1 sorting problems and electre methods

8/17/2019 1 Sorting Problems and Electre Methods

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Sorting problem statementor ‘problematic’

A(t)

A referencemodel

A can be an evolutive set, A(t)

The actions are uncomparable and each action has to be consideredindipendently from the others and its intrinsic value has to be

defined

The DM needs more

guarantees on the quality of an

action.a absolute versus relative

judgement

Several possible issues

A(t)


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ELECTRE and the problematic

Eight criteria, the evaluations of a1 on the criteria (profile of

a1 ), the evaluation profile of a positive reference action

(i.e. the acceptability condition) (- - - -)

Is a1 acceptable?

a1

ar


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And is this action acceptable?

ELECTRE and the problematic


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Bipolar reference

Absolute acceptability (ra) Absolute refutability (rr)

a1


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Bipolar reference

a2



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Bipolar reference

a3



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Bipolar reference

ra

rr

a1

a1S rr a1S ra rr S a1 ra S a1


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Processes and problem situations in which

actions have to be assigned to specific categories

Project selection, evaluation of applicants for loans

or grants, medical diagnosis, ...Risk assessment and management (in financial

ambits – e.g. business failure risk, in enterprise

risk management, in relation to the environmentand the territory problems, ...)

Management processes (maintenance problems,

supplier control, client management, monitoring,human resources, ... )


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Structure of the methods ELECTRE

for the sorting ptoblems• Input: a finite set A or an evolutive one A(t), F, R = , p j

• Phase O: definition of the assignment rules (structure andcategory number, nature of R - the reference set -, rules toassign the candidates to the categories), preference

modeling, calibration of the parameters and R validation• Phase I: Outranking model (to compare (a,r) and (r,a))

• Phase II: application of the rules to assign the candidates to

the categories

• Result analysis

r


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The trichotomic segmentation method with

a multi-profile intersecting reference set

• Input: A(t), F, R = , p j

• Phase O: preference and assignment rules modeling,

verification of the reference set consistency andcalibration of the parameters

• Phase I: Outranking model of ELECTRE II or I

• Phase II: application of the rules to assign thecandidates to the categories of Accetable or Refutable

• Result analysis

BUC


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Bipolar reference

rarr

a1

a1S rr a1S ra rr S a1 ra S a1


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Multiple not hierarchical references: a trichotomic

segmentation method with a multi-profile intersecting

reference set

(C)(B1 e B2)

an

Absolute acceptability Absolute refutability


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Phase II: decision situations and rules for a

bipolar reference

a5

cb

a3

cb

a2

cba1

cb

cba4

ca6b

ca7b

ca8b

b ca11

ca9b

ca10b cba12


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δ(a, a’) b1 b2 c1 c2 c3

b1 - 0,14 0,90 0,87 0,45

b2 0 - 0,81 0,70 1

c1 0 0,67 - 0,19 0,71

c2 0,46 0,66 0,42 - 0,70c3 0,58 0,54 0,67 0,41 -

c2

c1

c3

b1

b2

Thresholdδ*= 0.65

Verification of the reference set (R) consistency


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Trichotomic segmentation method with a

multi-profile intersecting reference setPhase II: 4 situations

a1

c1b1

c2

c3

b2

a1c1b1

c2

c3

b2

a1c1

b1

c2

c3b2

a1c1

b1

c2

c3

b2


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Phase II: applications of the

assignment rules

a1c1b1

c2

c3

b2

a1c1b1

c2

c3

b2

a1c1

b1

c2

c3

b2

a1c1

b1

c2

c3

b2


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Multiple hierarchical reference

Categories of sequential assignment for the candidate actions(in terms of risk, urgency,adequacy,..........)

C1

C2

C3

C4

C5

am


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Category but also strategy of action which is associated to the category, inrelation to a specific activity (of monitoring and control, design, activation of

new processes,....)

C1

C2

C3

C4

C5

ELECTRE TRI: a sorting method with a multiple

hierarchical reference set

Reference profiles (i.e. combinations of values on the family of

criteria) which represent the theoretical limits between the categories

C j


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ELECTRE Tri: a sorting method with a multiple

hierarchical reference set

• Input: A(t), F, R = , p j

• Phase O: preference and assignment rules modeling,

calibration of the parameters (ELECTRE TriAssistant)

• Phase I: Outranking model of ELECTRE III

• Phase II: the pessimistic (or conjunctive) assignment procedure and the optimistic (or disjunctive) one toassign the candidates to the sequential categories

• Result analysis

r


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Ordered categories defined by limit profiles

b b b b b

C C C

g

g

. . .

1

g2

n

0 1 2 k-1 k

1 2 k

g3


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Imprecision, uncertainty and ill determination of the data

require discrimination thresholds that identify the limitsbetween situations of indifference and strict preference

b b b b b

C C C

g

g

. . .

1

g2

n

0 1 2 k-1 k

1 2 k

g3


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Partial concordance index c j(a,b)

1

g(b) (b)(b) (b) (b)p q

c

g g

g

j j

- j

- j j

j

j

(a, b)

(a)

Direction of preference


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1

g(b)(b) (b)(b)(b) pq

c

ggg

j j

+ j

+ j j

j

j

(b, a)

(a)

Direction of preference


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1

g(b)(b) (b)(b) (b) pg gg

j j j j j

j

j(a)

Senso di preferenza(a, b)D

- v -

1

g(b) (b) (b) (b)(b) pg g g j j j j j

j

j(a)

Senso di preferenza

D (b, a)

+ v+


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Phase II: example

Degrees of credibility of the outranking relation

σs(ai,b1) σs(b1,ai) σs(ai,b2) σs(b2,ai) σs(ai,b3) σs(b3,ai)

a1 0.70 0.50 0.56 0.75 0.21 0.81

a2 0.95 0.32 0.65 0.70 0.29 0.98

bmin b1 b2 b3 bmax|-----------|--------|--------|----------|

C1

C2

C3

C4

Cuting level λ = 0.60

a1 - pessimistic procedure

a1 /Sb3 (a1 does not outrank b3)

a1 /Sb2a1Sb1Then a1∈C2

a1 - optimistic procedure

b1 / a1 (b1 is not preferred to a1)b2 a1Then a1 C2

1 sorting problems and electre methods

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