modern operational research and its mathematical methods. introduction
DESCRIPTION
AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.TRANSCRIPT
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Modern Operational Research and Its Mathematical Problems
4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
1August 7, 2009
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber ** and Baand Başşak Aktekeak Akteke--ÖztürkÖztürk
Institute of Applied Mathematics Institute of Applied Mathematics Middle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey
** Faculty of Economics, Management and Law, Universi ty of Siegen, GermanyFaculty of Economics, Management and Law, Universi ty of Siegen, Germany
Center for Research on Optimization and Control, Univ ersity of Aveiro, Portugal
and Its Mathematical Problems
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4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
1. Introduction into Operational Research and Its Mathematical Problems
2. Methods from Mathematical Data Mining (Supported by Optimization)2.1. Clustering Theory2.3. Classification Theory2.3. Regression Theory
2August 7, 2009
2.3. Regression Theory
3. Further Advanced Methods from Mathematical Optimization3.1 On Foundations of Continuous Optimization3.2 Nonsmooth Optimization3.3 Elements of Semi-Infinite Optimization
4. Applications of Mathematical Operational Research4.1 Quality Control and Improvement in Manufacturing4.2 Prediction of Credit Default and of Financial Processes4.3 Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks
5. Conclusion
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Introduction into Operational Research and Its Mathematical Problems
4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
3August 7, 2009
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber ** and Baand Başşak Aktekeak Akteke--ÖztürkÖztürk
Institute of Applied Mathematics Institute of Applied Mathematics Middle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey
** Faculty of Economics, Management and Law, Universi ty of Siegen, GermanyFaculty of Economics, Management and Law, Universi ty of Siegen, Germany
Center for Research on Optimization and Control, Univ ersity of Aveiro, Portugal
and Its Mathematical Problems
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HELP!!
Making Better
What is OR?
4August 7, 2009
Making Better Decisions
London, 15/2/2005
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What is OR?
Improving Communication
5August 7, 2009
Communication
Bruxelles, 28/1/2005
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Take Initiatives
What is OR?
6August 7, 2009
Milano, 28/4/2005
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Solving
hard problems
What is OR?
7August 7, 2009
hard problems
München, 1/7/2005
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Opening
new horizons
What is OR?
8August 7, 2009
new horizons
Istanbul, 5/7/2005
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History of OR
9August 7, 2009
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History of OR
10August 7, 2009
George B. DantzigJohn von Neumann Harry M. Markowitz
and many other ones
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• Railway freight transport has a market share of 20%.
• 100,000 Mil. ton km, of which:
� 45% inland traffic,
� 45% cross-border traffic,
� 10% transit traffic.
• Deutsche Bahn offers whole trains (~30 cars) and individual cars.
• Several individual cars with different destinations
Routing Cars in Rail Freight Service
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• Several individual cars with different destinationsare grouped to trains at classification yards.
• At the next classification yard, the cars are re-grouped, until they reached their destinations.
• Main question: what is the “best” path for each car?
A. Fügenschuh, H. Homfeld, A. Martin, H. SchülldorfRouting Cars in Rail Freight Service,OVERSYS, June 25, 2009.
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• Railway network length: 38,200 km
• 5000 trains per day, 150,000 cars
• Terminal stations: 2,200
• Classification yards:
� Large („Rangierbahnhöfe“): 11
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� Large („Rangierbahnhöfe“): 11
� Medium („Knotenbahnhöfe“): 30
� Small („Satellitenbahnhöfe“): 200
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Routing Cars in Rail Freight Service
Classification Yards
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• Disintegration of trains
• Sorting the cars (with the help of gravity)
• Assembling of new trains
Routing Cars in Rail Freight Service
Classification Yards
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entry tracks hump sorting tracks exit tracks
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Optimization Problem
• Minimize the total costs for all trains, cars, and the used infrastructure.
