FOUNDATIONS
Vol. 37
OF
COMPUTING AND
(2012)
DECISION
SCIENCES
No.
10.2478/v10209-011-0010-0
EVOLUTIONARY MULTI-OBJECTIVE OPTIMIZATION
FOR INFERRING OUTRANKING MODEL’S PARAMETERS
UNDER SCARCE REFERENCE INFORMATION AND EFFECTS
OF REINFORCED PREFERENCE
Eduardo FERNANDEZ *
Jorge NAVARRO **
Gustavo MAZCORRO ***
Abstract. Methods based on fuzzy outranking relations constitute one of the main
approaches to multiple criteria decision problems. The use of ELECTRE methods require
the elicitation of a large number of parameters (weights and different thresholds); but direct
eliciting is often a demanding task for the decision-maker (DM). For handling intensity-ofpreference effects on concordance levels, a generalized concordance model was proposed
by Roy and Slowinski which is more complex than previous outranking models. In this
paper, an evolutionary multi-objective-based indirect elicitation of the complete ELECTRE
III model-parameter set is proposed. The evolutionary multi-objective inference method is
successfully extended to inferring reinforced-preference model parameters. Wide
experimental evidence is provided to support the proposal, which performs well even
working on small size reference sets.
Keywords: Multiple criteria analysis, Fuzzy outranking relations, Parameter inference,
Evolutionary algorithms.
1. Introduction
Many practical decisions can be modeled by using multi-criteria decision analysis. Multicriteria methods entail a decision-maker (DM) reflecting his/her preferences in a prespecified mathematical structure. Hence, obtaining preference information from the DM
and formalizing such information into preferential parameters is a crucial aspect in building
a multi-criteria decision model ( 4 ). The development of these models can be based on
direct or indirect elicitation procedures. In the first case, the DM must specify preferential
parameters through an interactive process guided by a decision analyst ( 7 ). Usually, the
DMs reveal difficulties when they are inquired to assign values to parameters whose
meanings are barely clear for them. On the other hand, indirect procedures, which compose
the so-called preference-disaggregation analysis (PDA), use regression-like methods for
*
Corresponding author, Autonomous University of Sinaloa.
Autonomous University of Sinaloa.
***
National Technical Institute, UPIICSA
**
Unauthenticated
Download Date | 6/16/17 11:35 PM
DM
DM
DM
PDA
PDA
MCDA
UTA
MCDA
DM
DM
DM
PDA
DM
DM
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
165
DM
DM
DM
DM
DM
2. Assumptions and notations
G= g ,… gn
A A
x
Assumption 1:
a,b
a
DM
ab
T
T
A
x y
y
A A
DM
b
not a
b
(x,y)
A.
x
x,y)
y
Unauthenticated
Download Date | 6/16/17 11:35 PM
P
x
Assumption 2:
x,y,P
y
DM
DM
P*
T
x,y
x,y
x,y
x,y
x,y
x,y P*
x,y P*
x,y P*
x,y P*
x,y P*
S
P
Q
I
R
y,x P*
y,x P*
y,x P*
y,x P*
3. Parameter inference by using a multi-criteria error measure
xy
T
x,y P*
x
x
y
x,y P*
y
x is at least as good as y
x is at least as good as y
good as x
xy
T
xP
y
x
y
xQ
y
x
y
y
xI
x
y
x
xS
y is not at least as
y
y
x
y
x is at
least as good as y
x y
x,y P*
x y
P
not x y
x y
Q
not x
x y
I
x y
y
not x
y
not xS
y
P
Q
,I
,
x,y P*
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
x
DP
x y
P
DQ
x y
Q
DI
x y
I
D
not x
nP nQ nI
y
y
not x
x y
167
y
not x
y
not xS
y
n
P
Minimize nP nQ nI n
P RF
RF
DM
nP nQ nI n
P
nI
nQ
n Q nI
DM
n
n
nP nQ
P
nI n
P
DM
P
DM
Unauthenticated
Download Date | 6/16/17 11:35 PM
DM’
4. Inference of ELECTRE III parameters by evolutionary multiobjective optimization
4.1. Brief outline of ELECTRE III
