FlowSort : a sorting method for group-decision

FlowSort : a sorting method for group-decision

Ph. Nemery

Multiple Criteria Sorting Workshop Université Paris Dauphine

19/02/2008

Ph. Nemery

Plan:

1. Introduction 2. Ranking with flows 3. FlowSort : sorting with flows 4. FlowSort for Group Sorting 5. Conclusions

Ph. Nemery

2

Introduction

1. Introduction

Ph. Nemery

3

Introduction

General

In multi-criteria decision aid (MCDA) we generally distinguish 4 different decision problems (Roy):

- choice - description - ranking - sorting

Ph. Nemery

ai

a1

an

4

A

Introduction

General

In multi-criteria decision aid (MCDA) we generally distinguish 4 different decision problems (Roy):

- choice - description - ranking - sorting

Ph. Nemery

Ab

ai

a1

an

5

A

Introduction

General

In multi-criteria decision aid (MCDA) we generally distinguish 4 different decision problems (Roy):

- choice - description - ranking - sorting

Ph. Nemery

ai

a1

an

6

A

Introduction

General

In multi-criteria decision aid (MCDA) we generally distinguish 4 different decision problems (Roy):

- choice - description - ranking - sorting

Ph. Nemery

ai

a1

an n

...

a1

ai

7

2

1

an al

A

Introduction

General

In multi-criteria decision aid (MCDA) we generally distinguish 4 different decision problems (Roy):

- choice - description - ranking - sorting

Ph. Nemery

ai

a1

an

8

A

Introduction

Sorting Problem

Sorting Problematic: The aim is to assign a set of alternatives A = {a1 , . . . , an } evaluated on the basis of q criteria ϕk (k = 1, . . . , q) to K predefined ordered classes or categories. The K categories are completely ordered : where C1 is “preferred” to

ai

a1

Ph. Nemery

A

CK ! . . . ! Cj ! . . . ! C1 9

Introduction

Labelize

Sorting Problem: e.g. ai

a1

A Recommend

CK ! . . . ! Cj ! . . . ! C1

Examples: - a credit demand at a bank non-thrustful clients - more information - credit worthy clients - evaluation of the innovative character of some enterprises (France) passive - reactive - pre-active - proactive enterprises Ph. Nemery

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Plan:

2. PROMETHEE : Ranking Method based on Flows

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Preference Function

Promethee (I & II) methodology:

pair-wise comparison between all the actions to be ranked on the basis of the q criteria

ai

Pk (aj , ai )

ϕ1

1

aj

Ph. Nemery

ϕq

0

qk

12

pk

ϕk (aj ) − ϕk (ai )

Preference Function

Promethee (I & II) methodology: generalization

pair-wise comparison between all the actions to be ranked on the basis of the q criteria

ai

ϕ1

aj

P1 (aj , ai )

1

2

3

1

0

0

0

2

0.25

0

0

3

0.9

0.6

0

ϕq

ϕ1 (aj )

Ph. Nemery

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ϕ1 (ai )

Preference Degree

Promethee (I & II) methodology:

pair-wise comparison between all the actions to be ranked on the basis of the q criteria

π(aj , ai ) =

ai

ϕ1

Aggregation

π(ai , aj ) =

q !

k=1 q !

k=1

aj

Ph. Nemery

Pk (aj , ai ) ∗ ωk Pk (ai , aj ) ∗ ωk

0 ≤ π(aj , ai ) + π(ai , aj ) ≤ 1

ϕq

14

Preference Matrix

Promethee (I & II) methodology:

pair-wise comparison between all the actions to be ranked on the basis of the q criteria

ai

ϕ1

0

πij

πji

0

Aggregation

aj

Ph. Nemery

ϕq

15

Aggregation

Promethee (I & II) methodology:

pair-wise comparison between all the actions to be ranked on the basis of the q criteria which are then aggregated

ai

aj

ai

aj

0

πij

πji

Aggregation

0

ai

pair-wise comparisons between the actions Ph. Nemery

n n-1

16

...

3

2

1

aj

complete ranking

Positive Flows (I)

Promethee (I & II) methodology:

a1

aj

ai

ai

ai

0

aj

!

πij

an

πij = φ+ (ai )

j

aj

πji

0

n n-1

...

ai Ph. Nemery

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3

aj

2

1

φ (.) − +

Negative Flows (I)

Promethee (I & II) methodology:

a1

ai

ai

aj

ai

0

πij

aj

πji

0

an

n n-1

!

