The optimum gradient technique is reviewed and then used to optimize a shipboard reactor shield system consisting of a water-lead primary and a concrete-lead-polyethylene secondary shield. The shield is optimized to the point where the cost of further reductions in weight exceeds the worth (K) of these reductions to the ship, and subject to five dose point constraints. Plots of eight thicknesses as a function of K are given. For high K values (essentially weight optimization) the concrete thicknesses are zero. As K decreases and cost becomes more and more important in comparison to the weight, concrete is added, and the more expensive lead and polyethylene are subtracted.