List of Contents
Multi Objective Combinatorial
Optimization: Theory and Application
Horst W. Hamacher
Stefan Ruzika
• Multi Objective Combinatorial Optimization (MOCO)
– Application 1: Cancer Radiation
• Representative Sets for Pareto Solutions
– Application 2: Acquisition Problems in the Car
Industry
• The Adjacency Myth for Solving MOCO
– Application 3: Evacuation Planning
• 2-Phase Method and Ranking of CO Problems
– Application 4: Implementation of Sustainable
Sanitation in existing urban areas
• Research Ideas
MOCO and Applications,
Page 3
Horst W. Hamacher and Stefan Ruzika
Horst W. Hamacher and Stefan Ruzika
Multi Objective Optimization
Multi Objective Optimization
MOCO and Applications,
Page 2
(Eulen nach Athen tragen – to carry coals to Newcastle - llevar agua al mar)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
1
Multi Objective Combinatorial Optimization
(MOCO)
Multi Objective Combinatorial Optimization
(MOCO)
• X = feasible solution of combinatorial optimization problem
• fi(x) = cix linear, integer function, i=1,…,p
Multi Objective …Problem
•
•
•
•
•
•
•
•
•
Spanning Tree
Matroid Base
Minimum Cost Flow
Shortest Path
Knapsack
Assignment
Matching
Linear Programming
…
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
Multi Objective Combinatorial Optimization
(MOCO)
MOCO and Applications,
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Application 1: Cancer Radiation Therapy
Intensity Modulated Radiation Therapy (IMRT)
Most pictures and films
in this part are taken
from:
General References
• Matthias Ehrgott:
Multicriteria Optimiazation, Springer, 2nd edition, 2005
Wolfgang Schlegel
and Andreas Mahr:
• Stefan Ruzika:
On Multiple Objective Combinatorial Optimization,
Ph.D. Thesis, 2007
MOCO and Applications, Magdeburg, Nov 1, 2007
Horst W. Hamacher and Stefan Ruzika
Page 7
Horst W. Hamacher and Stefan Ruzika
3D Conformal
Radiation Therapy:
A multimedia introduction
to methods and
techniques,
2001 (Springer Book and
CD ROM)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
2
Three Optimization Problems in IMRT
Target
volume
Intensity Problem
2D transaxial slice of the body
?
?
?
?
Beam head
?
Which intensity
profile gives
the best
conformal
picture of
tumor?
?
?
?
?
?
Organs at risk
Geometry Problem: Where does the gantry stop?
Intensity Problem: How much radiation is sent off ?
Realization Problem: How is the radiation modulated?
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Intensity Problem:
Multi Criteria LP Model
Page 10
Calculation of Dose Distribution
Discretize
• radiation beams into beam
elements (bixels)
Conflicting criteria:
Horst W. Hamacher and Stefan Ruzika
beam elements
• high radiation in target volume (cancer cells should be destroyed)
• low radiation in organs at risk (organs should stay functional)
Use multicriteria LP model
(with two groups of objectives)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
• body parts into volume
elements (voxels)
In
volume elements
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
3
Calculation of Dose Distribution
• P(i,j) = dose in voxel i irradiated from bixel j under unit
intensity
• dose volume
D=Px
(xj = (unknown!) radiation intensity in bixel j)
• partitioned into
target volume (k=1)
and
organs at risk (k=2,...,K):
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
Find: Intensity vector x  0 satisfying the system of linear
inequalities
(target condition)
Dk = Pk x ≤ Uk e, k = 2, ... , K (risk conditions)
