In the control of environmental contaminants, it is often useful to sample at preselected locations to determine concentrations and their means. These locations might be on a surface, throughout a room, or outdoors. Applications include air and water pollution control, industrial hygiene, and contamination control in industry. Contamination is a major cause of yield and reliability losses in the microelectronics industry. Sampling the cleanroom environment or sampling the product surfaces can help diagnose and prevent contamination problems, but sampling is becoming increasingly expensive. One wants to use sampling resources effectively to achieve desired low levels of uncertainly. We assume that the locations to be sampled have been selected, perhaps as described by Cooper et al. We show how to calculate the optimal number of samples to be taken at each location so as to minimize the uncertainty in the mean over the entire region under study, subject to a cost constraint. We consider two distinet criteria for measuring this uncertainty. We also address the optimal allocation for minimizing the cost, subject to an upper bound on the standard error. We also discuss the differences between these approximate solutions and the true solutions, which are integers.

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