How Many Stops Fit in a Delivery Round? A Time Bound for a Single Rural Postman

A delivery round is planned by drawing a boundary on a map, but what actually limits it is time. This paper bounds the number of stops a round can contain: from a 360-minute working round, fixed overhead takes 75 minutes, each stop takes 1.6 minutes of service, and travel between stops in a 6.0 km2 area follows the Beardwood-Halton-Hammersley square-root estimate at bicycle speed. The bound is 116 stops, using 98.9 minutes of travel and 185.6 minutes of service; one more stop would take the round to 361.6 minutes. On foot the same area supports 59 stops and by motorbike 144. Tables for areas from 2 to 15 km2 show travel, not service, becoming the binding term as the area grows — which is why enlarging a round's territory fails before enlarging its stop count does. SIMULATED, NOT MEASURED.