Friday, October 12, 2007

A little operations research into TV programming

Consider the following idealized model of a TV channel faced with the problem of allocating its resources in the optimum way to achieve as high an audience as possible: The TV scheduling grid is divided into n consecutive slots, each of which has a total audience ai. The competition has distributed its resources (be it money or some intangible goods correlated with the appeal of the broadcast material) so that the i-th slot is devoted ci resource units, the total amount of resources of the competition being C = ∑ci. If we have M resource units to compete and allocate them as mi, i=1,...,n, the following is a reasonable model of the audience we obtain at the i-th slot:

s(mi)= ai mi/(mi+ci),

which is simply a measure of the resources we have put compared with the total resources allocated to the slot (ours and the competition's). The problem is then to find the scheduling {mi} with ∑mi = M and mi ≥ 0 for i=1,...,n, that maximizes our global audience S({mi}) = ∑s(mi).

In later entries we'll derive an analytical solution to this problem.