1 Optimizing depot locations based on a public transportation timetable Marjan van den Akker, Han Hoogeveen Marcel van Kooten Niekerk, QBuzz
Feb 25, 2016
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Optimizing depot locations basedon a public transportation
timetable
Marjan van den Akker,Han Hoogeveen
Marcel van Kooten Niekerk, QBuzz
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Outline
Problem description Vehicle scheduling Clustering heuristic Integer linear programming Computational results
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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
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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.
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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
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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
speed
% calc duration ≤ real duration
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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
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Two approaches
Clustering heuristic using K-means algorithm Depot location ILP
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Clustering heuristic (with K-means)
1. Generate vehicle tasks using linear programming VSLP with unknown depot locations
2. Generate N depot locations3. Assign start- and endpoints of vehicle tasks to nearest
depot.4. Optimize depot locations based on start and endpoints
assigned in step 3.5. If assignment has changed repeat steps 3 and 4, otherwise
go to step 66. Regenerate vehicle schedules with VSLP with current depot
locations.
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Step 2: generating N depot locations
a. Randomly from uniform distribution on smallest rectangle containing all start and end points.
b. Randomly from uniform distribution on convex hull of start and end points
c. Facility location ILP on raster of 1 km
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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:
n
1jsjyyxmin
s
m
1j ij
m
1j ij
j
1i
yx1
yxx
y
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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
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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
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Further research
Combine DP-ILP with clustering
Thank you for your attention!!!