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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!!!