Abstract A novel sphere packing problem is introduced. A maximum number of spheres of different radii should be placed such that the spheres do not overlap and their centers fulfill a quasi-containment condition. The latter allows the spheres to lie partially outside the given cuboidal container. Moreover, specified ratios between the placed spheres of different radii must be satisfied. A corresponding mixed-integer nonlinear programming model is formulated. It enables the exact solution of small instances. For larger instances, a heuristic strategy is proposed, which relies on techniques for the generation of feasible points and the decomposition of open dimension problems. Numerical results are presented to demonstrate the viability of the approach.

This paper addresses R&D portfolio selection in social institutions, state-owned enterprises, and other nonprofit organizations which periodically launch a call for proposals and distribute funds among accepted projects. A nonlinear discontinuous bicriterion optimization model is developed in order to find a compromise between a portfolio quality measure and the number of projects selected for funding. This model is then transformed into a linear mixed-integer formulation to present the Pareto front. Numerical experiments with up to 25 000 projects competing for funding demonstrate a high computational efficiency of the proposed approach. The acceptance/rejection rules are obtained for a portfolio using the rough set methodology.