs \[x_1, \: x_2, \: x_3, \: x_4, \: x_5 \]
of each department are unknown, and the direct monthly cost of each department are shown in the third column. The fourth, fifth and sixth columns shown the allocation of charges for the services of \[S_1, \: S_2, \: S_3\]
to the various departments. The problem is to find the total cost for each department.| Department | Total Cost | Direct Cost | \[S_1\] |
\[S_2\] |
\[S_3\] |
\[S_1\] |
\[x_1\] |
600 | 0.25 \[x_1\] |
0.15 \[x_2\] |
0.15 \[x_3\] |
\[S_2\] |
\[x_2\] |
1100 | 0.35 \[x_1\] |
0.20 \[x_2\] |
0.25 \[x_3\] |
\[S_3\] |
\[x_3\] |
600 | 0.10 \[x_1\] |
0.10 \[x_2\] |
0.35 \[x_3\] |
\[P_1\] |
\[x_4\] |
2100 | 0.15 \[x_1\] |
0.25 \[x_2\] |
0.15 \[x_3\] |
\[P_2\] |
\[x_5\] |
1500 | 0.15 \[x_1\] |
0.30 \[x_2\] |
0.10 \[x_3\] |
| Total | \[x_1\] |
\[x_2\] |
\[x_3\] |
\[x_1=600+0.25x_1+0.15x_2+0.15x_3\]
\[x_2=1100+0.35x_1+0.20x_2+0.25x_3\]
\[x_3=600+0.10x_1+0.10x_2+0.35x_3\]
We can write this as
\[\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= \begin{pmatrix}600\\1100\\600\end{pmatrix}+ \left( \begin{array}{ccc} 0.25 & 0.15 & 0.15 \\ 0.35 & 0.20 & 0.25 \\ 0.10 & 0.10 & 0.35 \end{array} \right) \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\]
, or\[(\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) - \left( \begin{array}{ccc} 0.25 & 0.15 & 0.15 \\ 0.35 & 0.20 & 0.25 \\ 0.10 & 0.10 & 0.35 \end{array} \right)) \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= \begin{pmatrix}600\\1100\\600\end{pmatrix} \]
Hence
\[ \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}={(\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) - \left( \begin{array}{ccc} 0.25 & 0.15 & 0.15 \\ 0.35 & 0.20 & 0.25 \\ 0.10 & 0.10 & 0.35 \end{array} \right) )}^{-1} \begin{pmatrix}600\\1100\\600\end{pmatrix}=\begin{pmatrix}1638\\2569\\1562\end{pmatrix} \]
with these values the table becomes
| Department | Total Cost | Direct Cost | \[S_1\] |
\[S_2\] |
\[S_3\] |
\[S_1\] |
1638 | 600 | 409.50 | 385.35 | 234.30 |
\[S_2\] |
2569 | 1100 | 573.30 | 523.80 | 390.50 |
\[S_3\] |
1562 | 600 | 163.80 | 256.90 | 546.70 |
\[P_1\] |
\[322.25\] |
2100 | 245.70 | 542.25 | 234.30 |
\[P_2\] |
\[2672.60\] |
1500 | 245.70 | 770.70 | 156.20 |