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Multi-Criteria Decision Support using ELECTRE methods Roman Słowiński Poznań University of Technology, Poland.

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Prezentacja na temat: "Multi-Criteria Decision Support using ELECTRE methods Roman Słowiński Poznań University of Technology, Poland."— Zapis prezentacji:

1 Multi-Criteria Decision Support using ELECTRE methods Roman Słowiński Poznań University of Technology, Poland

2 2 Weak points of preference modelling using utility (value) function Utility function distinguishes only 2 possible relations between actions: preference relation: a b U(a) > U(b) indifference relation: a b U(a) = U(b) is asymmetric (antisymmetric and irreflexive) and transitive is symmetric, reflexive and transitive Transitivity of indifference is troublesome, e.g. In consequence, a non-zero indifference threshold q i is necessary An immediate transition from indifference to preference is unrealistic, so a preference threshold p i q i and a weak preference relation are desirable Another realistic situation which is not modelled by U is incomparability, so a good model should include also an incomparability relation ? ?

3 3 Four basic preference relations and an outranking relation S Four basic preference relations are: {,,, ?} Criterion with thresholds p i (a)q i (a)0 is called pseudo-criterion The four basic situations of indifference, strict preference, weak preference, and incomparability are sufficient for establishing a realistic model of Decision Makers (DMs) preferences gi(b)gi(b) gi(a)gi(a) g i (a)-p i (b)g i (a)-q i (b)g i (a)+q i (a)g i (a)+p i (a) ab b a a b preference

4 4 Axiom of limited comparability (Roy 1985) : Whatever the actions considered, the criteria used to compare them, and the information available, one can develop a satisfactory model of DMs preferences by assigning one, or a grouping of two or three of the four basic situations, to any pair of actions. Outranking relation S groups three basic preference relations: S = {,, } – reflexive and non-transitive aSb means: action a is at least as good as action b For each couple a,bA: aSb non bSa a b a b aSb bSa ab non aSb non bSa a?b Four basic preference relations and an outranking relation S

5 5 ELECTRE Is: choice problem (P ) A x x x x x x x x x x x x x x x x x x x x x Chosen subset A of best actions Rejected subset A\A of actions

6 6 ELECTRE Is Input data: finite set of actions A={a, b, c, …, h} consistent family of criteria G={g 1, g 2, …, g n } Preferential information (i=1,…,k) intracriteria: – indifference thresholds q i (a)= i qg i (a)+ i q – preference thresholds p i (a)= i pg i (a)+ i p intercriteria: – importance coefficients (weights) of criteria k i – veto thresholds v i (a)= i vg i (a)+ i v 0q i (a)p i (a)v i (a) are functions of a worse evaluation of the two being compared The preference model, i.e. outranking relation S in set A is constructed via concordance and discordance tests

7 7 ELECTRE Is - concordance test Checks if the coalition of criteria concordant with the hypothesis aSb is strong enough (for each couple a,bA) Concordance coefficient C i (a,b) for each criterion g i : gi(b)gi(b) gi(a)gi(a) g i (a)-p i (b)g i (a)-q i (b)g i (a)+q i (a)g i (a)+p i (a) preference Ci(a,b)Ci(a,b) 0 1 gain-type gi(b)gi(b) gi(a)gi(a) g i (a)-p i (a)g i (a)-q i (a)g i (a)+q i (b)g i (a)+p i (b) preference Ci(a,b)Ci(a,b) 0 1 cost-type

8 8 ELECTRE Is - concordance test Aggregation of concordance coefficients for a,bA: C(a,b) [0, 1] Concordance test is positif if: C(a,b) where is a cutting level, such that

9 9 ELECTRE Is - discordance test and outranking relation Checks if among criteria discordant with the hypothesis aSb there is a strong opposition against aSb (for each a,bA such that C(a,b) ) For a discordant criterion g i, the opposition is strong if: (for gain-type criterion) g i (b) – g i (a) v i (a) (for cost-type criterion) g i (a) – g i (b) v i (a) Conclusion: aSb is true if and only if C(a,b) and there is no criterion strongly opposed to the hypothesis In result, for each couple a,bA, one obtains relation S either true (1) or false (0) The preference structure in set A can be represented by a graph where nodes represent actions and arcs represent relation S