Such an x does in general not exist !
Page 15
• Target volume :
50 Gray (lower bound)
Target volume
• Organs at risk :
20 Gray (upper bound)
15 Gray (upper bound)
10 Gray (upper bound
O rgans at risk
Dk = Pk x
Given: Dose bounds L1  0 and Uk  0, k =2, ... , K
MOCO and Applications,
Find intensity profile x such that dosage D=Px controls cancer
without destroying healthy organs !
D1 = P1 x
Ideal Intensity Profile
D1 = P1 x ≥ L 1 e
Objective
MOCO and Applications,
Page 14
Approximate Intensity Profile using Penalties
minimize µ1F1(x) + ... + µKFK(x) for given weights µ1,..., µK > 0
• Bortfeld, Schlegel, Brahme, Gustafsson ...
F1 (x): = || L1e - P1 x ||2
Fk (x): = || (Pk x - Uk e)+ ||2 , k = 2, ... , K
„Least square approach “
• Holmes, Mackie, Burkard, ...
F1 (x): = || (L1 e - P1 x)+ ||
Fk (x): = || (Pk x - Uk e)+ || , k = 2, ... , K
„Minimax approach “
Disadvantage:
Horst W. Hamacher and Stefan Ruzika
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
time consuming
unsatisfying results
Page 16
Horst W. Hamacher and Stefan Ruzika
4
Intensity Profile using
Multicriteria Optimization
Multi Criteria Data Base:
Graphical Search Tool
Hamacher, Küfer: Discrete Applied Mathematics, 2002
Too many
Pareto solutions !
minimize ( t1, t2, ... , tK )
such that
• P1 x + t1 L1 e  L1 e
• Pk x – tk Uk e  Uk e ,
Patent: ITWM - Trinkaus, 2002
least
tolerable
Representation of radiation in target volume
k = 2, ... ,K
• t = ( t1, t2, ... , tK )  0
• x0
no risk
Compute a representative set of 100-1,000 Pareto solutions and
search within the data base using graphical search tool
MOCO and Applications,
Page 17
most
tolerable
ideal
Horst W. Hamacher and Stefan Ruzika
most
tolerable
Representation of radiation in organs at risk
MOCO and Applications,
Multi Criteria Data Base:
Graphical Search Tool
Page 18
Horst W. Hamacher and Stefan Ruzika
Multi Criteria Data Base:
Graphical Search Tool
Example:
Spine
Heart
Spine
Heart
Possible radiation plan
taken from data base
Tumor
MOCO and Applications,
Tumor
Page 19
Horst W. Hamacher and Stefan Ruzika
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Horst W. Hamacher and Stefan Ruzika
5
Multi Criteria Data Base:
Graphical Search Tool
Spine
Heart
Multi Criteria Data Base:
Graphical Search Tool
Spine
Heart
Radiation plan
with improved heart radiation
Radiation plan
with improved spine radiation
Tumor
Tumor
Page 21
Horst W. Hamacher and Stefan Ruzika
Graphical Search Tool:
Most Recent Version
Page 22
Horst W. Hamacher and Stefan Ruzika
Three Optimization Problems in IMRT
Target
volume
2D transaxial slice of the body
?
?
Spider Web:
Applied in DSS_Evac_Logistic
?
?
Beam head
?
?
?
?
?
?
Organs at risk
Geometry Problem: Where does the gantry stop?
Intensity Problem: How much radiation is sent off ?
Realization Problem: How is the radiation modulated?
MOCO and Applications,, Nov 2014
Page 23
Horst W. Hamacher and Stefan Ruzika
Page 24
Horst W. Hamacher and Stefan Ruzika
6
Application of C1-Dec: MLC
Multileaf Collimator (MLC)
+
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Horst W. Hamacher and Stefan Ruzika
MLC in Action - a Simple Example
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
1
1
2
3
3
3
3
3
3
2
1
1
2
3
4
4
4
4
3
2
1
1
2
3
4
5
5
4
3
2
1
1
2
3
4
5
5
4
3
2
1
1
2
3
4
4
4
4
3
2
1
1
2
3
3
3
3
3
3
2
1
1
2
2
2
2
2
2
2
2
1
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Horst W. Hamacher and Stefan Ruzika
Decomposition Example
 2 5 3
A

