41 Counterexamples to Property (B) of the Discrete Time Bomber Problem


The discrete time "bomber problem" has been one of the longest standing open problems in operations research. In particular, the validity of one of the natural monotonicity conjectures ― known as property (B) ― has been an unresolved issue since 1968. In this paper we report 41 counterexamples to property (B) of this problem. We have found them by computing the exact solutions for nearly one million pairs of parameter values utilizing the GNU Multiple Precision (GMP) Arithmetic Library. All our counterexamples can readily be verified using a simple Mathematica program included in this paper.


The discrete time bomber problem, Error-free methods, GNU Multiple Precision (GMP) Arithmetic Library, Stochastic dynamic programming

2010 Mathematics Subject Classifications

62L05, 93E20


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