Identification of a 2-additive bi-capacity by using mathematical programming

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Identification of a 2-additive bi-capacity by using mathematical programming

Julien Ah-Pine^?, Brice Mayag^] and Antoine Rolland^? University of Lyon^? and University of Paris Dauphine^]

ADT 2013 - Bruxelles - 14th of november 2013

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Motivations

Multi-criteria Decision Making (MCDM) using aggregation operators

The Choquet integral (CI) wrt a capacity is a versatile approach :

I A capacity allows modelling the interactions between criteria

I A capacity assigns non-negative weights to all subsets of criteria : 2ⁿ−1 values to set

I The scores should be in a commensurable unipolar (non-negative) scale

In many situations, human decision makings are easier using bipolar scales (negative, neutral and positive parts) :

I Bi-capacities (BC) allows extending CI to bipolar scales

I But the nb of weights to set is bigger : 3ⁿ−1

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Motivations

Multi-criteria Decision Making (MCDM) using aggregation operators The Choquet integral (CI) wrt a capacity is a versatile approach :

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Motivations

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Motivations

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Motivations

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Motivations

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Motivations

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Motivations

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Contributions

We use the bipolar Choquet integral (BCI) as an aggregation operator in MCDM

Due to the combinatorial complexity, we restrict ourselves to 2-additive BC (2A-BC) which require 2n²+ 1 weights to set

Then, we propose to elicit the 2A-BC modelling the preference model of a decision maker. The elicitation process is based on already judged alternatives (either fictitious or real)

We propose optimization problems to elicit the values of a 2A-BC Our framework presents several common points with supervised machine learning tasks and we underline these points

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Contributions

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Contributions

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Contributions

We propose optimization problems to elicit the values of a 2A-BC

Our framework presents several common points with supervised machine learning tasks and we underline these points

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Contributions

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Outline

1 Bi-capacities and bipolar Choquet integrals

2 Identifying a 2-additive bi-capacity

3 An illustrative example

4 Relationships with supervised learning tasks in machine learning

5 Conclusion and future work

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Bi-capacities and bipolar Choquet integrals

Outline

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Notations

N ={1, . . . ,n} is a finite set ofn criteria

x is an alternative defined by its score distributionx = (x₁, . . . ,x_n) wherexi ∈Ris the value of x for the criterion i

2^N :={S ⊆N} is the set of subsets ofN

3^N :={(A,B)∈2^N ×2^N :A∩B=∅}is the set of couples of subsets ofN with an empty intersection

On 3^N, the relation vis such that

(A1,A2)v(B1,B2)⇔A1 ⊆B1∧B2 ⊆A2

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Notations

(A1,A2)v(B1,B2)⇔A1 ⊆B1∧B2 ⊆A2

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Notations

(A1,A2)v(B1,B2)⇔A1 ⊆B1∧B2 ⊆A2

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Notations

(A1,A2)v(B1,B2)⇔A1 ⊆B1∧B2 ⊆A2

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Notations

(A1,A2)v(B1,B2)⇔A1 ⊆B1∧B2 ⊆A2

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Examples of 2

^N

, 3

^N

and instances of v

N ={1,2,3}

2^N ={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} 3^N is given by pairs indicated by√

in the following indicator matrix :







(A,B) ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

∅ √ √ √ √ √ √ √ √

{1} √

. √ √

. . √

.

{2} √ √

. √

. .

{3} √ √ √

. √

. . .

{1,2} √

. . √

. . . .

{1,3} √

. √

. . . . .

{2,3} √ √

. . . .

{1,2,3} √

. . . .







({1,2},{3})v({1,2},{∅})v({1,2,3},{∅})

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Examples of 2

^N

, 3

^N

and instances of v

N ={1,2,3}

2^N ={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

3^N is given by pairs indicated by√







(A,B) ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

∅ √ √ √ √ √ √ √ √

{1} √

. √ √

. . √

.

{2} √ √

. √

. .

