1. Analyzing the Problem (Outranking Methods) Y. İlker TOPCU, Ph.D. www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info www.facebook.com/yitopcu twitter.com/yitopcu

2. Dominance vs. MAVT Dominance of a over b translates a sort of agreement for all points of view in favor of a: v j (a)>v j (b) where at least one of the inequalities is strict v j (a): performance value of alternative a w.r.t. attribute j Methods based on multi attribute value theory lead to a function allowing the ranking of all alternatives from best to worst Dominance relation is quite poor because very few pairs of alternatives verify it – Multi attribute value function is very rich because it introduces very strong mathematical hypotheses and necessitates very complicated questions to be asked to the decision maker (DM)

3. Development of Outranking Methods One may wonder whether it is always necessary to go that far for constructing a function in the frame of decision aid The underlying idea for the development of the outranking methods is to reveal a relation in between the dominance relation (too poor to be useful) and the multi attribute value function (too rich to really be reliable) What is attempted in outranking methods is to enrich the dominance relation by some elements Preference aggregation based; most outranking methods are non-compensatory

4. Incomparability When a DM must compare two alternatives, s/he will react in one of the three following ways: preference for one of them indifference between them refusal or inability to compare them Two alternatives can perfectly remain incomparable without endangering the decision aid procedure A conclusion of incomparability between some alternatives may also be quite helpful since it puts forward some aspects of the problem which would perhaps deserve a more thorough study

5. Outranking Relation A binary relation S is defined in the set of alternatives such that aSb if there are enough arguments to decide that a is at least as good as b, while there is no essential reason to refute that statement (given what is known about the DM’s preferences and given the quality of the valuations of the alternatives and the nature of the problem)

6. Main Steps of Outranking Methods 1. Building the outranking relation 2. Exploitating the outranking relation with regard to the chosen statement of the problem

7. PROMETHEE Preference Ranking Organization METHod for Enrichment Evaluation (Brans & Vincke, 1985) PROMETHEE I yields a partial preorder PROMETHEE II yields a unique complete preorder

8. Main Steps 1. Building the outranking relation DM chooses a generalized criterion and fixes the necessary parameters related to the selected criterion: a preference function is defined for each attribute Multicriteria preference index is defined as the weighted average of the preference functions This preference index determines a valued outranking relation on the set of alternatives.

9. Main Steps 2. Exploitating the outranking relation with regard to the chosen statement of the problem For each alternative, a leaving and an entering flow are defined. A net flow is also considered A partial preorder (PROMETHEE I) or a complete preorder (PROMETHEE II) can be proposed to the DM

10. Usual Criterion P k (a i,a j ) = Quasi Criterion P k (a i,a j ) = Crit. with Linear Pref. P k (a i,a j ) = Level Criterion P k (a i,a j ) = Crit. With Indifference Area. P k (a i,a j ) = Gaussian Criterion P k (a i,a j ) = Recommended Generalized Criteria p: preference threshold, q:indifference threshold

11. Generalized Criteria Criterion I 1 Pk(ai,aj)Pk(ai,aj) d qkqk Criterion II 1 Pk(ai,aj)Pk(ai,aj) d Pk(ai,aj)Pk(ai,aj) pkpk Criterion III 1 d Criterion IV qkqk pkpk 0.5 1 Pk(ai,aj)Pk(ai,aj) d qkqk pkpk Criterion V 1 Pk(ai,aj)Pk(ai,aj) d kk Criterion VI 1 Pk(ai,aj)Pk(ai,aj) d

12. Necessary Calculations Multiattribute Preference Index  (a i,a j ) = Leaving Flow Entering Flow Net Flow

13. PROMETHEE I Two complete preorders are built: Ranking the alternatives following the decreasing order of leaving flows Ranking the alternatives following the increasing order of entering flows The intersection of the preorders yields the partial preorder

14. PROMETHEE II A unique complete preorder is built: Ranking the alternatives following the decreasing order of net flows

15. Example for PROMETHEE Building the Relation Criterion V (Indifference area) is selected. Indifference and preference thresholds are fixed:

16. Example for PROMETHEE Exploiting the Relation Preference indices and flows

17. PROMETHEE I Results Two complete preorders: Graph of alternatives (partial preorder) a3a3 a7a7 a1a1 a2a2 a6a6 a5a5 a4a4

18. PROMETHEE II Results a3a3 a7a7 a1a1 a2a2 a6a6 a5a5 a4a4

19. ELECTRE Roy designed a method for choice problems in 1968: “Elimination et choix traduisant la réalité” (elimination and choice that translates reality) (Vincke, 1992; Yoon & Hwang, 1995; Evren & Ülengin, 1992)

20. Main Steps 1. Building the outranking relation A concordance index and a discordance index (if applicable) are associated to each ordered pair of alternatives A concordance threshold and a discordance threshold (or a discordance set) are defined An outranking relation is defined for each ordered pair of alternatives

21. Main Steps 2. Exploitating the outranking relation Outranking relations are represented by a graph The kernel of the graph is determined The alternatives in the kernel are selected and proposed to DM

22. Concordance Index Measures the arguments in favor of the statement “a i outranks a j ”: c(a i, a j ) = ( ) / W where w k is the relative importance of attribute k and W is the total importance of all attributes Concordance and Discordance indices take values between 0 and 1

23. Discordance Index Among the attributes in favor of a j, some may have some doubt upon the statement “a i outranks a j ”. This phenomenon is represented by a discordance index: d(a i, a j ) = where r is the normalized performance value and  is the maximum difference between the normalized performance values of any two alternatives w.r.t. all attributes

24. Discordance Set If performance values are qualitative for some attributes, a discordance set will be constructed. For each attribute k, a discordance set D k made of ordered pairs of performance values (  k,  k ) is defined (where  k is better than  k ): If x ik =  k and x jk =  k then the outranking of a j by a i is refused.

