Operations Research -

Jacek Błażewicz

bridging gaps between Manufacturing and Biology

Presentation of our region

Presentation of our region

Presentation of our region

Siegen

GRAPHSOne of the main concepts used in Computer Science and Operations Research.

Nodes

Arcs

Used to present different processes.

Poznań

Jacek Błażewicz

Jan Węglarz

`

Poznań

Siegen

Erwin Pesch

Poznań

Siegen Clausthal-Zellerfeld

Klaus Ecker

Poznań

Siegen

Saarbrücken

Clausthal-Zellerfeld

Günter Schmidt

Poznań

Siegen

Jacek BłażewiczMałgorzata Sterna

Erwin Pesch

Poznań

Siegen

Redmond

Poznań

Siegen

Redmond

Livermore

Jacek Błażewicz

Erwin Pesch

Poznań

Siegen

Redmond

Livermore

Bartosz

NowierskiŁukasz

Szajkowski

Bartosz

Nowierski

Łukasz

Szajkowski

Bartosz Nowierski

Łukasz Szajkowski

Poznań

Siegen

Redmond

Livermore

Jacek Błażewicz

Erwin Pesch

FLEXIBLE MANUFACTURING SYSTEM

HPC CENTER in POZMAN

Scheduling problems (deterministic)

1. A set of m processorsP1 , P2 , ... , Pm

2. A set of n tasksT1 , T2 , ... , Tn

3. Each task is characterized by

- processing time - pj

4. Precedence constraints Ti Tj

5. Preemptions

6. Criterion

- Cmax = max{Cj}

t

P1

P2

Tj

Tk

0 Cj Cmax

Tj

Tl

Partial order Ti Tj

Types of precedence graphs

Independent tasks

Dependent tasks task – on – node

chains

 in-trees opposing forest

 out-trees

Ti Tj

general graphs

  task – on - arc

uniconnected activity network

 

uan

1

2

4

3

T1 T4

T3T2

T5

Pm│pmtn,uan │Cmax

 a) 

1

2

4

3

T1 T4

T3

T5

T2

Uniquely ordered event nodes.

b)        

An example of a simple uniconnected activity network (a) and the corresponding precedence graph (b).

T1 T4

T3

T2 T5

Now LP formulation:

  Minimize

Subject to   

j=1,2,...,n xj ≥ 0

 Complexity K = O(nm) - a number of variables, thus for a fixed m the problem can be solved in polynomial time [Khachiyan, Karmarkar]. [J.Błażewicz, W.Cellary, R.Słowiński, J.Węglarz, 77]

K

1iimax xC

px ji

i

Qj

In practice:

Polynomial time = easy(solvable in practice)

NP-hard = difficult(not solvable in practice)

Theorem 1

Let G be an activity network (task-on-arcgraph). G is uniconnected if and only if G has aHamiltonian path.

Original graph G HamiltonianPrecedence graph H ?

Molecular biology

Chemical foundations of life Information coded in chemical molecules

Computational biology

Met

hod

s Met

hod

sPro

blem

s Pro

blem

s

Operations Research

Molecular Biology

DNA recognition

Human genome pairs of bases 3% nucleotides coding an information

9103

Human genome 3000 books

(valid information 90 books)

1 cell bacteria 20 books

Some flies 5000 books

Analyzed structures

One dimensional structures Analysis of DNA chains (and an information they carry on)

Two dimensional structures Analysis (and recognition) of substructures formed by consecutive subchains (e.g. Α-helix, β-harmony)

Three dimensional structures Analysis of 3-dimensional helix (NMR experiment)

A C G A T G C AG . . . . .

One dimensional structures

1.Reading DNA chains

2.Understanding an information contained in DNA

sequence alignment finding motifs in sequences assigning functions to subsequences (or motifs)

Levels

Sequencing

up to 700 nucleotides

combinatorial exact methods

Assembling

up to 1000000 nucleotides

heuristics

Mapping

greater than 1000000 nucleotides

search in data bases

CGGACACCGACGTCATTCTCATGTGCTTCTCGGCACA

Chromosome

Clones

Sequencing

(works on 103-104 bp range)

Assembling

(works on 105-106 bp range)

Genetic linkage map

(works on 107-108 bp range)

The different scales at which the human genome is studied

Hybridization Experiment

A C G T A C G T A C G T A C G T

Round 1

Round 2

A AC

ACG

ACGT

A A C A C G A C G T

1. Making a DNA chip

Round 3

A C G TACGT

A A A A

... and so on ... DNA chip

Full libraryof tetranucleotides

0,4mm

0,4mm 25m site per probe

44 – 0.0016 cm2

48 – 0.4096 cm2

410 – 6.5536 cm2

AAAA AACA AAGAAAAC AACC AAGCAAAT AACG AAGGAAAT AACT AAGTACAA ACCA

DNA chip TCCACTG... Many labeled copies of an original sequence

. .. . . . .

spectrum

Hybridization Experiment –cont.

2. Hybridization reaction

3. Reading results

Fluorescence image of the chip

Spectrum – a set of oligonucleotides complementary to fragments of original sequence

A hybridization reaction between a probe of known sequence (l-mer) and an unknown sequence (n-mer):

n-mer - . . . A A C T A G A C C T . . .

l-mer - G A T

C T A

A sequence complementary to the probe exists in the target

DNA sequencing without errors

The original sequence: AACTAGACCT

Spectrum = {AAC,ACT,CTA,TAG,AGA,GAC,ACC,CCT}

(Two possible solutions: AACTAGACCT, AACCTAGACT)

Lysov (1988)

A graph is based on l-mers (graph H)

Finding a Hamiltonian path – NP-hard

AAC

CCT

ACTCTA

TAG

ACC GACAGA

Pevzner (1989)

AAC AA AC

A graph based on (l-1)-mers (graph G):

AAAC

CT

TA

CC GAAG

Finding an Eulerian path – polynomially solvable

A problem of equivalence

A problem of uniqueness

Equivalence problem

The above class of directed labeled graphs –DNA graphs.

Characterization and recognition of these graphs and finding conditions for which the above transformation is possible.

J.Błażewicz, A.Hertz, D.Kobler, D.de Werra, On some properties of DNA graphs, Discrete Applied Math., 1999.

Definition

The directed line graph H = (V,U) of graph G = (X,V) is the graph with vertex set V and such that there is an arc from vertex x to vertex y in H if and only if the terminal endpoint of arc x in G is the initial endpoint of arc y in G.

Graph G – Pevzner graph

Directed line graph H – Lysov graph

Theorem 2

Let H be the directed line-graph of a graph G. Then

there is an Eulerian path in G if and only if there is a

Hamiltonian path in H.

Back to scheduling.

Original graph G Hamiltonian

Its directed line-graph H ?

J.Błażewicz, D.Kobler

European Journal of Operational Research, 2002

Theorem 3

Original graph G uan Hamiltonian

Its directed line-graph H interval order.

AB

CD

A B

C D

Intervals

Interval order (graph)

Theorem 4

Pm | pmtn, interval order | Cmax

is solvable in polynomial time.

Ich danke Ihnen ganz herzlich für diese hohe und besondere Auszeichnung. Ich freue mich darüber sehr und hoffe, dass die bestehende sehr gute Zusammenarbeit in der Zukunft noch weiter intensiviert wird.

Diese Auszeichnung ist dann sicherlich ein weiterer Anreiz.