...

ai

πji = φ− (ai )

j

Ph. Nemery

aj

18

3

aj

2

1

φ− (.)

Net Flows (II)

Promethee (I & II) methodology:

ai

aj

ai

0

πij

aj

πji

0

φ(ai ) = φ (ai ) − φ (ai ) −

+

n n-1

...

3

2

φ(.)

ai

aj

Ranking methods for valued preference relations: A characterization of a method based on leaving and entering flows D. Bouyssou and P. Perny (1991, EJOR) Ph. Nemery

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Plan:

3. FlowSort: a Sorting method based on flows

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Introduction

Sorting Problem: well-known methods Doumpos, Zopounidis

Several methods have been proposed since several decades: 1. Statistical & Econometric techniques Discriminant Analysis Logit & Probit Analysis 2. Non-Parametric Techniques Neural Networks Machine Learning Fuzzy Sets Rough Sets 3. MCDA Methods UTADIS AHP Outranking Methods Ph. Nemery

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Introduction

Sorting Problem: well-known outranking methods

Several approaches (with outranking methods) are proposed where the categories can be defined by: - limiting profiles (Electre-Tri, Yu) K categories --> K±1 limiting profiles

R = {r1 , . . . , rK+1 }

∀i = 1, . . . , q : ϕi ; wi

- central profiles (Doumpos & al) K categories --> at least K central profiles

rK+1

r1

r2

rK ai

C1

Ph. Nemery

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Introduction

Sorting Problem: well-known outranking methods

Several approaches (with outranking methods) are proposed where the categories can be defined by: - limiting profiles (Electre-Tri, Yu) K categories --> K±1 limiting profiles

R = {r1 , . . . , rK+1 }

∀i = 1, . . . , q : ϕi ; wi

- central profiles (Doumpos & al) K categories --> at least K central profiles

rK+1

r1

r2

rK ai

C1

Ph. Nemery

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Introduction

Sorting Problem: well-known outranking methods

Several approaches (with outranking methods) are proposed where the categories can be defined by: - limiting profiles (Electre-Tri, Yu) K categories --> K±1 limiting profiles

R = {r1 , . . . , rK+1 }

∀i = 1, . . . , q : ϕi ; wi

- central profiles (Doumpos & al) K categories --> at least K central profiles

rK+1

r1

r2

rK ai

ϕ1

C1 ϕq Ph. Nemery

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Introduction

FlowSort:

We will tackle the sorting problematic where the categories are either defined by limiting profiles or central profiles, by a ranking approach. Let us note: : the set of reference profiles - ai ∈ A : an action to be sorted : the set of reference profiles & the action to sort we will apply a ranking method on the set Ri use of the Promethee I & II methodology for the sorting problematic

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Main hypothesis

FlowSort:

Since the reference profiles define completely ordered categories, we will impose the following conditions: ∀k > 0, ∀j : ϕj (ri+k ) ≤ ϕj (ri )

rK+1

r1

r2

rK

⇓

C1

π(ri , ri+k ) > 0 & π(ri+k , ri ) = 0 Ph. Nemery

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ϕ1

ϕq

Proposition

FlowSort:

Proposition: Under the condition of the dominance of the reference profiles, the order of the flows of the reference profiles is invariant with respect to any action to assign.

Ph. Nemery

rK+1 rK ... rj ... r2 ! ! !+ ! ! ! + ∀k > 0 & ∀i : φRi (ri+k ) ≤ φRi (ri ) ai ai ai ai ai ai + ! ! ! ! ! ! φRi (.) rK+1 rK rj r 2 r1 !"#$ !"#$ C1 ! φ− (.) ! ! ! r1 r2 r3 ai 26 ! φnet (.) ! ! ! !

r1 ! ! ai

Assignment Rules

FlowSort:

Computation of the leaving, entering and net flows for every action of Ri φ+ Ri (ai )

! 1 = π(ai , x) |Ri | − 1

φ+ Ri (rj )

x∈Ri

! 1 = π(rj , x) |Ri | − 1 x∈Ri

! 1 − π(x, ai ) φRi (ai ) = |Ri | − 1

! 1 π(x, rj ) φ− Ri (rj ) = |Ri | − 1

− φRi (ai ) = φ+ (a ) − φ Ri i Ri (ai )

− φRi (rj ) = φ+ (r ) − φ Ri j Ri (rj )

x∈Ri

x∈Ri

Ph. Nemery

+ + r . . . r . .h.) r2 Cφ+ (ai ) = Ch , ifrK+1 φ+ (r ) < φ (a ) ≤ φ h+1 Ri K Ri ij Ri (r ! ! ! ! ! ! h+1 ai ai ai ai ai ai h+2 h K+2 2 1 ! ! ! ! ! ! φ+ Ri (.) rrj r 2 r1 K+1 rK rrh+1 h

!