 3 5 2
1
1
1
1
1
1
1
1
1
1
 1 1 1  0 1 1   0 1 0 
 2
  1
  2

 1 1 1  1 1 0   0 1 0 
Page 27
back
Horst W. Hamacher and Stefan Ruzika
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Horst W. Hamacher and Stefan Ruzika
7
C1 Matrix
0

1
Y 
0

0
1
  2, 4 

1,3 

  4, 6 


 3, 6 
1 1 0 0

1 0 0 0
0 0 1 1

0 1 1 1 
2
3
4
5
C1-Decomposition of Integer Matrices
Non-negative integer MxN matrix  = (amn)
Find:
„Good“ decomposition of into C1 matrices
A    t Yt
t
with t > 0 for all t.
 ... index set of all C1P matrices
or
... subset of this index set
6
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Horst W. Hamacher and Stefan Ruzika
Decomposition Time
minimize
Given:
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Horst W. Hamacher and Stefan Ruzika
Complexity of Decomposition Time Problem
DT(  ) :   t
DT polynomially solvable:
t '
such that
 Y
t '
t
t
A
Boland, Hamacher, Lenzen (Networks 2004)
t  0
Yt C1  matrix
Ahuja, Hamacher (Networks 2005)
‘  
 1 1 1  0 1 1   0 1 0 
A  2
  1
  2

 1 1 1  1 1 0   0 1 0 
MOCO and Applications,
Page 31
DT()=5
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
8
Complexity of Decomposition Cardinality
Problem
Decomposition Cardinality
minimize
DC(  ) :  t :  t  0
DC is strongly NP-hard:
such that
 Y
t '
t
t
A
t  0
Yt C1  matrix
Baatar, Ehrgott, Ham, Woeginger, Discrete Applied Mathematics, 2005
‘  
 1 1 1  0 1 1   0 1 0 
A  2
  1
  2

 1 1 1  1 1 0   0 1 0 
MOCO and Applications,
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DC()=3
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 34
Objectives are contradictory
Decomposition Sequence
minimize
DS(  ) :

t 1,...,T
t
t 1
( t) (t 1)
A=
 t Yt  A
t  0
Yt C1  matrix
e.g. s(t)(t+1) = dist(Yt,Yt+1) (e.g. = maximal “movement” of ones)
 1 1 1  0 1 1   0 1 0 
A  2
  1
  2

