### The fighter problem: optimal allocation of a discrete commodity

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 15 March 2011

https://projecteuclid.org/euclid.aap/1300198515

Digital Object Identifier
doi:10.1239/aap/1300198515

Mathematical Reviews number (MathSciNet)
MR2761150

Zentralblatt MATH identifier
1250.90106

#### Citation

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. https://projecteuclid.org/euclid.aap/1300198515

#### References

• Bartroff, J. (2011). A proof of the Bomber problem's spend-it-all conjecture. Sequent. Anal. 30, 52–57.
• 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.
• 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.
• Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.
• Klinger, A. and Brown, T. A. (1968). Allocating unreliable units to random demands. In Stochastic Optimization and Control (Proc. Adv. Sem., Madison, 1967), ed. H. F. Karreman, John Wiley, New York, pp. 173–209.
• Samuel, E. (1970). On some problems in operations research. J. Appl. Prob. 7, 157–164.
• Shepp, L. A., Simons, G. and Yao, Y.-C. (1991). On a problem of ammunition rationing. Adv. Appl. Prob. 23, 624–641.
• Simons, G. and Yao, Y.-C. (1990). Some results on the bomber problem. Adv. Appl. Prob. 22, 412–432.