Robust ordinal regression for outranking methods

Robust ordinal regression

for outranking methods

Salvatore Greco

1

Miłosz Kadzi ´nski

2

Vincent Mousseau

3

Roman Słowi ´nski

2

1

_{Faculty of Economics, University of Catania, Italy}

2

_{Institute of Computing Science, Pozna ´n University of Technology, Poland}

_{Laboratoire de G ´enie Industriel, Ecole Centrale Paris, France}

1

_Introduction

2

_{Robust ordinal regression for outranking methods}

3

_{Robust ordinal regression for group decision}

4

_{Extreme ranking analysis}

5

_{Representative set of parameters}

Multiple criteria problems

Characteristics

Actions

described by evaluation vectors

Family of criteria is supposed to satisfy

the

consistency conditions

Ranking

Rank

the actions

from the best to the

worst

according to DM’s preferences

Ranking can be

complete

or

partial

Choice

Outranking relation

Definition

Outranking relation S

groups three basic preference

relations: S = {∼, ., }

aSb

means

“action a is at least as good as action b”

Non-compensatory preference model used

in the

ELECTRE

family of MCDA methods

Accept incomparability

,

no completeness nor transitivity

Outranking relation on set of actions A is constructed via

concordance

and

discordance tests

Outranking relation

Concordance and discordance

Concordance test

: checks if the coalition

of criteria concordant with the hypothesis aSb

is strong enough:

C(a, b) =

P

j=1

k

· C

(a, b)/

P

k

=

[k

C

(a, b) + . . . + k

C

(a, b)]/(k

+ . . . +

k

)

Coalition is composed of two subsets of criteria:

these being clearly in favor of aSb, i.e.

C

(a, b) = 1, if g

(a) ≥ g

(b) − q

,

these that do not oppose to aSb, i.e. such

that b . a

Outranking relation

Concordance and discordance

Since (k

+ . . . +

k

) =

1, we can consider:

C(a, b) = ψ

(a, b) + . . . + ψ

(a, b),

where ψ

(a, b) = k

· C

(a, b), j = 1, . . . , m,

is a

monotone, non-decreasing function

w.r.t. g

(a) − g

(b)

Concordance test is

positive

if: C(a, b) ≥ λ,

where λ is a cutting level (concordance threshold)

Cutting level λ is required to be not less than 0.5

Outranking relation

Concordance and discordance

Discordance test

: checks if among criteria

discordant with the hypothesis aSb there is

a

strong opposition

against aSb:

g

(b) − g

(a) ≥ v

(for gain-type criterion)

g

(a) − g

(b) ≥ v

(for cost-type criterion)

Conclusion

: aSb is true if and only if C(a, b) ≥ λ

and there is no criterion strongly opposed

(making veto) to the hypothesis

For each couple (a, b) ∈ A × A, one obtains

relation S

either

true

(1) or

false

(0)

Outranking methods

Two major problems raised in the literature

Elicitation of preference information

:

Rather technical parameters, precise numerical values

Intra-criterion parameters vs. inter criteria parameters

Disaggregation-aggregation procedures

Mainly in terms of sorting problems

(e.g., ELECTRE TRI)

Robustness analysis

:

Examination of the impact of each parameter

on the final outcome

Indication of the solutions which are good (bad) for different

instances of a preference model

Robust ordinal regression

Main assumptions

Take into account

all instances of a preference model

compatible with the preference information given by the DM

Supply the DM with two kinds of results:

necessary results

specify recommendations worked out on

the basis of

all

compatible instances of a preference model

considered jointly

possible results

identify all possible recommendations

made by

at least one

compatible instance of a preference

model considered individually

Methods that use

value function

as a preference model:

UTA

GMS

, GRIP, UTADIS

GMS

(additive value function),

robust ordinal regression applied to Choquet integral

Robust ordinal regression for outranking methods

Questions

Does a outrank b for

all

compatible outranking models?

Does a outrank b for

at least one

compatible outranking

model?

