Math A-lympiad: Preliminary 1999-2000

GLAZED FROST IN ZEIST

Introduction
Glazed frost is a special phenomenon in the Netherlands. In wintertime, it sometimes rains while the ground is still frozen. The rain then freezes the moment it touches the earth. Immediately, every road is covered with glazed frost. These situations are life-threatening, moreover because it can happen so abruptly.
If glazed frost (or: an ice-storm) is threatening, immediate action should take place to prevent the roads from getting icy. That can be done by salting the roads. Salt lowers the freezing-point of water, so that rain wonít freeze. Trucks carrying salt, the salt-trucks, will sprinkle salt on the roads. Because an ice-storm can come up so suddenly, action should take place as soon as possible.

The problem as stated in Zeist
If an ice-storm is threatening, the roads in Zeist should be salted as efficient as possible. In Zeist, a discussion is taking place on how an efficient salting-plan should look like in the current situation, and if there are ways to improve the efficiency.

Data and conditions
Zeist is a town near Utrecht (which is a big city). Most people who live in Zeist are commuters. On the enclosed map you can see that Zeist is bounded by two different motorways (the A28 and the A12). The area in between (except for the ënormal streetsí in Bunnik, in the left-down-corner of the map), is the area to be salted. In this area the ëmain roadsí are grey, f.e. GRIFTLAAN in B4. The ënormal streetsí are white. A full salt-truck has an average speed of 60 km/h on a motorway, 30 km/h on a main road and 15 km/h on a normal street. A non-salting salt-truck has average speeds of resp. 80 km/h, 40 km/h and 20 km/h.
A full salt-truck can salt about 10 kilometers of road.
An empty salt-truck can be loaded from a salt-dump. The filling of a salt-truck takes 10 minutes.
Zeist has two salt-trucks. They are stationed at an industrial area. (All shaded areas on the map are industrial areas). The salt-dump can be found at the same industrial area where the trucks are stationed; this particular area can be found on the map in the under-right-corner of C2.

Preparatory assignment: The current situation
The moment an ice-storm is threatening, a salting-plan must be ready. This salting-plan must contain (among other things) a route-description (on the map or in words) for the drivers.
For the local authorities, it is important that the salting-plan is efficient. Formulate what you understand by an efficient salting-plan and come up with a plan that is as efficient as possible according to your definition. Also describe the process of making this plan.

The plan shouldnít be effected by trial and error; the intention is that you come up with a plan by using a well-considered and smart way of working. (Make realistic assumptions based on the data given and based on common sense).

Main assignment: An extended advice
The local authorities of Zeist want an advice on the possibilities of improving the efficiency on salting. The authorities want to use this advice in the discussion on the possible extension of the number of trucks, or the extension of the number of salt-dumps. Maybe there are yet more measures you can take to increase the efficiency.
Your team is consulted to write this advice in a report.
Self-evidently, the arguments on which your report is based, are important.

The report should contain at least the following parts:
ï Analysis of the current situation
In this part, the results of your prepatory assignment are presented.
The bottlenecks (if there are any) should also be indicated.
ï Suggestions for improvement of the efficiency
In this part of the report, you will give advice on possible adjustments of the situation, through which the efficiency could be improved.
Indicate the different interests that play a role in possible measures that could be taken.