PROMETHEE & GAIA methods. Prof. Dr. Yves De Smet, Department CoDE-SMG, ULB
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1 PROMETHEE & GAIA methods Prof. Dr. Yves De Smet, Deprtment CoDE-SMG, ULB 1
2 Outlne Introducton A pedgogcl exmple PROMETHEE I & II rnngs Propertes A few words bout rn reversl GAIA Softwre demonstrton Concluson 2
3 Hstorcl bcground Prof. Jen-Perre Brns (VUB, Solvy School Prof. Phlppe Vnce (ULB, Engneerng Fculty Prof. Bertrnd Mreschl (ULB, Solvy Brussels School of Economcs nd Mngement 3
4 Applctons Behzdn, M.; Kzemzdeh, R.B.; Albdv, A.; Aghds, M. (2010 «PROMETHEE: A comprehensve lterture revew on methodologes nd pplctons», EJOR, Vol.200(1, > 200 ppers publshed n > 100 ournls Topcs: Envronmentl mngement, hydrology nd wter mngement, fnnce, chemstry, logstcs nd trnsportton, energy mngement, helth cre, mnufctorng nd ssembly, sports, 4
5 5
6 Let us gree on few ponts Multcrter decson problems re lldefned (no optml solutons; Decson d versus decson mng; «The Truth s Out There» (X-Fles; «The purpose of models s not to ft the dt but to shrpen the questons», Smuel Krln 6
7 Let us strt wth eductonl exmple! 7
8 An eductonl exmple A plnt locton problem 6 possble loctons 6 crter Unts 10MW 10 6 $ 10 6 $ Engneers Power Cost Mntennce Vllge Securty Itly ,4 8 5 Belgum ,7 1 1 Germny ,2 4 7 Sweden , Austr Frnce ,
9 Mn prncple: pr-wse comprsons Engneers Power Cost Mntennce Vllge Securty Itly ,4 8 5 Belgum ,7 1 1 Germny ,2 4 7 Sweden , Austr Frnce ,6 5 6 Concernng the cost, Germny s better thn Austr! How cn we quntfy ths dvntge? 200? Wht does t men? 9
10 preference Uncrteron preference functon dfference 10
11 Step 1: compute uncrteron preference degree for every pr of lterntves Germny ,2 4 7 Engneers Power Cost Mntennce Vllge Securty Austr
12 Step 2: compute globl preference degree for every pr of lterntves? 0.25 Germny ,2 4 7 Engneers Power Cost Mntennce Vllge Securty Weghts Austr P(Austr, Germny=1* * * * *0.2 = 0.49 P(Germny,Austr= 0.25*0.2=
13 Preference mtrx How cn we explot ths mtrx? n order to obtn rnng (complete or prtl? 13
14 Step 3: compute postve, negtve nd net flow scores Austr Austr Germny Belgum Sweden Germny Belgum Sweden Frnce Itly Frnce Itly ( Germny ( Germny ( Germny ( Germny ( Germny
15 PROMETHEE II 15
16 PROMETHEE I 16
17 Formlzton PROMETHEE Preference Rnng Orgnston METHod for Enrchment Evlutons 17
18 Formlzton A fnte set of lterntves A={ 1, 2,.., n } A set of (quntttve or qulttve crter F={f 1,f 2,..,f q } W.l.g. these crter hve to be mxmzed 18
19 Step 1: un-crteron preferences 19 ], ( [, ( ( (, ( :, d P f f d A
20 Step 2: Compute preference mtrx As consequence: 20, (., ( :, 1 q w A (, 0 (, 0 (, (, 1
21 Step 3: compute flow scores 1 ( (, n 1 1 ( (, n 1 ( ( ( A A Mxmum number of prmeters: 3.q-1 21
22 PROMETHEE II Complete rnng bsed on the net flow score. P ( ( I ( ( 22
