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

Dominance vs. MAVT

• Dominance of a over b translates a sort of agreement for all points of view in favor of a: vj(a)>vj(b) where at least one of the inequalities is strict

vj(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)

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

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

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)

Main Steps of Outranking Methods

1. Building the outranking relation

2. Exploitating the outranking relation with regard to the chosen statement of the problem

PROMETHEE

• Preference Ranking Organization METHod for Enrichment Evaluation (Brans & Vincke, 1985)

• PROMETHEE I yields a partial preorder• PROMETHEE II yields a unique complete

preorder

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.

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

• Usual Criterion

Pk(ai,aj) =

• Quasi Criterion

Pk(ai,aj) =

• Crit. with Linear Pref.

Pk(ai,aj) =

• Level Criterion

Pk(ai,aj) =

• Crit. With Indifference Area.

Pk(ai,aj) =

• Gaussian Criterion

Pk(ai,aj) =

0 1

0 0

d

d

k

k

qd

qd

1

0

Recommended Generalized Criteria

k

kk

pd

pdpd

d

1

0 /

0 0

k

kk

k

pd

pdq

qd

1

5.0

0

k

kkkk

k

k

pd

pdqqp

qd

qd

1

0

0 )/2exp(1

0 0 22 dd

d

kp: preference threshold, q:indifference threshold

Generalized Criteria

Criterion I

1

Pk(ai,aj)

d

qkCriterion II

1

Pk(ai,aj)

d

Pk(ai,aj)

pkCriterion III

1

d

Criterion IV qk pk

0.5

1

Pk(ai,aj)

d

qk pkCriterion V

1

Pk(ai,aj)

d

skCriterion VI

1

Pk(ai,aj)

d

Necessary Calculations

• Multiattribute Preference Index

p(ai,aj) =

• Leaving Flow

• Entering Flow

• Net Flow

k

jikk aaPw ),(

Aa

jii

j

aaa ),()(

Aa

iji

j

aaa ),()(

)()()( iii aaa

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

PROMETHEE II

• A unique complete preorder is built:• Ranking the alternatives following the

decreasing order of net flows

Example for PROMETHEE

Building the Relation • Criterion V (Indifference area) is selected.• Indifference and preference thresholds are fixed:

Price Comfort Perf. Design

q k 49 0,49 0,49 0,49

p k 100 1 1 1

Example for PROMETHEE

Exploiting the Relation • Preference indices and flows

(a i ,a j ) a 1 a 2 a 3 a 4 a 5 a 6 a 7

a 1 0.2 0.2667 0.4667 0.4667 0.2667 0.6667 2.3333

a 2 0.0068 0.2667 0.4667 0.2667 0.2667 0.4667 1.7401

a 3 0.0068 0.2 0.2 0.2 0.2667 0.6667 1.5401

a 4 0.3333 0.2068 0.0068 0.2 0.2667 0.4667 1.4803

a 5 0.3333 0.0068 0.0068 0.2 0.2667 0.4667 1.2803

a 6 0.3333 0.2068 0.0068 0.2 0.2 0.4 1.3469

a 7 0.3333 0.3333 0.3333 0.3333 0.3333 0.3333 2

1.3469 1.1537 0.8871 1.8667 1.6667 1.6667 3.1333

PROMETHEE I Results

• Two complete preorders:

• Graph of alternatives (partial preorder)

a 1 a 2 a 3 a 4 a 5 a 6 a 7

1 3 4 5 7 6 23 2 1 6 4½. 4½. 7

a3

a7a1

a2

a6 a5

a4

PROMETHEE II Results

a 1 a 2 a 3 a 4 a 5 a 6 a 7

0.9864 0.5864 0.6531 -0.3864 -0.3864 -0.3197 -1.1333Rank 1 3 2 5½. 5½. 4 7

a3 a7a1 a2 a6

a5

a4

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)

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

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

Concordance Index

• Measures the arguments in favor of the statement “ai outranks aj”:

c(ai, aj) = ( ) / W

where wk 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

jkik xxk

kw:

Discordance Index

• Among the attributes in favor of aj, some may have some doubt upon the statement “ai outranks aj”. This phenomenon is represented by a discordance index:

d(ai, aj) =

where r is the normalized performance value and

d is the maximum difference between the normalized performance values of any two alternatives w.r.t. all attributes

otherwise ,/)(max(

, if ,0

ikjkk

jkik

rr

kxx

Discordance Set

• If performance values are qualitative for some attributes, a discordance set will be constructed.

• For each attribute k, a discordance set Dk made of ordered pairs of performance values (ak, bk) is defined (where bk is better than ak):

If xik = ak and xjk = bk then the outranking of aj by ai is refused.

