A small (‘cut price’) car rental company, renting one type of car, has depots in Glasgow, Manchester, Birmingham and Plymouth.[26] There is an estimated demand for each day of the week except Sunday when the company is closed. These estimates are given in Table 25.1. It is not necessary to meet all demand.
Table 25.1:
|
Glasgow |
Manchester |
Birmingham |
Plymouth |
|
|---|---|---|---|---|
|
Monday |
100 |
250 |
95 |
160 |
|
Tuesday |
150 |
143 |
195 |
99 |
|
Wednesday |
135 |
80 |
242 |
55 |
|
Thursday |
83 |
225 |
111 |
96 |
|
Friday |
120 |
210 |
70 |
115 |
|
Saturday |
230 |
98 |
124 |
80 |
Cars can be rented for one, two or three days and returned to either the depot from which rented or another depot at the start of the next morning. For example, a 2-day rental on Thursday means that the car has to be returned on Saturday morning; a 3-day rental on Friday means that the car has to be returned on Tuesday morning. A 1-day rental on Saturday means that the car has to be returned on Monday morning and a 2-day rental on Tuesday morning.
Table 25.2:
|
From |
To |
|||
|---|---|---|---|---|
|
Glasgow |
Manchester |
Birmingham |
Plymouth |
|
|
Glasgow |
60 |
20 |
10 |
10 |
|
Manchester |
15 |
55 |
25 |
5 |
|
Birmingham |
15 |
20 |
54 |
11 |
|
Plymouth |
8 |
12 |
27 |
53 |
Table 25.3:
|
From |
To |
|||
|---|---|---|---|---|
|
Glasgow |
Manchester |
Birmingham |
Plymouth |
|
|
Glasgow |
— |
20 |
30 |
50 |
|
Manchester |
20 |
— |
15 |
35 |
|
Birmingham |
30 |
15 |
— |
25 |
|
Plymouth |
50 |
35 |
25 |
— |
The rental period is independent of the origin and destination. From past data, the company knows the distribution of rental periods: 55% of cars are hired for one day, 20% for two days and 25% for three days. The current estimates of percentages of cars hired from each depot and returned to a given depot (independent of day) are given in Table 25.2.
The marginal cost, to the company, of renting out a car (‘wear and tear’, administration etc.) is estimated as follows:
|
1-Day hire |
£20 |
|
2-Day hire |
£25 |
|
3-Day hire |
£30 |
The ‘opportunity cost’ (interest on capital, storage, servicing, etc.) of owning a car is £15 per week.
It is possible to transfer undamaged cars from one depot to another depot, irrespective of distance. Cars cannot be rented out during the day in which they are transferred. The costs (£), per car, of transfer are given in Table 25.3.
Ten percent of cars returned by customers are damaged. When this happens, the customer is charged an excess of £100 (irrespective of the amount of damage that the company completely covers by its insurance). In addition, the car has to be transferred to a repair depot, where it will be repaired the following day. The cost of transferring a damaged car is the same as transferring an undamaged one (except when the repair depot is the current depot, when it is zero). Again the transfer of a damaged car takes a day, unless it is already at a repair depot. Having arrived at a repair depot, all types of repair (or replacement) take a day.
Only two of the depots have repair capacity. These are (cars/day) as follows:
|
Manchester |
12 |
|
Birmingham |
20 |
Having been repaired, the car is available for rental at the depot the next day or may be transferred to another depot (taking a day). Thus, a car that is returned damaged on a Wednesday morning is transferred to a repair depot (if not the current depot) during Wednesday, repaired on Thursday and is available for hire at the repair depot on Friday morning.
The rental price depends on the number of days for which the car is hired and whether it is returned to the same depot or not. The prices are given in Table 25.4 (in £).
Table 25.4:
|
Return to |
Return to |
|
|---|---|---|
|
Same Depot |
Another Depot |
|
|
1-Day hire |
50 |
70 |
|
2-Day hire |
70 |
100 |
|
3-Day hire |
120 |
150 |
There is a discount of £20 for hiring on a Saturday so long as the car is returned on Monday morning. This is regarded as a 1-day hire.
For simplicity, we assume the following at the beginning of each day:
Customers return cars that are due that day
Damaged cars are sent to the repair depot
Cars that were transferred from other depots arrive
Transfers are sent out
Cars are rented out
If it is a repair depot, then the repaired cars are available for rental.
In order to maximise weekly profit, the company wants a ‘steady state’ solution in which the same expected number will be located at the same depot on the same day of subsequent weeks.
How many cars should the company own and where should they be located at the start of each day?
This is a case where the integrality of the cars is not worth modelling. Rounded fractional solutions are acceptable.