Pay phones, parking meters, vending machines, and optimal Bayesian decisions on collection times

Payphones, parking meters, and vending machines illustrate the modem busine ss practice of substituting machinery for manpower. They do nor eliminate m anual labor completely, because full coin-boxes still must be replaced and vending machines still must be stocked. Deciding when to replace a coin-box is important, with unequal losses resulting from underestimation and overe stimation. This article derives optimal methodology for this problem by inc orporating collection history and specifying common prior distributions ove r average daily fill rate and standard deviation at each box. The approach is implemented and analyzed on collection records from 11,308 pay phones ov er a large geographical region. When the loss from overestimation is 19 tim es that from underestimation, our methods outperform the one in current use at least 69.9% of the time, translating into average potential collection- cost reductions exceeding 21%.