Talk Fahp 10 13 05 2-1

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005.

Defuzzifier

MAX operators

MIN operators

Fuzzifier

Y

Fuzzifier

Fuzzy logic X

Decision Support Systems, AHP & Fuzzy AHP

out

Fuzzy Rules

hidden layer #1

hidden layer #2 output layer

http://husky. if.uidaho.edu/ http://husky.if.uidaho.edu/pubTalks.html +1

+1

+1

Dr. Milos Manic, University of Idaho Idaho Falls ([email protected] )

Decision Support Systems – AHP & FAHP Today’s presentation 1.

Analytic Hierarchy Process (AHP), by Saaaty • phases & algorithm • relative & absolute measurement – examples • scales of measurement • problems

2.

Computational Intelligence in Decision Making • Fuzzy Logic (FL) (brief overview) • Fuzzy Decision Support Systems (FDSS), examples • Artificial Neural Networks (brief overview), examples

3.

Discussion on possible applications • •

DSS & spatial data? DSS for control?

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

1

Decision Support Systems (DSSs) - Intro Traditional DSS Algorithms • Promethee [Brans 85] • Electre [Roy 68, 78 96] • AHP [Saaty 80]

Computational Intelligence Enhancements • Fuzzy Logic Enhancements • of standard DSS techniques is used to accommodate vague, linguistic expert’s descriptions of alternatives/criteria • intangible properties possible to describe

• Artificial Neural Networks • approach for classification of complex, high-dimensional scenarios

© Dr. M. Manic, University of Idaho

Page 3

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

DSS - Analytic Hierarchy Process (AHP) AHP, Saaty ‘80 • early work on prioritization, hierarchies, eigenvalue analysis ’72, ’75… • simple & robust (four axioms) • reciprocal comparison ( reciprocal preferences, comparisons) • homogeneity ( bounded scale preference (7±2)) • independence of criteria with respect to alternatives • and expectations of complete hierarchic structure • three phases • problem decomposition (hierarchic) • evaluation phase (comparative judgment – pairwise comparisons) • synthesis of alternatives (ranking) © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Analytic Hierarchy Process (AHP) - Phases Three phases of AHP • Hierarchic (de)composition • incorporates intuitive understanding of the apportionment of the whole into its parts. • hierarchy levels can be inserted or eliminated as needed (sharp en or focus certain parts of the system). • typically goes from top (global character) to more specific at the bottom.

• Evaluation phase • based on the concept of paired comparisons as to their importance to a given criterion that occupies the level immediately above the elements being compared (relative or absolute measurements) • yields a relative scale of priorities which is the relative standing with respect to a criterion independently of any other criterion. • these relative weights sum to unity. • Ranking • Synthesis of pairwise comparison tables © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

AHP – Algorithm, Steps

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

3

Analytic Hierarchy Process (AHP) - Steps The theory of AHP 1. Hierarchic decomposition • A hierarchy does not need to be complete (an element attribute for all the elements in the level below) • There can be multiple levels (a different cut at the problem) • Levels can be inserted or eliminated as needed (sharpen the focus) Satisfaction with house

age

neighborhood

condition

size

financing

House A

House B

© Dr. M. Manic, University of Idaho

House C

Page 7

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Steps The theory of AHP 2. Paired comparisons • compare a single property from a level above with those from level below • pairwise comparison – for a pair of attributes, the “smaller” of each pair is used as the unit, and the larger one is measured in terms of multiplies of the smaller one. • A following matrix is obtained: Satisfaction with house

 w1 / w1 w1 / w2 w / w w / w 2 1 2 2 A=   M M   wn / w1 wn / w2

L w1 / wn  L w2 / wn   L M   L wn / wn 

age

neighborhood

condition

size

House A

House B

financing

House C

• Weight vector w = [w1, w2 , L, wn ] is obtained by eigenvector method:

T

Aw = n w © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Analytic Hierarchy Process (AHP) - Steps Satisfaction with house

