Advances in Applied Probability

The fighter problem: optimal allocation of a discrete commodity

Jay Bartroff and Ester Samuel-Cahn

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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.

Article information

Adv. in Appl. Probab., Volume 43, Number 1 (2011), 121-130.

First available in Project Euclid: 15 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 62L05: Sequential design

Bomber problem continuous ammunition discrete ammunition concavity


Bartroff, Jay; Samuel-Cahn, Ester. The fighter problem: optimal allocation of a discrete commodity. Adv. in Appl. Probab. 43 (2011), no. 1, 121--130. doi:10.1239/aap/1300198515.

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