∀j

!

ai

!

! φ− (.)

r1 ! ! ai

Assignment Rules

FlowSort:

Computation of the leaving, entering and net flows for every action of Ri φ+ Ri (ai )

! 1 = π(ai , x) |Ri | − 1

φ+ Ri (rj )

x∈Ri

! 1 = π(rj , x) |Ri | − 1 x∈Ri

! 1 − π(x, ai ) φRi (ai ) = |Ri | − 1

! 1 π(x, rj ) φ− Ri (rj ) = |Ri | − 1

− φRi (ai ) = φ+ (a ) − φ Ri i Ri (ai )

− φRi (rj ) = φ+ (r ) − φ Ri j Ri (rj )

x∈Ri

x∈Ri

rK+1 rK . . . + rj ... r2 + + ! ) π(p, d) ⇒ P1 (d, p) > P1 (p, d) ⇒ g1 (d) > g1 (p) − q 2

φRic) (rh−1 d∈ C1 : π(c, p) < π(p, ⇒ )P1 (c, p)
"1 (p, d) ⇒ c ∈ C2 : π(d, p) > π(p, d) ⇒ P1 (d, p) > P g1 (d) > g1 (p) − q Ch−1

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! j=1 Motivation de la Classification Ordinale Multicritère Rappels+Propriétés π(r , a ) = w P (r , a ) k i j j k i j i k j i j k w P (a , r ) π(a , r ) = i k j j i k Electre-Tri : introduction FlowSort ?=? UTADIS P (r , a ) = f (g (r ) − g (a )) ∀j = 1, ..., q j=1 j P k (ai , r ) =jf (g k (a )j− g i (r )) ∀j = 1, ...,Comparaison UTADIS : introduction U,E,F j=1 q j i k j i j k P (r , a ) = f (g (r ) − g (a )) ∀j = 1, ..., q j. k r˜ i j k j i FlowSort Extensions r ˜ r ˜ . . . r ˜ . . r ˜ K+1 K j 2 1 r ˜ r ˜ . . . r ˜ . . . r ˜ r ˜ Extension d’ Electre-Tri Future ? 1 rK+1 et a I !r2)) j ∀j ! ) = !f (g ! i (r !j!k = i1, ! ...,!q! ! j 2!k 3 I! k!ai)K P (a , r (a − g ! ! ! ! j i i k jClassification k IClustering Classification Ordinale: cas général j r3 iI ai etj ai& r2 Multicritère aπ(r a a a a a ai , ) = w P (r , a ) i k ii a i a j i ja k i (r ,ia )a= ia P aigj (ai )) ∀j = 1, ..., q i i f (ga j i(rk ) − i i i ij k i i 2 pj3 = q ∗ 2; ∀j = 1, ..., q, j=1 j !

P (a , r ) = f (g (a ) − g (r )) ∀j = 1, ..., q

P (r , a ) = f (g (r ) − g (a )) ∀j = 1, ..., q

r I a et a I r ! qaj g∗et(a 2; ∀j = 1,= ...,1,q,..., q rj3k= Pj (rk , ai ) = f (gj p(r )I − i j aii))I r∀j 2 FlowSort et UTADIS identiques ? q,q Pj (a:i , des rk ) =méthodes f (gjg(a ) − g (r )) ∀j = 1, ..., p = q ∗ 2; ∀j = 1, ..., i j k (a ) = g (r ) + q ; ∀j = 1, ..., q j ! 3 j [ ] [ ] jj [g i](aj) = r3 I ai et ai I r2 g[ j2;(r ) +=qj1,! ; ..., ∀j q, = 1, ..., q i] q ∗ [ ] pj [ = ] 3∀j

