Freie Universität Berlin
Bus Scheduling and Multicommodity Flows
Ralf Borndörfer
2015 Workshop on Combinatorial Optimization with Applications in
Transportation and Logistics
Beijing, 28.07.2015
1 Bus Scheduling and Multicommodity Flows | CO∈TL 2015
Freie Universität Berlin Outline
Optimal Assignments
Single Depot Vehicle Scheduling
Multiple Depot Vehicle Scheduling
Lagrangean Relaxation
Multicriteria Optimization
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 2
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 3
The Assignment Problem
The Problem
Input: 3 sources, 3 destinations, costs
Output: cost minimal assignment
1 2 3
3 4
3 3
7 6 7
8 10
1 2 3
solution cost = 20
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 4
solution cost = 17
A Heuristic
The Greedy Heuristic
heuretikos (gr.): inventive
heuriskein (gr.): to find
1 2 3
3 4
3 3
7 6 7
8 10
1 2 3
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 5
A Heuristic
The Greedy Heuristic
heuretikos (gr.): inventive
heuriskein (gr.): to find
1 2 3
3 4
3 3
7 6 7
8 10
1 2 3
solution cost = 16
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 6
A Lower Bound
A "Relaxation"
1 2 3
3 4
3 3
7 6 7
8 10
1 2 3
11 12 13 14 15 16 17 18 19 20 21 22
upper bound (solution) lower bound duality gap
bound cost = 13 solution
cost = 17 guarantee 4/17=23% 4/13=30%
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 7
optimum cost = 15
A Proof of Optimality
The "primal problem"
Minimum cost assignment
The "dual problem"
Maximum profit sales
5 4 0
3 4
3 3
7 6 7
8 10
7 8 9
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 8
0 0 0
3 3
4 4 3
3
3 3 7
7
6 6
7 7 8
8 10 10
0 0 0
An Exact Algorithm
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 9
0 0 0
3 3
4 4 3
3
3 3 7
7
6 6
7 7 8
8 10 10
0 0 0
Bound Costs: 0
Partial Sol. Costs: 0
0 0 0 0
0 0 0
An Exact Algorithm
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 10
0 0 0
3 0
4 0 3
0
3 0 7
4
6 2
7 4 8
5 10 6
0 0 0
Bound Costs: 10 Partial Sol. Costs: 3
0 0 0 0
0 0 0
+3 +4 +3
+0 +0 +0
An Exact Algorithm
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 11
An Exact Algorithm
0 0 0
3 0
4 0 -3
0
3 0 7
4
6 2
7 4 8
5 10 6
3 3 4
Bound Costs: 10 Partial Sol. Costs: 3
0 0 0 0
0 0 0
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 12
An Exact Algorithm
0 0 0
3 0
4 0 -3
0
3 0 7
4
6 2
7 4 8
5 10 6
3 3 4
Bound Costs: 10 Partial Sol. Costs: 6
0 0 0 0
0 0 0
+0 +0 +0
+0 +0 +0
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 13
An Exact Algorithm
0 0 0
-3 0
4 0 3
0
-3 0 7
4
6 2
7 4 8
5 10 6
3 3 4
Bound Costs: 10 Partial Sol. Costs: 6
0 0 0 0
0 0 0
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 14
An Exact Algorithm
0 0 0
-3 0
4 0 3
0
-3 0 7
4
6 2
7 4 8
5 10 6
3 3 4
Bound Costs: 15 Partial Sol. Costs: 15
0 0 0 0
0 0 0
+5 +5 +4
+5 +4 +0
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 15
An Exact Algorithm
5 4 0
3 0
-4 0 3
1
-3 0 7
3
6 1
7 0 -8
0 10 1
7 8 9
Bound Costs: 15 Partial Sol. Costs: 15
0 0 0 0
0 0 0
The "successive shortest path" algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 16
An Exact Algorithm
5 4 0
3 0
4 0 3
1
3 0 7
3
6 1
7 0 8
0 10 1
7 8 9
Bound Costs: 15 Solution Costs: 15
Guaranty: 0% (Optimal)
The "successive shortest path" algorithm
computes a shortest path for every source node,
i.e., does n shortest path calculations.