S
xSy x
y
DM
x
y
xSy
C x y)
Dxy
C x,y
C xSy
C yQx
gj G
gj G
gj x
gj y
gj y
qj
pj
gj x
gj y
qj
pj
D x,y
qj
Q
j pj qj
C yPx
j G
g y
g x
pj
P
cxy
C xy
xSy
xy
c x y Nd x y
Nd x y
cxy
c xy
j
kj
cj x y
cj x y
gj x
gj y
G
kjcj x y
k + k + ... + kN =
pj
p j – qj
j
j C yPx
j C yQx
j C xSy
v
g y) - g x
x,y
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
169
min
dj x,y
j C yPx
Nd x,y
v
dj x,y
u
v
u
u
v
u
g y) - g x
u
xy
A,
*
4.2 Constraints in Problem 3
DM
DM
DM
DM
DM
SI
LI
MI
MI
G
G G
g m gj
G G
g m gj
G G
g m gj
LI
SI
k
gmMIgj
gmLIgj
gm
gm
gm
k
k
k
k
n
gj
gj
gj
k
n
DM
For j= ,…n
0 qj min qj qj max
Unauthenticated
Download Date | 6/16/17 11:35 PM
pj min pj pj max qj max pj min
uj min uj uj max pj max uj min
vj min vj vj max uj max vj min
DM
vj vl vi
DM
p, v
vj- pj
vj + p j / – u i
ql
qj
pl
u
pj
i= ,…n
DM
DM
4.3 Description of the evolutionary approach for inferring the model
parameters
NSGA-II
NSGA-II
K’
K’
K’
NSGA-II
i, j
i
i
j
j
i
i
j
NSGA-II
Generate random population (size K’)
Evaluate objective values
Generate non-dominated fronts
Assign to these fronts rank based on Pareto dominance
Keep the best front (rank) in the population memory
Generate offspring population
Binary tournament selection
Crossover and mutation
For i = 1 to number of generations
Unauthenticated
Download Date | 6/16/17 11:35 PM
j
Evolutionary multi-objective optimization for inferring ...
171
With parent and offspring population
Generate non-dominated fronts
Assign to these fronts rank based on Pareto dominance
Loop (inside) by adding solutions to next generation
Starting from the best front until K’ individuals found
Calculate crowding distance between points on each front
Update the population memory
Select points (elitist) on the better front (with better rank)
and which are outside a crowding distance
Form next generation
Binary tournament selection
Crossover and mutation
Increment generation index
End of loop
n
n+
L
pjmin pjmax
b
b
L
n
L
qj
n+
n
j=L
a
n
j = n –L
pj
a
vj + p j 2 – x
B= x
L
n+
a
n
j
L
vjmin; vjmax
b
b
a
vj- pj
n-L
B
n
x
L = n+
b
j= n-L
vj
a
vj + p j 2 – x
n
uj
B
b
uj
n+
qjmin qjmax
a
vj pj
uj
kj
n
0 a
– ai-
L = n
a
an 1
k
k +k
kn =
a
a
Unauthenticated
Download Date | 6/16/17 11:35 PM
ai
K’
pm
pc
Population size
Number of generations=
Mutation probability=
Crossover probability
4.4. Final formalization and discussion
n P nQ nI n
a b
DM
DM
n P nQ nI n
P
nP nQ nI n
DM
P
DM
P
P
DM
P
P
DM
P
DM
P
P
P
P
P
P
P
P
P
P
P
P
P
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
173
DM
DM
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
DM
P
P
DM
T
P
P
P
DM
DM
DM
nP n Q n I n
P
P
DM
nP n Q n I n
P
5. Handling reinforced preference on the credibility of outranking
x y
x
k =k =k
p p p
xS y
y
v1=v2=v3
xy
yx
DM
x
x y
xS
y
Unauthenticated
Download Date | 6/16/17 11:35 PM
not
c xy
c xy
kj
gj
C xSy
gj
g y
g x
rpj
kj
gj
cxy
wjkj
wj
g x
C xRPy
C yQx
g y
rpj
cxy
c xy
j c
RP
wjkj
j c
j c
RP
S
wjkj
gj x
c
RP
j G c
gj y
pj qj
wj;
wj wj
rpj
kj
pj
j c Q
k
j j
kj
RP
pj
kj
rpj
uj
rpj
For j= ...n
uj rpj rpjmax
wj wjmax
5.1. Description of the evolutionary approach for inferring the model
parameters
n
n+
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
L
L
n
n
L
n
n
pj
pjmax
b
qLmin; qLmax
a
a
qL
B
x
b
L
L
vjmin; vjmax
b
b
n
B
n
L
n
a
L
j
pjmin;
a
x
vj- pj
uj
n-L
b
B
b
uj
x
rpjmin; rpjmax
j= n-L
a
vj p j
n
n
175
j n-L+
vj
a
vj + pj /2 – x
a
vj- pj
uj
n
j= n-L
a
n
j= n-L
a
uj
n
wjmin; wjmax
L= n
n
kj
n
0
k
a
ai – ai-
L = n+
a
an-
a
a
6. Some illustrative examples
6.1. First example: Inferring the ELECTRE III model parameters
