1. Simulation An Example

2. A garage hires out cars. It currently owns four cars which are hired out on a daily basis for a number of days. The demand pattern in the past has been as follows Number of cars 01234 Demand per day probability 0.150.250.300.200.10

3. The length of rental is shown below Length of rental (days) 123 Probability0.600.300.10

4. The company charges £40 a day for hire. It estimates that the total fixed overheads attributed to this operation are £10 a day. If the car is hired out the extra repair and maintenance costs generated are estimated at £6 a car a day.

5. If the demand for hired cars exceeds the current number of four cars they then hire from another garage at additional cost of £24 a day. Carry out a simulation to forecast the overall demand pattern for car hire and estimate the annual contribution to overheads and profits from the car hire business.

6. The first thing to do is to construct cumulative frequency tables to decide which random numbers refer to a particular scenario. Number of cars 01234 Demand per day probability 0.150.250.300.200.10 Cumulative probability 0.150.400.700.901.00 Random numbers 1-1516-4041-7071-9091-00

7. This means, for example, that a random number of 17 would interpret as a demand of 1 on that particular day, whilst a value of 95 would imply a demand of 4. The same process is carried out for length of hire.

8. The length of rental is shown below Length of rental (days) 123 Probability0.600.300.10 Cumulative probability 0.600.901.00 Random Numbers 1-6061-9091-00

9. We will use the following random numbers (from a book of tables) 84 42 56 73 87 75 18 91 76 66 64 83 97 11 69 41 80 92 38 75 28 87 77 03 57 09 85 86 46 86 40 15 31 81 78 91 30 22 88 58 Generate 10 days of business

10. DayDemand Total Demand Hire Car Number 1 2 3 4 Other Garage 5 6 7 8 1 2 3 4 5

11. Day 1 Random Number is 84 This means a demand of 3

12. DayDemand Total Demand Hire Car Number 1 2 3 4 Other Garage 5 6 7 8 133 X X X 2 3 4 5

13. Day 1 We now need to work out how long each car is out for

14. Length of Rental Car 1 Next Random Number is 42 This corresponds to a duration of 1 day

15. DayDemand Total Demand Hire Car Number 1 2 3 4 Other Garage 5 6 7 8 133 X X X 2 3 4 5

16. Length of Rental Car 2 Next Random Number is 56 This corresponds to a duration of 1 day Car 3 Next Random Number is 73 This corresponds to a duration of 2 days

17. DayDemand Total Demand Hire Car Number 1 2 3 4 Other Garage 5 6 7 8 133 X X X 2 X 3 4 5

18. Day 2 Random Number is 87 This means a demand of 3

19. DayDemand Total Demand Hire Car Number 1 2 3 4 Other Garage 5 6 7 8 133 X X X 234 X X X X 3 4 5

20. Length of Rental The next three random numbers are 75, 18 and 91 meaning rental durations of 2, 1 and 3 respectively

21. DayDemand Total Demand Hire Car Number 1 2 3 4 Other Garage 5 6 7 8 133 X X X 234 X X X X 3 X X 4 X 5

22. This can be carried on for a ten day period and the costs calculated.