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 17
An Exact Algorithm
5 4 0
3 0
4 0 3
1
3 0 7
3
6 1
7 0 8
0 10 1
7 8 9
Bound Costs: 15 Solution Costs: 15
Guaranty: 0% (Optimal)
Theorem: The assignment problem can be
solved in polynomial time of O(n3).
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 19
Algebraic Model
Graphen theoretic model Algebraic Model "Integer Program"
1 2 3
3 4
3 3
7 6 7
8 10
1 2
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 21 31
12 22 32
13 23 33
11 33
11 33
min 3 3 4
3 7 6
7 8 10
s.t. 1
1
1
1
1
1
, , 0
, , {0,1}
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x
x x
3
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 20
LP-Relaxation
Integer Program Linear Program
"LP-Relaxation" (here: integer)
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 21 31
12 22 32
13 23 33
11 33
11 33
min 3 3 4
3 7 6
7 8 10
s.t. 1
1
1
1
1
1
, , 0
, , 1
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x
x x
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 21 31
12 22 32
13 23 33
11 33
11 33
min 3 3 4
3 7 6
7 8 10
s.t. 1
1
1
1
1
1
, , 0
, , {0,1}
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x
x x
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 21
LP-Relaxation
Linear Program
"LP-Relaxation"
Eliminate x11
Eliminate x12, x13, x21, x31
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 21 31
12 22 32
13 23 33
11 33
11 33
min 3 3 4
3 7 6
7 8 10
s.t. 1
1
1
1
1
1
, , 0
, , 1
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x x
x x
x x
12 13 12 13
21 22 23
31 32 33
12 13 11
21 22 23
31 32 33
12 13 21 31
12 22 32
13 23 33
12 13 33
12 13 33
min 3(1 ) 3 4
3 7 6
7 8 10
s.t. 1
1
1
1 1
1
1
1 , , 0
1 , , 1
x x x x
x x x
x x x
x x x
x x x
x x x
x x x x
x x x
x x x
x x x
x x x
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 22
LP-Relaxation
Linear Program "LP-Relaxation"
"Polyhedron"
22 23 32 33
22 23 32 33
22 23
32 33
22 23 32 33
min 4 2 1 2 14
s.t. 1
1
1
, , , 0
x x x x
x x x x
x x
x x
x x x x
x22+x23
x32+x33
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 23
Mathematical Models
x32=1 x21=1 x13=1
"Polyhedron"
1 2 3
3 4
3 3
7 6 7
8 10
1 2 3 x22+x23
x32+x33
1 2 3
3 4
3 3
7 6 7
8 10
1 2 3
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 24
Linear Programming
Min x1 + 2x2
x1 + x2 2
x1 - x2 ≤ 1
x2 ≤ 3
x1 0
x2 0
x1
x2
Polyhedron
Simplex Algorithm
Freie Universität Berlin
Linear Programming 1987-2000 (Bixby, Solving Real-World Linear Programs: A Decade and More of Progress. Oper. Res. 50(1) 3-15, 2002)
Hardware
Software
"A Model that might have taken a year to solve 10 years ago, can now solve in less
than 10 seconds."
