Early and Late Cost Interdependence

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Figure 9.1 illustrates a cost-time relationship for a single delivery that has both early and late costs. Cost for an early delivery declines as it approaches the earliest desired arrival time (tmin), at which point cost is zero. After the latest desired arrival time (tmax), cost increases as time increases.

The relationship becomes more complex when you consider two visits, V1 and V2, such that V2 is directly after V1 on the route. Figure 9.2 depicts the cost function for these visits based upon arrival time at V1, where the time window for V1 is [2, 5]; the time window for V2 is [4, 8]; and the transit time from V1 to V2 is 1.

Under certain circumstances, a vehicle might need to wait between V1 and V2, causing an effective increase in the transit time. Figure 9.3 depicts the same two visits, V1 and V2, but with a transit time of 3.

• Just as for a standard PDP, delivery trucks will pick up parcels en route and deliver them on the same route. As a consequence, capacity constraints on the vehicles may reach a limit along the route, rather than when departing the depot.
• Pickups as well as deliveries have constraints on the time windows when the parcel can be picked up or delivered. Pickup or delivery activities outside these time windows incur additional costs for the delivery.