Aggregation functions for multicriteria decision aid

Aggregation Functions for

Multicriteria Decision Aid

Jean-Luc Marichal

The aggregation problem

Combining several numerical values into a single one

Example (voting theory)

Several individuals form quantifiable judgements about the measure of an object.

area(box 2)

area(box 1) = ?

box 1 box 2

x₁, . . . , x_{n −→} M (x₁, . . . , x_n) = x where M = arithmetic mean

geometric mean median

...

Decision making (voters _{→ criteria)}

Aggregation in multicriteria decision making

• Alternatives A = {a, b, c, . . . , } • Criteria N = {1, 2, . . . , n}

• Profile a ∈ A −→ (xa₁, . . . , xa_n) _{∈ R}n

. &

commensurable partial scores

• Aggregation function M : Rn → R

M : En _{→ R} (E _{⊆ R)}

Alternative crit. 1 _{· · · crit. n} global score a xa_{1 · · ·} xa_n M (xa₁, . . . , xa_n)

b xb_{1 · · ·} xb_n M (xb₁, . . . , xb_n)

... ... ... ...

Example :

math. physics literature global

student a 18 16 10 ?

student b 10 12 18 ?

Non-commensurable scales :

price consumption comfort global

(to minimize) (to minimize) (to maximize)

car a $10,000 0.15 `pm good ?

car b $20,000 0.17 `pm excellent ? car c $30,000 0.13 `pm very good ?

car d $20,000 0.16 `pm good ?

Scoring approach

For each i _{∈ N, one can define a net score :} S_i(a) = _{|{b ∈ A | b 4i} a_{}| − |{b ∈ A | b <i} a_}|

s_i(a) = Si(a) + (|A| − 1)

Aggregation properties

• Symmetry. M(x1, . . . , xn) is symmetric.

• Increasingness. M(x1, . . . , xn) is nondecreasing in each argument.

• Idempotency. M(x, . . . , x) = x for all x

• Internality. min xi ≤ M(x1, . . . , xn) ≤ max xi Note : id. + inc. _{⇒ int. ⇒ id.}

Quasi-arithmetic means

Theorem 1 (Kolmogorov-Nagumo, 1930)

The functions M_n : En _{→ R are symmetric,} continu-ous, strictly increasing, idempotent, and decompos-able if and only if there exists a continuous strictly monotonic function f : E _{→ R such that}

Proposition 1 (Marichal, 2000)

Symmetry can be removed in the K-N theorem.

Theorem 2 (Fodor-Marichal, 1997)

The functions M_n : [a, b]n _{→ R are symmetric,} contin-uous, increasing, idempotent, and decomposable if and only if there exist α and β fulfilling a _{≤ α ≤} β _{≤ b and a continuous strictly monotonic function} f : [α, β] _{→ R such that, for any n ≥ 1,}

M_n(x) =                      F_n(x) if x _{∈ [a, α]}n, G_n(x) if x _{∈ [β, b]}n, f−1h1 n n X i=1 f [median(α, x_i, β)]i otherwise where F_n and G_n are defined by...

Theorem 3 (Acz´el, 1948)

The function M : En _{→ R is symmetric, continuous,} strictly increasing, idempotent, and bisymmetric if and only if there exists a continuous strictly mono-tonic function f : E _{→ R such that}

M (x) = f−1h1 n n X i=1 f (x_i)i

When symmetry is removed :

There exist ω₁, . . . , ω_n > 0 fulfilling P

Theorem 4 (Acz´el, 1948)

The functions M_n : En _{→ E are continuous, strictly} increasing, and associative if and only if there exists a continuous and strictly monotonic function f : E _→ R such that M_n(x) = f−1h n X i=1 f (x_i)i + idempotency : _∅

Open problem : replace str. increasing by increasing

Theorem 5 (Fung-Fu, 1975)

The functions M_n : En _{→ R are symmetric,} continu-ous, increasing, idempotent, and associative if and only if there exists α _{∈ E such that}

Without symmetry :

Theorem 6 (Marichal, 2000)

The functions M_n : En _{→ R are continuous,} increas-ing, idempotent, and associative if and only if there exist α, β _{∈ E such that}

M_n(x) = (α_∧x1)_∨ n _ i=1 (α_{∧β ∧xi})_{∨(β ∧xn})_∨ n ^ i=1 x_i

Without symmetry and idempotency :

Theorem 7 (Marichal, 2000)

The functions M_n : [a, b]n _{→ [a, b] are continuous,} in-creasing, associative, and have a and b as idempotent elements if and only if there exist α, β _{∈ E such that}

M_n(x) =                  F_n(x) if x _{∈ [a, α ∧ β]}n, G_n(x) if x _{∈ [α ∨ β, b]}n, (α _{∧ x1}) _{∨ . . .} otherwise