 1 1 1  1 1 0   0 1 0 
MOCO and Applications,
(Marc Nussbaum, 2006)
T 1
T
  s
t 1
such that
Page 35
Horst W. Hamacher and Stefan Ruzika
DS()=5+(1+1)
Horst W. Hamacher and Stefan Ruzika
8 5 6
DT DC DS
5 3 6
= 3
1 0 0
1 1 1
1 1 1
0 0 1
+2
+3
+1
0 0 1
1 0 0
1 1 1
0 1 1
9
4
15
= 5
1 1 0
0 0 1
1 0 0
+6
+3
1 0 0
0 0 1
0 1 0
14
3
17
= 2
0 0 1
1 0 0
1 1 0
1 1 1
+4
+1
+3
1 1 1
1 0 0
0 1 0
0 0 1
10
4
14
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
9
Representative Sets for Pareto Solutions
Quality of Representation
•
Idea:
•
Identify a „small“ set of Pareto solutions „representing“
all Pareto solutions „well“
•
•
•
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
Approximation Algorithm:
Sandwich Procedure
Approximation Algorithm:
Sandwich Procedure
Burkard, Hamacher, Rote (1989)
f2(x)
f2(x)
Complexity vs. Accuracy:
# of evaluations = O ( accuray-2)
min w1f1(x) + w2f2(x)
subject to x 2 P
f1(x)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
f1(x)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
10
Non-Dominated Solutions
(non-supported)
f2(x)
Approximation Algorithm
Box Method
f2(x)
:=lex min (f1(x),f2(x)
s.t. x 2 X
Hamacher, Pedersen, Ruzika (2005)
lex min (f2(x),f1(x)
s.t.
f1(x) · 1
x2X
:= lex min (f2(x),f1(x)
s.t. x 2 X
f1(x)
MOCO and Applications,
Page 41
Horst W. Hamacher and Stefan Ruzika
f1(x)
MOCO and Applications,
Page 42
Approximation Algorithm
Horst W. Hamacher and Stefan Ruzika
Approximation Algorithm
f2(x)
f2(x)
f1(x)
f1(x)
MOCO and Applications,
Page 43
Horst W. Hamacher and Stefan Ruzika
Theorem:
There are no non-dominated points in R2, R3 , R4 , and R5
MOCO and Applications,
Page 44
Horst W. Hamacher and Stefan Ruzika
11
A Posteriori-Algorithm
Approximation Algorithm
f2(x)
f2(x)
f1(x)
Theorem:
The remaining area to be considered is reduced by
at least 50%
MOCO and Applications,
Page 45
Horst W. Hamacher and Stefan Ruzika
A Priori-Algorithm
f1(x)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
Quality Features of the Representation System
Algorithms:
f2(x)
• Accuracy:
• Size of
Representation:
• Cluster Density:
f1(x)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
12
MOCO and Applications,
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MOCO and Applications,
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MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
•
•
•
•
•
MOCO and Applications,
13
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Representative Systems:
Page 54
Horst W. Hamacher and Stefan Ruzika
Application 2: Acquisition Problem
Quality features for more than two criteria?
Decision support system for
KEIPER (car seats)
Crit.
…K
Crit.
1…
Hamacher, Ruzika,Tanatmis (2010)
…
Crit.
3-4
Crit.
1-2
Crit.
1
MOCO and Applications,
…
Crit.
2
Crit.
3
Apply iteratively
scalarization,
sandwich or
box approximation
…
Crit.
4
Page 55
Crit.
(k-1)-1
Crit.
K-1
Crit.
K
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
14
Product Portfolio
Markets
Structures
2000 Products
• Front seats
• Components
• Second row
• Structures
• Third row
