Abstract
In portfolio selection problems the items often depend on each other, and their synergies and redundancies need to be taken into account. We consider the knapsack problem in which the objective is modelled as the Choquet integral with respect to a supermodular capacity which quantifies possible synergies. We provide various formulations which lead to the standard linear mixed integer programs, applicable to small and large portfolios. We also study scalability of the solution methods and compare large problems defined with respect to 2-additive capacities which model pairwise interactions, and linear knapsack with respect to the Shapley values of these capacities.
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