The theory of AHP 2. Paired comparisons

age

• Weight vector w = [w1, w2 , L, wn ] is obtained by eigenvector method:

neighborhood

condition

size

financing

T House A

House B

House C

Aw = n w

• or

 w1 / w1 w / w  2 1  M   wn / w1

w1 / w2 w2 / w2 M wn / w2

L w1 / wn   w1   w1     w  L w2 / wn  w2  ⋅ = n⋅  2  M  L M  M       L wn / w n  wn  wn 

•Weight vector is normalized weight vector (elements divided by their sum) thus weight vector always sums up to 100%. •Matrix A has all positive elements and is reciprocal and consistent: aij = 1 / a ji

aik = aija jk

© Dr. M. Manic, University of Idaho

i , j , k = 1,..., n Page 9

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Steps Satisfaction with house

The theory of AHP

size

3. Comparison table synthesis into overall ranking table

age

House A

neighborhood condition House B

fina ncin

House Cg

• yields a ration scale capturing the order inherent in the judgments

4. Consistency Analysis • consistency is necessary but not sufficient for a good decision • consistency can be captured by Consistency Index (CI) as:

CI =

λmax − n n −1

A w=λmax w

where λmax − n is the deviation of the judgments from the consistent approximation. Consistency Index (CI) is the negative average of the other roots of the characteristic polynomial of A. • Consistency Ratio (CR) acceptable if below 10%

CR < 10% © Dr. M. Manic, University of Idaho

Page 10

CR =

CI CI average

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

5

SCALE OF MEASUREMENTS

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Scales of Measurement – 1 to 9? Scale - Miller’s 7±2 Myth! • [Miller 56] Miller, George A. The Magical Number Seven, Plus or Minus Two, The Psychological Review, vol. 63, pp. 81-97, 1956. • asymptotical limit – number of bits needed to describe ‘graspable’ information. • Absolute Judgments - channel capacity of observer • 4 different tones, • 6 different auditory pitches – channel capacity of 2.5bits, • 5 discriminable auditory loudness alternatives – channel cap. of 2.3bits • taste intensities – channel capacity of 1.9bits • pointer position in a linear interval – channel capacity of 3.25bits • size, hue, brightness, curvature, etc. • How do we function at all - Multidimensional Stimuli • can raise the channel capacity for a few orders of magnitude! Keep the number of items presented as an unstructured group under 7, use different codings to encode the same information, structure complexity through recursive grouping! © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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AHP – Example Relative Measurement – House Ranking Example

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Example AHP - Example 1. Hierarchic decomposition • A hierarchy does not need to be complete (an element attribute for all the elements in the level below) • There can be multiple levels (a different cut at the problem) • Levels can be inserted or eliminated as needed (sharpen the focus) Satisfaction with house

age

neighborhood

condition

size

financing

House A

© Dr. M. Manic, University of Idaho

House B

House C

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

7

Analytic Hierarchy Process (AHP) - Example AHP - Example 1. Fundamental scale (9 point scale)

© Dr. M. Manic, University of Idaho

Page 15

Taken from [Saaty 90]

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Example AHP - Example 2. Pairwise comparison of criteria

© Dr. M. Manic, University of Idaho

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Taken from [Saaty 90]

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Analytic Hierarchy Process (AHP) - Example AHP - Example 3. Alternatives against each criterion

© Dr. M. Manic, University of Idaho

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Taken from [Saaty 90]

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Example AHP - Example 3. Synthesis in overall ranking table

Taken from [Saaty 90]

• composite (global) priorities of the houses • first row – relative weights of criteria, each column relative weights for each criterion • ranks should sum to unity (100%) • largest weight indicates the preferred alternative © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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AHP – Example Absolute Measurement – Employee Evaluation Example

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Example AHP – Absolute Measurement Example Establish a scale of priorities for the criteria (subcriteria if any) • first row criteria ratings (grades), sum to unity, columns are intensities (=1)

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Analytic Hierarchy Process (AHP) - Example AHP – Absolute Measurement Example Establish a scale of priorities for the criteria (subcriteria if any) • first row criteria ratings (grades), sum to unity, columns are intensities (=1)