{3} √ √ √

. √

. . .

{1,2} √

. . √

. . . .

{1,3} √

. √

. . . . .

{2,3} √ √

. . . .

{1,2,3} √

. . . .







({1,2},{3})v({1,2},{∅})v({1,2,3},{∅})

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Examples of 2

^N

, 3

^N

and instances of v

N ={1,2,3}

2^N ={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}







(A,B) ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

∅ √ √ √ √ √ √ √ √

{1} √

. √ √

. . √

.

{2} √ √

. √

. .

{3} √ √ √

. √

. . .

{1,2} √

. . √

. . . .

{1,3} √

. √

. . . . .

{2,3} √ √

. . . .

√







({1,2},{3})v({1,2},{∅})v({1,2,3},{∅})

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Examples of 2

^N

, 3

^N

and instances of v

N ={1,2,3}

2^N ={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}







(A,B) ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

∅ √ √ √ √ √ √ √ √

{1} √

. √ √

. . √

.

{2} √ √

. √

. .

{3} √ √ √

. √

. . .

{1,2} √

. . √

. . . .

{1,3} √

. √

. . . . .

{2,3} √ √

. . . .

{1,2,3} √

. . . .







({1,2},{3})

v({1,2},{∅})v({1,2,3},{∅})

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Examples of 2

^N

, 3

^N

and instances of v

N ={1,2,3}

2^N ={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}







(A,B) ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

∅ √ √ √ √ √ √ √ √

{1} √

. √ √

. . √

.

{2} √ √

. √

. .

{3} √ √ √

. √

. . .

{1,2} √

. . √

. . . .

{1,3} √

. √

. . . . .

{2,3} √ √

. . . .

√







v({1,2,3},{∅})

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Examples of 2

^N

, 3

^N

and instances of v

N ={1,2,3}

2^N ={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}







(A,B) ∅ {1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

∅ √ √ √ √ √ √ √ √

{1} √

. √ √

. . √

.

{2} √ √

. √

. .

{3} √ √ √

. √

. . .

{1,2} √

. . √

. . . .

{1,3} √

. √

. . . . .

{2,3} √ √

. . . .

{1,2,3} √

. . . .







({1,2},{3})v({1,2},{∅})v({1,2,3},{∅})

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Bi-capacity (BC)

A set functionν : 3^N →Ris a bi-capacity (BC)on 3^N if it satisfies the following two conditions [Grabisch and Labreuche, 2005b],

[Grabisch and Labreuche, 2008] :

I Null weight for empty sets :

ν(∅,∅) = 0

I Monotonicity :∀(A1,A2),(B1,B2)∈3^N :

(A1,A2)v(B1,B2)⇒ν(A1,A2)≤ν(B1,B2)

Ex. 1 :ν({1,2},{3})≤ν({1,2},{∅}) (“less criteria in the - part”) Ex. 2 :ν({1,2},∅)≤ν({1,2,3},{∅}) (“more criteria in the + part”) ν is said to be normalized if in addition, it holds :

ν(N,∅) = 1∧ν(∅,N) =−1

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Bi-capacity (BC)

ν(∅,∅) = 0

(A1,A2)v(B1,B2)⇒ν(A1,A2)≤ν(B1,B2)

ν(N,∅) = 1∧ν(∅,N) =−1

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Bi-capacity (BC)

ν(∅,∅) = 0

(A1,A2)v(B1,B2)⇒ν(A1,A2)≤ν(B1,B2)

ν(N,∅) = 1∧ν(∅,N) =−1

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Bi-capacity (BC)

ν(∅,∅) = 0

(A1,A2)v(B1,B2)⇒ν(A1,A2)≤ν(B1,B2)

Ex. 1 :ν({1,2},{3})≤ν({1,2},{∅}) (“less criteria in the - part”)

Ex. 2 :ν({1,2},∅)≤ν({1,2,3},{∅}) (“more criteria in the + part”) ν is said to be normalized if in addition, it holds :

