## Advances in Applied Probability

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

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

Jay Bartroff and Ester Samuel-Cahn

#### 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 - *q*^{j}, 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1239/aap/1300198515

**Mathematical Reviews number (MathSciNet)**

MR2761150

**Zentralblatt MATH identifier**

1250.90106

**Subjects**

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

Secondary: 62L05: Sequential design

**Keywords**

Bomber problem continuous ammunition discrete ammunition concavity

#### 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