Muli-Crieria Decision Suppor using ELECTRE mehods Roman Słowiński Poznań Universiy of Technology, Poland
Weak poins of preference modelling using uiliy (value) funcion Uiliy funcion disinguishes only 2 possible relaions beween acions: preference relaion: a f b U(a) > U(b) indifference relaion: a b U(a) = U(b) f is asymmeric (anisymmeric and irrefleive) and ransiive is symmeric, refleive and ransiive Transiiviy of indifference is roublesome, e.g.? In consequence, a non-zero indifference hreshold q i is necessary An immediae ransiion from indifference o preference is unrealisic, so a preference hreshold p i q i and a weak preference relaion > are desirable Anoher realisic siuaion which is no modelled by U is incomparabiliy, so a good model should include also an incomparabiliy relaion? 2
Four basic preference relaions and an ouranking relaion S Four basic preference relaions are: {, >, f,?} preference afb a>b a b b>a bfa (a) (a)-p i (b) (a)-q i (b) (a)+q i (a) (a)+p i (a) (b) Crierion wih hresholds p i (a) q i (a) 0 is called pseudo-crierion The four basic siuaions of indifference, sric preference, weak preference, and incomparabiliy are sufficien for esablishing a realisic model of Decision Maker s (DM s) preferences 3
Four basic preference relaions and an ouranking relaion S Aiom of limied comparabiliy (Roy 1985): Whaever he acions considered, he crieria used o compare hem, and he informaion available, one can develop a saisfacory model of DM s preferences by assigning one, or a grouping of wo or hree of he four basic siuaions, o any pair of acions. Ouranking relaion S groups hree basic preference relaions: S = {, >, f} refleive and non-ransiive asb means: acion a is a leas as good as acion b For each couple a,b A: asb non bsa a>b afb asb bsa a b non asb non bsa a?b 4
ELECTRE Is: choice problem (P α ) A Chosen subse A of bes acions Rejeced subse A\A of acions 5
ELECTRE Is Inpu daa: finie se of acions A={a, b, c,, h} consisen family of crieria G={g 1, g 2,, g n } Preferenial informaion (i=1,,k) inracrieria: indifference hresholds q i (a)=α iq (a)+β q i preference hresholds p i (a)=α ip (a)+β p i inercrieria: imporance coefficiens (weighs) of crieria k i veo hresholds v i (a)=α iv (a)+β v i 0 q i (a) p i (a) v i (a) are funcions of a worse evaluaion of he wo being compared The preference model, i.e. ouranking relaion S in se A is consruced via concordance and discordance ess 6
ELECTRE Is - concordance es Checks if he coaliion of crieria concordan wih he hypohesis asb is srong enough (for each couple a,b A) Concordance coefficien C i (a,b) for each crierion : 1 C i (a,b) preference gain-ype 0 1 C i (a,b) (a) (a)-p i (b) (a)-q i (b) (a)+q i (a) (a)+p i (a) preference cos-ype (b) 0 (a) (a)-p i (a) (a)-q i (a) (a)+q i (b) (a)+p i (b) (b) 7
ELECTRE Is - concordance es Aggregaion of concordance coefficiens for a,b A: C ( a,b) = n i = 1 k C i i n i = 1 ( a,b) k i C(a,b) [0, 1] Concordance es is posiif if: C(a,b) λ where λ is a cuing level, such ha 0. 5 λ 1 min i = 1,...,n n { ki} i = 1 k i 8
ELECTRE Is - discordance es and ouranking relaion Checks if among crieria discordan wih he hypohesis asb here is a srong opposiion agains asb (for each a,b A such ha C(a,b) λ) For a discordan crierion, he opposiion is sronf: (for gain-ype crierion) (b) (a) v i (a) (for cos-ype crierion) (a) (b) v i (a) Conclusion: asb is rue if and only if C(a,b) λand here is no crierion srongly opposed o he hypohesis In resul, for each couple a,b A, one obains relaion S eiher rue (1) or false (0) The preference srucure in se A can be represened by a graph where nodes represen acions and arcs represen relaion S 9