• Mr. X = 0.061*0.604+0.196*0.731+0.043*0.199+0.071*0.750+0.162*0.188+0.466*0. 750 = 0.623 © Dr. M. Manic, University of Idaho

Page 21

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Example AHP – Absolute Measurement Example Establish a scale of priorities for the criteria (subcriteria if any) • first row criteria ratings (grades), sum to unity, columns are intensities (=1)

• Mr. X = 0.061*0.604+0.196*0.731+0.043*0.199+0.071*0.750+0.162*0.188+0.466*0. 750 = 0.623 • Mr. Y=0.369; Mr. Z=0.478; • Absolute measurement needs standard (student admission, employee promotion, etc.) • Agreement on standard needed to be used later on for rating of alternatives • Overall ranks do not necessarily sum to unity!!! Can renormalize though! © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

11

AHP – Relative vs. Absolute Measurement (1) Relative Measurement •

Steps: 1. 2. 3.

structuring of the problem as a hierarchy; elicitation of pairwise comparison criteria (both criteria among themselves as well as alternatives against each criteria in a level above); establish composite (global) priorities of the alternatives

Absolute Measurement •

Steps: 1. 2. 3.

establish two scales (global priority vector of criteria, and table of grades for each criterion, both sum up to 1); each alternative is assigned a grade for each criterion. establishee the final priorities through combination of global priorities with grades to produce a final ratio scale for each of the alternatives.

© Dr. M. Manic, University of Idaho

Page 23

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

AHP – Relative vs. Absolute Measurement (2) Relative Measurement • •

should be used in most new situations (we do not have sufficient understanding to compare intensities). may be used with confidence in well-understood situations.

Absolute Measurement • • • • • •

useful in areas where there is fairly good agreement on standards alternatives can be added or taken out during the process. each alternative will achieve certain “total score” (do not need to sum up to one) rank can not be reversed (involvement or deleting of alternatives does not influence the total score sum) no notion of final relative merit in case of absolute measurement. there can be equal or nearly equal alternatives.

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

12

AHP Deficiencies?!

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Analytic Hierarchy Process (AHP) - Problems Rank preservation and reversal • Relative measurement • addition or deletion of a new alternative can change existing ranking! • presence of a copy or near copy? • previous knowledge on uniqueness of the alternatives needed!

• Absolute measurement • rank is always preserved

• Sensitivity analysis • how sensitive is the overall decision to changes of individ. weights? • solution – experiment! slightly vary the values of the weights and observe • identify those weights that the decision is most sensitive to © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

13

A few words on fuzzy logic…

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Fuzzy Logic (FL) Fuzzy systems • Developed by Lofti Zadeh • http://www.cs.berkeley.edu/~zadeh/

• Over 15 honorary doctorates • His work was cited in over 30,000 publications

• Boolean Logic • Multivalued Logic •

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Briefly on Fuzzy Logic (FL) Fuzzy systems

• • •

Inputs can be any value from 0 to 1. The basic fuzzy principle is similar to Boolean logic. Max and min operators are used instead of AND and OR. The NOT operator also becomes 1 - #. A ∩ B ∩ C ⇒ min{A, B, C} − smallest value of A, B or C

A ∪ B ∪ C ⇒ max{A, B, C} − largest value of A, B or C ⇒ 1- A

A

− one minus A

Boolean

A∩B

Fuzzy

A∪B

complement

A∩B

A∪B

0

0

0

0

0

0

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0.3

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conjunction © Dr. M. Manic, University of Idaho

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disjunction

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Fuzzy Logic (FL) Fuzzy systems

© Dr. M. Manic, University of Idaho

Page 30

fuzzy

defuzzyfier

fuzzy

max operators

fuzzy

min operators

analog inputs

fuzzyfier

Block diagram of Zadeh fuzzy controller

analog output

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

15

Briefly on Fuzzy Logic (FL) Data representation using fuzzy sets µ(x)

µ (x)

1

1

0

1

0

0

x

domain

µ(x)

x

domain

domain

µ(x)

µ(x)

µ(x)

1

1

1

0

x

domain

x

0

x

domain

0 domain

x

‘Bell’ shaped, triangular, trapezoidal, shouldered fuzzy sets © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Fuzzy Logic (FL) Fuzzy sets connected by a Zadeh AND operator µ (x) Low 1