ELECTRE

GKMS

- Preference Information

Pairwise comparisons

Set of pairwise comparisons of reference actions

(a, b) ∈ B

_{⊂ A}

_{× A}

aSb

or

aS

_b

Intra-criterion preference information

[q

,q

]

- the range of

indifference

threshold

values allowed by the DM

[p

,p

]

- the range of

preference

threshold

values allowed by the DM

a ∼

b ⇔ “the difference between g

(a) and g

(b)

is

not significant

for the DM”

a 

b ⇔ “the difference between g

(a) and g

(b)

ELECTRE

GKMS

- Compatibility and Results

Compatibility

An outranking model is called

compatible

, if it is able to restore

all pairwise comparisons from B

R

_{for provided imprecise}

intra-criterion preference information

The necessary and the possible

In result, one obtains

two outranking relations

on set A, such

that for any pair of actions (a, b) ∈ A × A:

1

_a

_{necessarily outranks}

_{b (aS}

N

_b)

if a outranks b for

all

compatible outranking models

2

_a

_{possibly outranks}

_{b (aS}

P

_b)

ELECTRE

GKMS

- Compatible outranking model (1)

Set of concordance indices C(a, b), cutting levels λ,

indifference q

, preference p

, and veto thresholds v

,

j = 1, . . . , m, satisfying the foll. set of constraints E

:

If aSb for (a, b) ∈ B

C(a, b) =

P

ψ

(a, b) ≥ λ

g

(b) − g

(a) + ε ≤ v

,

j = 1, . . . , m

If aS

_{b for (a, b) ∈ B}

C(a, b) =

P

ψ

(a, b) + ε ≤ λ + M

(a, b)

g

(b) − g

(a) ≥ v

− δM

(a, b)

M

(a, b) ∈ {0, 1}, j = 0, . . . , m,

P

M

(a, b) ≤ m

where δ is a big given value





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

























ELECTRE

GKMS

- Compatible outranking model (2)

1 ≥ λ ≥ 0.5, v

≥ p

+ ε

v

≥ g

(b) − g

(a) + ε, v

≥ g

(a) − g

(b) + ε if a ∼

b

normalization:

P

ψ

(a

,

a

) =

1

monotonicity:

for all a, b, c, d ∈ A and j = 1, . . . , m :

ψ

(a, b) ≥ ψ

(c, d ) if g

(a) − g

(b) > g

(c) − g

(d )

ψ

(a, b) = ψ

(c, d ) if g

(a) − g

(b) = g

(c) − g

(d )









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



E

ELECTRE

GKMS

- Compatible outranking model (3)

partial concordance:

for all (a, b) ∈ A × A and j = 1, . . . , m :

[1] ψ

(a, b) = 0 if g

(a) − g

(b) ≤ −p

[2] ψ

(a, b) ≥ ε if g

(a) − g

(b) > −p

[3] ψ

(a, b) + ε ≤ ψ

(a

,

a

)

if g

(a) − g

(b) < −q

[4] ψ

(a, b) = ψ

(a

,

a

)

if g

(a) − g

(b) ≥ −q

[1] ψ

(a, b) = 0 if b 

a

[4] ψ

(a, b) = ψ

(a

,

a

), ψ

(b, a) = ψ

(a

,

a

)

if a ∼

b































































E

ELECTRE

GKMS

- Preference information

Extensions of the preference model

Consideration of

thresholds dependent on g

_j

(a)

(e.g., affine functions)

Pairwise comparisons of reference actions in terms of

relations of preference, indifference, or incomparability:

a  b, aIb,

or

a?b

Inter-criteria preference information

:

Interval weights of the criteria k

j

(e.g., k

1

>

0.2, k

2

<

0.5)

Pairwise comparisons of the weights of the criteria

Interval cutting level λ ∈ [λ

∗

, λ

∗

]

(e.g., λ > 0.75)

ELECTRE

GKMS

- Checking the truth of S

P

Idea

Prove that

aSb is possible

in the set of all compatible outranking models

aS

_{b ⇔ ε}

>

0

where:

ε

=

max ε

E

C(a, b) =

P

ψ

(a, b) ≥ λ

g

(b) − g

(a) + ε ≤ v

,

j = 1, . . . , m















Result

If ε

>

0 and the set of constraints is feasible, then a outranks b

for

at least one

compatible outranking model (

aS

_b

₎

ELECTRE

GKMS

- Checking the truth of S

N

Idea

Prove that

aS

_{b is not possible}

_{in the set of all compatible outranking}

model

aS

b ⇔ ε

≤ 0

where:

ε

=

max ε

E

C(a, b) =

P

ψ

(a, b) + ε ≤ λ + M

(a, b)

g

(b) − g

(a) ≥ v

− δM

(a, b)