ν(N,∅) = 1∧ν(∅,N) =−1

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Bi-capacity (BC)

ν(∅,∅) = 0

(A1,A2)v(B1,B2)⇒ν(A1,A2)≤ν(B1,B2)

Ex. 1 :ν({1,2},{3})≤ν({1,2},{∅}) (“less criteria in the - part”) Ex. 2 :ν({1,2},∅)≤ν({1,2,3},{∅}) (“more criteria in the + part”)

ν is said to be normalized if in addition, it holds : ν(N,∅) = 1∧ν(∅,N) =−1

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Bi-capacity (BC)

ν(∅,∅) = 0

(A1,A2)v(B1,B2)⇒ν(A1,A2)≤ν(B1,B2)

ν(N,∅) = 1∧ν(∅,N) =−1

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Bipolar M¨ obius transform (BMT) of a BC

A BCν can be associated to itsbipolar M¨obius transform (BMT) denoted b and defined by [Fujimoto, 2004, Fujimoto, 2007] :

b(A1,A2) := X

B₁⊆A₁ B₂⊆A₂

(−1)^|A¹^\B¹^|+|A²^\B²^|ν(B1,B2)

= X

(∅,A₂)v(B₁,B2)v(A₁,∅)

(−1)^|A¹^\B¹^|+|A²^\B²^|ν(B₁,B₂)

There is a one-to-one relation between ν and b and the converse relation is given by :

ν(A₁,A₂) := X

B₁⊆A₁ B₂⊆A₂

b(B₁,B₂)

B1 Note that the propertyν(∅,∅) = 0 translates into : b(∅,∅) = 0

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Bipolar M¨ obius transform (BMT) of a BC

b(A1,A2) := X

B₁⊆A₁ B₂⊆A₂

(−1)^|A¹^\B¹^|+|A²^\B²^|ν(B1,B2)

= X

(∅,A₂)v(B₁,B2)v(A₁,∅)

(−1)^|A¹^\B¹^|+|A²^\B²^|ν(B₁,B₂) There is a one-to-one relation between ν and b and the converse relation is given by :

ν(A₁,A₂) := X

B₁⊆A₁ B₂⊆A₂

b(B₁,B₂)

B1 Note that the propertyν(∅,∅) = 0 translates into : b(∅,∅) = 0

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Bipolar M¨ obius transform (BMT) of a BC

b(A1,A2) := X

B₁⊆A₁ B₂⊆A₂

(−1)^|A¹^\B¹^|+|A²^\B²^|ν(B1,B2)

= X

(∅,A₂)v(B₁,B2)v(A₁,∅)

(−1)^|A¹^\B¹^|+|A²^\B²^|ν(B₁,B₂) There is a one-to-one relation between ν and b and the converse relation is given by :

ν(A₁,A₂) := X

B₁⊆A₁ B₂⊆A₂

b(B₁,B₂)

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k -additivity

A BC requires 3ⁿ−1 values to be set

To reduce this complexity, we can use the concept ofk-additivity [Grabisch and Labreuche, 2005a, Grabisch and Labreuche, 2005b] and [Grabisch and Labreuche, 2008]

Given a positive integerk <n, a BCν is k-additive iff the two following conditions are satisfied [Fujimoto, 2007] :

B2 b= 0 if the nb of criteria involved in (A1,A2) is greater thank :

∀(A1,A2)∈3^N:|A1∪A2|>k ⇒b(A1,A2) = 0

B3 b6= 0 for at least one pair such that the nb of criteria involved isk :

