Optimal Group Testing with Processing Times and Incomplete Identification

We consider the group testing problem for a finite population of possibly defective items with the objective of sampling a prespecified demanded number of nondefective items at minimum cost. Group testing means that items can be pooled and tested together; if the group comes out clean, all items in it are nondefective, while a contaminated group is scrapped. Every test takes a random amount of time and a given deadline has to be met. If the prescribed number of nondefective items is not reached, the demand has to be satisfied at a higher (penalty) cost. We derive explicit formulas for the distributions underlying the cost functionals of this model. It is shown in numerical examples that these results can be used to determine the optimal group size.