M

(a, b) ∈ {0, 1}, j = 0, . . . , m,

P

M

(a, b) ≤ m



























Result

If ε

_{≤ 0 or the set of constraints is infeasible, then a outranks b}

for

all

compatible outranking model (

aS

_b

₎

ELECTRE

GKMS

- Properties of relations S

P

and S

N

Properties

S

P

and S

N

are reflexive, intransitive, and incomplete

S

P

⊇ S

N

aS

N

_{b ⇔ not(aS}

CP

_{b) and aS}

P

_{b ⇔ not(aS}

CN

_b)

From S

N

and S

P

, one can

obtain

indifference I, preference

= {P ∪ Q}, and incomparability R,

in a usual way

, e.g.:

if aS

N

b and bS

N

b, then aI

N

b

if aS

P

b and not(bS

P

b), then a 

P

b

Possible relations

between actions a and b for a single

instance of compatible outranking model conditioned by the

truth or falsity of S

N

and S

P

, e.g.:

if aS

N

b and bS

N

a, then aI

N

b

ELECTRE

GKMS

- Partial conclusions

Main distinguishing features

Taking into account

all instances of the outranking model

compatible with the provided preference information

Considering the marginal concordance functions

as

general

non-decreasing ones, defined in the

“spirit”

of ELECTRE

methods

Handling of preference information composed of

pairwise

comparisons

and of imprecise intra-criterion preference

information

Constructing two relations on the set of actions:

necessary and possible

ELECTRE

GKMS

- Illustrative example

Problem statement and given data

Actions

: 10 buses originally considered by the team

of Professor Jacek Zak (FMT, PUT)

Criteria

: 5 criteria: Price (th. euro), Exploitation costs

(th. zl/100k km), Comfort (pts), Safety (pts), Modernity (pts)

Evaluation table

:

Price

Exploit.

Comfort

Safety

Modern.

Bus name

[euro]

costs [zl]

[pts]

[pts]

[pts]

Autosan

209

87.5

7.64

9.04

7.8

Bova Futura

231

88

7.74

8.39

8.8

Ikarus EAG

207

92

5.67

4.44

5.6

Jelcz T

102

79.7

2.75

5.23

3.9

Setra S315

266

89.8

5.08

7.62

6.1

MAN Lion’s

239

83.4

5.18

7.07

4.8

...

...

...

...

...

...

ELECTRE

GKMS

- Illustrative example

Example

Step1:

Ask the DM for preference information

The DM provides imprecise intra-criterion preference information:

q

q

p

p

Price

5

10

20

40

Exploitation

3

5

10

16

Comfort

0.3

0.6

1.4

2.8

Safety

0.2

0.4

1.0

2.0

Modernity

0.3

0.7

1.8

2.8

... and pairwise comparisons:

MAN S Volvo, Neoplan S Bova

Volvo S

_{Autosan, Setra S}

_MAN





ELECTRE

GKMS

- Illustrative example

Example

Step 2:

Determine the necessary and possible outranking relations

Possible outranking matrix

:

S

P

₁

A

B

I

J

T

M

E

N

R

V

A

1

1

1

0

1

1

1

1

1

1

B

1

1

1

0

1

1

1

1

1

1

I

1

1

1

0

1

1

1

1

1

1

J

1

1

1

1

1

1

1

1

1

1

T

0

1

1

0

1

0

1

1

1

1

M

1

1

1

1

1

1

1

1

1

1

E

1

1

1

1

1

1

1

1

1

1

N

1

1

1

1

1

1

1

1

1

1

R

1

1

1

0

1

1

1

1

1

1

V

0

1

1

0

1

1

1

1

1

1

ELECTRE

GKMS

- Illustrative example

Example

Step 2:

Determine the necessary and possible outranking relations

Necessary outranking matrix

-

converg. index = |S

P

₌

_S

N

_{|% = 0.42}

_:

S

N

1

A

B

I

J

T

M

E

N

R

V

A

1

1

1

0

1

0

1

1

1

1

B

0

1

0

0

1

0

1

0

1

1

I

0

0

1

0

0

0

0

0

0

0

J

0

0

0

1

0

0

0

0

0

0

T

0

0

0

0

1

0

0

0

0

0

M

0

0

0

0

1

1

0

0

0

1

E

0

1

0

0

1

1

1

0

0

1

N

0

1

0

0

1

1

0

1

1

1

R

0

0

0

0

0

0

0

0

1

0

V

0

0

0

0

1

0

0

0

0

1

ELECTRE

GKMS

- Illustrative example

Example

Step 3:

