Final 1990-1991

Math A-lympiad: Final 1990-1991

'De Berkenplas' Amusement Park

The map you see above is of ‘De Berkenplas’ amusement park. This was originally a nature park located by the river A, a few small lakes, and a hill with the sonorous name ‘De Berkenberg’. The ‘Berkenplas’ restaurant formed part of the park.

A few years ago a businessman bought the nature park in order to turn it into an amusement park. As its natural state could not be disturbed to any great extent, the number of major attractions had to be kept limited. There is however space for adding a number of small attractions.
When planning the lay out of the amusement park, the fact that some people want spectacular attractions (which often don’t last very long), while others will come for quiet recreation was taken into account. Because the distances in the park are rather large, and in order to give people the opportunity to easily see much of the landscape, it was decided to build a small railway line. Because of the natural obstacles of the landscape, this railway had to be a single track railway. It goes without saying that there are stations at the major attractions. In addition to the railway lines there are also footpaths (see the map).

The major attractions to bring in the people are:

A go-cart circuit.
The ‘Cinemax’ theatre.
The ‘Octopus’ roller coaster.
The ‘Berkenplas’ restaurant.
The ‘A’ boat trip.

The proprietor wants the best synchronization possible between the flows of visitors to and from the attractions and the train timetable. He wants to keep the irritation of visitors caused by waiting for a train or for an attraction to a minimum. Satisfied visitors are a good advert for friends and acquaintances. That is why he wants a train standing ready at the entrance as often as possible so that people can get on as soon as they arrive.

Railway information

Three antique locomotives are used.
It is a single track railway.
There are five stations: Entrance, A, Circuit, Cinemax and Octopus.
Ten carriages are available each with a capacity of fifty people.
One locomotive can draw a maximum of four carriages.
The locomotives go at around ten kilometers per hour.
The stopping time at a station is a minimum of three minutes.
There are sufficient shunting facilities at the Entrance station.

Attraction information

The Cinemax show is 40 minutes and can accommodate 350 people at a time.
Octopus can take 30 people every three minutes.
Circuit has places for 40 people every six minutes.
A, the boat trip: leaves every quarter of an hour, lasts 30 minutes. 100 people can be taken each time.
The number of places in the restaurant is 350.

Park information

The gates open at 10:00 a.m.
After 21:00 no further visitors are allowed. The park closes at 22:00.
Peak times (expectation):
11:00 – 12:00
14:00 – 15:00
18:00 – 19:00.
Management expects twice as many people during peak hours as during the other times.
Number of visitors (expectation): 5000 per day.

Exercise 1
Design the optimum timetable (for one day) for the three trains in the park. Take all the above information into account and also other aspects that you consider relevant. Also specify the starting times of the attractions.
You can also indicate where to place the minor attractions in order better to control the flows of visitors. Give clear justification in your solutions.

Exercise 2
Management is considering (while keeping the timetable unchanged) taking measures to be able to better handle the people at the entrance preparing for future growth. They are thus considering building an extra landing stage for the boat (at the entrance) or buying two new carriages for the train but they are also open to other suggestions.
Advise the management in this respect.

Study

After one year a study was done among a representative group of ‘active’ visitors. These are visitors who want to go on the four major attractions (the restaurant is left out of consideration here). They were asked which three attractions they would most like to visit and in what order they should be visited in. The result is given in the table below.

(Cinemax = Cx, Circuit = C, Boat trip= A, Octopus = O)
Cx-C-A	60	C-Cx-A	81	A-Cx-C	8	O-Cx-C	84
Cx-C-O	10	C-Cx-O	9	A-Cx-O	22	O-Cx-A	36
Cx-A-C	17	C-A-Cx	30	A-C-Cx	18	O-C-Cx	36
Cx-A-O	18	C-A-O	30	A-C-O	27	O-C-A	24
Cx-O-C	147	C-O-Cx	75	A-O-Cx	38	O-A-Cx	10
Cx-O-A	98	C-O-A	75	A-O-C	37	O-A-C	10
(F.e..: 60 people first preferred Cx, then C, then A)						Total	1000

Exercise 3

The above data can be processed into ‘transition matrices’. These matrices have the following form:

		From
		I	Cx	C	A	O
To	Cx
	C
	A
	O

As you can see we do not assume that after the third attraction people will go back to the Entrance (exit). Try completing such a matrix in a sensible way and show the significance of such matrices for management when they as much as possible want to take the desires of the public into account, of course without discarding economic considerations. (With thanks to the ‘Nationaal Veenpark Barger-Compascuum).