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 25
Old Computer New Computer Speedup
Sun 3/50 Pentium 4, 1.7 GHz 800
Sun 3/50 Compaq Server ES 40, 667 MHz 900
Intel 386, 25 MHz Compaq Server ES 40, 667 MHz 400
IBM 3090/108S Compaq Server ES 40, 667 MHz 45
Old Code New Code Estimated Speedup
XMP Cplex 1.0 4.7
Cplex 1.0 Cplex 5.0 22,0
Cplex 5.0 Cplex 7.1 3.7
XMP Cplex 7.1 960
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 26
Linear Programming
Min x1 + 2x2
x1 + x2 2
x1 - x2 ≤ 1
x2 ≤ 3
x1 0
x2 0
x1
x2
Polyhedron
Simplex Algorithm
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 27
Integer Programming
Min x1 + 2x2
x1 + x2 2
x1 - x2 ≤ 1
x2 ≤ 3
x1 0
x2 0
x1, x2 integer
x1
x2
Branch-and-Bound
x1 ≤ 1 x1 2
Polyhedron
Freie Universität Berlin
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 28
Integer Programming
Min x1 + 2x2
x1 + x2 2
x1 - x2 ≤ 1
x2 ≤ 3
x1 0
x2 0
x1
x2
Cutting Planes
Polyhedron
Freie Universität Berlin
MIP-Speedup 1991-2010 (Bixby, Lecture on Mixed-Integer Programming, TU Berlin, 20.01.2010)
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1
10
100
1000
10000
100000
0
1
2
3
4
5
6
7
8
9
10
1.2→2.1 2.1→3 3→4 4→5 5→6 6→6.5 6.5→7.1 7.1→8 8→9 9→10 10→11
Cu
mu
lati
ve
Sp
ee
du
p
Ve
rsio
n-t
o-V
ers
ion
Sp
ee
du
p
CPLEX Version-to-Version Pairs
V-V Speedup Cumulative Speedup
Mature Dual Simplex: 1994
Mined Theoretical Backlog: 1998 29530x
Freie Universität Berlin
Sea Freight (Koopmans [1965], 7 sources, 7 sinks, all sea links)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 30
Freie Universität Berlin
Optimal Allocation of Scarce Resources (Nobel Price in Economics 1975)
Leonid V. Kantorovich
Tjalling C. Koopmans
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 31
Freie Universität Berlin LP Solution via Shadow Prices
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 32
Freie Universität Berlin Planning Problems in Public Transit
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 33
Service Design
Operational Planning
Operations Control
Freie Universität Berlin Vehicle Scheduling
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 34
too short to turn too long to wait
best choice
Freie Universität Berlin "Camel Curve"
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 35
68
Freie Universität Berlin
"Camel Curve" (Ario, Böhring & Mojsilovic [1980])
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 36
Freie Universität Berlin
Assignment Approach (Single Depot Vehicle Scheduling)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 37
1
3
4
6
5 5
2 5
D D
4
6
1
3
5 2
Freie Universität Berlin
Assignment Approach (Single Depot Vehicle Scheduling)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 38
4
6
1
3
5 2
1
3
4
6
5 5
2 5
D D
Freie Universität Berlin
Single Depot Vehicle Scheduling (Assignment Approach)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 39
D D 1 2 3
D D 1 2 3
4
4
D D 1 2 3
D D 1 2 3
4
4
1
1
2
2
3
3
D
D
4
4
1
1
2
2
3
3
D
D
4
4
Freie Universität Berlin Vehicle Scheduling Problem
Input
Timetabled and deadhead trips
Vehicle types and depot capacities
Vehicle costs (fixed and variable)
Output
Vehicle rotations
Problem
Compute rotations to cover all timetabled trips
Goals
Minimize number of vehicles
Minimize operation costs
Minimize line hopping etc.
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 40
Freie Universität Berlin Multicommodity Flow Model
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 42
Freie Universität Berlin Vehicle Scheduling (VS-OPT)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 43 43
Freie Universität Berlin
yIntegralit,}1,0{
Capacities
Trips1
Flow Aggregate0
Flow Vehicle,0
min
dijdijx
dd
j
ddjx
j
d
dijx
j
d d k
djkx
i
dijx
dj
i k
djkxd
ijx
d ij
dijx
dijc
Integer Programming Model (Multi-Commodity Flow Problem)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 44
Freie Universität Berlin Theoretical Results
Observation: The LP relaxation of the Multicommodity Flow Problem is in general not integer.
Theorem: The Multicommodity Flow Problem is NP-hard.
Theorem (Tardos et. al.): There are pseudo-polynomial time approximation algorithms to solve the LP-relaxation of Multicommodity Flow Problems which are faster than general LP methods.
