The Fighter Problem: Optimal Allocation of a Discrete Commodity

TL;DR

This paper analyzes the optimal missile allocation strategy for a fighter aircraft facing enemy intercepts modeled as a Poisson process, revealing conditions under which intuitive properties of the strategy hold or fail.

Contribution

It establishes the validity of certain intuitive properties of the optimal missile usage function and provides counterexamples where these properties do not hold.

Findings

Property C (saving for future encounters) holds for all u.

Property A (more usage closer to destination) holds universally.

Property B (more missiles lead to more usage) holds only for u=1 and fails for u=0.

Abstract

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., $a(j) = 1-q^j, \; \, 0 < q < 1$. If the enemy is not destroyed, the enemy shoots the fighter down with known probability $1-u$, where $0 \le u \le 1$. 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 $0 \le u \le 1$, 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.