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