µ(x) 0.5

Low 1 0.5 25 45 Temperature (domain)

x

µ(x)

0

Low 1

15

25 Alarm (domain)

x

0.75

100

1200 Pressure (domain)

© Dr. M. Manic, University of Idaho

x Page 32

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Briefly on Fuzzy Logic (FL) Zadeh min-max rule µ(x) Low

OK

1

µ (x) 0.5

Low

Medium

1 0.1

0.5

x

25 45 Temperature (domain)

µ(x)

0.1 0

Low

15

1

x

25 Alarm (domain)

0.75

100

x

1200 Pressure (domain)

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Fuzzy Logic (FL) Fuzzy systems Block diagram of Zadeh fuzzy controller

© Dr. M. Manic, University of Idaho

Defuzzifier

MAX operators

MIN operators

Y

Fuzzifier

X

Fuzzifier

Fuzzy logic

out

Fuzzy Rules Page 34

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Briefly on Fuzzy Logic (FL)

Input 1 i cold cool normal warm hot

Input 2 g cold A A A A B

Y

Defuzzifier

MAX operators

Fuzzifier

cool A A B B C

MIN operators

Y

Fuzzifier

X

Controller takes temperature reading from two inputs

Fuzzifier

X

Fuzzy logic

out

Fuzzifier

Fuzzy systems - Rule Evaluation - Zadeh fuzzy tables

Defuzzification

Rule selection cells min-max operations

out

Fuzzy Rules

normal A B C C D

warm B C C D E

hot A B C D E

10 8 6 4 2 0 30 25

30

20 15

15

10 5

© Dr. M. Manic, University of Idaho

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5

20

25

10

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Fuzzy Logic (FL) Fuzzy Operators • Different classifications of operators: • Cox has defined two initial classes: • general algebraic operators (Zadeh, Mean, Mean2, Mean1/2, Product and Bounded Sum), and • functional compensatory operators (Yager, Zimmerman, Dubois/Prade, as also as negation operators - Yager’s, Sugeno’s, Threshold, Cosine, etc.).

Operators on fuzzy sets should be adapted to a problem • certain operator can, depending on compensatory parameter value, have to a certain extant properties of both intersection and union. So, the strength of one operator can be modified from max to min (in some degree

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Briefly on Fuzzy Logic (FL) Fuzzy Operators Table 1. General algebraic operators of intersection and union µa ( x) I µb ( y)

Table2.Functionalcompensatoryoperators’classes

min (µa (x), µ b (y)) ( µa (x) + µ b (y))/ 2

(µa (x)+ µb(y)−µa (x)∗µb(y)-min(µa (x), µb(y), 1-k))

Mean2

mean(int) 2 (intensified mean) mean (int)1/2 (diluted mean)

µa(x)I µb(y) (fuzzy and) Dubois& (µa(x)∗ µb(y)) / Prade max( µa(x), µb(y), k)

γ

1.0

1.0

1.0

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0.4 XAxis

0.6

0.8

XAxis

0.0

0.6

0.2 0.8

0.4 XAxis

0.4 60.

0.8

s

0.00.

XAxis

© Dr. M. Manic, University of Idaho

) (

)

0.8 0.0 0.0

0.6 0.2

0.4

0.4 XAxis

0.6

0.8

0.2 0.0

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-0.5

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0.6 xis

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ZAxis

Z Axis

ZAxis

)

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0.6 0.2

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(

YAxis

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1 1 Generali  2p-1 2p-1  2p-1 2p-1  zed max0, µa(x) +µb(y) −12p−1 1−max0, 1−µa(x) + 1-µb(y) 2p−1   • +   pand p

0.6 0.2

0.2 0.8

0.0

1.0

xi

0.6 0.2

A

0.0 0.0

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YAx is

0.4

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0.2 0.0

80. -1.0 0.0

0.6 0.2

0.4

0.4 X Axsi

YAxis

)

