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

Bartroff, Jay and Goldstein, Larry and Rinott, Yosef and Samuel-Cahn, Ester (2010) On Optimal Allocation of a Continuous Resource Using an Iterative Approach and Total Positivity. [Discussion Paper]. p. 21. Discussion Paper Series (No. 530).

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## Abstract/Index

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 <= y <= 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 [C] in general, and that [A] holds for one special case and [B] for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (TP2) in the Bomber and Fighter Problems, respectively. The TP2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves TP2 and other properties.

Item Type: Report / Paper (Discussion Paper) Working Papers > Refereed Working Papers / of international relevance Department of Business and Management The paper has been published in "Advances in Applied Probability", (2010) Vol. 42(3), Pages 795-815. Area 13 - Economics and Statistics > SECS-P/05 Econometrics Maria Teresa Nistico 20 Dec 2010 10:00 21 Apr 2015 23:14 http://eprints.luiss.it/id/eprint/858