The quadratic knapsack problem with setup

The Quadratic Knapsack Problem is a well-known generalization of the classical 0-1 knapsack problem, in which any pair of items produces a pairwise profit if both are selected. Another relevant generalization of the knapsack problem is the Knapsack Problem with Setup, in which the items are partitioned into classes, the items of a class can only be inserted into the knapsack if the corresponding class is activated, and activating a class involves a setup cost and a setup capacity reduction. Despite a rich literature on these two problems, their obvious generalization, i.e., the Quadratic Knapsack Problem with Setup, was never investigated so far. We discuss applications, mathematical models, deterministic matheuristic algorithms, and computationally evaluate their performance. • First study on the Quadratic Multiple Knapsack Problem with Setups. • Mathematical models and reformulations. • Matheuristic algorithms. • Greedy heuristics and local search. • Computational experiments.