• Subject to
Routing Cars in Rail Freight Service
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• Subject to� Each order is routed through the network,
� The maximal transportation time is not exceeded,
� The trains are neither too long nor too heavy,
� The hump capacities are respected,
� The number of sorting tracks is not exceeded,
� The cars are routed according to the DB operation rules („Leitwege“).
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Bundling Effect
• Cost induced by cars
100
1040
0
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100
6030
30
40
0
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• Cost induced by cars
100
1040
0
Bundling Effect
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100
6030
30
40
0
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• Cost induced by cars
100
1040
200
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
200
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• Cost induced by cars
100
1040
200
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
200
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• Cost induced by trains
100
1040
0
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
0
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• Cost induced by trains
100
1040
0
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
0
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• Cost induced by trains
100
1040
70
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
70
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• Cost induced by trains
100
1040
70
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
70
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• Cost induced by trains
100
1040
190
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
130
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• Cost induced by trains
100
1040
190
Bundling Effect
Routing Cars in Rail Freight Service
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100
6030
30
40
130
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• Cost induced by trains
100
1040
230
Bundling Effect
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100
6030
30
40
170
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• Cost induced by trains
1040
230
Bundling Effect
Routing Cars in Rail Freight Service
100
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30
30
40
170
100
60
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Modes of Operation
• Three ways of sending cars from origin to destination:� Individual car routing
• Assign a sequence of yards to each car
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200 200
Ex.: Individual Car Routing
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300
200 200
200200
100
0
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200 200
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200 200
Ex.: Individual Car Routing
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Modes of Operation
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Arc Flow Model
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Arc Flow Model
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Arc Flow Model
Routing Cars in Rail Freight Service
Improving the Arc Flow Model
Heuristic Cuts: Hierarchy Constraints
•
•
•
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Refing the Model: Turnover Times
Computational Results
•
•
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Multi-Class Queueing Networks
1 2
6
3
{1,..., }
{ ( ), 0}k
K
Q t t
Κ =>
Queues/Classes
6K =
Routing Processes
(0)kQ k ∈ΚInitial Queue Levels
' ( ) , 'Φ ∈Κkk n k k
Resources
4I =
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5 4
( ) ∈ΚkS t k
Processing Durations
{1,..., }
{ } {0,1}I K ik ik
I
A A A×
Ι == ∈
Resource Allocation (Scheduling)
( )
( )
(0) 0
( ) ( )∈Κ
=
− ≤ − <∑
k
k
ik k kk
T t
T
A T t T s t s s t
Network Dynamics
' ' ''
( ) (0) ( ( )) ( ( ( )))∈Κ
= − + Φ∑k k k k k k k kk
Q t Q S T t S T t
Transient Fluid Solutions and Queueing Networks with Infinite Virtual Queues Infinite Virtual Queues
Y. Nazarathy, G. Weiss
14th INFORMS Applied Probability Conference,Eindhoven , July 9, 2007
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Ex.: network
Server 1Server 2
1
2
33
10
( )T
kk
Q t dt=∑∫
Attempt to minimize:
Multi-Class Queueing Networks
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Sta
cked
Que
ue L
evel
s
time T
Q1
Q2Q3
Trajectory of a single job
Finished Jobs
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Fluid formulation
1 2 3
0
1 1 1 1
0
2 2 1 1 2 2
min ( ( ) ( ) ( ))
( ) (0) ( )
( ) (0) ( ) ( )
µ
µ µ
+ +
= −
= + −
∫
∫
∫ ∫
T
t
t t
q t q t q t dt
q t q u s ds
q t q u s ds u s ds
such that
Server 1Server 2
1
2
3
Multi-Class Queueing Networks
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2 2 1 1 2 2
0 0
3 3 2 2 3 3
0 0
1 3
2
( ) (0) ( ) ( )
( ) (0) ( ) ( )
( ) ( ) 1
( ) 1
( ), ( ) 0
µ µ
µ µ
= + −
= + −
+ ≤≤
≥
∫ ∫
∫ ∫t t
q t q u s ds u s ds
q t q u s ds u s ds
u t u t
u t
u t q t
(0, )t T∈
This is a Separated Continuous Linear Program (SCLP).
