Cours PROMETHEE GAIA.pdf
Transcript of Cours PROMETHEE GAIA.pdf
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PROMETHEE & GAIA methods
Prof. Y. De Smet
CoDE-SMG, Université Libre de Bruxelles
May 9, 2012
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Outline
• Introduction
• A pedagogical example
• PROMETHEE I & II rankings
• GAIA
• Software demonstration: D-SIGHT
• A few words about rank reversal
• Preference elicitation
• Conclusion
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Historical background
Prof. Jean-Pierre Brans
(VUB, Solvay School)
Prof. Philippe Vincke
(ULB, Engineering Faculty)
Prof. Bertrand Mareschal
(ULB, Solvay Brussels School of
Economics and Management)
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Applications
Behzadian, M.; Kazemzadeh, R.B.; Albadvi, A.;
Aghdasi, M. (2010) « PROMETHEE: A comprehensive
literature review on methodologies and
applications », EJOR, Vol.200(1),198-215
4
• > 200 papers published in > 100 journals
• Topics: Environmental management, hydrology and
water management, finance, chemistry, logistics and
transportation, energy management, health care,
manufactoring and assembly, sports,…
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Let us agree on a few points
• Multicriteria decision problems are ill-defined (no
optimal solutions);
• Decision aid versus decision making;
• « The purpose of models is not to fit the data but to
sharpen the questions », Samuel Karlin;
• Parameters value:
– Interpretation;
– robustness
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Let us start with a
educational example !
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An educational example
• A plant location problem
– 6 possible locations
– 6 criteria
7
Engineers Power Cost Maintenance Village Security
Italy 75 90 600 5,4 8 5
Belgium 65 58 200 9,7 1 1
Germany 83 60 400 7,2 4 7
Sweden 40 80 1.000 7,5 7 10
Austria 52 72 600 2 3 8
France 94 96 700 3,6 5 6
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Main principle: pair-wise
comparisons
8
Engineers Power Cost Maintenance Village Security
Italy 75 90 600 5,4 8 5
Belgium 65 58 200 9,7 1 1
Germany 83 60 400 7,2 4 7
Sweden 40 80 1.000 7,5 7 10
Austria 52 72 600 2 3 8
France 94 96 700 3,6 5 6
• Concerning the cost, Germany is better than Austria !
• How can we quantify this advantage ? 200 ?
• What does it mean ?
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Unicriterion preference
function
9
difference
pre
fere
nce
200
0.25
1
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Step 1: compute unicriterion
preference degree for every
pair of alternatives
0.25
-200
Germany 83 60 400 7,2 4 7
Engineers Power Cost Maintenance Village Security
Austria 52 72 600 2 3 8
-31 12 -5.2 -1 1
1 0.75 1 0.3 0.63
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Step 2: compute global
preference degree for every
pair of alternatives
11
0.25
Germany 83 60 400 7,2 4 7
Engineers Power Cost Maintenance Village Security
Weights 0.2 0.2 0.2 0.1 0.15 0.15
Austria 52 72 600 2 3 8
0.5 0.75 1 0.3 0.63
P(Austria, Germany)=1*0.1+ 0.75* 0.2 +1*0.1+0.3*0.15+0.63*0.15
= 0.489
P(Germany,Austria)= 0.25*0.4=0.05
?
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Preference matrix
12
• How can we exploit this matrix ?
• … in order to obtain a ranking (complete or partial) ?
