Optimal randomized and non-randomized procedures for multinomial selection problems
Tollefson, Eric Sander
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Multinomial selection problem procedures are ranking and selection techniques that aim to select the best (most probable) alternative based upon a sequence of multinomial observations. The classical formulation of the procedure design problem is to find a decision rule for terminating sampling. The decision rule should minimize the expected number of observations taken while achieving a specified indifference zone requirement on the prior probability of making a correct selection when the alternative configurations are in a particular subset of the probability space called the preference zone. We study the constrained version of the design problem in which there is a given maximum number of allowed observations. Numerous procedures have been proposed over the past 50 years, all of them suboptimal. In this thesis, we find via linear programming the optimal selection procedure for any given probability configuration. The optimal procedure turns out to be necessarily randomized in many cases. We also find via mixed integer programming the optimal non-randomized procedure. We demonstrate the performance of the methodology on a number of examples. We then reformulate the mathematical programs to make them more efficient to implement, thereby significantly expanding the range of computationally feasible problems. We prove that there exists an optimal policy which has at most one randomized decision point and we develop a procedure for finding such a policy. We also extend our formulation to replicate existing procedures. Next, we show that there is very little difference between the relative performances of the optimal randomized and non-randomized procedures. Additionally, we compare existing procedures using the optimal procedure as a benchmark, and produce updated tables for a number of those procedures. Then, we develop a methodology that guarantees the optimal randomized and non-randomized procedures for a broad class of variable observation cost functions -- the first of its kind. We examine procedure performance under a variety of cost functions, demonstrating that incorrect assumptions regarding marginal observation costs may lead to increased total costs. Finally, we investigate and challenge key assumptions concerning the indifference zone parameter and the conditional probability of correct selection, revealing some interesting implications.