Problems with Police Cars and Pricing
26th February 2016
In the last week, STOR-i has put on two Probelm Solving Days. These are opportunities for all of the STOR-i students
to spend a day on a problem faced by a business or industry. The problems can be quite varied in topic and we have had
various companies come to ask for some ideas on how to tackle the issue. As it is only a few hours long, we are rarely
expected to have solved the problem and it is more about considering methods and approaches that could pave the way
towards a solution. The guest companies have ranged from EDF to West Yorkshire Police Service. Towards the end of the
day, the groups come back together to present their ideas. It is important that when presenting the ideas, they are
accessible to the people we are resenting to
Last Friday, it was West Yorkshire Police who came with their
problem of how to decide how many Police cars, and of what
type, should their Force have as part of their fleet. The fleet managers explained to us that a large proportion of the
time, Police cars are parked up at the yard. In these times of austerity, the Police want to make sure that they make the
most of any assets they have, and so they were wondering how many cars they actually needed. The data given to us
included the maximum number of cars out at any time per day over three months.
After we had been given the problem, we split into groups and began to think of solutions. I would like to describe some
of these. The majority of the groups went about this problem in terms of how many marked cars would the Police need to cover a
day that occurred once every $N$ years. This is when extreme value theory (as discussed in Tails,
Droughts and Extremes) come into play. As the data is the maximum values every day, the approximation was made that
each day was independent and identically distributed. After this, the Generalised Extreme Value distribution was fitted
and estimates for the number of cars needed to cover a day that occurred with probability $p$. These showed that, even for
days that occur once every 100 years, the 95% confidence intervals for the number of cars was well below the current number
of cars! This suggests that the West Yorkshire Police could reduce the fleet size of marked cars by quite a lot.
The approximation made was pointed out by Professor Jon Tawn, and he said
he was surprised that it would hold. That at least made me feel a little better about not using it in our group.
Another group took quite a different approach indeed. They applied queueing theory to the situation. For this, they used the
number of incidents a day and as a Poisson process with a certain probability of the incident being classed as an emergency,
priority or standard. Each of the classes has a different expected number of cars going out to an incident and so would take
up a certain number of 鈥渟ervers鈥�. The idea was to use an infinite server queue, which will always have enough cars to cope
with the demand, and to compare this with a system with $C$ cars. Once the number of cars $C$ was big enough to be a good
approximation to the infinite server queue, this $C$ was taken as the number of cars required by the Police. This method
also suggested that the number of cars could be reduced considerably. More surprisingly though, it actually produced a number
not dissimilar to that of the Extreme value theory method.