An Allocation Problem with Applications to Operations Research and Statistics

Samuel Zahl
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Samuel Zahl: Cambridge Air Force Research Laboratories, Bedford, Mass

Operations Research, 1963, vol. 11, issue 3, 426-441

Abstract: A considerable number of problems in operations research and statistics have the following form maximize (integral) f [ x , y ( x )] dx subject to (integral) g [ x , y ( x )] dx = constant with respect to bounded y ( x ). We give a necessary and sufficient condition for a maximizing function under fairly weak restrictions and prove its existence. The solution is applied to a general version of B. O. Koopman's search problem, and to the Neyman, Pearson lemma of statistics. We also show that in the discrete version of this problem, where x is replaced by an index and sums replace integrals, our condition is sufficient but not necessary and give, as illustration of the sufficiency, a solution to an assignment problem.

Date: 1963
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