Incremental specification of pairwise comparisons:

Analyze the necessary (S

_{and S}

_{) and possible (S}

_{and S}

_{) results}

In the following iterations state aSb or aS

_{b for pairs (a, b) ∈ A × A,}

for which the possible relation S

_{(or S}

_{) was true, but not the}

necessary S

_{(or S}

_{) one}

The DM provides additional pairwise comparisons:

Mercedes S Ikarus, Ikarus S MAN

Bova S

_Neoplan





ELECTRE

GKMS

- Illustrative example

Example

Step 4:

Determine the necessary and possible outranking relations

Possible outranking matrix S

₁

P

⊇ S

P

2

:

S

P

2

A

B

I

J

T

M

E

N

R

V

A

1

1

1

0

1

1

1

1

1

1

B

1

1

1

0

1

1

1

0

1

1

I

0

0

1

0

1

1

1

0

1

1

J

1

1

1

1

1

1

1

1

1

1

T

0

1

1

0

1

0

0

0

1

0

M

1

1

1

1

1

1

1

1

1

1

E

1

1

1

1

1

1

1

1

1

1

N

1

1

1

1

1

1

1

1

1

1

R

1

1

1

0

1

1

1

1

1

1

V

0

1

1

0

1

1

1

1

1

1

ELECTRE

GKMS

- Illustrative example

Example

Step 4:

Determine the necessary and possible outranking relations

Necessary outranking matrix S

₁

N

⊆ S

N

2

-

converg. index = 0.56

:

S

N

2

A

B

I

J

T

M

E

N

R

V

A

1

1

1

0

1

1

1

1

1

1

B

0

1

0

0

1

1

1

0

1

1

I

0

0

1

0

0

1

0

0

0

0

J

0

0

0

1

0

0

0

0

0

0

T

0

0

0

0

1

0

0

0

0

0

M

0

0

0

0

1

1

0

0

0

1

E

0

1

1

0

1

1

1

0

0

1

N

0

1

0

0

1

1

0

1

1

1

R

0

1

0

0

1

0

0

0

1

1

V

0

0

0

0

1

0

0

0

0

1

ELECTRE

GKMS

- Basic exploitation procedures

Recommendation in case of choice problems

Identify

kernel K

N

of the necessary outranking graph S

N

Identify such a ∈ A : ∀b ∈ A, b 6= a it holds not(bS

P

a)

Recommendation in case of ranking problems

ELECTRE

GKMS

- Illustrative example

Step 4:

Final recommendation

ELECTRE

GKMS

- Extensions

Analysis of incompatibility

Associate a

binary variable v

a,b

with each couple of reference

actions (a, b) ∈ B

R

_:

aSb ⇔ C(a, b) =

P

m

j=1

ψ

j

(a, b) + M

v

a,b

≥ λ

and g

j

(b) − g

j

(a) + ε ≤ v

j

(a) + M

v

a,b

,

j = 1, . . . , m,

where M is a big positive value (transform E

A

into E

A

v

)

If

v

a,b

=

1

, then the corresponding constraint is

always satisfied

Identify a

minimal subset

of troublesome exemplary decisions:

min f =

P

(a,b)∈B

v

a,b

,

subject to E

A

v

ELECTRE

GKMS

- Gradual confidence levels

Valued possible and necessary outranking relations

B

1

⊆ B

⊆ . . . ⊆ B

- embedded sets of pairwise comparisons

S

1

⊇ S

R

2

⊇ . . . ⊇ S

R

s

- sets of compatible outranking models

Let θ

be the

confidence level

assigned to pairwise comparisons

concerning pairs ((a, b) ∈ B

t

and (a, b) /

∈ B

), t = 1, . . . , s :

1 = θ

> θ

> . . . > θ

>

0

S

val

:

A × A → {θ

, θ

, . . . , θ

,

0}:

if

∃t

: aS

t

b, then S

(a, b) =

max

{θ

:

aS

b, t = 1, . . . , s}

if

@t

: aS

b, then S

(a, b) =

0

S

val

:

A × A → {1 − θ

,

1 − θ

, . . . ,

1 − θ

,

1}:

if

∃t

: aS

t

b, then S

(a, b) =

min

{1 − θ

:

not(aS

b), t = 1, . . . , s}

if

∀t

: aS

ELECTRE

GKMS

- Illustrative example

The ELECTRE

GKMS

− GROUP method

Characteristics

Several DMs

D = {d

1

, . . . ,

d

s

}

cooperate in a decision problem

DMs

share the same “description”

of the decision problems

The collective results (ranking or subset

of the best actions) should account for

preferences expressed by each DM

Avoid discussions of DMs

on technical parameters

Reason in terms of

necessary

and possible relations and coalitions

The ELECTRE

GKMS

− GROUP method

Characteristics

For each d

h

∈ D

0

⊆ D who expresses her

individual preferences as in ELECTRE

GKMS

,

calculate the necessary and possible

outranking relations

With respect to all DMs

four situations are

considered

:

aS

b : aS

b for

all

d

∈ D

aS

b : aS

b for

all

d

∈ D

aS

b : aS

h

b for

at least one

d

∈ D

0

aS

b : aS

h

b for

at least one

d

∈ D

ELECTRE

GKMS

− GROUP - Extensions

Consider preferences of DMs individually

A is small, DMs have outlook of the whole set A,

interrelated preferences

Analyze statements of DMs

individually

Examine the

spaces of agreement and disagreement

Consider preferences of DMs simultaneously

A is numerous,

DMs are experts only w.r.t. to its small disjoint subsets

Combine

knowledge of DMs into preference information

of a single fictitious DM

ELECTRE

GKMS

− GROUP - Extensions

Consider preferences of all DMs simultaneously

Suppose that S

D

A

of compatible outranking

models is

not empty

One obtains two outranking relations:

S

N

D

and

S

P

D

Difference between

S

N

D

and

S

N,N

D

aS

D

b

⇔ aSb for

all

outranking models

compatible with

all preferences of all DMs

from D

aS

b

⇔ aSb for

all

compatible

outranking models of

each DM

from D

If S

D

A

6= ∅, then for all a, b ∈ A,

aS

_D

N,N

b ⇒ aS

N

D

b and and aS

D

N

b ⇒ aS

P,N

ELECTRE

GKMS

− GROUP - Extensions

Consider preferences of all DMs simultaneously

Suppose that S

D

A

of compatible outranking

models is

not empty

One obtains two outranking relations:

S

N

D

and

S

P

D

Difference between

S

P

D

and

S

P,P

D

aS

D

b

⇔ aSb for

at least one

outranking

model compatible with

all preferences of all

DMs

from D

aS

b

⇔ aSb for

at least one

compatible

outranking model of

at least one DM

from D

If S

D

A

6= ∅, then for all a, b ∈ A,

aS

P

D

b ⇒ aS

P,P

D

b and aS

D

P

b ⇒ aS

P,N

D

b

PROMETHEE - Main principles

Preference function and degree

Compute

unicriterion preference degree

for every pair of

actions: π

j

(a, b), j = 1, . . . , m

Compute

global preference degree

for every pair of actions

(a, b) ∈ A × A:

π(a, b) =

P

m

j=1

k

j

· π

j

(a, b),

PROMETHEE - Main principles

Outranking flows

The

positive

outranking flow Φ

_(a):

Φ

_{(a) = 1/(n − 1)}

P

b∈A

π(a, b)

The

negative

outranking flow Φ

(a):

Φ

(a) = 1/(n − 1)

P

b∈A

π(b, a)

Net outranking flow

:

Φ(a) = Φ

_{(a) − Φ}

_(a)

PROMETHEE-II

:

aPb if Φ(a) > Φ(b)

PROMETHEE-I

:

aPb if Φ

_{(a) ≥ Φ}

_{(b) and Φ}

_{(a) ≤ Φ}

_(b)

PROMETHEE

GKS

- Compatible outranking model (1)

Pairwise comparisons (constr.):

if aS

b for (a, b) ∈ B

_{: π(a, b) ≥ π(b, a)}

Pairwise comparisons (exploit.):

if aS

b for (a, b) ∈ B

: Φ(a) ≥ Φ(b)

Normalization:

P

j=1

π

(a

,

a

) =

1

Monotonicity:

for all a, b, c, d ∈ A and j = 1, . . . , m :

π

(a, b) ≥ π

(c, d ) if g

(a) − g

(b) > g

(c) − g

(d )

π

(a, b) = π

(c, d ) if g

(a) − g

(b) = g

(c) − g

(d )



















































E

PROMETHEE

GKS

- Compatible outranking model (2)

Partial preference:

for all (a, b) ∈ A × A and j = 1, . . . , m :

[1] π

(a, b) = 0 if g

(a) − g

(b) ≤ q

[2] π

(a, b) ≥ ε if g

(a) − g

(b) > q

[3] π

(a, b) + ε ≤ π

(a

,

a

)

if g

(a) − g

(b) < p

[4] π

(a, b) = π

(a

,

a

)

if g

(a) − g

(b) ≥ p

[1] π

(a, b) = 0, π

(b, a) = 0 if a ∼

b

[4] π

(a, b) = π

(a

,

a

)

if a 

b































































E

Extreme ranking analysis

Motivation

Binary relations vs. ranking

Complete

rankings are intuitive, easy to understand, and popular

The DM is interested in

ranks and scores

of the actions

Examine

how different are all rankings

compatible

with preferences of the DM

Compute

the highest and the lowest rank

attained by each action

Extreme ranking analysis

The highest rank in robust multiple criteria ranking

Assume that a ∈ A is in the

top

of the ranking

Identify the minimal subset of actions that are

simultaneously not worse

than a:

min f

=

X

v

E

Φ(a)

> Φ(b) − Mv

, ∀b ∈ A \ {a}







E

where M is a big positive value

P

_(a)

_{of action a is indicated by (f}

Extreme ranking analysis

The lowest rank in robust multiple criteria ranking

Assume that a ∈ A is in the

bottom

of the ranking

Identify the minimal subset of actions that are

simultaneously not better

than a:

min f

=

X

b∈A\{a}

v

E

Φ(b) >

Φ(a)

− Mv

, ∀b ∈ A \ {a}







E

where M is a big positive value and ε is a small

positive value

Extreme ranking analysis

Extreme net outranking flows

The

highest

outranking net flow: Φ

∗

(a) =

max

Φ(a), s.t. E

A

S

The

lowest

outranking net flow: Φ

∗

(a) =

min

Φ(a), s.t. E

A

S

Extreme ranking analysis - Extensions

Incremental specification of preference information

S

t

⊆ S

R

t−1

, for all t = 2, . . . , s,

P

(a) ≥ P

(a) and P

(a) ≤ P

(a)

Φ

(a) ≤ Φ

(a) and Φ

(a) ≥ Φ

(a)

Interval orders

Preference 

and

indifference ∼

relations

w.r.t. the intervals

of ranking positions [P

(a), P

(a)], e.g.:

a 

b ⇔ P

(a) < P

(b) and P

(a) < P

(b)

a ∼

b ⇔ [P

(a), P

(a)] ⊂ [P

(b), P

(b)] or

Extreme ranking analysis - Extensions

Application to multiple criteria choice problems

Define

additional conditions

for being in B

The

most appealing

approach: B = {a ∈ A : P

_{(a) = 1}}

Possible approach: B = {a ∈ A : P

(a) ≤ 3 and P

(a) ≤ |A|/2}

Group actions:

indifference classes

which stem

from the ranking based on the best (or the worst) positions

Application to multiple criteria problems of different type

Group decision

in the spirit of “necessary and possible”:

P

(a) =

\

P

(a) and P

(a) =

[

P

(a)

Set of parameters corresponding to the extreme ranks

PROMETHEE

GKS

- Illustrative example

Problem statement and given data - Liveability of cities

Actions

: 15 cities originally considered by Financial Times

Criteria

: 5 criteria: stability (g

), healthcare (g

), culture (g

),

education (g

), infrastructure (g

)

Evaluation table

City

g

g

g

g

g

Vancouver

95.0

100.0

100.0

100.0

96.4

Vienna

95.9

100.0

96.5

100.0

100.0

Melbourne

95.0

100.0

95.1

100.0

100.0

Toronto

100.0

100.0

97.2

100.0

89.3

Calgary

100.0

100.0

89.1

100.0

96.4

Osaka

90.0

100.0

93.5

100.0

96.4

Tokyo

90.0

100.0

94.4

100.0

92.9

Hong Kong

95.0

87.5

85.9

100.0

96.4

. . .

