Technical Lab · Synthetic demonstration

Single-vehicle routing with time windows

One depot, one vehicle and eight synthetic customers. The objective is travel time. A window or the capacity limit can reject the trip that looks shortest on the map.

All technical examples

Synthetic demonstration · Independent educational implementation. Not the doctoral routing study.

Visit every customer from a depot and return, without missing a delivery window or exceeding the vehicle capacity.

A shorter route that arrives after the window, or that overfills the vehicle, is not a usable plan. The constraint changes the decision, not only the picture.

See which constraint binds

Point at or tap a customer for its demand, window and arrival.

Route, arrows give direction
Delivery window
Service starts

Travel time is the objective. Windows and capacity decide which tours are allowed. Depot in the centre, customers A–H, and the minimum-travel route that respects the selected windows and capacity. The square is the depot. Letters are customers; small numbers give the visit order. An open dot and a dotted line mean the vehicle arrived early and waited.

Standard windows, capacity 32. The exact search visits C–B–A–H–G–F–E–D and returns to the depot. Travel time is 204 minutes and the route waits 12 minutes for windows to open. Load is 31 of 32. The objective is travel time; service time does not change the ranking of complete routes.

Solver: exact enumeration for this small teaching instance

Schedule

Times are minutes from departure. A wait means the vehicle arrived before the window opened.

Customer windows and planned arrivals
Stop	Window	Demand	Arrival	Wait
Customer A	0–150	4	98	0
Customer B	20–180	3	68	0
Customer C	40–220	5	28	12
Customer D	60–260	4	245	0
Customer E	80–300	3	217	0
Customer F	90–330	4	187	0
Customer G	70–300	5	157	0
Customer H	15–190	3	126	0

About this example

Method: Exact exhaustive search for one vehicle and eight stops. A branch is dropped when the load will not fit or the vehicle would arrive after the window closes.
What the code does: It minimises travel time, waits if it arrives early, and records the arrival, the wait and the load. Service time is included in the schedule. It does not change which complete route is shortest.
Assumptions: Travel minutes equal straight-line distance. There is one vehicle, one depot and no road network. Customers A–H, their demand and their windows are invented.
Result: Relaxed windows allow a short tour with no waiting. Standard windows keep a short tour but the vehicle waits. Tight windows force a longer tour. Capacity 18 cannot carry every order, so there is no route.
Limitations: This is one vehicle and one depot, not a general multi-vehicle solver, a city network or a driver’s shift. It is not the doctoral delivery study. A larger routing problem would go to a solver such as OR-Tools.

Implementation

Objective Minimise travel time back to the depot. Constraints Visit each customer once. Arrive no later than the window end. Load never exceeds capacity.

src/lib/lab/routing.ts View source on GitHub: src/lib/lab/routing.ts

A visit is rejected before it is added to the route.src/lib/lab/routing.ts

function canVisit(problem: RoutingProblem, stop: RouteStop, time: number, load: number, x: number, y: number): { start: number; travel: number } | null {
  if (load + stop.demand > problem.capacity) return null;
  const travel = travelMinutes({ x, y }, stop, problem.speed);
  const arrival = time + travel;
  if (arrival > stop.due) return null;
  return { start: Math.max(arrival, stop.ready), travel };
}

The doctoral case study is a separate, published simulation. This map does not reproduce it.