Download - Operations Research -
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Operations Research -
Jacek Błażewicz
bridging gaps between Manufacturing and Biology
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Presentation of our region
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Presentation of our region
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Presentation of our region
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Siegen
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GRAPHSOne of the main concepts used in Computer Science and Operations Research.
Nodes
Arcs
Used to present different processes.
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Poznań
Jacek Błażewicz
Jan Węglarz
`
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Poznań
Siegen
Erwin Pesch
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Poznań
Siegen Clausthal-Zellerfeld
Klaus Ecker
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Poznań
Siegen
Saarbrücken
Clausthal-Zellerfeld
Günter Schmidt
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Poznań
Siegen
Jacek BłażewiczMałgorzata Sterna
Erwin Pesch
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Poznań
Siegen
Redmond
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Poznań
Siegen
Redmond
Livermore
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Jacek Błażewicz
Erwin Pesch
Poznań
Siegen
Redmond
Livermore
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Bartosz
NowierskiŁukasz
Szajkowski
Bartosz
Nowierski
Łukasz
Szajkowski
Bartosz Nowierski
Łukasz Szajkowski
Poznań
Siegen
Redmond
Livermore
Jacek Błażewicz
Erwin Pesch
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FLEXIBLE MANUFACTURING SYSTEM
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HPC CENTER in POZMAN
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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
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5. Preemptions
6. Criterion
- Cmax = max{Cj}
t
P1
P2
Tj
Tk
0 Cj Cmax
Tj
Tl
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Partial order Ti Tj
Types of precedence graphs
Independent tasks
Dependent tasks task – on – node
chains
in-trees opposing forest
out-trees
Ti Tj
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general graphs
task – on - arc
uniconnected activity network
uan
1
2
4
3
T1 T4
T3T2
T5
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Pm│pmtn,uan │Cmax
a)
1
2
4
3
T1 T4
T3
T5
T2
Uniquely ordered event nodes.
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b)
An example of a simple uniconnected activity network (a) and the corresponding precedence graph (b).
T1 T4
T3
T2 T5
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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
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In practice:
Polynomial time = easy(solvable in practice)
NP-hard = difficult(not solvable in practice)
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Theorem 1
Let G be an activity network (task-on-arcgraph). G is uniconnected if and only if G has aHamiltonian path.
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Original graph G HamiltonianPrecedence graph H ?
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Molecular biology
Chemical foundations of life Information coded in chemical molecules
Computational biology
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Met
hod
s Met
hod
sPro
blem
s Pro
blem
s
Operations Research
Molecular Biology
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DNA recognition
Human genome pairs of bases 3% nucleotides coding an information
9103
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Human genome 3000 books
(valid information 90 books)
1 cell bacteria 20 books
Some flies 5000 books
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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 . . . . .
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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)
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Levels
Sequencing
up to 700 nucleotides
combinatorial exact methods
Assembling
up to 1000000 nucleotides
heuristics
Mapping
greater than 1000000 nucleotides
search in data bases
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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.
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Back to scheduling.
Original graph G Hamiltonian
Its directed line-graph H ?
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J.Błażewicz, D.Kobler
European Journal of Operational Research, 2002
Theorem 3
Original graph G uan Hamiltonian
Its directed line-graph H interval order.
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AB
CD
A B
C D
Intervals
Interval order (graph)
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Theorem 4
Pm | pmtn, interval order | Cmax
is solvable in polynomial time.
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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.