DM
DM
DM
DM
Unauthenticated
Download Date | 6/16/17 11:35 PM
U
V
F
V
xy
F
T x y
F
F
Vx
F F F F
F F F F
Vy
V
V
V
Fi
V
V
V
V
vj
k
k
For j
qj
pj
uj
vj
vj + pj
m
ui
j
m
vj pj
DM
DM
D
x y
not x y
D
DM
DM
x y
not x y
P
P
P
P
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
177
P
V
P
DM
P
P
f
xS
y
x y
f
C
P
Vx
x y
xS y
C C’
C
f
f
P
Vy
not xS
xy
f
f
y
C’’
C
f
V
C’
C’’
P
V
P
6.2 Second example: Inferring reinforced preference model parameters
k
k
For j
qj
pj
uj
vj
wj
rpj
vj + pj
m
ui
j
m
vj pj
x y
not x y
P
P
P
Unauthenticated
Download Date | 6/16/17 11:35 PM
P
P
x y
S
C
C C’
C
xy
f
P
xS( )y
C
f
f
C’’
P
f
f
f
x
y
not xS
y
Partial conclusions:
P
P
f
q p u, v k
f
P
P
f
6.3 The preference information comes from an outranking model
6.3.1 A four criteria problem
U
g g g,g
gi
xy
q
p
u
v
rp
w
k
xy
U U
x
y x y
xy
P
P
P
P
P
Unauthenticated
Download Date | 6/16/17 11:35 PM
*
Evolutionary multi-objective optimization for inferring ...
179
P
P
6.3.2 A ten criteria problem
g g
U
,g
gi
xy
q
p
u
v
rp
w
k
k
k
For j
qj
pj
uj
vj
wj
rpj
vj + pj
m
ui
j
m
vj pj
P
P
P
P
6.4 The preference information comes from a real DM
R&D
B
Unauthenticated
Download Date | 6/16/17 11:35 PM
g
DM
x
x
x
k
k
For j
qj
pj
uj
vj
wj
rpj
vj + pj
y
m
D
T
ui
y
y
ui
j
DM
m
vj pj
B
ab
ab
D D a b
a
b
P
S
f
f
C
xy
T
B B x
y
BD
f
f
x y
card C
P
7. Concluding remarks
DM’s
DM
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
181
P
DM
DM
DM
x
y
y
x
DM
NSGA-II
DM-
Unauthenticated
Download Date | 6/16/17 11:35 PM
DM
DM
Acknowledgements
References
European Journal of Operational Research 103
Evolutionary Algorithms for
Solving Multi_Objective Problems
Multi-Objective Optimization using Evolutionary Algorithms
European Journal of
Operational Research 138
European Journal of Operational Research, 170
Multicriteria Decision Aid Classification Methods
European Journal of Operational Research, 199
European Journal of Operational Research, 185
European
Journal of Operational Research, 198
Annals of Operations
Research, 120
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
183
European Journal of
Operational Research, 191
Annals of Operations Research,
185
European Journal of
Operational Research, 10
Decision with multiple objectives: preferences and value
tradeoffs
IEEE
Transactions on Information Theory, 14
71 Meeting of the Euro Working Group Multiple Criteria
Decision Aiding
Annals of Operations Research, 120
Genetic Algorithms + Data Structures = Evolution Programs
Journal of Global Optimization, 12
European Journal
of Operational Research, 130
European Journal of Operational
Research, 156
Annals of Operations Research, 80
Management Science, 20
71 Meeting of the Euro Working Group Multiple Criteria
Decision Aiding
Annals
of Operations Research, 138
Reading in Multiple Criteria Decision Aid
Unauthenticated
Download Date | 6/16/17 11:35 PM
European Journal of Operational Research, 188
Psichometrika, 38,
Received April, 2012
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
185
Figures
dj x,y
dj x,y
gj x
uj
gj x
vj
gj y
p1
u1
v1
…
…
pn
un
vn
k1
k2
…
kn
q1
q2
…
qn
p1
u1
v1
…
…
pn
un
vn
k1
k2
…
kn
q1
q2
…
qn
vn
rpn
w1
… wn
q1
… qn
p1
u1
q1
…
p1
u1
qn
v1
v1
rp1
rp1
… pn
…
pn
un
un
vn
rpn
w1
…
wn
k1
k1
… kn
…
kn
Unauthenticated
Download Date | 6/16/17 11:35 PM
Unauthenticated
Download Date | 6/16/17 11:35 PM
nP nQ nI n
k
q
p
u
Tables
Table 1. Potential Pbest* (Example of 6.1, first experiment)
v
u
p
q
k
Table 2. Standard deviations from the example of 6.1
v
Evolutionary multi-objective optimization for inferring ...