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Fluid solution
Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).
The Optimal Solution:
20
3( )q t
Multi-Class Queueing Networks
61August 7, 2009
0 10 20 30 40
0
5
10
15
2( )q t
1( )q t
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VILLAGE STUDIES IN CENTRAL ANATOLIA
The changing face of the Anatolian Village
62August 7, 2009
The changing face of the Anatolian Village
Indigenous v/s contempory
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The Anatolian VillageFrancoise Summers and Soofia T. Elias-ÖzkanDepartment of Architecture, METU, Ankara, Turkey
Gerhard-Wilhelm Weber Institute of Applied Mathematics, METU, Ankara, Turkey
63August 7, 2009
A common scene in Central Anatolia portraying abandoned traditional village houses among which new ones with pyramidal clay tile roofs are built.
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The Anatolian Village
64August 7, 2009
A common scene in Central Anatolia portraying abandoned traditional village houses among which new ones with pyramidal clay tile roofs are built.
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Energy and Comfort
65August 7, 2009
Stove used in villages.
Solar water heater.
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Environmental Performance of Buildings
66August 7, 2009
Poster for the PLEA Conference 2003
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Krep: 19 cm hollow clay blocks with concrete slab supporting a clay tile roof, NW facing window;
Kgue: 30 cm sandwich hollow clay blocks with insulation and insulated aerated concrete low pitch roof, NE facing window;
Blue: 19 cm hollow brick with
Environmental Performance of Buildings
67August 7, 2009
Blue: 19 cm hollow brick with timber rafters and ceiling
supporting a pyramidal claytile roof, SE (shaded by balcony roof) and NE facing windows;
Baba: a traditional stone and mudbrick house with flat mud roof, south facing window shaded by balcony roof.
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Temperature
68August 7, 2009
Comparative Graphs
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Environmental Studies
69August 7, 2009
Ecotect for analysis
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Bioclimatic Buildings
ECO MUDBRICK AND STRAWBALE GREEN HOUSETEMPERATURE CHART(16 SEPT -10 OCT 2005)
35
40
70August 7, 2009
0
5
10
15
20
25
30
18/0
9/20
05
19/0
9/20
05
20/0
9/20
05
21/0
9/20
05
23/0
9/20
05
24/0
9/20
05
25/0
9/20
05
26/0
9/20
05
28/0
9/20
05
29/0
9/20
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30/0
9/20
05
01/1
0/20
05
03/1
0/20
05
04/1
0/20
05
05/1
0/20
05
Time
Tem
pera
ture
(C
)
exterior mudbrick interior greenhouse interior
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sustainable living
sun
water
cowproducts
(milk, meat)plant
enlargedbasis for
marketability
investment
Balaban Valley
D. DeTombe, A., I., H. Gökmen, T. Bali,
S. Belen, J. Körezlioğlu, H. Önder , H. Tuydes, W.
Applications in the Energy Sector – Development
71August 7, 2009
heat(dunk)
wastebiogas reactorheat or
electricalenergy
high density energy(tractor fuel, electricity)
fertilizer
cost savings and increasedexpendable income and
savingsinvestment
expenditure on energy consumption
socio-econo – environment networks
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Gene-Environment Networks
72August 7, 2009
if gene j regulates gene i
otherwise
,i iξ ζl
1:
0i jχ =
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Earth Warming
73August 7, 2009
Mean temperature anomalies during the period 1995 to 2004 with respect to the average temperatures from 1940 to 1980.
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Networks and Dynamics
74August 7, 2009
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Sequence Data(cDNA, Genome,Genbank, etc.)
Laser Scan of the Array
Test Material Control Material
mRNA-Isolation
cDNA-Synthesisand Labeling
Comp. Bio. & Med.
75August 7, 2009
Selection or Design andSynthesis of the Probes
Array Production
Picture AnalysisHybridization
Array Preparation Sample Preparation Data Analysis
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Comp. Bio. & Med.