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Step 3: compute positive,
negative and net flow
scores
13
Germany
Austria
Belgium
Sweden
France
Italy
0.05
0.188
0.3
0.437
0.215
Germany
Austria
Belgium
Sweden
France
Italy
0.489
0.4
0.333
0.225
0.225
( ) 0.238Germany ( ) 0.334Germany
( ) ( ) ( ) 0.1Germany Germany Germany
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PROMETHEE II
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PROMETHEE I
15
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Formalization
PROMETHEE
Preference Ranking Organisation METHod for
Enrichment Evaluations
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Formalization
• A finite set of alternatives:
A={a1,a2,..,an}
• A set of criteria:
F={f1,f2,..,fq}
• W.l.g. these criteria have to be maximized
17
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Step 1: uni-criterion
preferences
)],([),(
)()(),(:,
jikkjik
jkikjikji
aadPaa
afafaadAaa
18
This operation has to be meaningful:
INTERVAL SCALE
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Preference functions
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Step 2: Compute preference
matrix
As a consequence:
20
),(.),(:,1
jik
q
k
kjiji aawaaAaa
( , ) 0
( , ) 0
( , ) ( , ) 1
i i
i j
i j j i
a a
a a
a a a a
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Step 3: compute flow scores
Maximum number of parameters: 3.q-1
21
)()()(
),(1
1)(
),(1
1)(
iii
Ab
ii
Ab
ii
aaa
abn
a
ban
a
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PROMETHEE II
Complete ranking based on the net flow score.
22
)()(
)()(
jiji
jiji
aaIaa
aaPaa
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PROMETHEE I
Partial ranking based on both the positive and negative
flow scores.
23
otherwiseJaa
aaaaIaa
aaaaPaa
aaaaPaa
ji
jijiji
jijiji
jijiji
,
)]()([)]()([
)]()([)]()([
)]()([)]()([
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The net flow score: a recipe ?
• From local to global information !
24
( , )i ja a
ai
aj
( , )i ja a
Si=S(ai),S(aj)=Sj
• One could expect that:
ij ji i js s
ill-defined problem
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Property
Intuition: “The PROMETHEE multicriteria net flow (ai) is
the centred score si (i=1,…,n) that minimizes the sum of
the squared deviations from the pair-wise comparisons of
the actions”
25
2
1 1
n n
i j ij ji
i j
Q s s
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Proof (1):
26
2
1
1 1 1
, , ,n n n
n i j ij ji i
i j i
L s s s s s
1, , ,0
n
i
L s s
s
1, , ,0
nL s s
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Proof (2):
27
2
1
1 1 1
, , ,n n n
n i j ij ji i
i j i
L s s s s s
n
ijj
ijiiji an
n
ns
1
)(1
)(1
])(.[4
)]().1.[(4
)]()[(.4
,1
,1 ,1
,1
n
ijj
jiiji
jiij
n
ijj
n
ijj
ji
n
ijj
jiijji
i
sn
ssn
sss
L
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Preferential independance (1)
28
J F is preferentially independent within G if a, b,
c, d A such that
We have a P b c P d
Jjdfbf
Jjcfaf
Jjdfcf
Jjbfaf
jj
jj
jj
jj
),()(
),()(
),()(
),()(
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Preferential independance (2)
29
Dish Sauce
a French fries Bolognese
b Spaghetti Bolognese
c French fries Mayonnaise
d Spaghetti Mayonnaise
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Prerential independance (3)
30
Jj
jj
Jj
jj
Jj
jj
Jj
jj
Jj
jj
q
j Jj
jjjj
bbwbwawbw
awawawa
)()(.)(.)(.)(.
)(.)(.)(.)(1
)()( baaPb
Jj
jj
Jj
jj bwaw )(.)(.
Jjdfbf
Jjcfaf
Jjdfcf
Jjbfaf
jj
jj
jj
jj
),()(
),()(
),()(
),()(
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31
Prerential independance (5)
Jj
jj
Jj
jj bwaw )(.)(.
)()(.)(.
)(.)(.)(.)(.
)(.)(.)(.)(1
ddwdw
bwdwawdw
cwcwcwc
Jj
jj
Jj
jj
Jj
jj
Jj
jj
Jj
jj
Jj
jj
Jj
jj
q
j Jj
jjjj
Jjdfbf
Jjcfaf
Jjdfcf
Jjbfaf
jj
jj
jj
jj
),()(
),()(
),()(
),()(
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GAIA
Geometrical Analysis for
Interactive Assistance
32
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GAIA (1)
• We have:
• Where
• In other words, every alternative can be
represented by a vector:
33
1 1
1 1
1 1( ) . ( , ) . ( , )
1 1
1. ( , ) ( , ) . ( )
1
q q
i k k i k k i
b A k b A k
q q
k k i k i k k i
k b A k
a w a b w b an n
w a b b a w an
),(),()( ik
Ab
ikik abbaa
)](),..,(),([ 21 ikiii aaa
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GAIA (2)
34
1
2
3
…
q dimensions
)](),..,(),([ 21 ikiii aaa
2
3
1
2 dimensions
Principal component analysis
D value
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GAIA (3)
• Idea:
35
u
i
pi
0
n
i
n
i
iii OpMaxpMin1 1
22 ||||
'|| uOp ii
Max
1'.
