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ABCs of the bomber problem and its relatives

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  • Richard Weber

Abstract

In a classic Markov decision problem of Derman et al. (Oper. Res. 23(6):1120–1130, 1975 ) an investor has an initial capital x from which to make investments, the opportunities for which occur randomly over time. An investment of size y results in profit P(y), and the aim is maximize the sum of the profits obtained within a given time t. The problem is similar to a groundwater management problem of Burt (Manag. Sci. 11(1):80–93, 1964 ), the notorious bomber problem of Klinger and Brown (Stochastic Optimization and Control, pp. 173–209, 1968 ), and types of fighter problems addressed by Weber (Stochastic Dynamic Optimization and Applications in Scheduling and Related Fields, p. 148, 1985 ), Shepp et al. (Adv. Appl. Probab. 23:624–641, 1991 ) and Bartroff et al. (Adv. Appl. Probab. 42(3):795–815, 2010a ). In all these problems, one is allocating successive portions of a limited resource, optimally allocating y(x,t), as a function of remaining resource x and remaining time t. For their investment problem, Derman et al. (Oper. Res. 23(6):1120–1130, 1975 ) proved that an optimal policy has three monotonicity properties: (A) y(x,t) is nonincreasing in t, (B) y(x,t) is nondecreasing in x, and (C) x−y(x,t) is nondecreasing in x. Theirs is the only problem of its type for which all three properties are known to be true. In the bomber problem the status of (B) is unresolved. For the general fighter problem the status of (A) is unresolved. We survey what is known about these exceedingly difficult problems. We show that (A) and (C) remain true in the bomber problem, but that (B) is false if we very slightly relax the assumptions of the usual model. We give other new results, counterexamples and conjectures for these problems. Copyright Springer Science+Business Media, LLC 2013

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  • Richard Weber, 2013. "ABCs of the bomber problem and its relatives," Annals of Operations Research, Springer, vol. 208(1), pages 187-208, September.
    • Handle: RePEc:spr:annopr:v:208:y:2013:i:1:p:187-208:10.1007/s10479-011-1015-z
    DOI: 10.1007/s10479-011-1015-z
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    References listed on IDEAS

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    1. Woonghee Tim Huh & Chandra Kiran Krishnamurthy & Richard Weber, 2011. "Concavity and monotonicity properties in a groundwater management model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(7), pages 670-675, October.
    2. Oscar R. Burt, 1964. "Optimal Resource Use Over Time with an Application to Ground Water," Management Science, INFORMS, vol. 11(1), pages 80-93, September.
    3. Knapp Keith C. & Olson Lars J., 1995. "The Economics of Conjunctive Groundwater Management with Stochastic Surface Supplies," Journal of Environmental Economics and Management, Elsevier, vol. 28(3), pages 340-356, May.
    4. C. Derman & G. J. Lieberman & S. M. Ross, 1975. "A Stochastic Sequential Allocation Model," Operations Research, INFORMS, vol. 23(6), pages 1120-1130, December.
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    Cited by:

    1. Takashi Kamihigashi, 2017. "41 Counterexamples to property (B) of the discrete time bomber problem," Annals of Operations Research, Springer, vol. 248(1), pages 579-588, January.
    2. Krishna Kalyanam & David Casbeer & Meir Pachter, 2020. "A sequential partial information bomber‐defender shooting problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(3), pages 223-235, April.

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