. . .

. . .

. . .

. . .

. . .

PROMETHEE

GKS

- Illustrative example

Preference infromation

The DM provides imprecise intra-criterion preference information:

q

q

p

p

Stability

2

2

5

5

Healthcare

3

5

6

8

Culture

0

1

2

3

Education

3

5

6

8

Infrastracture

0

1

2

3

... and pairwise comparisons:

Vancouver  Vienna

Tokyo  Hong Kong

London  Seoul

Lagos  Port Moresby



























PROMETHEE

GKS

- Illustrative example

Extreme ranking results for PROMETHEE

City

P

_(a)

_P

∗

(a)

Φ

(a)

Φ

(a)

Vancouver

1

2

13.999

6.727

Vienna

2

4

10.727

6.000

Melbourne

3

5

9.307

4.500

Toronto

1

6

11.799

2.000

Calgary

5

9

7.999

−7.999

Osaka

4

8

7.999

0.500

Tokyo

4

8

7.999

1.500

Hong Kong

8

11

2.118

−4.999

Singapore

8

12

1.571

−7.999

London

3

11

10.999

−3.999

New York

8

11

2.499

−3.999

Seoul

11

12

−0.500

−6.999

Lagos

13

14

−10.538

−12.999

Port Moresby

14

15

−11.000

−13.999

Harare

14

15

−10.000

−13.999

Representative set of parameters

Motivation

Assign

precise values

to variables of the model

Identify

the representative set of parameters

without

loosing the advantage

of knowing all compatible outranking

models

Extend robust ordinal regression in its capacity of

explaining the final output

Get

synthetic representation

of the robust results

Exploit outranking relation for these parameters in order to

obtain

representative results

Representative set of parameters

Question

Which set of parameters is

representative

for the set of all outranking

models compatible with the preference infromation?

Representativeness

In the sense of

robustness preoccupation

“One for all, all for one”

Representative set of parameters

Procedure

Pre-defined targets

built on results of robust ordinal

regression and extreme ranking analysis

Enhancement of differences between actions

Emphasize the evident advantage

of some actions over the others

e.g., C(a, b) ≥

N

λ, C(a, b) <

N

λ,

Φ(a) >

N

Φ(b), P

∗

(a) < P

∗

(b)

Reduce the ambiguity

in the statement of such advantage

e.g., C(a, b) ≥

P

_λ

_{and C(a, b) <}

P

_λ,

Φ(a) >

P

_{Φ(b) and Φ(b) >}

P

_Φ(a),

P

∗

(a) < P

∗

(b) and P

∗

(a) > P

∗

(b)

Representative set of parameters

Procedure

Targets consist in

maximization

or

minimization

of the

difference between scores of actions, e.g.:

max ε

, such that Φ(a) ≥ Φ(b) + ε, if Φ(a) >

N

_Φ(b)

min δ

, such that |Φ(a) − Φ(b)| ≤ δ,

if Φ(a) >

P

_{Φ(b) and Φ(b) >}

P

_Φ(a)

DM is left

the freedom of assigning priorities

to targets

Targets may be attained:

One after another

, according to a given priority order

So as to find a

compromise

solution, e.g.:

max ε, such that Φ(a) − Φ(b) ≥ |Φ(c) − Φ(d )| + ε

if Φ(a) >

N

_{Φ(b) and Φ(c) >}

P

_{Φ(d ) and Φ(d ) >}

P

_Φ(c)

Representative set of parameters

Innovation of this proposal

Dealing with consequences of preference information

provided by the DM

Involving the DM in the process of specifying the targets

General framework for considering a few targets in

different configurations

General monotonic functions taking all criteria values as

coordinates of characteristic points

Autonomous method vs. confrontation with results of

robust ordinal regression

Summary

New family of outranking-based methods

The preference information is used within a robust ordinal

regression approach to build a complete set of compatible

outranking models

Identification of possible and necessary consequences of

provided information

Identification of extreme ranking results

Representative set of parameters built on relations defined

w.r.t. the whole set of compatible preference models

Future research

The necessary and the possible for robust multiple criteria

sorting for outranking methods

Eliciting and selecting compatible instances of the