Unauthenticated
Download Date | 6/16/17 11:35 PM
187
Table 3. Comparison using other random sets (example of 6.1)
Sample se t
p be st*
f1
f2
Table 4. Other results from the example of 6.1
Second experiment
Third experiment
nP nQ nI n
p
nP , nQ , nI , n
p
p
p
k
q
p
u
v
Unauthenticated
Download Date | 6/16/17 11:35 PM
p
q
rp
w
k
nP nQ nI n
Table 5. Some potential Pbest* from the example of 6.2
u
v
Evolutionary multi-objective optimization for inferring ...
Unauthenticated
Download Date | 6/16/17 11:35 PM
189
Table 6. Comparison using other random sets (Example of 6.2)
Sample se t
p be st*
f1
f2
Table 7. Other results from the example of 6.2
n n n n
Pbest*
Pbest*
k
w
rp
q
p
u
v
Unauthenticated
Download Date | 6/16/17 11:35 PM
Solution
1
2
3
4
5
6
7
8
9
10
11
Unauthenticated
Download Date | 6/16/17 11:35 PM
MCP
w
rp
q
p
u
v
(nP , nQ , nI , n )*=(0,0,0,0)
(0.246,0.257,0.243,0.254) (1.960,2.713,2.057,1.986) (3.455,2.849,3.313,3.294) (0.164,0.153,0.148,0.151) (0.764,0.696,0.706,0.696) (2.143,2.043,2.074,2.044) (3.614,3.471,3.513,3.476)
k
Table 8. Some potential Pbest* from the example of 6.3.1
Evolutionary multi-objective optimization for inferring ...
191
Subset
1
2
3
4
5
6
7
8
9
10
11
Unauthenticated
Download Date | 6/16/17 11:35 PM
k
w
rp
q
p
u
Table 9. Standard deviations from the example of 6.3.1
v
Evolutionary multi-objective optimization for inferring ...
Table 10. Comparison using random sets (Example of 6.3.1)
Sample se t
p be st*
f1
f2
Table 11. Other results from the example of 6.3.1
Second experiment
p
p
k
w
rp
q
p
u
v
Unauthenticated
Download Date | 6/16/17 11:35 PM
193
Table 12. Some potencial Pbest* from the example of 6.3.2
k
w
rp
q
p
u
v
k
w
rp
q
p
u
v
k
w
rp
q
p
u
v
k
w
rp
q
p
u
v
k
w
rp
q
p
u
v
Unauthenticated
Download Date | 6/16/17 11:35 PM
Evolutionary multi-objective optimization for inferring ...
Table 13. Final solutions from the example of 6.3.2
P
Q
I
p
p
k
w
rp
q
p
u
v
Table 14. Comparison using random sets (Example of 6.3.2)
Sample se t
p be st*
f1
f2
Table 15. Reference assignments (example of 6.4)
Unauthenticated
Download Date | 6/16/17 11:35 PM
195
Unauthenticated
Download Date | 6/16/17 11:35 PM
197
U
p
Q
rp
w
k
np,nQ,nI,n
Table 16. Some non-dominated solutions from the example of 6.4
v
Evolutionary multi-objective optimization for inferring ...
Unauthenticated
Download Date | 6/16/17 11:35 PM