Optimizing depot locations based on a public transportation timetable

1. Optimizing depot locations basedon a public transportation timetable Marjan van den Akker, Han Hoogeveen Marcel van Kooten Niekerk, QBuzz

2. Outline • Problem description • Vehicle scheduling • Clustering heuristic • Integer linear programming • Computational results

3. Problem description Given: • Timetable= Collection R trips with: • Given start and finishing time • Given start en finishing location • Collection of buses • Assumption: one type of bus • Collection S of depots • Number of depots N to be opened

4. Problem description (2) • Goal: • Find set of depot locations • Find feasible assignment of trips to busses • Minimize total cost • Such that: • Each trip is performed by exactly one vehicle • Depot capacity is not exceeded • Number of buses starting at a depot equals the number of trip ending at the depot.

5. Total cost • Time is money!! • cost units are minutes • Fixed costs of the depots: • Neglected with fixed number of depots • Fixed costs per vehicle • 1000 units • Variable vehicle and driver costs: • 120 units per hour for a driving bus • 60 units per hour for a bus standing still outside the depot

6. Estimating duration of deadhead trips • With unknown depot locations many possible deadhead trips • Approximation: time to drive Euclidean distance with constant speed • 20 km/h then for 80 % of calculated duration upper bound on real duration % calc duration ≤ real duration speed

7. VSLP: Scheduling vehicle tasks • Linear program • Decision variables: • Xij = 0/1 signals if trip i and j are performed consecutively • Xsi = 0/1 signals if vehicle goes from depot s to trip i • Xis = 0/1 signals if vehicle goes from trip i to depot s • Reduce number of variables by allowing mid day parking at depots • Minimize total cost Subject to: • Every trip exactly one successor • Every trip exactly one predecessor • Number of buses leaving depot = number of buses returning to depot • Number of buses leaving parking = number of buses returning to parking

8. Two approaches • Clustering heuristic using K-means algorithm • Depot location ILP

9. Clustering heuristic (with K-means) • Generate vehicle tasks using linear programming VSLP with unknown depot locations • Generate N depot locations • Assign start- and endpoints of vehicle tasks to nearest depot. • Optimize depot locations based on start and endpoints assigned in step 3. • If assignment has changed repeat steps 3 and 4, otherwise go to step 6 • Regenerate vehicle schedules with VSLP with current depot locations.

10. Step 2: generating N depot locations • Randomly from uniform distribution on smallest rectangle containing all start and end points. • Randomly from uniform distribution on convex hull of start and end points • Facility location ILP on raster of 1 km

11. Step 4: Optimize depot locations based on start and end points • Given a set x1,x2,…,xmof start and end points for depot ys • Geometric median: • Approximation:

12. DLIPL: Depot location ILP • Extension of VSLP • Ys= 0/1 if depot s is closed/opened • Additional constraints: • Depot is only used when opened • Number of depots equals N

13. Computational results • 4 real-life instances from the Netherlands, • 200-1700 trips, • 20-150 vehicles • 2,3,...,8 depots • Clustering with random points: 106 runs • Cost of solution: DL-ILP ≤ Cluster FL ≤ Cluster convex ≤ Cluster rectangle • Computation time: DL-ILP >> Cluster 1 % sligthly sligthly

14. Further research • Combine DP-ILP with clustering Thank you for your attention!!!