ELECTRE Is eploiaion of he ouranking graph The ouranking graph S represens a preference srucure in A S a b c d e a 1 0 1 1 1 a b b 1 1 1 1 0 c 0 0 1 0 1 d 0 0 0 1 0 e c e 0 0 1 0 1 d Wha are he bes acions? Kernel K of he ouranking graph: acions (nodes) belonging o K do no ourank each oher (no arc beween hem) each acion no belonging o K is direcly ouranked by a leas one acion from K 10
ELECTRE Is search for he kernel In order o have a single kernel, he graph should be acyclic a b a b cycle {c,e} e c f d d f={c,e} Cycles should be firs eliminaed in one of wo ways: replacing he cycle by a dummy node represening an equivalence class (clique) 11
ELECTRE Is search for he kernel In order o have a single kernel, he graph should be acyclic a b a b cycle {c,e} e d c e d c Cycles should be firs eliminaed in one of wo ways: replacing he cycle by a dummy node represening an equivalence class (clique) cuing he cycle = eliminaiong he weakes arcs (ouranking relaions) 12
Algorym wykrywania cykli w grafie α. Uworzyć ablicę z jednym wejściem. W każdym wierszu wpisać indeksy bezpośrednich nasępników danego wierzchołka i (zw. lisę ZA). β. Poszukać linię pusą i skreślić numer ej linii wszędzie am gdzie wysępuje w ablicy. Linie z samymi skreślonymi indeksami rakować jak puse. Skreślanie konynuować aż do wyczerpania możliwości. γ. Jeśli wszyskie indeksy mogły być skreślone, o graf nie ma cykli. W przeciwnym razie ma co najmniej jednej cykl. δ. Wybrać indeks dowolnej linii niepusej a w niej dowolny nie skreślony indeks i a e skreślić go. Skreślanie konynuować do momenu, gdy sekwencja skreślonych indeksów uworzy cykl. b d c α. β. γ. δ. a c b - c d d a e b, c a c b - c d d a e b, c nie wszyskie indeksy mog y by skre lone wybieramy wiersz a i indeks c a c b - c d d a e b, c a c b - c d d a e b, c a c b - c d d a e b, c id. a do a c b - c d d a e b, c cykl worz : c, d i a b α. β. γ. e f orzymany graf jeszcze raz przepuszczamy przez algorym b - e b, f f - b - e b, f f - b - e b, f f - b - e b, f f - b - e b, f f - b - e b, f f - wszyskie indeksy mog y by skre lone wi c graf jes acykliczny 13
Algorym wykrywania cykli w grafie α. Uworzyć ablicę z jednym wejściem. W każdym wierszu wpisać indeksy bezpośrednich nasępników danego wierzchołka i (zw. lisę ZA). β. Poszukać linię pusą i skreślić numer ej linii wszędzie am gdzie wysępuje w ablicy. Linie z samymi skreślonymi indeksami rakować jak puse. Skreślanie konynuować aż do wyczerpania możliwości. γ. Jeśli wszyskie indeksy mogły być skreślone, o graf nie ma cykli. W przeciwnym razie ma co najmniej jednej cykl. δ. Wybrać indeks dowolnej linii niepusej a w niej dowolny nie skreślony indeks i a e skreślić go. Skreślanie konynuować do momenu, gdy sekwencja skreślonych indeksów uworzy cykl. α. β. γ. δ. b d c a c b d c d d a e b, c nie ma pusych linii wi c przechodzimy do osaniego kroku wybieramy wiersz a i indeks c a c b d c d d a e b, c a c b d c d d a e b, c a c b d c d d a e b, c id. a do a c b d c d d a e b, c cykl worz : c, d i a b α. β. γ. e f orzymany graf jeszcze raz przepuszczamy przez algorym b f e b, f f - b f e b, f f - b f e b, f f - b f e b, f f - b f e b, f f - b f e b, f f - wszyskie indeksy mog y by skre lone wi c graf jes acykliczny 14
Algorym znajdowania jądra w grafie acyklicznym α. Uworzyć ablicę z jednym wejściem. W każdym wierszu wpisać indeksy bezpośrednich poprzedników danego wierzchołka i (zw. lisę PRZED). β. Zaznaczyć linię pusą (znaczkiem ). Skreślić całe linie, w kórych wysępuje wierzchołek z indeksem zaznaczonej linii pusej. Skreślić indeks skreślonego wiersza gdziekolwiek pojawia się w ablicy. Każda linia z samymi skreślonymi indeksami jes uznawana z pusą. Ierować aż do wyczerpania możliwości. γ. Jeśli wszyskie indeksy mogły być zaznaczone lub skreślone, o graf ma pojedyncze jądro złożone z wierzchołków odpowiadających zaznaczonym ( ) wierszom. a e b d c α. β. γ. a - b d c a, d, e d a e b a - b d c a, d, e d a e b a - b d c a, d, e d a e b a - b d c a, d, e d a e b a - b d c a, d, e d a e b a - b d c a, d, e d a e b wszyskie wiersze mog y zosa zaznaczone lub skre lone, wi c graf ma pojedyncze j dro: N = { a, b} 15
