A model of lung retention of radioactive particles is considered in this work. When a leak of radiation happens in facilities with workers the accident is detected only at the end of the workers' shift when the lters are cheked. The actual time when the leak happens is not konwn and therefore the design space of times to perform bioassays on the workers should be considered as a random set. Optimal designs for di erent possible times of the accident are computed and they are compared with the worst case, when the accident happens at the beginning of the shift. Di erent distributions of the probability are assumed as the distribution of the moment of the accident and designs computed for those situations.
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