I have an linear algebra problem to solve. Currently I have n number of customer lists and m number of offers.
I have matrix (P) with dimension m x n. Elements of P indicates probability of responding to the relevant offer for each customers.
I need to find a matrix (N) with binary numbers (0,1) indicating whether an offer is made to a particular customer. It also dimension m x n
R is a column vector (mx1) representing revenues that can be made from each offers and C is also a column vector (mx1) representing costs generated from each offers.
How do I solve for the N which can give the maximum profit ?
In summary I have P = probability (m x n) R = Revenue (m x 1) C = Cost (m x 1) I need to find N = binary variable (m x n) which will give the largest profit. I think the profit is P(T) x N x (R-C) where P(T) is transpose of P
This is an example of 0-1 Knapsack problem. It's NP-complete, but there are a plenty heuristic and/or approximate algorithms for solving it.