A Decision System Using ANP and Fuzzy Inputs

Jaroslav Ramík

Silesian University Opava School of Business Administration Karviná

Czech Republice-mail: ramik@opf.slu.cz

FUR XII, Rome, June 2006

Content

• Problem -AHP

• Dependent criteria – ANP

• Solution

• Case study

• Conclusion

Problem- AHP

• MADM problem – AHP

• AHP- supermatrix

• AHP- limiting matrix

Content

MADM problem – AHP

Goal: find the best variantGoal: find the best variant

C1C1 C2

C2 CnCn

V1V1 V2

V2 VmVm

…

…

- Criteria

- Variants

Content

AHP – supermatrix

IW0

00W

000

W

32

21

Supermatrix:

)(

)( 1

21

nCw

Cw

W

),(),(

),(),(

1

111

32

mnm

n

VCwVCw

VCwVCw

W

1,1

m

rir CVw 1

n

iiCw

1

Content

AHP- limiting matrix

IWWW

000

000

W

322132

Limiting matrix:k

kWW

lim

2132WWZ

- vector of evaluations of variants (weights)

Content

Dependent criteria – ANP

• Dependent evaluation criteria – ANP

• Dependent criteria – supermatrix

• Dependent criteria – limiting matrix

• Uncertain evaluations

• Uncertain pair-wise comparisons

Content

Dependent evaluation criteria – ANP

Content

GOAL

CRITERIA

VARIANTS

Feedback

Dependent criteria – supermatrix

Supermatrix:

IW0

0WW

000

W

32

221 2

),(),(

),(),(

1

111

22

nnn

n

CCwCCw

CCwCCw

W

- matrix of feedback between the criteria

Content

Dependent criteria – limiting matrix

Limiting matrix: k

kWW

lim

IWIWWWIW

000

000

W1

2232211

2232

211

2232 WWIWZ

21322

2222232 ...)( WWWWIWZ

- vector of evaluations of variants (weights)

Content

Uncertain evaluations

);;(~ UML aaaa

0 aL aM aU

Triangular fuzzy number

1

Content

Uncertain pair-wise comparisons

);;(~ UML aaaa )

1;

1;

1(

1~

LMU aaaa

0 ¼ 1/3 ½ 1 2 3 4

Reciprocity

Content

Solution• Fuzzy evaluations

• Fuzzy arithmetic

• Fuzzy weights and values

• Defuzzyfication

• Algorithm

Content

Fuzzy evaluations

• Fuzzy values (of criteria/variants):

• Triangular fuzzy numbers: , k = 1,2,...,r

• Normalized fuzzy values:

rvvv ~,...,~,~21

);;(~ Uk

Mk

Lkk vvvv

);;(~ Uk

Mk

Lkk wwww

S

v

S

v

S

vw

Uk

Mk

Lk

k ;;~

j

MjvS

Content

Fuzzy arithmetic

• Addition:

• Subtraction:

• Multiplication:

• Division:

• Particularly:

);;(~~~ UUMMLL babababa

);;(~~~ UUMMLL babababa

)*;*;*(~

*~~ UUMMLL babababa

)/;/;/(~

/~~ LUMMUL babababa

);;(~ UML aaaa );;(~ UML bbbb aL > 0, bL > 0

)1

;1

;1

(1~

LMU aaaa

Content

Fuzzy weights and values

• Triangular fuzzy pair-wise comparison matrix (reciprocal):

• approximation

of the matrix:

)1;1;1()1

;1

;1

()1

;1

;1

(

);;()1;1;1()1

;1

;1

(

);;();;()1;1;1(

~

222111

222111111

111111111

Ln

Mn

Un

Ln

Mn

Un

Un

Mn

LnLMU

Un

Mn

Ln

UML

aaaaaa

aaaaaa

aaaaaa

A

r

rrr

r

r

w

w

w

w

w

w

w

w

w

w

w

ww

w

w

w

w

w

~

~

~

~

~

~

~

~

~

~

~

~~

~

~

~

~

~

~

21

2

2

2

1

2

1

2

1

1

1

W

Content

Fuzzy weights and values

Solve the optimization problem:

subject to

Solution:

min;log,log,logmax,

222

ji

UijL

j

UiM

ijMj

MiL

ijUj

Li a

w

wa

w

wa

w

w

0 Li

Mi

Ui www i = 1,2,...,r

);;(~ Uk

Mk

Lkk wwww

r

i

rr

j

Mij

rr

j

Skj

Sk

a

a

w

1

/1

1

/1

1},,{ UMLS

Logarithmic method

Content

Defuzzyfication

• Result of synthesis: Triangular fuzzy vector, i.e.

