Sterrett Procedure for the Generalized Group Testing Problem

Group testing is a useful method that has broad applications in medicine, engineering, and even in airport security control. Consider a finite population of $N$ items, where item $i$ has a probability $p_i$ to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem. The optimum procedure, with respect to the expected total number of tests, is unknown even in case when all $p_i$ are equal. \cite{H1975} proved that an ordered partition (with respect to $p_i$) is the optimal for the Dorfman procedure (procedure $D$), and obtained an optimum solution (i.e., found an optimal partition) by dynamic programming. In this paper, we investigate the Sterrett procedure (procedure $S$). We provide close form expression for the expected total number of tests, which allows us to find the optimum arrangement of the items in the particular group. We also show that an ordered partition is not optimal for the procedure $S$ or even for a slightly modified Dorfman procedure (procedure $D^{\prime}$). This discovery implies that finding an optimal procedure $S$ appears to be a hard computational problem. However, by using an optimal ordered partition for all procedures, we show that procedure $D^{\prime}$ is uniformly better than procedure $D$, and based on numerical comparisons, procedure $S$ is uniformly and significantly better than procedures $D$ and $D^{\prime}$.