∃(A1,A2)∈3^N :|A1∪A2|=k∧b(A1,A2)6= 0

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k -additivity

∀(A1,A2)∈3^N:|A1∪A2|>k ⇒b(A1,A2) = 0

∃(A1,A2)∈3^N :|A1∪A2|=k∧b(A1,A2)6= 0

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k -additivity

∀(A1,A2)∈3^N:|A1∪A2|>k ⇒b(A1,A2) = 0

∃(A1,A2)∈3^N :|A1∪A2|=k∧b(A1,A2)6= 0

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k -additivity

∀(A1,A2)∈3^N:|A1∪A2|>k ⇒b(A1,A2) = 0

∃(A1,A2)∈3^N :|A1∪A2|=k∧b(A1,A2)6= 0

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k -additivity

∀(A1,A2)∈3^N:|A1∪A2|>k ⇒b(A1,A2) = 0

∃(A1,A2)∈3^N :|A1∪A2|=k∧b(A1,A2)6= 0

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2-additive BC

1-additive BC assume the criteria are independent and boil down to linear aggregation operators

Thus we deal with 2-additive BC (2A-BC) to have a richer aggregation function. Henceforth,b is the BMT of a 2A-BC ν In this case, only subsets with at most two criteria matter To lighten the notations we will take :

b_i|=b({i},∅) ; b_ij|=b({i,j},∅) ; b_i|j =b({i},{j})

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2-additive BC

Thus we deal with 2-additive BC (2A-BC) to have a richer aggregation function. Henceforth,b is the BMT of a 2A-BC ν

In this case, only subsets with at most two criteria matter To lighten the notations we will take :

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2-additive BC

Thus we deal with 2-additive BC (2A-BC) to have a richer aggregation function. Henceforth,b is the BMT of a 2A-BC ν In this case, only subsets with at most two criteria matter

To lighten the notations we will take :

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2-additive BC

Thus we deal with 2-additive BC (2A-BC) to have a richer aggregation function. Henceforth,b is the BMT of a 2A-BC ν In this case, only subsets with at most two criteria matter To lighten the notations we will take :

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Example of the BMT of a 2A-BC

N ={1,2,3}

We assumeb is a BMT of a 2A-BCν

√are (possibly) non null elements andx are (necessarily) null elements :

b =







(A,B) ∅ 1 2 3 12 13 23 123

∅ √ √ √ √ √ √ √

x

1 √

. √ √

. . x .

2 √ √

. √

. x . .

3 √ √ √

. x . . .

12 √

. . x . . . .

13 √

. x . . . . .

23 √

x . . . .







The nb of elements to be set forb reduces from 3ⁿ to 2n²+ 1 (27 vs 19 in this example)

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Example of the BMT of a 2A-BC

N ={1,2,3}

We assumeb is a BMT of a 2A-BCν

√are (possibly) non null elements andx are (necessarily) null elements :

b =







(A,B) ∅ 1 2 3 12 13 23 123

∅ √ √ √ √ √ √ √

x

1 √

. √ √

. . x .

2 √ √

. √

. x . .

3 √ √ √

. x . . .

12 √

. . x . . . .

13 √

. x . . . . .

23 √

x . . . .

123 x . . . .







The nb of elements to be set forb reduces from 3ⁿ to 2n²+ 1 (27 vs 19 in this example)

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2-additive BC (cont’d)

P1 Themonotonicity property reduces to [Mayag et al, 2012] :

∀(A,B)∈3^N,∀k ∈A:b_k|+X

j∈B

b_k_|j + X

i∈A\k

b_ik|≥0

∀(A,B)∈3^N,∀k ∈A:b|k+X

j∈B

b_j|k + X

i∈A\k

b|ik ≤0

P2 Thenormalization property reduces to [Mayag et al, 2012] : X

i∈N

b_i|+ X

{i,j}⊆N

b_ij_|= 1

X

i∈N

b_|i + X

{i,j}⊆N

b_|ij =−1

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2-additive BC (cont’d)

P1 Themonotonicity property reduces to [Mayag et al, 2012] :

∀(A,B)∈3^N,∀k ∈A:b_k|+X

j∈B

b_k_|j + X

i∈A\k

b_ik|≥0

∀(A,B)∈3^N,∀k ∈A:b|k+X

j∈B

b_j|k + X

i∈A\k

b|ik ≤0 P2 Thenormalization property reduces to [Mayag et al, 2012] :

