simulation an example. a garage hires out cars. it currently owns four cars which are hired out on a...

SimulationSimulation

An ExampleAn Example

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 Number of carsof cars

00 11 22 33 44

Demand Demand per day per day probabilityprobability

0.150.15 0.250.25 0.300.30 0.200.20 0.100.10

The length of rental is shown below

Length Length of rental of rental (days)(days)

11 22 33

ProbabilityProbability 0.600.60 0.300.30 0.100.10

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.

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.

The first thing to do is to construct cumulative frequency tables to decide which random numbers refer to a particular scenario.

Number Number of carsof cars

00 11 22 33 44

Demand Demand per day per day probabilityprobability

0.150.15 0.250.25 0.300.30 0.200.20 0.100.10

Cumulative Cumulative probabilityprobability

0.150.15 0.400.40 0.700.70 0.900.90 1.001.00

Random Random numbersnumbers

1-151-15 16-4016-40 41-7041-70 71-9071-90 91-0091-00

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.

The length of rental is shown below

Length of Length of rental rental (days)(days)

11 22 33

ProbabilityProbability 0.600.60 0.300.30 0.100.10

Cumulative Cumulative probabilityprobability

0.600.60 0.900.90 1.001.00

Random Random NumbersNumbers

1-601-60 61-9061-90 91-0091-00

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

DayDay DemandDemand Total Total DemandDemand

Hire Car NumberHire Car Number

1 2 3 41 2 3 4Other GarageOther Garage

5 6 7 85 6 7 8

11

22

33

44

55

Day 1Day 1

Random Number is 84Random Number is 84

This means a demand of 3This means a demand of 3

DayDay DemandDemand Total Total DemandDemand

Hire Car NumberHire Car Number

1 2 3 41 2 3 4Other GarageOther Garage

5 6 7 85 6 7 8

11 33 33 X X XX X X

22

33

44

55

Day 1Day 1

We now need to work out how long each We now need to work out how long each car is out forcar is out for

Length of RentalLength of Rental

Car 1Car 1

Next Random Number is 42Next Random Number is 42

This corresponds to a duration of 1 day This corresponds to a duration of 1 day

DayDay DemandDemand Total Total DemandDemand

Hire Car NumberHire Car Number

1 2 3 41 2 3 4Other GarageOther Garage

5 6 7 85 6 7 8

11 33 33 XX X X X X

22

33

44

55

Length of RentalLength of Rental

Car 2Car 2

Next Random Number is 56Next Random Number is 56

This corresponds to a duration of 1 day This corresponds to a duration of 1 day

Car 3Car 3

Next Random Number is 73Next Random Number is 73

This corresponds to a duration of 2 daysThis corresponds to a duration of 2 days

DayDay DemandDemand Total Total DemandDemand

Hire Car NumberHire Car Number

1 2 3 41 2 3 4Other GarageOther Garage

5 6 7 85 6 7 8

11 33 33 XX XX X X

22 XX

33

44

55

Day 2Day 2

Random Number is 87Random Number is 87

This means a demand of 3This means a demand of 3

DayDay DemandDemand Total Total DemandDemand

Hire Car NumberHire Car Number

1 2 3 41 2 3 4Other GarageOther Garage

5 6 7 85 6 7 8

11 33 33 XX XX X X

22 33 44 X X X X XX X X

33

44

55

Length of RentalLength of Rental

The next three random numbers are The next three random numbers are 75, 18 and 91 meaning rental durations of 75, 18 and 91 meaning rental durations of 2, 1 and 3 respectively2, 1 and 3 respectively

DayDay DemandDemand Total Total DemandDemand

Hire Car NumberHire Car Number

1 2 3 41 2 3 4Other GarageOther Garage

5 6 7 85 6 7 8

11 33 33 XX XX X X

22 33 44 X X XX XX X X

33 XX X X

44 XX

55

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