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Hard Mixed Integer Programsin Practice
Do cutting planes really help?
Alexander MartinTU Darmstadt
DIAMANT Workshop „Integer Programming Day“TU Eindhoven
January 27, 2006
A. Martin 2
Mixed Integer Program
p = n: Integer Programp = 0: Linear Program
c
A. Martin 3
The Branch-and-Cut Method
c
1P 2P
c
P
A. Martin 4
ReferenzenMIPs in Darmstadt
• Optimization of gas networks
• UMTS network planning
• Integrated planning of school buses in public mass transport
• Design of Clos networks
• Integral sheet metal construction of a higher bifurcation order
• Modelling the power consumption in public buildings
• Facility location problems for service companies
• Optimal Partitioning of Block Structured Grids
• Semidefinite and Polyedral Relaxations for Graph Partitioning
• Protein folding
A. Martin 5
1. The Problem
2. Modelling Non-linear Functions- with binary variables- with SOS constraints
3. Polyhedral Analysis
4. Computational Results
Outline: Optimization of Gas Networks
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- contracts- Physicalconstraints
Goal
Subject To
Minimize fuel gas consumption
Optimization of Gas Networks
A. Martin 7
Gas Networks in Detail
A. Martin 8
Gas Networks: Nature of the Problem
• Non-linear- fuel gas consumption of compressors- pipe hydraulics- blending, contracts
• Discrete- valves- status of compressors- contracts
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Pressure Loss in Gas Networks
stationarycase
horizontalpipes
pout
pin
q
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Approximation of Pressure Loss: Binary Approach
pin
pout
q
A. Martin 11
Approximation of Pressure Loss: SOS Approach
pin
pout
q
(1) must meet thetriangle condition
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Branching on the Triangle Condition
31=iλ
1=∑ iλ 1=∑ iλ
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The SOS Constraints: General Definition
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The SOS Constraints: Special Cases
• SOS Type 2 constraints
• SOS Type 3 constraints
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The Binary Polytope
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The Binary Polytope: Inequalities
21=iλ
21=iy
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The SOS Polytope
Pipe 1 Pipe 2
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|Δ| |Y| Vertices Facets Max. Coeff.
8 12 16 18 25
16 18 49 47 42
24 24 73 90 670
32 32 142 10492 50640
The SOS Polytope: Increasing Complexity
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The SOS Polytope: Properties
Theorem. There exist only polynomially manyvertices
• The vertices can be determined algorithmically• This yields a polynomial separation algorithm by
solving for given and
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The SOS Polytope: Generalizations
• Pipe to pipe with respect to pressure and flow• Several pipes to several pipes• Pipes to compressors (SOS constraints of Type 4)• General Mixed Integer Programs:
Consider Ax=b and a set I of SOS constraints of Type for such that each variable is contained in exactlyone SOS constraint. If the rank of A (incl. I) and are fixed then
has only polynomial many vertices.
