An empirical distribution-based approximation of PROMETHEE II's net flow scores

Transcription

1 imw January, 215 Brussels, Belgium An empirical distribution-based approximation of PROMETHEE II's net flow scores Stefan Eppe, Yves De Smet Computer & Decision Engineering Department Université libre de Bruxelles

2 the stake T (n) =O n 2 n actions

3 promethee II A = {a 1...a n } n actions: a i i 2 I = {1...n} m criteria: g h h 2 H = {1...m} g h (a i ) 2 [, 1], 8(i, h) 2 I H

4 promethee II 1 P h (a, b) q h p h 1 f h g(a, h (a, b) b) P h (a, b) = 8 >< >:, if g h (a, b) apple q h g h (a,b) q h p h q h,ifq h < g h (a, b) apple q h 1, if g h (a, b) >p h

5 promethee II P h (a, b) =P h (a, b) P h (b, a) h(a) = 1 n 1 X b2a P h (a, b) unicriterion net flow score (a) = X h2h w h h (a) net flow score

6 promethee II P h (a, b) =P h (a, b) P h (b, a) h(a) = 1 n 1 X b2a P h (a, b) unicriterion net flow score (a) = X h2h w h h (a) net flow score

7 promethee II s continuous extension h(a) = 1 n 1 X b2a P h (a, b) unicriterion net flow score small abuse of notations h (y) = Z 1 P h (y, ) f h ( ) d

8 promethee II s continuous extension h (y) = Z 1 P h (y, ) f h ( ) d

9 approximation models h (y) = Z 1 P h (y, ) f h ( ) d g h U [,1] f h ( ) =1 PLA piecewise linear approximation

10 piecewise linear approximation PLA h(y) h (a) 1 2 q h p h 1-p h 1-q h 1 y f h (a) A B C D E h (y) = Z 1 P h (y, ) d

11 piecewise linear approximation PLA PLA h (a) h(y) h 1 2 q h p h 1-p h 1-q h 1 y f h (a) A B C D E

12 approximation models h (y) = Z 1 P h (y, ) f h ( ) d g h U [,1] f h ( ) =1 g h? f h ( ) =? PLA piecewise linear approximation EDA empirical distribution-based approximation

13 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d integration by parts empirical distribution of evaluations (needs sorting) CDF by numerical integration Details

14 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d h(y) P h (y, 1) ˆF h (y) + 2 p h q h Fh (y q ) F h (y p )+F h (y + p ) F h (y + q ) F h (y) Z y ˆF h ( ) d approximation by Riemann sum

15 simulations experimental exploration

16 simulations - setup Parameter Value(s) Number of actions n 1, 1, 1, 1 Number of criteria m 5, 7, 1 Ex post approximation models Ex ante approximation models P3R PLA, EDA Runs per instance config. N trials 1 no correlation between criteria

17 simulations - setup fp(x) h (y) p(x) p(x) 1 x y y y 1 x 1 x p(x) p(x) y 1 x y 1 x no correlation between criteria

18 simulations results

19 simulations - results = EDA PLA P3R Pearson s correlation coefficient 1 actions / 7 criteria 1 runs mixed distribution

20 simulations - results Hit rate Time Complexity PROMETHEE II 1% 1% 21 n 2 1% EDA 99,9% 99%,77,3% n log n n PLA 84% 58%,3 n,1% 1. actions / 7 criteria 1 runs mixed distribution

21 conclusion seconds 3 23 P2 Time 15 8 PLA Hit rate EDA %

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23 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d integration by parts Z b a u( )v ( ) d = h i b u( ) v( ) a Z b a u ( )v( ) d ( u( ) = P h (y, ) v ( ) = f h ( )

24 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d h(y) = h i y P h (y, ) F h ( ) Z y d P h d (y, ) F h ( ) d

25 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d h(y) = h i y P h (y, ) F h ( ) Z y d P h d (y, ) F h ( ) d F h () = d P h d (y, ) = 8 < : p h 2 q h 1 y p y q y + q y + p 1 P h (y, ) =P h (y, ) y 1 P h (,y)

26 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d h(y) = P h (y, 1)F h (y)+ 2 p h q h Z yq y p F h ( ) d + Z y + p y + q F h ( ) d!

27 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d h(y) = P h (y, 1)F h (y)+ 2 p h q h Z yq y p F h ( ) d + Z y + p y + q F h ( ) d! empirical cumulated distribution function F h (y) ˆF h (y) = 1 n N h(y)

28 empirical distribution-based approximation PLA EDA h (y) = Z 1 P h (y, ) f h ( ) d h(y) = P h (y, 1)F h (y)+ 2 p h q h Z yq y p F h ( ) d + Z y + p y + q F h ( ) d! numerical integration function (Riemann sum) F h (y) = nx 1 i=1 ˆF h a h (i) fh a h (i+1) f h a h (i) Z y F h ( ) d

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