Components
Engineering
Services
• Concept
• Headrests
• Recliner
• Tracks
• Tilt adjusters
• Design and styling
surfacing
• Height adjusters
• Comfort
• Locking systems
• Safety and function
• Packaging and DMU
• Sampling
MOCO and Applications,
Page 57
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Event Phases
Potential
Acquisition
Page 58
Horst W. Hamacher and Stefan Ruzika
Acquisition Prioritization Problem
Development
Series
Inquiry
Offer
Concept
Product development and verification
Planning and verification of production
Product inspection of customer
Procurement of production resources
Production
Running time
Series
MOCO and Applications,
Page 59
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
15
Global Company Strategies
Political Strategies
Global Company Strategies
Financial Strategies
Political Strategies
Financial Strategies
Market
Market
12
16
8
15
4
20
5
Global
West. Europe
East. Europe
North America
South America
Africa
Asia
MOCO and Applications,
Product
4
12
11
10
8
9
Customer
Taumel 2000 man
Taumel 2000 CF
Taumel 2000 CF P2
Fold 2000® 3o
Fold 2000® 6o
Lift 2000 Power
Side Track 2000
17
20
8
19
7
12
10
11
Page 61
DaimlerChrysler
GM-Group
BMW-Group
Toyota
Hyundai
VW-Group
Ford-Group
Horst W. Hamacher and Stefan Ruzika
Yield on Sales
Product
Total Sales
Customer
Investment
Return on Investment
Amortization Time
Sales per Investment
MOCO and Applications,
Page 62
Acquisition Prioritization Problem
Horst W. Hamacher and Stefan Ruzika
Multi Objective Model
max
fx
px
 AC x  AB
 IC x  IB
 DC x  DB
t  1,, T
xi  x j
xi  {0,1}
i, j  Dl l  1,, k
i  1,, n
iP
max
iP
s.t.
iP
iP
iP
≤
≤
i i
i i
it i
it i
it i
t
t  1,, T
t
t
t  1,, T
≤
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
16
Keiper Sales Intelligence Tool
Performance Testing
• Setup of Parameters
• Preparation of
Data
• Assistence in
Parameter Entry
•
•
•
•
•
•
• Realization of
Box-Algorithm
Costs uniformly at random in [0,100]
Time horizon: 16
Accuracy: 0.1% of initial rectangle
Number of variables: around 50
Number of constraints: 48
100 instances per setup
• Measuring Performance when …
• Database
• Illustration of
Results
• Decision Support
MOCO and Applications,
1. … increasing number of variables and
2. … increasing number of constraints
while other parameters remain fixed.
Page 65
Horst W. Hamacher and Stefan Ruzika
Varying Number of Variables
MOCO and Applications,
Page 66
Horst W. Hamacher and Stefan Ruzika
The Adjaceny Myth for Solving MOCO
• number of
• representative points
1. Start with Pareto solution
(simple: e.g. lexicographic solution)
2. Go to „adjacent“ Pareto solution by Pareto exchange
3. Stop when no new solution can be obtained in this
way.
• number of
variables
MOCO and Applications,
Page 67
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
17
The Adjaceny Myth for Solving MOCO
The Adjaceny Myth for Solving MOCO
1. Start with Pareto solution
2. Go to „adjacent“ Pareto solution by Pareto
exchange
2. Go to adjacent Pareto solution by Pareto exchange
B
-1,2
A
-1,2
C
0,1
E
4,2