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 45
Freie Universität Berlin
yIntegralit,}1,0{
Capacities
Trips1
Flow Aggregate0
Flow Vehicle,0
min
dijdijx
dd
j
ddjx
j
d
dijx
j
d d k
djkx
i
dijx
dj
i k
djkxd
ijx
d ij
dijx
dijc
Integer Programming Model (Multi-Commodity Flow Problem)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 47
Freie Universität Berlin
Binary,}1,0{
Capacities
Trips1
Flow Agg.0
Lagrangeanminmax,
dijx
dx
jx
jxx
xxxc
d
ij
d
j
d
dj
d
d
ij
d d i
d
ji
i
d
ij
dj i i
d
ji
d
ij
d
j
d ij
d
ij
d
ij
Lagrangean Relaxation (Subproblem is a Min Cost Flow Problem)
Subproblem: Min-Cost Flow
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 48
Freie Universität Berlin Lagrangean Relaxation
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 49
min
0
T
b
Bx d
x
c x
Ax
min (
0
max )
Tb
d
x
c x Ax
Bx
(max min ) T
i ii
bc x Ax
( )max f
x1
x2
x3
x4
P(A,b)P(B,d)
f1
f2
f3
f4 f P(B,d)
Freie Universität Berlin
yIntegralit,}1,0{
Capacities
Trips1
Flow Aggregate0
Flow Vehicle,0
min
dijdijx
dd
j
ddjx
j
d
dijx
j
d d k
djkx
i
dijx
dj
i k
djkxd
ijx
d ij
dijx
dijc
Integer Programming Model (Multi-Commodity Flow Problem)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 50
Freie Universität Berlin
Binary,}1,0{
Capacities
Trips1
Flow Agg.0
Lagrangeanminmax,
dijx
dx
jx
jxx
xxxc
d
ij
d
j
d
dj
d
d
ij
d d i
d
ji
i
d
ij
dj i i
d
ji
d
ij
d
j
d ij
d
ij
d
ij
Lagrangean Relaxation (Subproblem is a Min Cost Flow Problem)
Subproblem: Min-Cost Flow
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 51
Freie Universität Berlin
Lagrangean Relaxation (Subproblem is a Min Cost Flow Problem)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 52
Freie Universität Berlin
Lester Randolph Ford Jr.
Delbert Ray Fulkerson
Network Flows
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 53
Freie Universität Berlin
Military Logistics (Ford & Fulkerson [1955], Schrijver [2002])
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 54
Freie Universität Berlin MCF
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 55
Freie Universität Berlin CINT 2006
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 56 Ralf Borndörfer
56
Freie Universität Berlin
Lagrangean Pricing Algorithm (Löbel [1997])
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 57
Freie Universität Berlin Lagrangean Relaxation II
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 58
0
min
0 , Vehicle flow
0 Aggregated flow
1 Timetabled trips
Depot capacities
{0,1} , Deadhead trips
d d
ij ij
d ij
d d
ij jk
i k
d d
ij jk
d i d k
d
ij
d i
d
j d
j
d
ij
c x
x x j d
x x j
x j
x d
x ij d
Freie Universität Berlin
Subproblem: Several independent Min-Cost-Flows (single-depot)
0
min
0 , Vehicle flow
0 Aggregated flow
Timetabled trips
Depot capacities
{0,1} , Deadhead trips
max 1
d d
ij ij
d ij
d d
ij jk
i k
d d
ij jk
d i d k
d
j d
j
d
ij
d
ij
d i
c x
x x j d
x x j
x d
x ij d
x
Lagrangean Relaxation II
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 59
Freie Universität Berlin Heuristics
Cluster First – Schedule Second
"Nearest-depot" heuristic
Lagrange Relaxation II + tie breaker