'
uu
uCu
where 'nC
)(...)(
.........
)(...)(
1
1
nkn
iki
aa
aa
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GAIA (4)
36
1'.
'
uu
uCuMax)1'..('),( uuuCuuL
1'.
..
uu
uuC
uuCu .' 1Max
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GAIA(3)
37
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GAIA(4): criteria
38
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GAIA(5): alternatives
39
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GAIA(6): alternatives /
criteria
40
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GAIA(7): Decision stick
41
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Software demonstration
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A few words about rank
reversal
43
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Rank reversal • We could have:
• In other words: a pairwise rank reversal …
• This opens a discussion about rank reversal …
– AHP: Belton and Gear (1983), Saaty and Vargas (1984),
Triantaphyllou (2001), Wang and Elhag (2006), Wijnmalen and
Wedley (2009)
– ELECTRE: Wang and Triantaphyllou (2005)
– PROMETHEE: De Keyser and Peeters (1996)
• The concept of rank reversal is not fully formalized (add
a copy of an alternative, deletion of a non discriminating
criterion, deletion of an alternative, …)
• A direct consequence of Arrow’s theorem
44
( ) ( )ij ji i ja a
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Deletion of a non
discriminating criterion
45
)('
)('
)(
)(
)()(
0)(
,1
,1
,1
1
ik
q
kjj
ijjk
q
kjj
ij
k
j
k
q
kjj
ijj
q
j
ijji
iik
aW
awW
aW
wW
aw
awa
Aaa
q
kjj
jk wW,1
where and k
j
j W
ww '
)(')(')()( jiji aaaa
f1 f2 … fk … fq
a1 f1(a1) f2(a1) … α … fq(a1)
a2 f1(a2) f2(a2) … α … fq(a2)
… … … … … … …
an f1(an) f2(an) … α … fq(an)
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• Let us assume that:
• Then:
Dominance
46
1
1
( ) ( ), 1,..
1( ) ( , ) ( , )
1
1( , ) ( , ) ( )
1
k i k j
q
i k k i k i
k b A
q
k k j k j j
k b A
f a f a k q
a w a b b an
w a b b a an
This result holds for any set A such that ai, aj A
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More general result (1)
47
Notations: ,
No RR
No RR (for any action removed) if
if if
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More general result (2)
48
RR can only occur if
Generalization: when k actions are removed
No RR if
refined threshold (depends on the sample and (a,b))
rough threshold (constant)
Mareschal, B., De Smet, Y. and Nemery, P. « Rank Reversal in the PROMETHEE II Method : Some New
Results », proceedings of de IEEE 2008 International Conference on Industrial Engineering and
Engineering Management, Singapore, (2008): 959-963
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More general result (3)
49
Statistical results relative to the «rough threshold» (for q = 2, DA=Unif)
Conclusion: The number of RR occurences is really small.
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More general result (4)
50
2/9
Verly, C. and De Smet, Y « Some considerations about rank reversal
occurrences in the PROMETHEE methods » to appear in the International
Journal of Multicriteria Decision Making.
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Related works for
PROMETHEE I
• No rank reversal will happen between ai
and aj if
51
1| ( ) ( ) |
1
1| ( ) ( ) |
1
i j
i j
a an
a an
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Rank reversal = risk of
manipulation • Joint work with Julien Roland and Céline Verly (to appear
in the proceedings of the IPMU 2012 conference)
• Aim: to quantify the likelihood of manipulation in a
simplified version of the PROMETHEE II ranking:
– Usual preference function and equal weights
– Copeland scores
• More formaly:
– A given decision maker has a perfect information on the
evaluation table;
– He may propose new alternatives in order to make
alternative ai the first one;
– Question: how many alternatives are necessary ?