1 1  p p p  p p p 1−min1, (1−µa(x)) +(1-µb(y))  min1,( µa(x) + µb(y) )  yand y  

0.8

0.6 0.2

YAx is

(

0.0

0.2

Y

YAxis

0.8

0.4

+

}

min µa (x ) + µb ( y ),1

ZAxis

)

0.6

Yager

{

max {µa ( x )+ µb ( y )-1 , 0}

vand v •

µa ( x) + µb ( y) − µa ( x ) • µb ( y )

µa(x)+µb(y) 1+µa (x)•µb(y) µa (x)$+µb(y)-(1-r) µa(x)•µb(y) r+(1−r)(1−µa (x)•µb(y) )

Hamache µa (x)•µb(y) r r +(1−r) µ (x)+ $ µb(y) a • +

(

(intensified mean)

mean (un)1/2 (diluted mean)

ZAxis

µa (x)•µb(y)

mean (un) 2

µa ( x ) • µb ( y)

ZAxis

+

Mean Product $• & +$ (probability) Bounded ⊗ &⊕

n   n  ∏µi (x) 1−∏(1−µi(x))  i=1   i=1 

εandε 1−(1−µa (x))( 1−µb(y)) •

2∗ ( max (µa (x), µb (y)))) / 3

/max1( µa (x), 1- µb(y), k) (1−γ)

(fuzzy o r)

max (µ a (x), µ b (y)) ( min ( µ a (x), µ b (y)) +

Z Axsi

Zimmerm an& Zysno Einstein

µa (x) U µb (y)

(fuzzy and)

µa(x) Uµb(y) (fuzzy or)

Zadeh Mean

0.6

0.8

0.2 0.0

Figure 2. Yager’s union operator when p =1, 10, 1000, 5000, 104 , 10 5, 11.10 4, 25.104

Page 37

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Fuzzy Logic (FL) IMPORTANT! Fuzzy reasoning is strongly influenced by the: • fuzzyfication phase • adequate choice of fuzzy operators • adequate choice of defuzzyfication algorithms

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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ANNs & FL as Approximators - Comparison Required surface 2 2 z = 0.9 ⋅ e −0. 003⋅( x − 20) − 0. 015⋅( y −6)

Surface approximated by 2-1 ANN 1 0.8 0.6

0.80-0.90

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S6

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inputs

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S19 S17 S15 S13 S11 S9 S7

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0.50 0.40 0.30

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0.70

output

0.70-0.80 0.60-0.70 0.50-0.60

3

0.80

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0.90

Surface approximated by 1-1-1 ANN +1

#1

0.9 0.8

0.8-0.9 0.7-0.8

0.7

+1

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#2

0.6-0.7 0.5-0.6 0.4-0.5 0.3-0.4 0.2-0.3 0.1-0.2 0-0.1

0.5

+1

output

0.4

#3

0.3 0.2 0.1

S15

© Dr. M. Manic, University of Idaho

19

13

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4

7

10

S8 1

inputs

0 S1

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

ANNs & FL as Approximators - Comparison Required surface 2 2 z = 0.9 ⋅ e −0. 003⋅( x − 20) − 0. 015⋅( y −6)

Surface approximated by Fuzzy System 10 9 8 7

0.90

9-10 8-9

6

0.80-0.90

0.80

7-8 5

0.70-0.80 0.60-0.70 0.50-0.60

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

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25

© Dr. M. Manic, University of Idaho

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101235 111346 323456 434567 545678 556789

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C B A B C D B B B D E D C D E 30

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Fuzzy Decision Making Fuzzy Decision Support Systems (FDSS)

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

(Fuzzy) Decision Support Systems •

Thomas Saaty AHP (Analytic Hierarchy Process) •

•

George Miller’s magical nubmer seven, plus or minus two ’56

Hierarchic structure from overall goal to criteria (subcriteria ), and alternatives • Criteria: criteria compared pairwise (multiple hierarchy levels possible) • Alternatives: pairwise comparison of alternatives for all alternatives, for each criterion • Result: ranking of alternatives Selection of optimum solution for water supply