23 PROMETHEE I Prtl rnng bsed on both the postve nd negtve flow scores. P ( ( ( ( I ( ( ( ( J otherwse 23
24 The net flow score: recpe? From locl to globl nformton! (, (, S =S(,S( =S ll-defned problem One could expect tht: s s 24
25 Property The PROMETHEE multcrter net flow f( s the centred score s (=1,,n tht mnmzes the sum of the squred devtons from the pr-wse comprsons of the ctons n n Q s s 25
26 Proof (1: n n n,,, 2 L s1 s s s s n L s,,, 1 sn 0 s L s,,, 1 sn 0 26
27 Proof (2: 1 n n L 2 2 s s s s,,, 2 1 n 1 s s 4 n n 4n 1 s s 1 1 n 4ns 0 1 n n n L s s s s s s 1 n n 1 ( 27 n 1 f( n
28 A few words bout rn reversl 28
29 Rn reversl We could hve: f( f( In other words: prwse rn reversl Ths opens dscusson bout rn reversl AHP: Belton nd Ger (1983, Sty nd Vrgs (1984, Trntphyllou (2001, Wng nd Elhg (2006, Wnmlen nd Wedley (2009 ELECTRE: Wng nd Trntphyllou (2005 PROMETHEE: De Keyser nd Peeters (1996 The concept of rn reversl s not fully formlzed (dd copy of n lterntve, deleton of non dscrmntng crteron, deleton of n lterntve, A drect consequence of Arrow s theorem 29
30 Deleton of non dscrmntng crteron 30 '( ( ' 1 ( 1 ( 1 1 ( 1 1 ( 0 ( 1, 1, 1, 1 q q q q W w n W W w n W w n w n A f f f f f f f q W w 1, where nd W w w ' '( '( ( ( f f f f f 1 f 2 f f q 1 f 1 ( 1 f 2 ( 1 α f q ( 1 2 f 1 ( 2 f 2 ( 2 α f q ( 2 n f 1 ( n f 2 ( n α f q ( n
31 Domnnce Let us ssume tht: f ( f (, 1,.. q Then: q 1 f( w (, b ( b, n ba q 1 w (, b ( b, f( n 1 ba Ths result holds for ny set A such tht, A 31
32 More generl result (1 Nottons:, No RR f No RR (for ny cton removed f 32
33 More generl result (2 RR cn only occur f refned threshold (depends on the smple nd (,b rough threshold (constnt Generlzton: when ctons re removed No RR f 33
34 More generl result (3 Sttstcl results reltve to the «rough threshold» (for q = 2, DA=Unf Concluson: The number of RR occurences s relly smll. 34
35 More generl result (4 2/9 Verly, C. nd De Smet, Y «Some consdertons bout rn reversl occurrences n the PROMETHEE methods» 35
36 Relted wors for PROMETHEE I No rn reversl wll hppen between nd f 1 f ( f ( n 1 1 f ( f ( n 1 36
37 GAIA Geometrcl Anlyss for Interctve Assstnce 37
38 GAIA (1 We hve: q q 1 1 ( w. (, b w. ( b, Where n1ba 1 n1ba 1 q q 1 w. (, b ( b, w. f ( 1 n 1 ba 1 ( (, b ( b, ba In other words, every lterntve cn be represented by vector: [ (, 2 (,.., ( 1 ] 38
39 GAIA (2 D vlue f 1 f 1 f 2 f 2 f 3 [ (, 2(,.., ( 1 ] f 3 q dmensons 2 dmensons Prncpl component nlyss 39
40 GAIA(3 40
41 GAIA(4: crter 41
42 GAIA(5: lterntves 42
43 GAIA(6: lterntves / crter 43
44 GAIA(7: Decson stc 44
45 Softwre demonstrton 45
46 Concluson A trl verson of the softwre cn be downloded for free t: Bblogrphc references: Don t hestte to contct me for further nformton: yves.de.smet@ulb.c.be 46
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