You are consulting for a trucking company that does a large amount of business s

Question

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

Explanation / Answer

Consider the solution given by the greedy algorithm as a sequence of packages, here represented by indexes: 1, 2, 3, ... n. Each package i has a weight, w_i, and an assigned truck t_i. { t_i } is a non-decreasing sequence (as the k'th truck is sent out before anything is placed on the k+1'th truck). If t_n = m, that means our solution takes m trucks to send out the n packages.

If the greedy solution is non-optimal, then there exists another solution { t'_i }, with the same constraints, s.t. t'_n = m' < t_n = m.

Consider the optimal solution that matches the greedy solution as long as possible, so or all i < k, t_i = t'_i, and t_k != t'_k.
t_k != t'_k => Either

1) t_k = 1 + t'_k
     i.e. the greedy solution switched trucks before the optimal solution.

     But the greedy solution only switches trucks when the current truck is full. Since t_i = t'_i i < k, the contents of the current truck after adding the k - 1'th package are identical for the greedy and the optimal solutions.
     So, if the greedy solution switched trucks, that means that the truck couldn't fit the k'th package, so the optimal solution must switch trucks as well.

     So this situation cannot arise.

   2) t'_k = 1 + t_k
      i.e. the optimal solution switches trucks before the greedy solution.

      Construct the sequence { t"_i } s.t.
        t"_i = t_i, i <= k
        t"_k = t'_i, i > k

      This is the same as the optimal solution, except package k has been moved from truck t'_k to truck (t'_k - 1). Truck t'_k cannot be overpacked, since it has one less packages than it did in the optimal solution, and truck (t'_k - 1)
      cannot be overpacked, since it has no more packages than it did in the greedy solution.

      So { t"_i } must be a valid solution. If k = n, then we may have decreased the number of trucks required, which is a contradiction of the optimality of { t'_i }. Otherwise, we did not increase the number of trucks, so we created an optimal solution that matches { t_i } longer than { t'_i } does, which is a contradiction of the definition of { t'_i }.

    So the greedy solution must be optimal.
   

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