Cost

Elevation Benefit

Reservoir on Lopatnica River

Water Quality

Reservoir on Gvozda~kaRiver

Water Quantity

System Organization

MPHS "Studenica"

Manic M., Muskatirovic J., Selection of optimum solution for wat er supply in fuzzy decision environment, proceedings, Hydroinformatics '98, Copenhagen, 24-26 August, (1998). © Dr. M. Manic, University of Idaho

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(Fuzzy) Decision Support Systems •

Thomas Saaty ’72 – AHP (Analytic Hierarchy Process)

• •

Fuzzy (linguistic) description of criteria? Intengible properties described in its own mathematical environment

1

Selection of optimum solution for water supply

Water Quality

Elevation Benefit

Cost

Water Quantity

System Organization

0.5

Reservoir on Lopatnica River

medium small

Reservoir on Gvozda~kaRiver

MPHS "Studenica"

large

1

0 0

5

10 Cost (1/10 6$)

15

20

0.5

Linguistic description of cost criterion

small

Linguistic description of water quantities criterion

large

medium

0 0

5

10

15

20

2

Water Quantities ( ×10 l/s) Manic M., Muskatirovic J., Selection of optimum solution for wat er supply in fuzzy decision environment, proceedings, Hydroinformatics '98, Copenhagen, 24-26 August, (1998). © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Decision Support Systems Based On Fuzzy Preference of Fuzzy Alternatives

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

22

Fuzzy Preference Approach for Computer Network Attack Detection (

) )

≥ Sγ A j > Ai otherwise

f

a)

(

)

Sγ Ai < A j =

min {x− γ , Jmax }

∑ ∑ ∑ ∑ I max

x= I min

y= J min

I max

Jmax

x = I min

y= Jm i n

µ Ai ( x ) Θµ A j ( y )

µ Ai ( x ) Θµ A j ( y )

Function F, generating a profile

µ( x )

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

A

1

0%

( (

when S γ Ai > A j ≥

)

Satisfaction function:

F r e q u e n c y

Function F, generating a profile

Function F, generating a profile

0

)

(

Orlovsky's fuzzy preference relation

F r e q u e n c y

f

F r e q u e n c y

f

(

A

)

S γ Ai > A j −   Rs Ai , A j =  − Sγ A j > Ai ,   0, 

b)

max%

0%

Figure 1. a) Two attack signatures, and their correspondent fuzzy value b) Example of comparison of two attack signatures

1

x

Preference

Zadeh - intersect. Zadeh - union Product Product - union

0

max Overlapping factor (%)

0

Figure 2. Lee's preference relation depending on an overlapping extent and shape of attack signatures, for the above test example

Manic, M. Wilamowski, D., Towards The Attack Signatures’ Comparison In Survivable Computer Networks, IECON'01 - 27 © Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

A few words on Artificial Neural Networks

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Briefly on Artificial Neural Networks (ANNs) Biological Neuron Artificial neuron x1 x2

∑

w1 w2

f(net)

Neuron

f(net) 1

xn

wn

net

f(net)- 1

Jacek Zurada

© Dr. M. Manic, University of Idaho

Page 47

Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Briefly on Artificial Neural Networks (ANNs) Activation functions out out

net

net

(a)

(b)

out

k

out

net

k

c)

o = f(net)=

o = f(knet)= tanh( knet) = 2 = -1 1+ exp (- 2knet) d)

1 1 + exp (- knet)

© Dr. M. Manic, University of Idaho

net

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

24

Briefly on Artificial Neural Networks (ANNs) BAM - Bi-directional Associative Memories Network oscillates between two patterns, recovering the other pattern…

Associations:

a

a b

• Military targets • Language translations • Etc.