25. Thresholds & Outranking Relation A (relatively large) concordance threshold ( ) and, if necessary, a (relatively small) discordance threshold ( ) are defined by the DM or by calculating the average value of the indices. Using the concordance threshold and discordance threshold or set, the outranking relation S is defined: a i S a j or a i S a j

26. Kernel Having outranking relations, which can be represented by a digraph, a subset of alternatives is sought such that: any alternative which is not in the subset is outranked by at least one alternative of the subset the alternatives of the subset are incomparable This type of set is called a kernel of the graph Remark: If the graph has no cycle, the kernel exists and is unique.Each cycle can be replaced by a unique element (considering the alternatives in the cycle as tied)

27. Example for Kernel a3a3 a5a5 a1a1 a2a2 a4a4 a7a7 a6a6 The kernel is subset {a 1, a 3, a 6 } (the set of preferred alternatives)

28. Further Example for Kernel a3a3 a8a8 a1a1 a2a2 a4a4 a5a5 a7a7 a6a6 Kernel ???

29. Further Example (ctd.) Kernel a5a5 a1a1 a2a2 a4a4 a7a7 a3a3 a8a8 a6a6 Cycle The kernel is subset {a 1, a 2, a 5 } (the set of preferred alternatives)

30. Example for ELECTRE Building the Relation: Concordance indices a 1 is better than a 2 w.r.t. acceleration (3), a 2 is better than a 1 w.r.t. price (5), a 1 is as good as a 2 w.r.t. comfort (4) and design (3) c(a 1, a 2 ) = (3+4+3)/15; c(a 2, a 1 ) = (5+4+3)/15

31. Building the Relation Discordance set The outranking of  by  is refused in the three following cases (stated by DM) Concordance threshold Assume concordance threshold as 12/15

32. Building the Relation Outranking relations Concordance indices which are greater than or equal to concordance threshold are found. Outranking relations are obtained for the ordered pairs associated by these indices if the pairs are not the element of the discordance set

33. Exploiting the Relation Representation of outranking relations by a graph The kernels are subsets {a 2, a 4, a 7 } and {a 2, a 5, a 7 } a3a3 a2a2 a1a1 a5a5 a4a4 a7a7 a6a6

34. ELECTRE METHOD FAMILY ELECTRE (I) is designed for choice problems ELECTRE II aims to rank the alternatives ELECTRE III concerns ranking problems involving quasi and/or pseudo criteria; bases upon a valued outranking relation ELECTRE IV ranks actions without introducing any weighting of criteria

35. ELECTRE II Roy and Bertier introduced some variations at ELECTRE I in 1971 (Vincke, 1992) : 1. Building the outranking relation Two concordance thresholds and a discordance threshold (or a discordance set) are defined A strong outranking relation (S F ) and a weak outranking relation (S f ) are built 2. Exploitating the outranking relation A complete preorder is obtained by calculating the degrees of the graph’s vertices (based on S F ) Ties are eliminated on the basis of S f

36. Outranking Relations Strong and weak outranking relations: a i S F a j a i S f a j >

37. The Degree of a Vertex The degree of an alternative p represented by a vertex: d(p): The difference between “the number of alternatives which are strongly outranked by the alternative” and “the number of alternatives which strongly outrank that alternative”

38. Example for ELECTRE II For the car purchase problem, assume that all inputs are same and second concordance index is 10/15 Representation of S F by a graph: a3a3 a2a2 a1a1 a5a5 a4a4 a7a7 a6a6

39. The Result d(a 1 ) = 0 – 1 = –1; d(a 2 ) = 2 – 0 = 2; d(a 3 ) = 0 – 3 = –3 d(a 4 )= 2 – 0 = 2; d(a 5 ) = 2 – 0 = 2; d(a 6 ) = 0 – 2 = –2 d(a 7 )= 0 – 0 = 0 a 2, a 4, a 5 a 7 a 1 a 6 a 3 Ties are eliminated on the basis of S f : The ranking of alternatives is as follows: a 5 – a 4 – a 2 – a 7 – a 1 – a 6 – a 3 a2a2 a5a5 a4a4

40. Complementary ELECTRE Instead of using critical threshold values, a net concordance and a net discordance index can be calculated for each alternative ( Yoon & Hwang, 1995). The net concordance of an alternative p (c p ): c p = The net discordance of an alternative p (d p ): d p = ––

41. Complementary ELECTRE Two complete preorders are built: Ranking the alternatives following the decreasing order of net concordance indices Ranking the alternatives following the increasing order of discordance indices The intersection of the preorders yields the partial preorder If complete preorder is desired as a result, average rank of alternatives can be used

42. Example c(a 1 ) = 60 – 55 = 5; c(a 2 ) = 58 – 59 = – 1; c(a 3 ) = 62 – 62 = 0; c(a 4 ) = 62 – 58 = 4; c(a 5 ) = 65 – 61 = 4; c(a 6 ) = 65 – 60 = 5; c(a 7 ) = 43 – 60 = –17 Ranking w.r.t. net concordance indices a 6, a 1 a 4, a 5 a 3 a 2 a 7