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 a_i. 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 c_i resource units, the total amount of resources of the competition being C = ∑c_i. If we have M resource units to compete and allocate them as m_i, i=1,...,n, the following is a reasonable model of the audience we obtain at the i-th slot:

s(m_i)= a_i m_i/(m_i+c_i),

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 {m_i} with ∑m_i = M and m_i ≥ 0 for i=1,...,n, that maximizes our global audience S({m_i}) = ∑s(m_i).

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