• Corresponding crisp (nonfuzzy) vector:

where

TUm

Mm

Lm

UMLTn zzzzzzzzZ );;(),...,;;(~,...,~~

11111

.3

Ui

Mi

Lig

i

zzzx

),...,,( 121ggg xxxx

zL zM xg zU

1/3

Content

Algorithm

Step 1: Calculate triangular fuzzy weights(of criteria, feedback and variants):

Step 2: Calculate the aggregating triangular fuzzy evaluations of the variants:

or

Step 3: Find the „best“ variant using a ranking method (e.g. Center Gravity)

rwww ~,...,~,~21

21

1

2232

~~~~~WWIWZ

21322

2222232

~~~~~~~~~WWWWIWZ

Content

Case study

• Case study - outline• Case study - criteria• Case study - variants• Case study - feedback• Case study - W32* and W22*

• Case study - synthesis• Case study - crisp case with fedback• Case study - crisp case NO fedback• Case study - comparison

Content

Case study - outline

• Problem: Buy the best product (a car)3 criteria 4 variants

• Data: triangular fuzzy pair-wise comparisons fuzzy weights

• Calculations: 1. with feedback

2. without feedback

• Crisp case: „middle values of triangles“

Case study

Case study - criteriaL M U L M U L M U

C = 1,000 1,000 1,000 2,000 3,000 4,000 4,000 5,000 6,000 - criterion 1: C10,250 0,333 0,500 1,000 1,000 1,000 3,000 4,000 5,000 - criterion 2: C20,167 0,200 0,250 0,200 0,250 0,333 1,000 1,000 1,000 - criterion 3: C3 - criterion 1: C1 - criterion 2: C2 - criterion 3: C3

L M UW21= 0,508 0,627 0,733 - criterion 1: C1economical criterion - v(C1)

0,231 0,280 0,345 - criterion 2: C2technical criterion - v(C2)0,082 0,094 0,111 - criterion 3: C3esthetical criterion - v(C3)

Case study

Case study - variantsL M U L M U L M U L M U

A1 = 1,000 1,000 1,000 2,000 3,000 4,000 4,000 5,000 6,000 6,000 7,000 8,000 - variant 1: V10,250 0,333 0,500 1,000 1,000 1,000 3,000 4,000 5,000 5,000 6,000 7,000 - variant 2: V20,167 0,200 0,250 0,200 0,250 0,333 1,000 1,000 1,000 4,000 5,000 6,000 - variant 3: V30,125 0,143 0,167 0,143 0,167 0,200 0,167 0,200 0,250 1,000 1,000 1,000 - variant 4: V4 - variant 1: V1 - variant 2: V2 - variant 3: V3 - variant 4: V4

L M U L M U L M U L M UA2 = 1,000 1,000 1,000 1,000 2,000 3,000 2,000 3,000 4,000 3,000 4,000 5,000 - variant 3: V3

0,333 0,500 1,000 1,000 1,000 1,000 1,000 2,000 3,000 2,000 3,000 4,000 - variant 2: V20,250 0,333 0,500 0,333 0,500 1,000 1,000 1,000 1,000 1,000 2,000 3,000 - variant 1: V10,200 0,250 0,333 0,250 0,333 0,500 0,333 0,500 1,000 1,000 1,000 1,000 - variant 4: V4 - variant 3: V3 - variant 2: V2 - variant 1: V1 - variant 4: V4

L M U L M U L M U L M UA3 = 1,000 1,000 1,000 3,000 4,000 5,000 6,000 7,000 8,000 7,000 8,000 9,000 - variant 2: V2

0,200 0,250 0,333 1,000 1,000 1,000 4,000 5,000 6,000 6,000 7,000 8,000 - variant 3: V30,125 0,143 0,167 0,167 0,200 0,250 1,000 1,000 1,000 5,000 6,000 7,000 - variant 1: V10,111 0,125 0,143 0,125 0,143 0,167 0,143 0,167 0,200 1,000 1,000 1,000 - variant 4: V4 - variant 2: V2 - variant 3: V3 - variant 1: V1 - variant 4: V4

L M U L M U L M UW32 = 0,450 0,547 0,636 0,113 0,160 0,233 0,088 0,100 0,114 - variant 1: V1

0,238 0,287 0,349 0,191 0,278 0,393 0,518 0,598 0,674 - variant 2: V20,103 0,121 0,144 0,330 0,467 0,587 0,229 0,266 0,309 - variant 3: V30,040 0,045 0,052 0,076 0,095 0,135 0,033 0,036 0,041 - variant 4: V4 - criterion 1: C1 - criterion 2: C2 - criterion 3: C3 Case study

Case study - feedbackL M U L M U

B1 = 1,000 1,000 1,000 2,000 3,000 4,000 - criterion 2: C20,250 0,333 0,500 1,000 1,000 1,000 - criterion 3: C3