ELECTRE Is final recommendaion The bes acions = he kernel: a b a b f c e d d K={b} K={b,e} Acions in he kernel can be: iniial (wihou predecessors) isolaed (incomparable o all ohers) inermediae (boh ouranking and ouranked) final (wihou successors) 16
ELECTRE Is - eample k PRICE =3 k TIME =5 k COMFORT =2 17
ELECTRE Is - eample λ=0.75 18
ELECTRE Is - eample 19
ELECTRE Is - eample 20
ELECTRE Is - eample 21
ELECTRE Is - eample 22
ELECTRE Is - eample Kernel= {RER} 23
ELECTRE TRI: soring problem (P β ) A... Class 1 Class 2 Class p Class 1 f Class 2 f... f Class p 24
ELECTRE TRI Inpu daa: finie se of acions A={a, b, c,, h} consisen family of crieria G={g 1, g 2,, g n } preference-ordered decision classes Cl, =1,,p Decision classes are caracerized by limi profiles b, =0,1,,p Cl 1 Cl 2 Cl p-1 Cl p b 0 b 1 b 2 b p-2 b p-1 b p a d g 1 g 2 g 3 g n The preference model, i.e. ouranking relaion S is consruced for each couple (a, b ), for every a A and =0,1,,p 25
ELECTRE TRI Preferenial informaion (i=1,,k) inracrieria: indifference hresholds q i for each class limi, =0,1,,p preference hresholds p i for each class limi, =0,1,,p inercrieria: imporance coefficiens (weighs) of crieria k i veo hresholds for each class limi v i, =0,1,,p 0 q i p i v i are consan for each class limi b, =0,1,,p Concordance and discordance ess of ELECTRE TRI validae or invalidae he asserions asb and b Sa 26
ELECTRE TRI - concordance es Checks how srons he coaliion of crieria concordan wih he hypohesis asb (for each couple (a,b ), a A and =0,1,,p) Concordance coefficien C i (a,b ) for each crierion : 1 C i (a,b ) preference gain-ype 0 g (b ) i (b )-p i (b )-q i (b )+q i (b )+p (a) i 1 C i (a,b ) preference cos-ype 0 (b ) (b )-p i (b )-q i (b )+q i (b )+p i (a) 27
ELECTRE TRI - concordance es Aggregaion of concordance coefficiens for a,b A: C ( a,b ) = n i = 1 k C i i n i = 1 ( a,b ) k i C(a,b ) [0, 1] 28
ELECTRE TRI - discordance es Checks how srons he coaliion of crieria discordan wih he hypohesis asb (for each couple (a,b ), a A and =0,1,,p) Discordance coefficien D i (a,b ) for each crierion : 1 D i (a,b ) C i (a,b ) preference gain-ype 0 1 C i (a,b ) (b )-v i (b ) (b )-p i (b )-q i (b )+q i (b )+p i D i (a,b ) (a) preference cos-ype 0 (b )-p i (b )-q i (b ) (b )+q i (b )+p i (b )+v i (a) 29
ELECTRE TRI credibiliy of he ouranking relaion Conclusion from concordance and discordance ess credibiliy of he ouranking relaion: σ 1 D ( ) ( ) i ( a,b ) a,b = C a,b 1 C( a,b ) i F F = { i : D ( a,b ) > C( a,b )} where i σ(a,b ) [0, 1] σ(a,b ) λ no yes σ(b,a) λ σ(b,a) λ no yes no yes a? b b {f >}a a{f >}b a b Wha is he assignmen of acions o decision classes? 30
ELECTRE TRI eploiaion of S and final recommendaion Assignmen of acions o decision classes based on wo ways of comparison of acions a A o limi profiles b, =0,1,,p Pessimisic: compare acion a successively o each profile b, =p-1,,1,0; if b is he firs profile such ha asb, hen a Cl +1 comparison of acion a o profiles b Cl 1 Cl 2 Cl p-1 Cl p b 0 b 1 b 2 b p-2 b p-1 b p a g 1 g 2 g 3 g n Pessimisic, because for zero hresholds, a Cl +1 (a) (b ) i, e.g. a Cl 1 31
ELECTRE TRI eploiaion of S and final recommendaion Assignmen of acions o decision classes based on wo ways of comparison of acions a A o limi profiles b, =0,1,,p Opimisic: compare acion a successively o each profile b, =1,,p; if b is he firs profile such ha b {f >}a, hen a Cl comparison of acion a o profiles b Cl 1 Cl 2 Cl p-1 Cl p b 0 b 1 b 2 b p-2 b p-1 b p a g 1 g 2 g 3 g n Opimisic, because for zero hresholds, a Cl (b )< (a) i, e.g. a Cl 2 32
ELECTRE TRI - eample k PRICE =3 k TIME =5 k COMFORT =2 λ=0.75 33
ELECTRE TRI - eample 34
ELECTRE TRI - eample λ=0.75 35