LECTURE 11-12. Course:

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LECTURE 11-12. Course Design of Systems
Structural Approach Dept. Communication
Networks Systems, Faculty of Radioengineering
Cybernetics Moscow Inst. of Physics and
Technology (University)
Mark Sh. Levin Inst. for Information
Transmission Problems, RAS
Email mslevin_at_acm.org / mslevin_at_iitp.ru
PLAN 1.Framework of multicriteria decision
making 2.Partitioning the procedure of
multicriteria decision making 3.Numerical
examples partitioning of initial problem
aggregation of results 4.Mapping
Sept. 25, 2004
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Framework of multicriteria decision making its
partitioning (parallelization)
DECISION MAKER DM
ALTERNATIVES
CRITERIA
METHODS
EXPERTS
SOLVING PROCESS
. . .
DECISIONS
APPROACHES FOR PARTITIONING OF DECISION MAKING
FRAMEWORK 1.By criteria
2.By alternatives
3.By experts
4.By methods
5.Hybrid approaches

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Parallel Processing Scheme
SOLVING PROCESS
DATA (VERSION 1)
PRELIMINARY DECISIONS
. . .
INITIAL DATA
PARTI- TIONING
RESULTANT DECISIONS
SOLVING PROCESS
AGGREGATION
DATA (VERSION i)
. . .
PRELIMINARY DECISIONS
SOLVING PROCESS
DATA (VERSION N)
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Example Partitioning by criteria groups
C1 C2 C3
C4 C5
C6
C7
TOTAL
A1 A2 A3 A4 A5
10 8 9
7 6 1
3 3 8 7
7 9 10
3 1
3 10 7 6
7 9 3
2 2 9
9 9 8
9 1 1
1 7 10 10
8 8
1 2 2
FINAL AGGREGATION
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Example Partitioning by alternative groups
PRELIMINARY STAGE
FINAL STAGE
C1 C2 C3
1st step
C1 C2 C3
A1 A2 A3 A4 A5 A6 A7 A8 A9
10 8 9
1 8 7 7
3 10 7 6
3 9 9 9
1 7 10 10
1 10 10 6
1 6 8
9 2 7 7
9 3 9 8
9 1
A1 A4 A5 A6 A9
10 8 9
2 9 9 9
3 7 10 10
1 10 10 6
1 9 8 9
4
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Example Joint partitioning by criteria groups
alternative groups
C1 C2 C3
C4 C5
A1 A2 A3 A4 A5 A6 A7 A8 A9

• 10 8 9
  7 6
• 8 7 7
  9 10
• 10 7 6
  7 9
• 9 9 9
  8 9
• 10 10 8
  8
• 10 10 6
  7 9
• 6 8 9
  10 8
• 7 7 9
  10 10
• 9 8 9
  9 9

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Possible schemes for joint partitioning by
criteria groups alternative groups
C1 C2 C3
C4 C5
A1 A2 A3 A4 A5 A6 A7 A8 A9
Ranking 1
Ranking 2
Ranking 3
Ranking 4
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Possible schemes for joint partitioning by
criteria groups alternative groups
SCHEME 1
Ranking 1
AGGREGATION (1 2)
Ranking 2
FINAL AGGREGATION
Ranking 3
AGGREGATION (3 4)
Ranking 4
Ranking 1
SCHEME 2
AGGREGATION (1 3)
Ranking 3
FINAL AGGREGATION
Ranking 2
AGGREGATION (2 4)
Ranking 4
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Example Integration tables (Glotov Paveljev)
S AB A(CD)
Scale for S
BCD

• 1 2 2
• 1 2 2 3
• 2 2 3 3
• 3 3 3 4

1 2 B 3 4
A
C
D
Scale for B

• 2 3 4
• A

• 1 2 3
• 2 2 3 3
• 3 3 4 4

1 2 D 3

• 2 3 4
• C

1 2 3 4
1 2 3 4
1 2 3
Scale for A
Scale for C
Scale for D
EXAMPLE Basic estimates are the following 4
for A, 3 for C, 1 for D
intermediate estimate for B is 2
resultant estimate for S is
3. NOTE multidimensional integration
tables are possible (and usefu l) too.
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Mapping Optimization Models
yf (x)
x ? R
f(x2)
f(x1)
x
x1
x2
y ? Y ? R
x ? X ? R
MAPPING yf (x)
Y
X
0
0
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Mapping Optimization Models
z
z f(x1,x2) ? R2
z f(x1,x2)
(x1,x2) ? X ? R2
f(x1,x2) ? Z ? R
x2
x2
x1
x1
GENERALLY
(x1, ,xm) ? X ? Rm
MAPPING
Y
X
(y1, ,yn) ? Y ? Rn
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Mapping Optimization Models
yf (x)
x ? X ? R
Optimal Point
X x1, x2 (admissible domain)
f (xo)
f (x) is objective function
max f (x) subject to x? X
x
max f (x) subject to x ?
x1 x ? x2
x1
x2
xo
yf (x)
Global Optimal Point
f (xo)
GENERALLY
max f (x) subject to ?1 (x) ?
0 . . . ?k (x) ?
0
x
xo
xo
Local Optimal Point
?j (x) is constraint function (1 ? j ? k)
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Illustration table for optimization models
Objective Constraint Type of
Method function f (x)
?j (x) model

Linear Linear Linear
simplex
ellipsoid

method
method of

Karmarkar Quadratic Linear
Quadratic simplex

ellipsoid Convex Linear Convex
gradient
method

ellipsoid
method . . .
. . . . . . . . .

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Integration (aggregation) approaches
NOTES

1.Objective function can be
examined as vector-like one too
(multi-objective optimization)


2.Constraints can be
examined as binary relations too
3. In discrete optimization
discrete spaces are examined
4.In stochastic optimization all
parameters / functions
can be stochastic ones

5.It is possible to take into
account uncertainty by two ways
stochastic parameters / functions

parameters / functions on the basis of
fuzzy sets