Purpose: To illustrate this tradeoff, a simple model is presented in which the optimal number of samples per lot depends on the prevalence of contaminated sample units and the relative costs of sampling a lot and of drawing and testing a sample unit from a lot.
Methods: The assumed objective is to maximize the number of contaminated lots that are rejected subject to an overall food safety sampling budget constraint. The optimization problem is solved using the Lagrangian and numerical methods.
Results: Under a budget constraint, the optimal sample size depends only on prevalence and the ratio of the cost per lot to the cost per sample unit, not on the size of the budget or number of lots inspected. If the ratio of the cost per lot to the cost per sample unit is substantial, the optimal number of samples per lot increases as prevalence decreases. However, if the ratio of the cost per lot to the cost per sample unit is sufficiently small, the optimal number of samples per lot reduces to one (i.e., simple random sampling), regardless of prevalence.
Significance: In practice, the cost per sample unit may be large relative to the cost per lot due to the expense of laboratory testing and the presence of natural bottlenecks in the food production and distribution system (e.g., ports of entry) through which many lots must pass. In the food safety domain, sampling plans with few samples per lot are commonly criticized for their lack of statistical rigor; however, the need to balance the tradeoffs between the number and size of clusters has long been appreciated in the field of experimental and survey design.