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The fighter problem: optimal allocation of a discrete commodity

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  01 July 2016

Jay Bartroff  and
Ester Samuel-Cahn
Jay Bartroff*
Affiliation:
University of Southern California
Ester Samuel-Cahn*
Affiliation:
The Hebrew University of Jerusalem
*
∗ Postal address: Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, KAP 108, Los Angeles, CA 90089-2532, USA. Email address: bartroff@usc.edu
∗∗ Postal address: Department of Statistics and Center for the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem, 91905, Israel.
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Abstract

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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.

Keywords

Bomber problemcontinuous ammunitiondiscrete ammunitionconcavity

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L05: Sequential design
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 121 - 130
Copyright
Copyright © Applied Probability Trust 2011 

References

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