Optimal parallel funding of research and development projects

To insure against the costs of failure, research managers often initiate several more-or-less independent research projects with the same target. But then they face the problem of how many parallel "teams" to fund in order to maximize the probability of a timely breakthrough. If there are too many teams, resources are stretched too thin, but if there are too few teams, an opportunity for exceptional achievement may be missed. Several possible project objectives are identified, including maximizing the achievement of the best team and maximizing the probability that at least one team will attain a threshold. General optimization problems based on these goals are then formulated in fixed, uncertain, and competitive environments. These problems are solved analytically and numerically for achievement distributions in a family based on the exponential distribution. The applicability of these solutions to funding R & D, and to other problems such as supporting athletes, is then discussed.