Schedule First – Cluster Second
Lagrange relaxation I
Schedule – Cluster – Reschedule
Schedule: Lagrange relaxation I
Cluster: Look at paths
Solve a final min-cost flow
Plus tabu search
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 60
61
IVU.plan / IVU.crew / IVU.vehicle System Overview with Optimization Modules
IVU.crew
IVU.vehicle IVU.plan
timetable
stops routes
vehicle roster
duty roster
depot management
payroll accounting
vehicle dispatch
personnel dispatch
vehicle
workings
duties
BS-OPT VS-OPT
Integrated
Plan
nin
g
IS-OP
T
DS-OPT
R-OPT AFD
APD WS-OPT
Freie Universität Berlin BVG (Berlin)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 62
Freie Universität Berlin Urban Scenarios
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 63
BVG HHA VHH
depots 10 14 10
vehicle types 44 40 19
timetabled trips 25 000 16 000 5 500
deadheads 70 000 000 15 100 000 10 000 000
cpu mins 200 50 28
Freie Universität Berlin
yIntegralit,}1,0{
Capacities
Trips1
Flow Aggregate0
Flow Vehicle,0
min
dijdijx
dd
j
ddjx
j
d
dijx
j
d d k
djkx
i
dijx
dj
i k
djkxd
ijx
d ij
dijx
dijc
Integer Programming Model (Multi-Commodity Flow Problem)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 64
Freie Universität Berlin Lagrangean Relaxation
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 65
min
0
T
b
Bx d
x
c x
Ax
min (
0
max )
Tb
d
x
c x Ax
Bx
(max min ) T
i ii
bc x Ax
( )max f
x1
x2
x3
x4
P(A,b)P(B,d)
f1
f2
f3
f4 f P(B,d)
Freie Universität Berlin Subgradient Method
max
𝑋 =conv 𝑥𝜇 polyhedral (piecewise linear)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 66
f
1
1f
2
f̂
3
T T( ) ( )f c x b Ax
ˆ ( ) : min ( )
k
kJ
f f
T T( ) : min ( )
x Xf c x b Ax
𝜆𝑘+1: = 𝜆𝑘 + 𝑢𝑘(𝑏 − 𝐴𝑥𝜇𝑘)
subgradient
Freie Universität Berlin
Bundle Method (Kiwiel [1990], Helmberg [2000])
max
𝑋 =conv 𝑥𝜇 polyhedral (piecewise linear)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 67
f
1
1f
2
f̂
3
T T( ) ( )f c x b Ax
2
1ˆ ˆargmax ( )
2k
k k k
uf
ˆ ( ) : min ( )
k
kJ
f f
T T( ) : min ( )
x Xf c x b Ax
Freie Universität Berlin Primal Approximation
Theorem:
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 68
( )k k Nx converges to a point : ,x x Ax b x X
T( ) ( )k k kf c x b Ax
ˆkf
z
f
1kf
1k
1
k
kJ
x x
1
1ˆ ( )
k
k kJ
b Axu
0 ( )kb Ax k
Freie Universität Berlin Quadratic Subproblem
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 69
2ˆ ˆmax ( )
2k
k k
uf (1)
2ˆmax
2
s.t. ( ), for all
kk
k
uv
v f J
(2)
2
1ˆmax ( ) ( )2
s.t. 1
0 1, for all
k k
k
kJ J
J
k
f b Axu
J
(3)
min
Freie Universität Berlin
Bundle Method (IVU41 838,500 x 3,570, 10.5 NNEs per column)
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 70
50
100
150
200
250
300
350
400
20 40 60 80 100 120 140 sec
450
bundle volume barrier
cascent
0
0
Freie Universität Berlin
Ralf Borndörfer 71
Heuristic Evaluation of f
f
f‘
4 1 2 3
4f̂
5f
Freie Universität Berlin Thank you for your attention
Ralf Borndörfer Freie Universität Berlin Zuse-Institute Berlin Takustr. 7 14195 Berlin-Dahlem Fon (+49 30) 84185-243 Fax (+49 30) 84185-269
[email protected] www.zib.de/borndoerfer
Bus Scheduling and Multicommodity Flows | CO∈TL 2015 94