Optimal Food Safety Sampling Under a Budget Constraint

Abstract

Much of the literature regarding food safety sampling plans implicitly assumes that all lots entering commerce are tested. In practice, however, only a fraction of lots may be tested due to a budget constraint. In such a case, there is a tradeoff between the number of lots tested and the number of samples per lot. To illustrate this tradeoff, a simple model is presented in which the optimal number of samples per lot depends on the prevalence of sample units that do not conform to microbiological specifications and the relative costs of sampling a lot and of drawing and testing a sample unit from a lot. The assumed objective is to maximize the number of nonconforming lots that are rejected subject to a food safety sampling budget constraint. 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. In practice, the cost per sample unit may be large relative to the cost per lot due to the expense of laboratory testing and other factors. Designing effective compliance assurance measures depends on economic, legal, and other factors in addition to microbiology and statistics.

Optimal Inspection of Imports to Prevent Invasive Pest Introduction

“…Our goal of minimizing expected slippage, or the expected number of accepted defective individuals, is an alternative to that of minimizing the expected number of accepted defective lots, which is the objective of some applications of acceptance sampling in public safety programs. (4,15,16) Even though our approach instructs inspectors to focus on larger lots with higher plant infestation rates (Table III), we expect that solutions that minimize the number of accepted defective lots will allocate sampling effort to lots with higher infestation rates with less regard to lot size. For example, in our case with an inspection capacity of 2,193 (Table III), only the two largest among five Codiaeum lots (which has the highest estimated infestation rate) are sampled (and rather intensively) under the objective of minimizing expected slippage.…”

“…(1) Although acceptance sampling is usually described in the context of manufacturing, it plays an important role in public safety programs by helping administrators attain the program quality standards. Acceptance sampling is used in programs to maintain food safety, (2)(3)(4) control grain purity, (5) monitor illegal drug use, (6,7) reduce the spread of animal diseases, (8) and prevent the introduction of damaging pests on agricultural imports. (9) The literature on acceptance sampling in public safety programs focuses on single sampling plans for individual lots.…”

“…Work is needed on this broader problem of allocating resources to inspection sampling based on externally imposed resource constraints. (4,10) We address the problem of allocating a targeted sampling budget among a set of lots that are received and inspected simultaneously and can differ in their sizes, composition, costs of sampling, and damages from defective items in accepted lots. The objective is to determine the single acceptance sampling plan for each lot to minimize the total damages caused by defective items across accepted lots subject to the budget constraint.…”

“…We use the term "targeted sampling budget" to represent the portion of a total sampling budget that is available for acceptance sampling. 5 We first derive a formula for the expected number 4 While the literature on acceptance sampling in public safety programs focuses on single sampling plans, other types of sampling plans, including double sampling, multiple sampling, sequential sampling, and skip-lot sampling, have been developed and standardized (e.g., see NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/ div898/handbook/pmc/section2/pmc22.htm). For example, in the context of import inspection sampling for invasive species, Batabyal and Yoo (11) study a random inspection scheme based on the "continuous sampling" plan first formulated by Dodge.…”

“…In recent work similar to ours, authors allocate a fixed inspection budget among lots or shipment pathways to minimize the expected number or cost of accepted nonconforming lots. (4,15,16) Our work departs from those models by deriving a formula for expected slippage (number of defective items in an accepted lot) and allocating the inspection budget to individual lots to minimize the aggregate cost of slippage across heterogeneous lots. Yamamura et al (10) address a similar problem of allocating a fixed number of samples across lots to minimize the total number of defective items admitted in accepted lots; however, they use simplifying assumptions that both defective rate and sample proportion of each lot are small.…”