A
0,3
C
0,1
-2,3
D
E
4,2
w(T‘) = w(T) + wDC – wED
= (3,8) + (-2,3) – (4,2)
= (-3,9)
w(T) = (3,8)
MOCO and Applications,
3. Stop when no new solution can be obtained
B
T‘ = T + (D,C) – (E,D)
0,3
Page 69
Myth: All efficient solutions are obtained this way
-2,3
D
Horst W. Hamacher and Stefan Ruzika
Pareto Spanning Trees are not Connected
MOCO and Applications,
Page 70
Horst W. Hamacher and Stefan Ruzika
Pareto Spanning Trees are not Connected
Ehrgott, Klamroth 1997
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
18
Continuous Pareto Network Flows
are Connected
Integer Pareto Network Flows
are not Connected
Przybylski, Gandibleux, Ehrgott (2005):
Isermann (1979)
Adjacency Graph
Adjacency Graph
basic efficient flows
efficient flows
change of flow along
fundamental cycle
MOCO and Applications,
Page 73
Horst W. Hamacher and Stefan Ruzika
Non-Connected Pareto Adjacency Graphs
change of flow
along cycle
MOCO and Applications,
Page 74
Horst W. Hamacher and Stefan Ruzika
Multiobjective Minimal Cost Network Flows
(MMCF)
Gorsky, Klamroth, Ruzika (2006)
Hamacher, Pedersen, Ruzika (2007, EJOR)
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
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Horst W. Hamacher and Stefan Ruzika
19
Multiobjective Minimal Cost Network Flows
(MMCF)
Existing Literature
MMCF
MMCF
Discrete
MMCIF
Continuous
MMCF
exact
Hamacher, Pedersen, Ruzika (2005)
approximate
exact
approximate
exact
approximate
Malhotra, Puri (1984)
Malhotra, Puri (1984)
Lee, Pulat (1991)
Lee, Pulat (1991)
Pulat, Huarng, Lee (1992)
Pulat, Huarng, Lee (1992)
Gonzalez-Martin (2000, 2003)
Page 77
Horst W. Hamacher and Stefan Ruzika
exact
MOCO and Applications,
Page 78
Existing Literature
Horst W. Hamacher and Stefan Ruzika
Existing Literature
Hamacher, Pedersen, Ruzika (2005)
Hamacher, Pedersen, Ruzika (2005)
MMCF
MMCF
Discrete
MMCIF
Continuous
MMCF
exact
approximate
exact
Burkard, Hamacher,
Rote (1989)
Ruhe (1988, 1991)
Fruhwirth, Burkard,
Rote (1989)
Ruhe, Fruhwirth (1990)
MOCO and Applications,
Page 79
approximate
Burkard, Hamacher,
Rote (1989)
Ruhe (1988, 1991)
Fruhwirth, Burkard,
Rote (1989)
Ruhe, Fruhwirth (1990)
Sedeno-Noda,
Sedeno-Noda,
Gonzalez-Martin (2000, 2003)
MOCO and Applications,
Discrete
MMCIF
Continuous
MMCF
Discrete
MMCIF
Continuous
MMCF
approximate
exact
approximate
exact
approximate
Lee, Pulat (1993)
Lee, Pulat (1993)
Gonzalez-Martin,
Sedeno-Noda (2001)
Gonzalez-Martin,
Sedeno-Noda (2001)
Przybylski, Gandibleux,
Ehrgott (2005)
Przybylski, Gandibleux,
Ehrgott (2005)
Figueira (2002)
Horst W. Hamacher and Stefan Ruzika
X
Figueira (2002)
MOCO and Applications,
Page 80
Horst W. Hamacher and Stefan Ruzika
20
Application 3: Evacuation Planning
Dynamic network flow and evacuation
Network representation of a building
Aircraft evacuation
all passengers and crew must be evacuated from the aircraft to
the ground within 90 seconds (Federal Aviation Administration)
(capacity, travel time)
exit
exit
107
Node
25
(4,1)
(8,1)
21
(4,4)
9
(12,1)
(9,2)
(9,2)
exit
0
(9,2)
(10,0)
(3,5)
0
13
exit
10
(9,2)
(16,2)
exit
4
Arc
(9,0)
(5,3)
(12,5)
1
(45,0)
(12,1)
(12,1)
(120,0)
14
11
18
0
(10,0)
(9,2)
exit
(13,3)
(9,2)
(18,2)
(9,2)
22
Building evacuation
107
(3,1)
(9,2)
3
24
0
(9,0)
(9,0)
It is never appropriate to use the elevator during a fire or similar
building emergency-even in a two-story building
(10,0)
(17,1)
8
20
15
(3,2)
(3,2)
(3,3)
(10,0)
0
Page 81
23
(9,2)
2
(9,2)
7
(40,0)
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 82
107
(3,3)
40
(12,1)
12
19
107
Horst W. Hamacher and Stefan Ruzika
Dynamic network flow and evacuation
Dynamic network flow and evacuation
Static Network Representation:
2
1

