How Long Should an Uncollected Parcel Wait? A Holding-Period Rule from the Decay of Collection

Every parcel office holds a shelf of the uncollected, and every office has a rule — a week, a fortnight, a month — for when a parcel goes back to its sender. The rule is rarely derived. This paper derives it from the shape of collection itself: recipients collect with a probability that decays over days (modelled as a Weibull curve with a stated scale of 4.2 days), a stated 6.0 percent never collect at all, a shelf day costs USD 0.09, and a return costs USD 3.80. The cost-minimising holding period is 26 days, by which 92.7 percent of parcels have been collected, at an expected cost of USD 0.855 per parcel — against USD 0.861 for the common month-long rule and USD 1.375 for a strict week. A reminder message on day three, modelled as quickening the remaining collection, shortens the optimal shelf life to 22 days: reminders do not just speed collection, they change how long the shelf should be. SIMULATED, NOT MEASURED.