WT

W

⇔ © Dr. M. Manic, University of Idaho

Page 49

(this is the other pattern associated with the butterfly…) Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Associative Memories •No I/O relation, just contractor behavior on input. •Attractor (1 state attracts the other). •8x7 pixels => 56 inputs for ANN

Character recognition (various noise level ) © Dr. M. Manic, University of Idaho

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Kohonen Networks problems Input pattern transformation on a hypersphere

x1 x2

z1 z2

. . .

xn

zn zn+1

xc1

xe1

xc3

R − x 2

2

xe3

1 0.5 0 xc2

-0.5

xe2

-1 10 5

10 5

0 -5

-5

0

-10 -10 Wilamowska, K., Manic, M. Unsupervised pattern clustering for data mining, IECON'01 – 27. Annual Conference of the IEEE Industrial Electronics Societ y, Denver, Colorado, Nov 29 to Dec 2, pp.1862-1867, 2001. © Dr. M. Manic, University of Idaho Page 51 Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Benefits, Applications, Further Reading

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

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Computational Technologies – DSS and FL/ANNs

Benefits? • • • •

FL – Working with imprecise/inaccurate data (DM) FL/ANNs - Universal, robust, adaptive approximations ANNs - Clustering of complex, multi-dimensional data ANNs - Supervised/unsupervised learning (outcome unknown)

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

DSS - Summary AHP • Useful with both new scenarios (relative), and standardized situations (absolute measurement. • Problems – rank preservation and sensitivity Possible compoutational intelligence extensions • Fuzzy Logic • • •

•

using imprecise/inacurate, linguistic descriptions - alternatives & criteria established logic (operators) for dealing with such descriptions Separate decision systems based on fuzzy sets & fuzzy preference relations

Artificial Neural Networks •

Universal, robust, adaptive, even unsupervised clustering

Possible applications? •

DSS & Spatial Data?

© Dr. M. Manic, University of Idaho

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Further Reading on DSSs, AHP, & FAHP • A few of Satty’s Books… • Saaty, T.L, The Analytic Hierarchy Process: Planning Setting Priorities, Resource Allocation, Mcgraw-Hill, 1980, ISBN: 0070543712 • Saaty, T.L, Multicriteria Decision Making: The Analytic Hierarchy Process (Analytic Hierarchy Process), RWS Publications; 2nd edition, 1990, ISBN: 0962031720 • Saaty, T.L, Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World, RWS Publications; 3rd Rev edition, 1999, ISBN: 096203178X • Saaty, T.L, Decision Making in Economic, Political, Social and Technological Environments With the Analytic Hierarchy Process, Rws Publications, 1994, ISBN: 0962031771 • Saaty, T.L, Fundamentals of Decision Making and Priority Theory With the Analytic Hierarchy Process, RWS Publications, 2000, ISBN: 0962031763 • Saaty, T.L, Prediction, Projection, and Forecasting: Applications of the Analytic Hierarchy Process in Economics, Finance, Politics, Games and Sports, Kluwer Academic Publishers, 1990, ISBN: 0792391047

© Dr. M. Manic, University of Idaho

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Guest Talk on FAHP, Dr. Allesi’s course, 10.13.2005

Further Reading on DSSs, AHP, & FAHP • A few papers to start with… • Saaty, T.L., An eigenvalue allocation model for prioritization and planning, Energy management and policy center, Univ. of Pennsylvania, 1972. • Saaty, T.L., Hierarchies & priorities, eigenvalue analysis, Univ. of Pennsylvania, 1975. • Saaty, T.L., A scaling method for priorities in hierarchical structures, J.Math.Psychology, vol.15, no.3, pp.234281, 1977. •Saaty, T.L., Exploring the interface between hierarchies, Multiple objectives and fuzzy sets, Fuzzy sets & systems, 1978. •[Saaty 90] Saaty, T.L., “How to make a decision: The Analytic Hierarchy Process”, European Journal of Operational Research 48, pp.9 -26, 1990 • [Saaty 87] Saaty, T.L., “Rank generation, preservation and reversal in the analytic hierarchy decision process”, Decision Science, Vol. 18, pp.157-177, 1987. • [Saaty 93] Saaty, T.L., Vargas, L.G., “Experiments on rank preservation and reversal in relative measurement”, Mathl . Comput. Modelling Vol. 17, No. 4/5, pp.13-18, PergamonPress LTD., 1993. • [Vargas 90] Vargas, G.L., “An overview of the Analytic Hierarchy Process and its applications”, European Journal of Operational Research 48, pp.2 -8, 1990. •[Trintaphyllou 97] Trintaphyllou, E., Sanchez, A., “A sensitivity analysis approach for some deterministric multicriteria decision making methods”, Decision Sciences, Vol.28, pp.151-194, 1997. • [Lindstedt 2001] Lindstedt, M.R.K., et. al., “Using intervals for global sensitivity analysis in multiattribute value trees”, Proc. Of the 15th Int.Conf . on Multiple Criteria Decision Making, Ankara, Turkey, July 10-14, 2000. •[Drake 98] Drage, P.R., “Using the analytic hierarchy process in engineering education”, Int.J.Engng Ed. Vol. 14, No.3, pp.191-196, 1998. • [Miller 56] Miller, George A. The Magical Number Seven, Plus or Minus Two, The Psychological Review, vol. 63, pp. 81-97, 1956. © Dr. M. Manic, University of Idaho