3
Time t= 0
2
 10  10
5
3 5 5 5 

2
 1 1 1 1  1
7
7
7
7
7
1, 1
5 1
3
10, 3
3
O
4
5, 2
7, 1
4
Occupancy in each room:
1 2 3 4
5 3 2 qi
Capacity and travel time:
1,2 2,3 2,4 3,4
10
5
1
7
3
2
1
1
Page 83
1

2
MOCO and Applications,
17
(20,0)
(16,0)
Emergency planning zones:
Immediate response zone,
Protective action zone, and precautionary zone
(17,1)
5
(12,1)
Region evacuation
Building:
(10,0)
(7,0)
(16,2)
107
(10,0)
maximum evacuation time is 60 minutes
MOCO and Applications,
16
6
0
Ship evacuation
Horst W. Hamacher and Stefan Ruzika
2

3

4

T=5
Occupancy in each room:
1 2 3 4
5 3 2 qi

Flow x(t) in the time expanded network
= dynamic flow in the static network
uij
ij
How many occupants can be
=
evacuated within a given time T ?
MOCO and Applications,
How much dynamic flow can be
sent through the network in T time periods?
Page 84
Horst W. Hamacher and Stefan Ruzika
21
Maximal vs. Earliest Arrival Flow
1
3, 6
1, 1
MDF
1
2
3
4
travel time, capacity
Ford and Fulkerson [62]
4
0, 5
1, 5
3, 6
6
MDF
EAF
2
5
Hoppe and Tardos [1994]
Fleischer and Tardos [1998]
Fleischer and Skutella [2002]
Hamacher and Tjandra [2003])
d
3
0
1, 1
1, 5 0, 5
s
Maximal Dynamic vs. Earliest Arrival Flow
7
0
1
2
3
4
5
6
7
0
1
2
3
4
EAF
5
6
7
0
s
s
s
s
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
d
d
d
d
2
2
2
7
MOCO and Applications,
1
Page 85
1
2
2
7
Horst W. Hamacher and Stefan Ruzika
2
MOCO and Applications,
Current KL-Research in Evacuation Planning
Simulation and optimization
2
2
7
Page 86
1
2
3
4
5
6
7
1
1
2
2
7
Horst W. Hamacher and Stefan Ruzika
Evacuation Drill, University of K-Town
e.g. cellular automata:
Simulating the movement behaviour
Simulation
• Nagel and Schreckenberg (1992)
• Klüpfel, König, Wahle, and
Schreckenberg (2000)
Better solution
Dynamic network flow:
Define the decision values
Optimization
Use
time-dependent multicriteria
dynamic flows and
shortest paths
MOCO and Applications,
• Reachability level at position at time t
Defined by the distance from the safety exit
• Fire level at time t
Distance from the source of fire
Page 87
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 88
Horst W. Hamacher and Stefan Ruzika
22
DSS VisualFlow in DSS_Evac_Logistic
2- Phase Method
Ulungu and Teghem (1994)
• Phase 1: Compute set of supported Pareto solutions
by weighted sum method.
MOCO and Applications, Magdeburg, Nov 1, 2007
Page 89
Horst W. Hamacher and Stefan Ruzika
2- Phase Method
MOCO and Applications,
Page 90
Horst W. Hamacher and Stefan Ruzika
2- Phase Method and Ranking Method
• Phase 1: Compute set of supported Pareto solutions
by weighted sum method
min {c() x = c1x + (1-) c2x : 0<<1 }
y
x1
x5
x2
• Phase 2: Search triangles for non-supported
solutions.
set of 5 best solutions of
c() x =  c1x + (1-) c2x
x4
x3
Find the k best solution of
min c()x
until c()x > c()y
Update triangles iteratively
MOCO and Applications,
Page 91
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 92
Horst W. Hamacher and Stefan Ruzika
23
Binary Ranking Method
Binary Ranking Method
1. Find C1 and C2, the best and second best cut in C
2. Choose arc e 2 C1 n C2
3. Set
• I1 := {e}, O1 :=;
• I2 := ;, O1 := {e}
• K := {1,2}
4. C3 is best of second best cuts in C(Ik,Ok), k 2 K
5. Iterate
Find the k best solution of min cx
Hamacher and Queyranne, 1985
Example: Ranking (s,t) cuts in a network:
C
C = set of all (s,t)-cuts
C(I,O) = set of all (s,t)-cuts containing all arcs of I and no arc of O
Alg2 (I,O) = Algorithm to compute the second best cut in C(I,O)
C1
C (I1,O1)
C1
MOCO and Applications,
Page 93
Horst W. Hamacher and Stefan Ruzika
Binary Ranking Method
MOCO and Applications,
C2
C (I2,O2)
C12
C2
Page 94
C22
Horst W. Hamacher and Stefan Ruzika
Implementation of Sustainable Sanitation in existing
urban areas
Inka Kaufmann, Marcel Kalsch, Tanja Meyer,
Horst Hamacher, Theo Schmitt, 2005-2006
MOCO and Applications,
Page 95
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 96
Horst W. Hamacher and Stefan Ruzika
24
Methods
Methods
initial
situation
initial
situation
objective
MOCO and Applications,
Page 97
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications, Magdeburg, Nov 1, 2007
OCO
d A li
Methods
Horst W. Hamacher and Stefan Ruzika
Methods
initial
situation
initial
situation
objective
MOCO and Applications,
Page 98
i
objective
Page 99
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 100
Horst W. Hamacher and Stefan Ruzika
25
Methods
Methods
initial
situation
initial
situation
objective
MOCO and Applications,