10 10 ELECTRE Is – exploitation of the outranking graph The outranking graph S represents a preference structure in A What are the best actions ? Kernel K of the outranking graph: actions (nodes) belonging to K do not outrank each other (no arc between them) each action not belonging to K is directly outranked by at least one action from K a e d c b Sabcde a10111 b11110 c00101 d00010 e00101

11 11 ELECTRE Is – search for the kernel In order to have a single kernel, the graph should be acyclic Cycles should be first eliminated in one of two ways: replacing the cycle by a dummy node representing an equivalence class (clique) a e d c b cycle {c,e} a d f b f={c,e}

12 12 abe, fdc

13 13 ELECTRE Is – search for the kernel In order to have a single kernel, the graph should be acyclic Cycles should be first eliminated in one of two ways: replacing the cycle by a dummy node representing an equivalence class (clique) cutting the cycle = eliminationg the weakest arcs (outranking relations) a e d c b cycle {c,e} a e d c b

14 14 Algorytm wykrywania cykli w grafie. Utworzyć tablicę z jednym wejściem. W każdym wierszu wpisać indeksy bezpośrednich następników danego wierzchołka x i (tzw. listę ZA)..Poszukać linię pustą i skreślić numer tej linii wszędzie tam gdzie występuje w tablicy. Linie z samymi skreślonymi indeksami traktować jak puste. Skreślanie kontynuować aż do wyczerpania możliwości..Jeśli wszystkie indeksy mogły być skreślone, to graf nie ma cykli. W przeciwnym razie ma co najmniej jednej cykl..Wybrać indeks dowolnej linii niepustej a w niej dowolny nie skreślony indeks i skreślić go. Skreślanie kontynuować do momentu, gdy sekwencja skreślonych indeksów utworzy cykl.

15 15 Algorytm wykrywania cykli w grafie. Utworzyć tablicę z jednym wejściem. W każdym wierszu wpisać indeksy bezpośrednich następników danego wierzchołka x i (tzw. listę ZA)..Poszukać linię pustą i skreślić numer tej linii wszędzie tam gdzie występuje w tablicy. Linie z samymi skreślonymi indeksami traktować jak puste. Skreślanie kontynuować aż do wyczerpania możliwości..Jeśli wszystkie indeksy mogły być skreślone, to graf nie ma cykli. W przeciwnym razie ma co najmniej jednej cykl..Wybrać indeks dowolnej linii niepustej a w niej dowolny nie skreślony indeks i skreślić go. Skreślanie kontynuować do momentu, gdy sekwencja skreślonych indeksów utworzy cykl.

16 16 Algorytm znajdowania jądra w grafie acyklicznym. Utworzyć tablicę z jednym wejściem. W każdym wierszu wpisać indeksy bezpośrednich poprzedników danego wierzchołka x i (tzw. listę PRZED)..Zaznaczyć linię pustą (znaczkiem ). Skreślić całe linie, w których występuje wierzchołek z indeksem zaznaczonej linii pustej. Skreślić indeks skreślonego wiersza gdziekolwiek pojawia się w tablicy. Każda linia z samymi skreślonymi indeksami jest uznawana z pustą. Iterować aż do wyczerpania możliwości..Jeśli wszystkie indeksy mogły być zaznaczone lub skreślone, to graf ma pojedyncze jądro złożone z wierzchołków odpowiadających zaznaczonym () wierszom.

17 17 ELECTRE Is – final recommendation The best actions = the kernel: Actions in the kernel can be: initial (without predecessors) isolated (incomparable to all others) intermediate (both outranking and outranked) final (without successors) a d f b K={b} a e d c b K={b,e}

18 18 ELECTRE Is - example k PRICE =3 k COMFORT =2 k TIME =5

19 19 ELECTRE Is - example =0.75

20 20 ELECTRE Is - example

21 21 ELECTRE Is - example

22 22 ELECTRE Is - example

23 23 ELECTRE Is - example

24 24 ELECTRE Is - example Kernel= {RER}

25 25 ELECTRE TRI: sorting problem (P ) Class 1... x x x x x x x x x x x x x x x x A Class 2 Class p Class 1 Class 2... Class p