Interval scales

Example : marks obtained by students - on a [0,20] scale : 16, 11, 7, 14

- on a [0,1] scale : 0.80, 0.55, 0.35, 0.70 - on a [-1,1] scale : 0.60, 0.10, -0.30, 0.40

Definition. M : Rn _{→ R is stable for the positive} linear transformations if

M (rx₁ + s, . . . , rx_n + s) = r M (x₁, . . . , x_n) + s

for all x₁, . . . , x_{n ∈ R and all r > 0, s ∈ R.}

Theorem 8 (Acz´el-Roberts-Rosenbaum, 1986) The function M : Rn _{→ R is stable for the positive} linear transformations if and only if

M (x) = S(x) Fx1 − A(x) S(x) , . . . , x_{n − A(x)} S(x)  + A(x) where A(x) = 1 n P i xi, S(x) = q P i[xi − A(x)]2, and F : Rn _{→ R arbitrary.}

Interesting unsolved problem :

Theorem 9 (Marichal-Mathonet-Tousset, 1999) The function M : En _{→ R is increasing, stable for} the positive linear transformations, and bisymmetric if and only if it is of the form

M (x) = _ i_∈S x_i or ^ i_∈S x_i or n X i=1 ω_ix_i where S _{⊆ N, S 6= ∅, ω1}, . . . , ω_{n ≥ 0, and} P i ωi = 1.

Theorem 10 The functions M_n : En _{→ R are} in-creasing, stable for the positive linear transforma-tions, and decomposable if and only if they are of the form M_n(x) = n _ i=1 x_i or n ^ i=1 x_i or x₁ or x_n or 1 n n X i=1 x_i

An illustrative example (Grabisch, 1996)

Evaluation of students w.r.t. three subjects: mathematics, physics, and literature.

Student M P L global

a 0.90 0.80 0.50 ?

b 0.50 0.60 0.90 ?

c 0.70 0.75 0.75 ?

(marks are expressed on a scale from 0 to 1)

Often used: the weighted arithmetic mean

WAM_ω(x) = n X i=1 ω_ix_i with P

i ωi = 1 and ωi ≥ 0 for all i ∈ N

Suppose we want to favor student c

Student M P L global

a 0.90 0.80 0.50 0.74 b 0.50 0.60 0.90 0.65 c 0.70 0.75 0.75 0.73

No weight vector (ω_M, ω_P, ω_L) satisfying

Definition (Choquet, 1953; Sugeno, 1974)

A fuzzy measure on N is a set function v : 2N _{→ [0, 1]} such that

i) v(_{∅) = 0, v(N) = 1} ii) S _{⊆ T ⇒ v(S) ≤ v(T )}

v(S) = weight of S

= degree of importance of S

A fuzzy measure is additive if

v(S _{∪ T ) = v(S) + v(T )} if S _{∩ T = ∅} → independent criteria

v(M, P) = v(M) + v(P) (= 0.70)

The discrete Choquet integral

Definition

Let v _{∈ FN}. The (discrete) Choquet integral of x _{∈ R}n w.r.t. v is defined by

Cv(x) := n

X

i=1

x_(i)[v(A_(i)) _{− v(A(i+1)})] with the convention that x_{(1) ≤ · · · ≤ x(n)}. Also, A_(i) = _{{(i), . . . , (n)}.}

Properties of the Choquet integral

• Linearity w.r.t. the fuzzy measures

There exist 2n functions f_T : Rn _{→ R (T ⊆ N)} such that

Cv =

X

T_⊆N

v(T ) f_T (v _{∈ FN}) Indeed, on can show that

Cv(x) = X T_⊆N v(T ) X K_⊇T (₋₁₎|K|−|T | ^ i_∈K x_i | {z } f_T(x)

• Stability w.r.t. positive linear transformations For any x _{∈ R}n, r > 0, s _{∈ R,}

Cv(r x1 + s, . . . , r xn + s) = r Cv(x1, . . . , xn) + s Example : marks obtained by students

- on a [0, 20] scale : 16, 11, 7, 14

- on a [0, 1] scale : 0.80, 0.55, 0.35, 0.70 - on a [_{−1, 1] scale : 0.60, 0.10, −0.30, 0.40}

• Increasing monotonicity For any x, x0 _{∈ R}n, one has

x_{i ≤ x}0_{i ∀i ∈ N ⇒ Cv}(x) _{≤ Cv}(x0)

• Cv is properly weighted by v

Cv(eS) = v(S) (S ⊆ N)

e_S = characteristic vector of S in _{{0, 1}}n Example : e_{1,3} = (1, 0, 1, 0, . . .)

