Explicit Exact Solution of Damage Probability for Multiple Weapons against a Unitary Target

Abstract

We study the damage probability when M weapons are used against a unitary target. We use the Carleton damage function to model the distribution of damage probabil- ity caused by each weapon. The deviation of the impact point from the aimpoint is attributed to both the dependent error and independent errors. The dependent error is one random variable affecting M weapons the same way while independent errors are associated with individual weapons and are independent of each other. We con- sider the case where the dependent error is significant, non-negligible relative to in- dependent errors. We first derive an explicit exact solution for the damage probabil- ity caused by M weapons for any M. Based on the exact solution, we find the optimal aimpoint distribution of M weapons to maximize the damage probability in several cases where the aimpoint distribution is constrained geometrically with a few free parameters, including uniform distributions around a circle or around an ellipse. Then, we perform unconstrained optimization to obtain the overall optimal aim- point distribution and the overall maximum damage probability, which is carried out for different values of M, up to 20 weapons. Finally, we derive a phenomenological approximate expression for the damage probability vs. M, the number of weapons, for the parameters studied here.