26 26 ELECTRE TRI Input data: finite set of actions A={a, b, c, …, h} consistent family of criteria G={g 1, g 2, …, g n } preference-ordered decision classes Cl t, t=1,…,p Decision classes are caracterized by limit profiles b t, t=0,1,…,p The preference model, i.e. outranking relation S is constructed for each couple (a, b t ), for every aA and t=0,1,…,p g1g1 g2g2 g 3 gngn b0b0 b1b1 b p-2 b p-1 bpbp b2b2 Cl 1 Cl 2 Cl p-1 Cl p a d

27 27 ELECTRE TRI Preferential information (i=1,…,k) intracriteria: – indifference thresholds q i t for each class limit, t=0,1,…,p – preference thresholds p i t for each class limit, t=0,1,…,p intercriteria: – importance coefficients (weights) of criteria k i – veto thresholds for each class limit v i t, t=0,1,…,p 0q i tp i tv i t are constant for each class limit b t, t=0,1,…,p Concordance and discordance tests of ELECTRE TRI validate or invalidate the assertions aSb t and b t Sa

28 28 ELECTRE TRI - concordance test Checks how strong is the coalition of criteria concordant with the hypothesis aSb t (for each couple (a,b t ), aA and t=0,1,…,p) Concordance coefficient C i (a,b t ) for each criterion g i : gi(bt)gi(bt) gi(a)gi(a) g i (b t )-p i t g i (b t )-q i t g i (b t )+q i t g i (b t )+p i t preference Ci(a,bt)Ci(a,bt) 0 1 cost-type gi(bt)gi(bt) preference Ci(a,bt)Ci(a,bt) 0 1 gain-type g i (b t )-p i t g i (b t )-q i t g i (b t )+q i t g i (b t )+p i t gi(a)gi(a)

29 29 ELECTRE TRI - concordance test Aggregation of concordance coefficients for aA and b t, t=0,1,…,p: C(a,b t ) [0, 1]

30 30 ELECTRE TRI - discordance test Checks how strong is the coalition of criteria discordant with the hypothesis aSb t (for each couple (a,b t ), aA and t=0,1,…,p) Discordance coefficient D i (a,b t ) for each criterion g i : gi(a)gi(a) 0 1 Ci(a,bt)Ci(a,bt) preference cost-type g i (b t )+v i t Di(a,bt)Di(a,bt) g i (b t )-p i t g i (b t )-q i t gi(bt)gi(bt) g i (b t )+q i t g i (b t )+p i t gi(a)gi(a) preference Di(a,bt)Di(a,bt) 0 1 gain-type Ci(a,bt)Ci(a,bt) g i (b t )-v i t gi(bt)gi(bt) g i (b t )-p i t g i (b t )-q i t g i (b t )+q i t g i (b t )+p i t

31 31 ELECTRE TRI – credibility of the outranking relation Conclusion from concordance and discordance tests – credibility of the outranking relation: (a,b t )[0, 1] What is the assignment of actions to decision classes ? (a,b t ) (b t,a) a b t b t { }a a ? b t a{ }b t not yes

32 32 ELECTRE TRI – exploitation of S and final recommendation Assignment of actions to decision classes based on two ways of comparison of actions aA to limit profiles b t, t=0,1,…,p Pessimistic: compare action a successively to each profile b t, t=p-1,…,1,0; if b t is the first profile such that aSb t, then aCl t+1 Pessimistic, because for zero thresholds, aCl t+1 g i (a)g i (b t ) i, e.g. aCl 1 g1g1 g2g2 g 3 gngn b0b0 b1b1 b p-2 b p-1 bpbp b2b2 Cl 1 Cl 2 Cl p-1 Cl p a comparison of action a to profiles b t

33 33 ELECTRE TRI – exploitation of S and final recommendation Assignment of actions to decision classes based on two ways of comparison of actions aA to limit profiles b t, t=0,1,…,p Optimistic: compare action a successively to each profile b t, t=1,…,p; if b t is the first profile such that b t { }a, then aCl t Optimistic, because for zero thresholds, aCl t g i (b t )>g i (a) i, e.g. aCl 2 g1g1 g2g2 g 3 gngn b0b0 b1b1 b p-2 b p-1 bpbp b2b2 Cl 1 Cl 2 Cl p-1 Cl p a comparison of action a to profiles b t

34 34 ELECTRE TRI - example k PRICE =3 k TIME =5 k COMFORT =2 =0.75

35 35 ELECTRE TRI - example

36 36 ELECTRE TRI - example =0.75


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