A. Martin 21
Binary versus SOS Approach
• Binary- more (binary) variables- more constraints- LP solutions with fractional y variablesand correct λ variables
• SOS+ no binary variables+ triangle condition can be incorporated
within branch & bound+ underlying polyhedra are tractable
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Computational Results
Nr of Pipes Nr of Compressors
Total lengthof pipes
Time (ε = 0.05)
Time (ε = 0.01)
11 3 920 1.2 sec 2.0 sec
20 3 1200 1.2 sec 9.9 sec
31 15 2200 11.5 sec 104.4 sec
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Example SOSε = 0.01
Delta Method 1)
Gap TimeLambda Method 2)
Gap Time
net 1 2.0 s 0 % 0h:2m:49s 0 % 17h:18m:24s
net 2 9.9 s 0 % 2h:5m:43s 68,9 % > 1 day
net 3 104.4 s > 1 day > 1 day
Computational Results: A Comparison
1) Wilson and Lee (2000)2) Text book approach
88
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• The Problem
• Modelling as a MIP
• The MIP Approach
• Heuristics
• Computational Results
Cooperation: EU-Projekt MOMENTUM
Operators: TNO, E-Plus, Vodafone Portugal
Vendors: Siemens Mobile
R&D: Atesio, TU Darmstadt, TU Lissabon, ZIB
Planning UMTS Networks
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© Digital Building Model Berlin (2002), E-Plus Mobilfunk GmbH & Co. KG, Germany
• ScenarioBerlin Alexander Platz
• Network- 16 potential sites- 3 antennas per site
• Demand / Trafficvoice - telephonyvideo - telephonyfile - downloadstreaming multimedia
Planning UMTS Networks
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• ScenarioBerlin Alexander Platz
• Network- 16 potential sites- 3 antennas per site
Planning UMTS Networks
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• ScenarioBerlin Alexander Platz
• Network- 16 potential sites- 3 antennas per site
• Demand / Trafficvoice - telephonyvideo - telephonyfile - downloadstreaming multimedia
Planning UMTS Networks
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Decisions– Sectorization– Antenna height– Antenna tilt– Antenna type– Pilot power
Planning Decisions
Question:Which sites should beconfigured in whichway to satisfy thedemand?
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• W-CDMA• Multi-service • CIR-target• Self Interference• Network Quality
voice uservoice uservoice uservoice uservoice uservoice user
videotelephony
user
UMTS – Universal Mobile Telecom. System
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interference
S
I
≥ R
• W-CDMA• Multi-service • SIR-target• Self Interference• Network Quality
voice uservoice uservoice uservoice uservoice uservoice user
videotelephony
user
UMTS – Universal Mobile Telecom. System
A. Martin 31
interference
S
I
≥ R
other cell interference
• W-CDMA• Multi-service • SIR-target• Self Interference• Network Quality
voice uservoice uservoice uservoice uservoice uservoice user
videotelephony
user
UMTS – Universal Mobile Telecom. System
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interference
C
I
≥ Rother cell int.
The Model: Variables
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interference
C
I
≥ Rother cell int.
The Model: Constraints
A. Martin 34
interference
C
I
≥ Rother cell int.
The Model: Constraints
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interference
C
I
≥ Rother cell int.
The Model: Constraints
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The complete Model
A. Martin 37
site
s
inst
alla
tions
pilo
t pow
ers
mobile assignmentUL powerDL power
traffic snapshot
The complete Model for one Snapshot
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assignment UL powerDL power
site
s
inst
alla
tions
pilo
t pow
ers
assignmentUL powerDL power
. . .
traffic snapshot
traffic snapshot
. . .The complete Model for several Snapshots
A. Martin 39
© Digital Building Model Berlin (2002), E-Plus Mobilfunk GmbH & Co. KG, Germany
Planning UMTS Networks
1) The MIP Approach- Preprocessing- Heuristic Cuts- Exploiting the MIPs
2) Heuristics- Installation Selection- Mobil Assignment- Power Control
3) Solutions
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inim xx ≤Best Client Cuts
jim zx −≤ 1Best Server Cuts
MIR Cuts
there are more …
0)( 1
≤+
− ↑
↑↑↑
mi
mim
im px m
η
αγμ
Complex polyedral structureHeuristic Cut
Heuristic Cuts
m
n
i
