## Modelling Average Unit Maintenance Cost

A certain factory uses crates to pack bottles of soda pop. The condition of crates falls into one of four categories - good, fair, poor, broken. If a crate is broken it must be repaired, which costs £2.50, takes a week, and costs £1.85 in lost production. The transition matrix is:
 From\To Good Fair Poor Broken Good 0 0.8 0.2 0 Fair 0 0.6 0.4 0 Poor 0 0 0.5 0.5 4 1 0 0 0
This means for example that there is a probability of 0.8 that a crate will move from the 'good' category to the category 'fair' from one week to the next.
The transition matrix is
$A= \left( \begin{array}{cccc} 0 & 0.8 & 0.2 & 0 \\ 0 & 0.6 & 0.4 & 0 \\ 0 & 0 & 0.5 & 0.5 \\ 1 & 0 & 0 & 0 \end{array} \right)$

Now use, for a smoothly running operation
$\mathbf{x}B=\mathbf{x}$
to get
$x_1=x_4$

$x_2=0.8x_1+0.6x_2$

$x_3=0.2x_1+0.4x_2+0.5x_3$

$x_4=0.5x_3$

Where
$x_1, \; x_2, \; x_3, \; x_4$
are the probabilities of a randomly selected box being in the category good, fair, poor or broken.
Solving these equations gives
$x_1=1/6, \; x_2=1/3, \; x_3=1/3, \; x_4=1/6$
.
The average weekly cost of maintaining a crate and lost production is is
$1/6(2.50+1.85)=£0.725$
. 