## Journal of Applied Probability

- J. Appl. Probab.
- Volume 50, Number 2 (2013), 403-418.

### Generalized bomber and fighter problems: offline optimal allocation of a discrete asset

Abba M. Krieger and Ester Samuel-Cahn

#### Abstract

The classical bomber problem concerns properties of the optimal
allocation policy of a given number, *n*, of anti-aircraft missiles,
with which an airplane is equipped. The airplane begins at a distance
*t* >0 from its destination and uses some of the anti-aircraft missiles
when intercepted by enemy planes that appear according to a homogeneous
Poisson process. The goal is to maximize the probability of reaching
its destination. The fighter problem deals with a similar situation,
but the goal is to shoot down as many enemy planes as possible. The
optimal allocation policies are dynamic, depending upon both the number
of missiles and the time which remains to reach the destination when
the enemy is met. The present paper generalizes these problems by
allowing the number of enemy planes to have any distribution, not just
Poisson. This implies that the optimal strategies can no longer be
dynamic, and are, in our terminology, offline. We show that properties
similar to those holding for the classical problems hold also in the
present case. Whether certain properties hold that remain open
questions in the dynamic version are resolved in the offline version.
Since `time' is no longer a meaningful way to parametrize the
distributions for the number of encounters, other more general
orderings of distributions are needed. Numerical comparisons between
the dynamic and offline approaches are given.

#### Article information

**Source**

J. Appl. Probab., Volume 50, Number 2 (2013), 403-418.

**Dates**

First available in Project Euclid: 19 June 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1371648949

**Digital Object Identifier**

doi:10.1239/jap/1371648949

**Mathematical Reviews number (MathSciNet)**

MR3102488

**Zentralblatt MATH identifier**

1301.60101

**Subjects**

Primary: 62L99: None of the above, but in this section

Secondary: 60K10: Applications (reliability, demand theory, etc.)

**Keywords**

Dynamic allocation invincible fighter frail fighter hazard rate dominance likelihood ratio dominance optimality properties

#### Citation

Krieger, Abba M.; Samuel-Cahn, Ester. Generalized bomber and fighter problems: offline optimal allocation of a discrete asset. J. Appl. Probab. 50 (2013), no. 2, 403--418. doi:10.1239/jap/1371648949. https://projecteuclid.org/euclid.jap/1371648949