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On Optimal Allocation of a Continuous Resource Using an Iterative Approach and Total Positivity

Abstract

We study a class of optimal allocation problems, including the well-known Bomber Problem, with the following common probabilistic structure. An aircraft equipped with an amount~\$x\$ of ammunition is intercepted by enemy airplanes arriving according to a homogenous Poisson process over a fixed time duration~\$t\$. Upon encountering an enemy, the aircraft has the choice of spending any amount~\$0\le y\le x\$ of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of \$y\$. Two different goals have been considered in the literature concerning the optimal amount~\$K(x,t)\$ of ammunition spent: (i)~Maximizing the probability of surviving for time~\$t\$, which is the so-called Bomber Problem, and (ii) maximizing the number of enemy airplanes shot down during time~\$t\$, which we call the Fighter Problem. Several authors have attempted to settle the following conjectures about the monotonicity of \$K(x,t)\$: [A] \$K(x,t)\$ is decreasing in \$t\$, [B] \$K(x,t)\$ is increasing in \$x\$, and [C] the amount~\$x-K(x,t)\$ held back is increasing in \$x\$. [A] and [C] have been shown for the Bomber Problem with discrete ammunition, while [B] is still an open question. In this paper we consider both time and ammunition continuous, and for the Bomber Problem prove [A] and [C], while for the Fighter we prove [A] and [C] for one special case and [B] and [C] for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (\$\mbox{TP}_2\$) in the Bomber and Fighter Problems, respectively. The \$\mbox{TP}_2\$ property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves \$\mbox{TP}_2\$ and other properties.Comment: 2 figure

Topics: Mathematics - Probability, Mathematics - Statistics Theory, Primary 60G40, Secondary 62L05, 91A60
Year: 2011
OAI identifier: oai:arXiv.org:1103.0308