Modern Operational Research and Its Mathematical Methods. Introduction

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


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

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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

Introduction into Operational Research and Its Mathematical Problems


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

HELP!!

Making Better

What is OR?

4August 7, 2009

Making Better Decisions

London, 15/2/2005

What is OR?

Improving Communication

5August 7, 2009

Communication

Bruxelles, 28/1/2005

Take Initiatives

What is OR?

6August 7, 2009

Milano, 28/4/2005

Solving

hard problems

What is OR?

7August 7, 2009

hard problems

München, 1/7/2005

Opening

new horizons

What is OR?

8August 7, 2009

new horizons

Istanbul, 5/7/2005

History of OR

9August 7, 2009

History of OR

10August 7, 2009

George B. DantzigJohn von Neumann Harry M. Markowitz

and many other ones

• 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.

• 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


Classification Yards

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• Disintegration of trains

• Sorting the cars (with the help of gravity)

• Assembling of new trains


Classification Yards

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entry tracks hump sorting tracks exit tracks

Optimization Problem

• Minimize the total costs for all trains, cars, and the used infrastructure.

• Subject to


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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“).

Bundling Effect

• Cost induced by cars

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Bundling Effect


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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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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


Improving the Arc Flow Model

Heuristic Cuts: Hierarchy Constraints

•

•

•

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Refing the Model: Turnover Times

Computational Results

•

•

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

Ex.: network

Server 1Server 2

1

2

33

10

( )T

kk

Q t dt=∑∫

Attempt to minimize:


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Sta

cked

Que

ue L

evel

s

time T

Q1

Q2Q3

Trajectory of a single job

Finished Jobs

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


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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).

Fluid solution

Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).

The Optimal Solution:

20

3( )q t


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0 10 20 30 40

0

5

10

15

2( )q t

1( )q t

VILLAGE STUDIES IN CENTRAL ANATOLIA

The changing face of the Anatolian Village

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The changing face of the Anatolian Village

Indigenous v/s contempory

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.

The Anatolian Village

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A common scene in Central Anatolia portraying abandoned traditional village houses among which new ones with pyramidal clay tile roofs are built.

Energy and Comfort

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Stove used in villages.

Solar water heater.

Environmental Performance of Buildings

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Poster for the PLEA Conference 2003

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

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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.

Temperature

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Comparative Graphs

Environmental Studies

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Ecotect for analysis

Bioclimatic Buildings

ECO MUDBRICK AND STRAWBALE GREEN HOUSETEMPERATURE CHART(16 SEPT -10 OCT 2005)

35

40

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0

5

10

15

20

25

30

18/0

9/20

05

19/0

9/20

05

20/0

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05

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01/1

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05

04/1

0/20

05

05/1

0/20

05

Time

Tem

pera

ture

(C

)

exterior mudbrick interior greenhouse interior

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

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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

Gene-Environment Networks

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if gene j regulates gene i

otherwise

,i iξ ζl

1:

0i jχ =

Earth Warming

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Mean temperature anomalies during the period 1995 to 2004 with respect to the average temperatures from 1940 to 1980.

Networks and Dynamics

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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.

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Selection or Design andSynthesis of the Probes

Array Production

Picture AnalysisHybridization

Array Preparation Sample Preparation Data Analysis

Comp. Bio. & Med.

Comp. Bio. & Med.

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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

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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

( ( ))P m P t P•

= −

•

•

•

Anticipatory Systems

logistic equation:

delay retarding - advancing anticipation

will

memory wish

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•

• •

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

delay retarding - advancing anticipation

will

memory wish•

•

•

Anticipatory Systems

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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

Impulsive Systems

quasilinear impulsive integrodifferential equations:

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nse

optimal control of response:

nse

object oriented

sports (and general) medicine

Further Systems

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object oriented

modules

DAE ODE

training (or reconvalescence) program

( , )⋅

= ⋅ Mx N v x K

Optimization

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I, K, L finite

• 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

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• This resemblance is often described by a distance function.

The disjoint subsets πi (S), i=1,…,k, are named clusters:

( ) and for .π π π= ∩ = ∅ ≠k

i i jS S , i jUUUU

Clustering

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1

( ) and for .π π π=

= ∩ = ∅ ≠i i ji

S S , i jUUUU

The second synthetic data set has the parameters

and σ = 0.3.5ˆ =k

Clustering

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The components are obviously overlapping in this case.

Stock Markets

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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

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Wiener process

(0, ) ( [0, ])∈tW N t t T

Ex.: price , wealth , interest rate , volatility

processes

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

-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

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- 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

Quality Control and Improvement

transmission cases

engine block

oil pan

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gearbox

oil pan

printed circuit boards

L curve :

5

5.5

5

5.5

5

5.5

5

5.5

Numerical Experience and Comparison


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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

desirability functions (quality opt.)

max-, min-type

continuous selections

normal forms smoothening


92August 7, 2009

stability

instabilitybilevel

problems

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.

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Brussels. • Representing today more than 10000 OR practitioners and academics all around Europe.

What is EURO?

• ΕπιχειρησιακήΕρευνα

• Añgerñargreining

94August 7, 2009

• Operačního Výzkumu

EURO

95August 7, 2009

EUEURORO

online.org/-http://www.euro

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.


EURO

97August 7, 2009

/description#description505543http://www.elsevier.com/wps/find/journaldescription.cws_home/

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)

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

EURO

100August 7, 2009


EURO

101August 7, 2009


EURO

102August 7, 2009


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

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)

Special Initiatives

• The Africa Project

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• Branding OR

Do we need to evolve?

- European job market

- European business

- European research funding

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- European research funding

- European training standards

- Europe in progress …

We have a role to play in this process

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

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

Education

Modern Operational Research and Its Mathematical Methods. Introduction