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The fighter problem: optimal allocation of a discrete commodity

Part of: Sequential methods Stochastic processes

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

Bartroff, J. (2011). A proof of the Bomber problem's spend-it-all conjecture. Sequent. Anal. 30, 52–57.Google Scholar
Bartroff, J., Goldstein, L. and Samuel-Cahn, E. (2010a). The spend-it-all region and small time results for the continuous Bomber problem. Sequent. Anal. 29, 275–291.Google Scholar
Bartroff, J., Goldstein, L., Rinott, Y. and Samuel-Cahn, E. (2010b). On optimal allocation of a continuous resource using an iterative approach and total positivity. Adv. Appl. Prob. 42, 795–815.CrossRefGoogle Scholar
Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.Google Scholar
Klinger, A. and Brown, T. A. (1968). Allocating unreliable units to random demands. In Stochastic Optimization and Control (Proc. Adv. Sem., Madison, 1967), ed. Karreman, H. F., John Wiley, New York, pp. 173–209.Google Scholar
Samuel, E. (1970). On some problems in operations research. J. Appl. Prob. 7, 157–164.CrossRefGoogle Scholar
Shepp, L. A., Simons, G. and Yao, Y.-C. (1991). On a problem of ammunition rationing. Adv. Appl. Prob. 23, 624–641.Google Scholar
Simons, G. and Yao, Y.-C. (1990). Some results on the bomber problem. Adv. Appl. Prob. 22, 412–432.Google Scholar