1. Optimizing depot locations based on 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!! Fixed costs of the depots:
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,…,xm of 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,
trips,
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. Thank you for your attention!!!
Further research
Combine DP-ILP with clustering
Thank you for your attention!!!