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Further Reading on DSSs, AHP, & FAHP • A few papers on fuzzy logic to start with… • Saaty , L.T., Exploring the interface between hierarchies, Multiple Objectives and Fuzzy Sets, Fuzzy Sets and Systems, January 1978. • [Cox 94] Cox, E., The Fuzzy systems Handbook, A practioner’s guide, Academic Press Inc. 1994. • [Hiirsalmi 00] Hiirsalmi, M., Kotsakiks, E., Pesonne, A., Wolski, A., “Discovery of fuzzy models from observation data, FUME Project, VTT Information Technology 2000. • [Hong 00] Hong, T.P., Wang, S.L, “Determining appropriate membership functions to simplify fuzzy induction”, Intelligent Data Analysis 4, pp.51-66, IOS Press, 2000. • [Li 95] Li, H.X., Yen, V.C., Fuzzy sets and fuzzy decision making, CRC Press, 1995. •[Lee 94] K.M.Lee, C.H.Cho, H.L.Kwang. "Ranking fuzzy values with satisfaction function". Fuzzy Sets and Systems 64, pp.295-309, 1994. •[Sun 94] Sun,C.T., "Rule -Base Structure Identification in an Adaptive-Network -Based Fuzzy Inference System", IEEE Trans. on Fuzzy Syst., vol.2, no.1, pp.64-73, Febr. 1994. •[Zahariev 91] S.Zahariev. "On Orlovsky's definition of nondomination". Fuzzy Sets and Systems 42, pp.229-235, 1991. •[Buckley 85] J.J.Buckley. "Ranking alternatives using fuzzy numbers". Fuzzy Sets and Systems 15, pp.21-31, 1985. •Manic, M. Frincke, D., Towards the Fault Tolerant Software: Fuzzy Extension of Crisp Equivalence Voters, IECON'01 - 27 •Manic, M. Frincke, D. Milutinovic, B., Towards the fuzzy logic in intrusion detection systems, Proceedings from South Eastern Europe Workshop on Computational Intelligence and Information Technologies, pp.33-40, June 2001. •Manic, M. Wilamowski, D., Fuzzy Preference Approach for Computer Network Attack Detection, International Joint INNS-IEEE Conference on Neural Networks, Washington DC, July 14-19, pp.1345-1349, 2001 •Manic, M., Fuzzy -Operators Weight Refinements, Proceedings, Annual Reliability & Maintainability Symposium, RAMS’99, from IEEE Reliability Society, Washington, DC USA, January 18-21 1999, pp.245-251, (1999). •Manic M., Milutinovic S., Refinements of fuzzy operators weights, proceedings, The Seventh Turkish Symposium on Artificial Intelligence and Neural Networks, TAINN’98, Bilkent University, Ankara, Turkey, June 24-46 1998, pp.205-213, (1998). •Manic M., Muskatirovic J., "Selection of optimum solution for water supply in fuzzy decision environment", proceedings, Hydroinformatics '98, Copenhagen, 24-26 August, (1998). •Manic M., Milutinovic S., Fuzzy preference relation depending on different operators and fuzzy numbers, proceedings, International Fuzzy Systems Association, IFSA '97, Prague, June 25-29, 1997, pp.64-69, (1997). © Dr. M. Manic, University of Idaho

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