Internet shopping with price sensitive discounts

A customer would like to buy a given set of products in a given set of Internet shops. For each Internet shop, standard prices for the products are known as well as a concave increasing discounting function of total standard and delivery price. The problem is to buy all the required products at the minimum total discounted price. Computational complexity of various special cases is established. Properties of optimal solutions are proved and polynomial time and exponential time solution algorithms based on these properties are designed. Two heuristic algorithms are suggested and computationally tested.