Comp. Bio. & Med.
76August 7, 2009
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:( )) ( ( ( ))= F Q E ts t
where
( ) ( ) ( )( ) ( ) ( )•
= + +s t s t s tE t E t E tM C D
Networks and Dynamics
77August 7, 2009
1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t=
,1( ) <i iE t θθθθ
,1 ,2( )< <i i iE tθ θθ θθ θθ θ
( )<ii,d iE tθθθθ
0 for
1 for( ( )) :
...
for
i
i
Q E t
d
=
θθ11,,11 θθ11,,22
θθ22,,11
θθ22,,22
hybrid systems
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( ( ))P m P t P•
= −
•
•
•
Anticipatory Systems
logistic equation:
delay retarding - advancing anticipation
will
memory wish
78August 7, 2009
•
• •
1444444442444444443( , ( ), ([ 1]))x f t x t x t
•= +( , ( ), ([ ]))x f t x t x t
•=
( , ( ), ([ ]), ([ 1]))x f t x t x t x t•
= +
memory wish
anticipation
extended Malthusian model
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delay retarding - advancing anticipation
will
memory wish•
•
•
Anticipatory Systems
79August 7, 2009
memory wish
anticipation •
• •
1444444442444444443( , ( ), ([ ]))x f t x t x t
•=
( , ( ), ([ ]), ([ 1]))x f t x t x t x t•
= +
economy( , ( ), ([ 1]))x f t x t x t
•= +
extended Malthusian model
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Impulsive Systems
quasilinear impulsive integrodifferential equations:
80August 7, 2009
nse
optimal control of response:
nse
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object oriented
sports (and general) medicine
Further Systems
81August 7, 2009
object oriented
modules
DAE ODE
training (or reconvalescence) program
( , )⋅
= ⋅ Mx N v x K
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Optimization
82August 7, 2009
I, K, L finite
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• An essential tool for “unsupervised” learning is cluster analysis which suggests categorizing data (objects, instances) into groups such that the likeness within a group is much higher than the one between the groups.
Clustering
83August 7, 2009
• This resemblance is often described by a distance function.
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The disjoint subsets πi (S), i=1,…,k, are named clusters:
( ) and for .π π π= ∩ = ∅ ≠k
i i jS S , i jUUUU
Clustering
84August 7, 2009
1
( ) and for .π π π=
= ∩ = ∅ ≠i i ji
S S , i jUUUU
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The second synthetic data set has the parameters
and σ = 0.3.5ˆ =k
Clustering
85August 7, 2009
The components are obviously overlapping in this case.
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Stock Markets
86August 7, 2009
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drift and diffusion term
( , ) ( , )= +t t t tdX a X t dt b X t dW
Stochastic Differential Equations
Financial Mathematics
87August 7, 2009
Wiener process
(0, ) ( [0, ])∈tW N t t T
Ex.: price , wealth , interest rate , volatility
processes
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Value-at-Risk (VaR) = α - percentile of distribution of random variable
(a smallest value such that probability that random variableis smaller or equals to this value is greater than or equal to α)
Financial Mathematics
88August 7, 2009
Random variable, ξ
Fre
qu
en
cy
1 − α
VaRProbability
Maximalvalue
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-Finding characteristics “critical-to-quality”-finding input variables that significantly affect quality output
- Predicting quality- quality output is a real-valued variable,- finding empirical models which relate input characteristics of quality to output ones,
Quality Analysis
89August 7, 2009
- finding empirical models which relate input characteristics of quality to output ones,-using such models to predict what the resulting quality characteristics will be for a
given set of input parameters
- Classification of quality- for nominal, binary or ordinal outputs- for a given set of input parameters, predicting the class of the quality output
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Quality Control and Improvement
transmission cases
engine block
oil pan
90August 7, 2009
gearbox
oil pan
printed circuit boards
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L curve :
5
5.5
5
5.5
5
5.5
5
5.5
Numerical Experience and Comparison
Quality Control and Improvement
91August 7, 2009
2(
)d
yψ
θ−
ψθ
−ψ
θ−
ψθ
−
0 0.2 0.4 0.6 0.8 1 1.2 1.42.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1 1.2 1.42.5
3
3.5
4
4.5
5
2Lθθθθ
2Lθθθθ
2(
)d
yψ
θ−
ψθ
−ψ
θ−
ψθ
−
0 0.2 0.4 0.6 0.8 1 1.2 1.42.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1 1.2 1.42.5
3
3.5
4
4.5
5
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desirability functions (quality opt.)