X

i∈N

b_i|+ X

{i,j}⊆N

b_ij_|= 1

X

i∈N

b_|i + X

{i,j}⊆N

b_|ij =−1

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Bipolar Choquet integral (BCI) wrt a 2A-BC

P3 Thebipolar Choquet integral (BCI) wrtb denoted C_b is given by : C_b(x) =

n

X

i=1

b_i|x_i⁺+

n

X

i=1

b_|i x_i⁻+

n

X

i,j=1

b_i|j (x_i⁺∧x_j⁻)

+ X

{i,j}⊆N

b_ij_|(x_i⁺∧x_j⁺) + X

{i,j}⊆N

b_|ij (x_i⁻∧x_j⁻)

wherex_i⁺ =

xi ifxi >0

0 ifx_i ≤0 andx_i⁻=

−x_i ifxi <0 0 ifx_i ≥0 anda∧b= min(a,b)

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Identifying a 2-additive bi-capacity

Outline

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Elicitation process

The preferences model of the DM is modeled by a 2A-BC ν represented by its BMTb

The DM provides a subset of alternativesX⁰ ⊆X and∀x∈X⁰ we are given :

I the scores distributionx = (x1, . . . ,xn)

I the overall scoreS(x) that is the aggregated value forx Let us denote by T the set of pairs{((x₁, . . . ,xn),S(x))}_x∈X⁰ No extra information is provided

No interaction loop with the DM is assumed

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Elicitation process

The DM provides a subset of alternativesX⁰ ⊆X and ∀x∈X⁰ we are given :

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Elicitation process

I the overall scoreS(x) that is the aggregated value forx

Let us denote by T the set of pairs{((x₁, . . . ,xn),S(x))}_x∈X⁰ No extra information is provided

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Elicitation process

I the overall scoreS(x) that is the aggregated value forx Let us denote by T the set of pairs{((x₁, . . . ,xn),S(x))}_x∈X⁰

No extra information is provided

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Elicitation process

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Elicitation process

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Taking into account preference relations

Our goal is to identify a 2A-BC that is consistent with T

Being consistent with T implies that we need to respect the preference relations provided byT as much as possible

C1 The BCI C_b wrtb should satisfy the given preference relations :

∀x,x⁰ ∈X⁰,x 6=x⁰ :S(x)−S(x⁰)≥0⇒ C_b(x)− C_b(x⁰)≥0

However it might happen that it is impossible to satisfy these constraints for all pairs :

I Our restriction to 2A-BC might be too strong and more general BC could better fit the problem

I The DM could provide scores and preferences that present incoherences and which are not fully representable by any BC

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Taking into account preference relations

Our goal is to identify a 2A-BC that is consistent with T Being consistent with T implies that we need to respect the preference relations provided byT as much as possible

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Taking into account preference relations

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Taking into account preference relations

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Taking into account preference relations

I Our restriction to 2A-BC might be too strong and more general BC

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Taking into account preference relations

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Taking into account preference relations (cont’d)

C2 If C1 cannot be satisfied, we replace it with more flexible constraints :

∀x,x⁰∈X⁰,x 6=x⁰ :S(x)−S(x⁰)≥0⇒ C_b(x)− C_b(x⁰)≥ −ξ_xx⁰ whereξ_xx⁰ ≥0 areslack variables of the model

ξ_xx⁰ allow inconsistencies :

I ifξxx⁰ = 0 thenb provides no incoherence with (x,x⁰)

I if 0< ξxx⁰ thenb cannot reproduce the preference relation for (x,x⁰)

L1 We could also imposeC_b to be bounded :

∀x ∈X⁰ :lb≤ C_b(x)≤ub

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Taking into account preference relations (cont’d)

I ifξxx⁰ = 0 thenb provides no incoherence with (x,x⁰)

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Taking into account preference relations (cont’d)

I ifξxx⁰ = 0 thenbprovides no incoherence with (x,x⁰)