52
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Linear mathematical program
53
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Results for 10 alternatives
and 3 criteria
54
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Comparison with the bound
55
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A few words about
preferences elicitation
56
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To our knowledge, only a
few contributions: • Özerol, G., Karasakal, E. (2007) « Interactive outranking approaches for
multicriteria decision-making problems with imprecise information » JORS, 59,
1253-1268
• Sun, Z. and Han, M. (2010) « Multi-criteria decision making based on PROMETHEE
method », in Proceedings of the 2010 international Conference on Computing,
Control and Industrial Engineering, 416-418
• Frikha, H., Chabchoub, H., Martel, J.-M. (2010) « Inferring criteria’s relative
importance coefficients in PROMETHEE II », International Journal of Operational
Research 7(2), 357-275
• Eppe, S., De Smet, Y., Stützle, T. (2011) « A bi-objective optimization model to
eliciting decision maker’s preferences for the PROMETHEE II method » Proceedings
of ADT (2011), 56-66
• Eppe, S., De Smet, Y. (2012) « Studying the impact of information structure in the
PROMTHEE II preference elicitation process: A simulation based approach » to
appear in the proceddings of the IPMU 2012 conference
• Eppe, S., De Smet, Y. « An experimental parameter space analysis of the
PROMETHEE II outranking method » ongoing work (1)
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General idea (1)
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w*,q*,p*
P2
R*
w,q,p
R
P2
PAC, WSR,ASR
Set of compatible
parameters !
Set of compatible
rankings !
Worst Kendall’s Tau
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General idea (2)
59
w*,q*,p*
P2
R*
w,q,p
R
P2
Set of compatible
parameters !
Set of compatible
rankings !
w*,q*,p*
W
WC
PAC, WSR,ASR
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Eppe, S., De Smet, Y., Stützle, T. (2011) « A bi-objective
optimization model to eliciting decision maker’s preferences for the
PROMETHEE II method » Proceedings of ADT (2011), 56-66
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• Main idea: quality and robustness
• Distinctive feature: the DM may communicate mistakes
(n=100,q=2)
NSGA-II
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Eppe, S., De Smet, Y. (2012) « Studying the impact of
information structure in the PROMTHEE II preference elicitation
process: A simulation based approach » to appear in the
proceddings of the IPMU 2012 conference
• Main idea: quantify information infrastructure …
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Eppe, S., De Smet, Y. (2012) « Studying the impact of
information structure in the PROMTHEE II preference elicitation
process: A simulation based approach » to appear in the
proceddings of the IPMU 2012 conference
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Eppe, S., De Smet, Y. (2012) « Studying the impact of
information structure in the PROMTHEE II preference elicitation
process: A simulation based approach » to appear in the
proceddings of the IPMU 2012 conference
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• Main idea: to overcome the limitations of the previous
approach; pairwise comparisons have to be « well-
chosen »
• q-Eval
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Eppe, S., De Smet, Y. « An experimental parameter space analysis
of the PROMETHEE II outranking method » ongoing work (1)
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Eppe, S., De Smet, Y. « An experimental parameter space analysis
of the PROMETHEE II outranking method » ongoing work (2)
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Eppe, S., De Smet, Y. « An experimental parameter space analysis
of the PROMETHEE II outranking method » ongoing work (3)
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Eppe, S., De Smet, Y. « An experimental parameter space analysis
of the PROMETHEE II outranking method » ongoing work (4)
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Eppe, S., De Smet, Y. « An experimental parameter space analysis
of the PROMETHEE II outranking method » ongoing work (4)
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Directions for future
research
1. Theoretical foundations of the
PROMETHEE & GAIA methods
– Rank reversal
2. Preferences’ elicitation
3. Current directions
– GIS, DEA, Uncertainty (see Mareschal et al.),
Sorting (see Nemery et al.), …
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