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The Hague
• Only one forth of the sites are needed
• Good Quality
• Running time less than 15 minutes
Solutions: The Hague
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1. The Problem
2. A MIP Model
3. Solution methods
4. Computational results
5. Implementation in Practice
Outline: Optimizing School buses
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The Planning of School Buses
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0102030405060708090
100
05:30 06:00 06:30 07:00 07:30 08:00 08:30 09:00
Busse (IST)
0
5
10
15
20
25
07:30
07:35
07:40
07:45
07:50
07:55
08:00
08:05
08:10
08:15
08:20
08:25
08:30
Schulen (IST)
The Planning of School Buses
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pupils
county
schools bus company
transports
pays
ZIV
convinces
negogiates
pays
negogiates
The Planning of School Buses
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Integrierte Optimierung der Schulanfangszeiten und des Nahverkehrsangebots
Integrated Optimization of School Starting Times and thePublic Mass Transport (IOSANA)
The Planning of School Buses
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8:09
8:05
8:04
0:05
0:02
0:04
Depot
start of tripα t1
bus schedules xt1t2
TRIPSt ∈2
7:26
7:28 7:30 7:35
7:36
7:377:41
7:518:00
t1 ∈ TRIPS
start of school sτ
A MIP Model: Variables
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6:30 - 6:457:00 - 7:300:12
0:08 0:260:49
0:30
8:00 - 8:40
Basic Model (without schools): VRP TW
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min Ct xΔ ,t + δt1t2
shift xt1t2t1 ,t2
∑t∑
α t1+ δt1
trip +δt1t2
shift ≤α t2+ M(1− xt1t2
)
xt1t2t2
∑ + xΔ ,t1=1
xt1t2t1
∑ + xt2 ,Δ =1
α t ≤α t ≤α t
• Goal: Minimizenumber of busesand deadhead trips
• Join trips to tours
• Synchronize times
• Attend to time windows
Basic Model (without schools): VRP TW
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8:00
7:26
7:35
7:377:41
7:51
8:00
0:05-0:45
7:28 7:30
7:36
α t
τ s
α t + δstschool +ω st
school ≤ 5τ s
α t + δstschool +ω st
school≥ 5τ s
VRP CTW: Extention to coupled time windows
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? 96?1
Solution Methods
• Preprocessing (exact)- Bound Strengthening- Variable fixing
• Primal- Parametrized greedy heuristic- Improvement heuristics
• Dual- Branch-and-Cut- Column generation
A. Martin 52
66 96
82 9671
501
Branch-and-Cut
parametrized greedylocal search
Greedy
Column generation
Computational Results
A. Martin 53
0102030405060708090
100
05:30 06:00 06:30 07:00 07:30 08:00 08:30 09:00
Busse (IST)Busse (PLAN)
0
5
10
15
20
25
07:30
07:35
07:40
07:45
07:50
07:55
08:00
08:05
08:10
08:15
08:20
08:25
08:30
Schulen (IST)
Schulen (PLAN)
Computational Results
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Stakeholders and conflicts– students, parents– teacher, head of school– county– bus company
Performance in Practice
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• Lotte/Tecklenburg von 8:00 auf 8:20 (Neue OZ, 13.3.2003)
– „Das Konzept ist klasse, aber wir wollen es nicht“[The concept is brilliant, but we don‘t want it]
– „Das geht auf Kosten der Familien“– „Die Mikrowelle wird der Freund der Familie“
[The micro wave will be the friend of the family]– Folge des BPI-Konzeptes sei es, durch den späteren Schulschluss in der sechsten Stunde
Unterricht im biologischen Lerntief erteilen zu müssen[As a consequence, lessons must be given in the biological low of learning]
– [Die] Benachteiligung der Schüler im ländlichen Raum - etwa durch lange Wege bei Museumsbesuchen - dürfe nicht noch verstärkt werden
– In keinen Betrieb könne man von außen hineinreden, aber in die Schule. „Wir sind schließlich die Fachleute für den pädagogischen Bereich. Gespart werden muss. Dabei sind unsere Kinder die falsche Adresse“, bekräftigte [der Rektor einer Tecklenburger Schule]
Performance in Practice
A. Martin 56
Summary
There are interesting and challenging MIPs in practice• The Gas Problem
- challenging: non-linearities- use SOS in the right way
• The UMTS Problem- challenging: size, numerics- heuristic cuts might help
• The Bus Problem- methods are available to some extend- challenging: Implementation in practice
But, for none of them (polyhedral) cuts really helped !?
