A Decision System Using ANP and Fuzzy Inputs
Jaroslav Ramík
Silesian University Opava School of Business Administration Karviná
Czech Republice-mail: [email protected]
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
References• Buckley, J.J., Fuzzy hierarchical analysis. Fuzzy Sets and Systems 17,
1985, 1, p. 233-247, ISSN 0165-0114.• Chen, S.J., Hwang, C.L. and Hwang, F.P., Fuzzy multiple attribute
decision making. Lecture Notes in Economics and Math. Syst., Vol. 375, Springer-Verlag, Berlin – Heidelberg 1992, ISBN 3-540-54998-6.
• Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, 1990, ISBN 0521305861.
• Ramik, J., Duality in fuzzy linear programming with possibility and necessity relations. Fuzzy Sets and Systems 157, 2006, 1, p. 1283-1302, ISSN 0165-0114.
• Saaty, T.L., Exploring the interface between hierarchies, multiple objectives and fuzzy sets. Fuzzy Sets and Systems 1, 1978, p. 57-68, ISSN 0165-0114.
• Saaty, T.L., Multicriteria decision making - the Analytical Hierarchy Process. Vol. I., RWS Publications, Pittsburgh, 1991, ISBN .
• Saaty, T.L., Decision Making with Dependence and Feedback – The Analytic Network Process. RWS Publications, Pittsburgh, 2001, ISBN 0-9620317-9-8.
• Van Laarhoven, P.J.M. and Pedrycz, W., A fuzzy extension of Saaty's priority theory. Fuzzy Sets and Systems 11, 1983, 4, p. 229-241, ISSN 0165-0114.
DĚKUJI VÁM(Thank You)