objective
Page 101
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 102
Horst W. Hamacher and Stefan Ruzika
#
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objective
#
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- problem of flooding
- rehabilitation demand
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deficits:
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- catchment data
- topographical data
- (sub)surface data
- infrastructure
 subcatchments and
simplified networks
!"15
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#
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Data analysis and
preparation:
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MOCO and Applications,
Page 103
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 104
Horst W. Hamacher and Stefan Ruzika
26
future state
choice of activities
simultaneous
• project scheduling problem
– find succession of activities
– challenge:
• future state and conditions
– example of future state for implementation
– not all given possible measures have to be chosen
– only those activities should be chosen to fulfil the
future state and with minimal costs
– storm water runoff and wastewater should not be mixed
any more
– decentralised treatment of greywater
– blackwater should be treated centrally in WWTP
• network flow problem
– different flows, e.g.. design flow or pollution
loads must fulfil constraints and conditions
– time-dependent network
 hypothetical, not necessarily best state
– constraints
– obliged values for resources protection (35 % direct
reduction of water consumption / 20 % greywater reuse)
– obliged values for water balance (55 % evaporation / 25 %
infiltration / 20 % effluent)
MOCO and Applications,
Page 105
Horst W. Hamacher and Stefan Ruzika
 complex model structure
MOCO and Applications,
Page 106
mathematical model structure
Horst W. Hamacher and Stefan Ruzika
results – activities
#
Y
!
#
Y
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#
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33
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Page 108Y
M57_1
Horst W. Hamacher and Stefan Ruzika
19
#
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#
Y
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#
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MOCO and Applications,
44
!"2 5
10
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#
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#
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Horst W. Hamacher and Stefan Ruzika
#
Y
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#
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27
results – activities
Research Ideas
period under consideration [years]
16
15
14
0
unp.
rainw. ut. 1
red. gw
red. bw
gw rec. t
bw c
CS47_1
nFS13_1
nFS47_1
nFS47_1
nFS47_1
CS12_1
CS13_1
CS14_1
unp.
ret sc
red. gw
red. bw
gw rec. r t
bw c
nFSRÜ 2
nFSRÜ 2
CS47_1
unp.
red. gw
red. bw
gw rec. r t
bw c
nFS50_1
CS48_1
5
10
15
20
25
30
35
40
45
50
55
• Box method requires more research in -constrained
scalarization:
– Constrained Knapsack
– Constrained Paths
– Trees
separating greywater and
blackwater
Ruzika, 2007
Ruzika, 2007
Henn, Ruzika, 2007
• Parametric polyhedral analysis is needed
separating rainwater and foul
water
f1(x) = 
X
unp. unpaving; ret (sc) – rainwater retention (semi centralised); rainw. ut.
rainwater utilization; red. gw / bw – reduction water consumption greywater /
blackwater; gw rec. (r) t – grywater recycling (and reuse) with technical device; bw
c – centralised blackwater treatment; CSxy sewer combined sewer system; nFSxy –
new foul sewer system
MOCO and Applications,



Page 109
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 110
Horst W. Hamacher and Stefan Ruzika
There is hardly any optimization problem which
is not multi objective !
Research Ideas
• Use MOCO or, more general, multi objective
optimization to solve single objective problems
– semi-simultaneous flows  Engau, Hamacher, 2006
– branch and bound
 Ruzika 2007
THE END
• „Reasonable“ adjacency definitions making the
adjacency graph connected.
MOCO and Applications,
Page 111
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 112
Horst W. Hamacher and Stefan Ruzika
28
References
Bilder von zitierten Autoren
•
•
•
Ulungu
•
Kouvelis
Teghem
Sayin
•
MOCO and Applications,
OCO
d A li
Page 113
Horst W. Hamacher and Stefan Ruzika
MOCO and Applications,
Page 114
Horst W. Hamacher and Stefan Ruzika
i
29