max-, min-type
continuous selections
normal forms smoothening
Quality Control and Improvement
92August 7, 2009
stability
instabilitybilevel
problems
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What is EURO?http://www.euro-online.org
• The Association of Operational Research Societies in Europe (within IFORS).
• Established the 29th of January, 1975, in Brussels.
93August 7, 2009
Brussels. • Representing today more than 10000 OR practitioners and academics all around Europe.
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What is EURO?
• ΕπιχειρησιακήΕρευνα
• Añgerñargreining
94August 7, 2009
• Operačního Výzkumu
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EURO
95August 7, 2009
EUEURORO
online.org/-http://www.euro
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EURO
EURO aims is to promote Operational Researchthroughout Europe.
http://www.ifors.org/
96August 7, 2009
EUROis also one currency.
Please visit the EURO Web Site Forumto send us comments and suggestions about this site.
online.org/-http://www.euro
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EURO
97August 7, 2009
/description#description505543http://www.elsevier.com/wps/find/journaldescription.cws_home/
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European Journal of Operational Research
• Established in 1975.
• At that time: 6 issues and 200 pages annually.
• In 2006: 24 issues and 9000 pages annually.
98August 7, 2009
• In 2006: 24 issues and 9000 pages annually.
• 35% acceptance rate
• 6251 cites in 2004 (2nd worldwide)
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0,6
0,7
0,8
0,9
EJOR Impact Factor
99August 7, 2009
0,3
0,4
0,5
0,6
1998 1999 2000 2001 2002 2003 2004
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EURO
100August 7, 2009
online.org/-http://www.euro
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EURO
101August 7, 2009
online.org/-http://www.euro
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EURO
102August 7, 2009
online.org/-http://www.euro
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EURO today is
• Annual budget of 200000€• Permanent office located in Brussels: Philippe Van Asbroeck, Véronique Bastin, Bernard Fortz and in Fribourg: Marino Widmer
• Executive Committee: Valerie Belton, Martine
103August 7, 2009
• Executive Committee: Valerie Belton, Martine Labbé, Gerhard Wäscher, Marc Sevaux, Bjarni Kristjansson, Jesper Larsen
• IFORS Vice President: Grazia Speranza• EURO Web Site: http://www.euro-online.org
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What EURO does?
• Organizes the EURO conferences.• Supports the EURO Working Groups.• Supports the young OR researchers through the ESWI and the ORP3
• Supports the colleagues from weak currency
104August 7, 2009
• Supports the colleagues from weak currency countries and in Africa.
• Discerns a number of awards: EDDA, EDSM, EEPA, EGM, MSSIP.
• Publishes EJOR (through Elsevier)
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Special Initiatives
• The Africa Project
105August 7, 2009
• Branding OR
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Do we need to evolve?
- European job market
- European business
- European research funding
106August 7, 2009
- European research funding
- European training standards
- Europe in progress …
We have a role to play in this process
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What are we looking for?
• Improve EURO structure and organization.
• Improve and increase EURO services.
• Increase and diversify funding of EURO.
• Increase and improve visibility and impact of
107August 7, 2009
• Increase and improve visibility and impact of Operational Research in Europe
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References
http://www.euro-online.org/display.php?pageid=102&
http://www.elsevier.com/wps/find/journaldescription.cws_home/505543/description
http://www.bookya.de/autor/Wolfgang+Domschke/ Bücher von Wolfgang Domschke
108August 7, 2009