Approximate solution of a profit maximization constrained virtual business planning problem

•We study a profit maximization constrained virtual business planning problem.•A generalization of the knapsack problem called Knapsack-of-Knapsacks is used.•We developed a fully polynomial time approximation scheme to solve this problem.

A virtual business problem is studied, in which a company-contractor outsources production to specialized subcontractors. Finances of the contractor and resource capacities of subcontractors are limited. The objective is to select subcontractors and distribute a part of the demanded production among them so that the profit of the contractor is maximized. A generalization of the knapsack problem, called Knapsack-of-Knapsacks (K-of-K), is used to model this situation, in which items have to be packed into small knapsacks and small knapsacks have to be packed into a large knapsack. A fully polynomial time approximation scheme is developed to solve the problem K-of-K.