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:

ai S aj or ai S aj

c

d

daad

caac

ji

ji

ˆ),(

ˆ),(

kDxx

caac

kjkik

ji

,),(

ˆ),(

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)

Example for Kernel

a3a5

a1

a2a4

a7a6

The kernel is subset {a1, a3, a6} (the set of preferred alternatives)

Further Example for Kernel

a3

a8

a1

a2

a4a5

a7

a6

Kernel ???

Further Example (ctd.)

Kernel

a5

a1

a2

a4

a7

a3 a8 a6

Cycle

The kernel is subset {a1, a2, a5} (the set of preferred alternatives)

Example for ELECTRE

• Building the Relation: Concordance indices

a 1 a 2 a 3 a 4 a 5 a 6 a 7

a 1 10/15 10/15 10/15 10/15 10/15 10/15

a 2 12/15 12/15 7/15 10/15 7/15 10/15

a 3 11/15 11/15 10/15 10/15 10/15 10/15

a 4 8/15 8/15 12/15 12/15 12/15 10/15

a 5 8/15 11/15 12/15 12/15 12/15 10/15

a 6 11/15 11/15 11/15 11/15 11/15 10/15

a 7 5/15 8/15 5/15 8/15 8/15 9/15

a1 is better than a2 w.r.t. acceleration (3), a2 is better than a1 w.r.t. price (5),a1 is as good as a2 w.r.t. comfort (4) and design (3)c(a1, a2) = (3+4+3)/15; c(a2, a1) = (5+4+3)/15

Building the Relation

• Discordance set

The outranking of b by a is refused in the three following cases (stated by DM)

• Concordance threshold

Assume concordance threshold as 12/15

Price Price Comfort 300 250 U 100 100 O

Building the Relation

• Outranking relationsConcordance 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

a 1 a 2 a 3 a 4 a 5 a 6 a 7

a 1 10/15 10/15 10/15 10/15 10/15 10/15

a 2 12/15 12/15 7/15 10/15 7/15 10/15

a 3 11/15 11/15 10/15 10/15 10/15 10/15

a 4 8/15 8/15 12/15 12/15 12/15 10/15

a 5 8/15 11/15 12/15 12/15 12/15 10/15

a 6 11/15 11/15 11/15 11/15 11/15 10/15

a 7 5/15 8/15 5/15 8/15 8/15 9/15

Exploiting the Relation

• Representation of outranking relations by a graph

The kernels are subsets {a2, a4, a7} and {a2, a5, a7}

a3

a2

a1

a5

a4a7

a6

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

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 (SF) and a weak

outranking relation (Sf) are built

2. Exploitating the outranking relation• A complete preorder is obtained by calculating the

degrees of the graph’s vertices (based on SF)• Ties are eliminated on the basis of Sf

Outranking Relations

• Strong and weak outranking relations:

ai SF aj

ai Sf aj

kDxx

ww

caac

kjkik

xxkk

xxkk

ji

jkikjkik

,),(

ˆ),(

::

1

kDxx

ww

caac

kjkik

xxkk

xxkk

ji

jkikjkik

,),(

ˆ),(

::

2

1c 2c>

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”

Example for ELECTRE II

• For the car purchase problem, assume that all inputs are same and second concordance index is 10/15

• Representation of SF by a graph:

a3

a2

a1

a5

a4a7

a6

The Result

d(a1) = 0 – 1 = –1; d(a2) = 2 – 0 = 2; d(a3) = 0 – 3 = –3

d(a4)= 2 – 0 = 2; d(a5) = 2 – 0 = 2; d(a6) = 0 – 2 = –2

d(a7)= 0 – 0 = 0

a2, a4, a5 a7 a1 a6 a3

• Ties are eliminated on the basis of Sf:

• The ranking of alternatives is as follows:

a5 – a4 – a2 – a7 – a1 – a6 – a3

a2

a5

a4

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 (cp):

cp =

• The net discordance of an alternative p (dp):

dp =

pkk

kpc ),( pkk

pkc ),(–

pkk

kpd ),( pkk

pkd ),(–

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

Example

a 1 a 2 a 3 a 4 a 5 a 6 a 7

a 1 10 10 10 10 10 10 60

a 2 12 12 7 10 7 10 58

a 3 11 11 10 10 10 10 62

a 4 8 8 12 12 12 10 62

a 5 8 11 12 12 12 10 65

a 6 11 11 11 11 11 10 65

a 7 5 8 5 8 8 9 43

55 59 62 58 61 60 60

c(a1) = 60 – 55 = 5; c(a2) = 58 – 59 = – 1; c(a3) = 62 – 62 = 0; c(a4) = 62 – 58 = 4; c(a5) = 65 – 61 = 4; c(a6) = 65 – 60 = 5; c(a7) = 43 – 60 = –17

Ranking w.r.t. net concordance indices

a6, a1 a4, a5 a3 a2 a7