Heuristic Algorithm for Multi-item Internet Shopping with Price Sensitive Discounts

We study an optimization aspect of Internet shopping with price sensitive discounts from customer perspective (which is a specific case of the Internet Shopping Optimization Problem [1, 5]). Specifically, we consider a problem in which a customer would like to buy products of a given set N = {1, . . . ,n} in a given set of Internet shops M = {1, . . . ,m} at the minimum total final price. There are the following given parameters and decision variables : di delivery price of all products from shop i to the customer ; pi j standard price of product j in shop i, pi j = p j if standard prices of product j are the same in all shops ; Ni subset of products of the set N in shop i (eligible products for shop i), Ni ⊆ N ; M j subset of shops in which product j can be bought (eligible shops for product j), M j ⊆M ; Si subset of products selected by the customer in shop i (basket of shop i, decision variable), N = ∪i=1Si and Si∩S j = φ, i 6= j, for a feasible solution ; Ti(Si) = di+∑ j∈Si pi j total delivery and standard price in shop i for a given set of products Si ⊆ Ni ; if there is no ambiguity, notation Si in Ti(Si) can be omitted ; fi(T ) discounting function for final price, a concave increasing differentiable or concave piecewise linear function of total delivery and standard price T in shop i at all points T > 0, fi(0) = 0. We denote the above problem as IS, where the abbreviation stands for Internet Shopping. Its mathematical program can be written as follows :