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March 2011 The fighter problem: optimal allocation of a discrete commodity
Jay Bartroff, Ester Samuel-Cahn
Adv. in Appl. Probab. 43(1): 121-130 (March 2011). DOI: 10.1239/aap/1300198515

## Abstract

In this paper we study the fighter problem with discrete ammunition. 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 is shown through a surprising counterexample, which is also valid for small u > 0 values.

## Citation

Jay Bartroff. Ester Samuel-Cahn. "The fighter problem: optimal allocation of a discrete commodity." Adv. in Appl. Probab. 43 (1) 121 - 130, March 2011. https://doi.org/10.1239/aap/1300198515

## Information

Published: March 2011
First available in Project Euclid: 15 March 2011

zbMATH: 1250.90106
MathSciNet: MR2761150
Digital Object Identifier: 10.1239/aap/1300198515

Subjects:
Primary: 60G40
Secondary: 62L05

Keywords: Bomber problem , concavity , continuous ammunition , discrete ammunition