- criterion 2: C2 - criterion 3: C3

L M U L M UB2 = 1,000 1,000 1,000 1,000 2,000 3,000 - criterion 1: C1

0,333 0,500 1,000 1,000 1,000 1,000 - criterion 3: C3 - criterion 1: C1 - criterion 3: C3

L M U L M UB3 = 1,000 1,000 1,000 3,000 4,000 5,000 - criterion 1: C1

0,200 0,250 0,333 1,000 1,000 1,000 - criterion 2: C2 - criterion 1: C1 - criterion 2: C2

L M U L M U L M UW22 = 0,000 0,000 0,000 0,471 0,667 0,816 0,693 0,800 0,894 - criterion 1: C1

0,612 0,750 0,866 0,000 0,000 0,000 0,179 0,200 0,231 - criterion 2: C20,217 0,250 0,306 0,272 0,333 0,471 0,000 0,000 0,000 - criterion 3: C3

- criterion 1: C1 - criterion 2: C2 - criterion 3: C3

Case study

Case study - W32* and W22*

w1 = 0,231 - variantsw2 = 0,769 - feedback

L M U L M U L M UW32* = 0,104 0,126 0,147 0,026 0,037 0,054 0,020 0,023 0,026 - variant 1: V1

0,055 0,066 0,081 0,044 0,064 0,091 0,120 0,138 0,155 - variant 2: V20,024 0,028 0,033 0,076 0,108 0,135 0,053 0,061 0,071 - variant 3: V30,009 0,010 0,012 0,017 0,022 0,031 0,008 0,008 0,009 - variant 4: V4

- criterion 1: C1 - criterion 2: C2 - criterion 3: C3

L M U L M U L M UW22* = 0,000 0,000 0,000 0,363 0,513 0,628 0,533 0,615 0,688 - criterion 1: C1

0,471 0,577 0,666 0,000 0,000 0,000 0,138 0,154 0,178 - criterion 2: C20,167 0,192 0,236 0,209 0,256 0,363 0,000 0,000 0,000 - criterion 3: C3

- criterion 1: C1 - criterion 2: C2 - criterion 3: C3

Case study

Case study - synthesisL M U

Z = 0,174 0,327 0,584 - V10,178 0,341 0,651 - V20,134 0,271 0,504 - V30,033 0,061 0,119 - V4

xgi variant rank0,362 V1 = 20,390 V2 = 10,303 V3 = 30,071 V4 = 4

Total fuzzy evaluation of variants

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,000 0,200 0,400 0,600 0,800 1,000

- V1

- V2

- V3

- V4

Case study

Case study - crisp case with fedback

Crisp case: aL = aM = aU L M U L M U L M U

W22 = 0,000 0,000 0,000 0,513 0,513 0,513 0,615 0,615 0,615 economical criterion0,577 0,577 0,577 0,000 0,000 0,000 0,154 0,154 0,154 technical criterion0,192 0,192 0,192 0,256 0,256 0,256 0,000 0,000 0,000 esthetical criterioneconomical criterion technical criterion esthetical criterion

L M U rankZ = 0,327 0,327 0,327 - V1 2

0,341 0,341 0,341 - V2 10,271 0,271 0,271 - V3 30,061 0,061 0,061 - V4 4

Total evaluation of variants - non-fuzzy solution with feedback (ANP)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,000 0,200 0,400 0,600 0,800 1,000

- V1

- V2

- V3

- V4

Case study

Case study - crisp case NO fedback

L M U rankZ = 0,397 0,397 0,397 - V1 1

0,314 0,314 0,314 - V2 20,231 0,231 0,231 - V3 30,058 0,058 0,058 - V4 4

Crisp case: aL = aM = aU, W22 = 0

Total evaluation of varianst - non-fuzzy solution without feedback (AHP)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,000 0,200 0,400 0,600 0,800 1,000

- V1

- V2

- V3

- V4

Case study

Case study - comparison

Total evaluation of varianst - non-fuzzy solution without feedback (AHP)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,000 0,200 0,400 0,600 0,800 1,000

- V1

- V2

- V3

- V4

Total evaluation of variants - non-fuzzy solution with feedback (ANP)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,000 0,200 0,400 0,600 0,800 1,000

- V1

- V2

- V3

- V4

Total fuzzy evaluation of variants

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,000 0,200 0,400 0,600 0,800 1,000

- V1

- V2

- V3

- V4

Case study

Conclusion

• Fuzzy evaluation of pair-wise comparisons may be more comfortable and appropriate for DM

• Occurance of dependences among criteria is realistic and frequent

• Dependences among criteria may influence the final rank of variants

• Presence of fuzziness in evaluations may change the final rank of variants

Case study

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DĚKUJI VÁM(Thank You)