j j r1 rh rK+1 ∈) − = , 3..., r= rj∀r r)hg= r(r gjk(a g{r )+ qj1,; }. ∀jq= 1, ..., q 1K+1 K+1 d 1(r iR ji )) P (r , a ) = f (g (a ∀j ..., j k i j ! [ ] [ ] [ ] pj = qj ∗ 2; ∀j = 1, ..., q, ∀r ∈R = g{r(r rK+1 }. = 1, ..., q 1 , ..., g (a ) = ) + q ; ∀j a&b : j i j 3 j a b c d [ ]r [ ] ! 15 rK+1 rh [ ] a&b : 1 r3 I ai et ai I r2

g2 (a) =∈gr2R (b)=et{rg11,(a) < g1 (b) b ∀r ..., r }. K+1 r r K+1 h 1 5 10 10 15 (a)q∀r=∈ g2R (b)=et g (a) < g 10 gj (ai ) = gj (r3 ) + qj ; ∀j g =11,g2..., 1, ..., r 1 (b) {r }. 1 K+1 a&b : pj = qj ∗ 2; ∀j = 1, ..., q, : et1g (a)1< g (b) Or :a&b 5 c 2= 2(b) g22 (a) g g 2 1 1 Or : ∀r ∈ R = {r1 , ..., rK+1 }. g2 (a) = g2 (b) et g1 (a) < g1 (b) a ga&b j (ai ): = gj (r3 ) + qj ; ∀j = 1, ..., b q∈ C2 : π(b, p) > π(p, b) ⇒ P1 (b, p) > P1 (p, b) ⇒ g1 (b) > g1 (p) − q g (a) 1= g (b) 2et g (a) < g (b) Orb :∈ C2 : π(b, p) > π(p, b) ⇒ P1 (b, p) > P1 (p, b) ⇒ g1 (b) > g1 (p) − q

2 1 1 a ∈ C1 Or : π(a, : p) < π(p, a) ⇒ P1 (a, p) < P1 (p, a) ⇒ g1 (a) < g1 (p) − q ∀r ∈ R = {r1 , ..., rK+1 }. ac&d ∈ C:1 : π(a, p) < π(p, a) ⇒ P1 (a, p) < P1 (p, a) ⇒ g1 (a) < g1 (p) − q C2 b ∈c&d C 12 : : π(b, p) > π(p, b) ⇒ P1 (b, p) > P1 (p, b) ⇒ g1 (b) > g1 (p) − q a&b : ! ! g2 (c) ! =bgC∈ g2p) (.) Or : g1 (c) g1π(p, (d) b) ⇒ P1 (b, p) > P1 (p, b) ⇒ g1 (b) > g1 (p) − q C!2 et :! π(b, 2 (d) g2 (a) = g2 (b) et g1 (a) < g1 (b) a ∈g2C :=2π(a, p)et g1 (p) − q 1 (p,gb) Or : 2 r a r2 = r et g1 (c) < g1 (d) Org= :2 (d) g2 (c) g211(d) et g⇒ < g1 !π(p,! g (.) 1 P (d, p) > P (p, d) ⇒ g (d) > g (p) − q d) ⇒ c&d : 2 1 1 1 1 r a r r 3 p) i> P12(p, b) ⇒ g11 (b) > g1 (p) b ∈ C2 : π(b, p) > π(p, b) ⇒ P1 (b, − q c ∈ C : π(d, p) > π(p, d) ⇒ P (d, p) > P (p, d) ⇒ g (d) > 2 p) < π(p, c) ⇒ P (c, p) 1 1 g (c) < g 1 (p) −gq1 (p) − q Or : d ∈ C : π(c, < P (p, c) ⇒ Or : 1 1 1 1 1 g (c) = g (d) et g (c) < g (d) 2 1 a 2∈ C1 : π(a, p) < π(p, a) ⇒1 P1 (a,dp) a)p) ⇒> π(p, d)d)⇒⇒P1P(d, p)p)>>PP11(p, − qq φc+∈cC∈2 C: 2π(d, : π(d, π(p, (p,d) d) ⇒ ⇒ gg11(d) (d) > g1 (p) − 1 (d, g2 (c)Or = :g2 (d) et g1 (c) < g1 (d) d ∈dC∈1 C: 1π(c, p) p) π(p, d) ⇒ P1 (d, p) > P g1 (d) > g1 (p) − q Ch−1

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