You are consulting for a trucking company that does a large amount of
business shipping packages between New York and Boston. The volume is
high enough that they have to send a number of trucks each day between
the two locations. Trucks have a fixed limit W on the maximum amount
of weight they are allowed to carry. Boxes arrive at the New York station
one by one, and each package i has a weight wi. The trucking station
is quite small, so at most one truck can be at the station at any time.
Company policy requires that boxes are shipped in the order they arrive;
otherwise, a customer might get upset upon seeing a box that arrived
after his make it to Boston faster. At the moment, the company is using
a simple greedy algorithm for packing: they pack boxes in the order they
arrive, and whenever the next box does not fit, they send the truck on its
way.
But they wonder if they might be using too many trucks, and they
want your opinion on whether the situation can be improved. Here is
how they are thinking. Maybe one could decrease the number of trucks
needed by sometimes sending off a truck that was less full, and in this
way allow the next few trucks to be better packed.
Prove that, for a given set of boxes with specified weights, the greedy
algorithm currently in use actually minimizes the number of trucks that
are needed. Your proof should follow the type of analysis we used for
the Interval Scheduling Problem: it should establish the optimality of this
greedy packing algorithm by identifying a measure under which it