evolutionary multi-objective optimization for inferring ... - Sciendo

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 **

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DM DM DM PDA PDA MCDA

UTA MCDA DM

DM DM PDA

DM

DM

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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

b

not a

b

(x,y)

A. x

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x,y) y

DM

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*

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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

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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

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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

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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

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j

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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-

a

an 1

k k +k

L = n

kn =

a a

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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

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P

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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

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xS

y

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

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j c

RP

wjkj j c

j c RP

S

wjkj

gj x

c

RP

kj

j G c

gj y pj qj

rpj

k

j j

kj

RP

pj

wj;

wj wj

j c Q

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+

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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

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n a

L

j

pjmin;

a x

vj- pj

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b

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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

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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

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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

f V

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

C’’

C

m

vj pj

x y

not x y P

P P

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C’’

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

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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

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*

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

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g DM x x x k k For j qj pj uj vj wj rpj vj + pj

y

ui

m

D T

ui

y

y

j

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B ab

ab

D D a b

a

b

P S f

f C

xy T

DM

B B x

y

BD f

f

x y card C

P

7. Concluding remarks

DM’s DM

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181

P

DM DM

DM x y

DM

NSGA-II

DM-

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y

x

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

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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

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European Journal of Operational Research, 188 Psichometrika, 38, Received April, 2012

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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

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…

wn

k1

k1

… kn

…

kn

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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

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187

Table 3. Comparison using other random sets (example of 6.1)

p be st*

Sample se t

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

k q p u v

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p

p q rp w k nP nQ nI n

Table 5. Some potential Pbest* from the example of 6.2

u

v

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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

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MCP

Solution 1 2 3 4 5 6 7 8 9 10 11

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

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Subset 1 2 3 4 5 6 7 8 9 10 11

k

w

rp

q

p

u

Table 9. Standard deviations from the example of 6.3.1

v

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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

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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

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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)

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195

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U p Q rp w k np,nQ,nI,n

Table 16. Some non-dominated solutions from the example of 6.4

v

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197