The Fighter Problem: Optimal Allocation of a Discrete Commodity

The Fighter problem with discrete ammunition is studied. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter they destroy an enemy plane with probability a(j), where a(0) = 0 and {a(j)} is a known, strictly increasing concave sequence, e.g., a(j) = 1−qj, 0 < q < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1− u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0, T ]. Let K(n, t) be the smallest optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: [A] The closer to the destination, the more of the n missiles one should use, [B] the more missiles one has, the more one should use, and [C] the more missiles one has, the more one should save for possible future encounters. We show that [C] holds for all 0 ≤ u ≤ 1, that [A] and [B] hold for the “Invincible Fighter” (u = 1), and that [A] holds but [B] fails for the “Frail Fighter” (u = 0), the latter through a surprising counterexample.