# OR-Notes

## J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

#### Markov processes tutorial class question

Ford are currently investigating the behaviour of car fleet buyers in switching between companies. Preliminary investigations have revealed that fleet buyers usually have a number of companies from whom they buy, but that they tend to keep in mind a target percentage to be bought from each company. Each year, as new models are announced, these buyers reassess their target percentage for each company.

Market research indicates that, for the purpose of preliminary analysis, a fleet buyer's target percentage for Ford cars can be regarded as being one of 100%, 70%, 50% or 20%.

An in-depth study of previous buying behaviour has produced the transition matrix shown below for the probability of switching each year between target percentages:

```                       To target %
100  70   50   20
From target % 100 | 0.6  0.3  0.1  -   |
70 | -    0.7  0.3  -   |
50 | -    0.4  0.4  0.2 |
20 | -    0.2  0.5  0.3 |```

The current situation is that, for every 100 fleet buyers, 5 have a target percentage for Ford cars of 100%, 30 a target percentage of 70%, 45 a target percentage of 50% and 20 a target percentage of 20%.

• draw the state-transition diagram
• what will be the percentage of fleet buyers having a target percentage for Ford cars of 50%
• in 2 years time; and
• in the long-run?
• would you expect the actual percentage of fleet buyers having a target percentage for Ford cars of 50% to approach the long-run figure calculated above or not (and why)?
• what advantages and disadvantages can you think of in using Markov theory to forecast fleet buyers behaviour in this way?