Independent criteria Dependent criteria

WAM_ω(e_{i}) = ω_{i Cv}(e_{i}) = v(i) WAM_ω(e_{i,j}) = ω_i + ω_{j Cv}(e_{i,j}) = v(i, j)

Example :

v(M, P) < v(M) + v(P)

k k k

Axiomatic characterization of the class of Choquet integrals with n arguments

Theorem (Marichal, 2000)

The operators M_v : Rn _{→ R (v ∈ FN}) are

• linear w.r.t. the underlying fuzzy measures v M_v is of the form

M_v = X T_⊆N

v(T ) f_T (v _{∈ FN}) where f_T’s are independent of v

• stable for the positive linear transformations M_v(r x₁ + s, . . . , r x_n + s) = r M_v(x₁, . . . , x_n) + s for all x _{∈ R}n, r > 0, s _{∈ R, and all v ∈ FN}

• increasing

• properly weighted by v

Back to the example

Assumptions :

- M and P are more important than L - M and P are somewhat substitutive

Non-additive model : _Cv v(M) = 0.35 v(P) = 0.35 v(L) = 0.30 v(M, P) = 0.60 (redundancy) v(M, L) = 0.80 (complementarity) v(P, L) = 0.80 (complementarity) v(_{∅) = 0} v(M, P, L) = 1

Student M P L WAM Choquet

a 0.90 0.80 0.50 0.74 0.71

b 0.50 0.60 0.90 0.65 0.67

c 0.70 0.75 0.75 0.73 0.74

Another example (Marichal, 2000) Student M P L global a 0.90 0.70 0.80 ? b 0.90 0.80 0.70 ? c 0.60 0.70 0.80 ? d 0.60 0.80 0.70 ?

Behavior of the decision maker :

When a student is good at M (0.90), it is preferable that he/she is better at L than P, so

a _b

When a student is not good at M (0.60), it is prefer-able that he/she is better at P than L, so

d _c Additive model : WAM_ω

a _{b ⇔ ωL} > ω_P d _{c ⇔ ωL} < ω_P

)

No solution !

Non additive model : _Cv

Particular cases of Choquet integrals

• Weighted arithmetic mean WAM_ω(x) = n X i=1 ω_ix_i , n X i=1 ω_i = 1, ω_{i ≥ 0} Proposition Let v _{∈ FN}.

The following assertions are equivalents i) v is additive

ii) _{∃ a weight vector ω such that Cv} = WAM_ω iii) _Cv is additive: _Cv(x + x0) = _Cv(x) + _Cv(x0) • Ordered weighted averaging (Yager, 1988)

OWA_ω(x) = n X i=1 ω_ix_(i) , n X i=1 ω_i = 1, ω_{i ≥ 0} with the convention that x_{(1) ≤ · · · ≤ x(n)}.

Proposition (Grabisch-Marichal, 1995) Let v _{∈ FN}.

The following assertions are equivalents i) v is cardinality-based

Ordinal scales

Example : Evaluation of a scientific journal paper on importance

1=Poor, 2=Below average, 3=Average, 4=Very Good, 5=Excellent

Values : 1, 2, 3, 4, 5

or : 2, 7, 20, 100, 246 or : -46, -3, 0, 17, 98

Numbers assigned to an ordinal scale are defined up to an increasing bijection ϕ : R _{→ R.}

Definition. A function M : En _{→ R is comparison} meaningful if for any increasing bijection ϕ : E _{→ E} and any x, x0 _{∈ E}n,

M (x₁, . . . , x_n) _{≤ M(x}0₁, . . . , x0_n) m

Means on ordered sets

Example. The arithmetic mean is a meaningless function. Consider

4 = 3 + 5 2 <

1 + 8

2 = 4.5

and any bijection ϕ such that ϕ(1) = 1, ϕ(3) = 4, ϕ(5) = 7, ϕ(8) = 8. We have 5.5 = 4 + 7 2 6< 1 + 8 2 = 4.5 Theorem 12 (Ovchinnikov, 1996)

The function M : En _{→ R is symmetric, continuous,} internal, and comparison meaningful if and only if there exists k _{∈ N such that}

M (x) = x_(k) (x _{∈ E}n)

Lattice polynomials

Definition. A lattice polynomial defined in Rn is any expression constructed from the variables x₁, . . . , x_n and the symbols _{∧, ∨.}

Example : (x_{2 ∨ (x1 ∧ x3})) _{∧ (x4 ∨ x2})

It can be proved that such an expression can always be put in the form

L_c(x) = _ T_⊆N c(T )=1 ^ i_∈T x_i

Without symmetry :

Theorem 13 (Marichal-Mathonet, 2001)

The function M : En _{→ R is continuous, idempotent,} and comparison meaningful if and only if there ex-ists a nonconstant set function c : 2N _{→ {0, 1}, with} c(_{∅) = 0, such that}

M (x) = L_c(x) (x _{∈ E}n)

Without symmetry and idempotency :

Theorem 14 The function M : En _{→ R is} non-constant, continuous, and comparison meaningful if and only if there exist a nonconstant set function c : 2N _{→ {0, 1}, with c(∅) = 0, and a continuous and} strictly monotonic function g : E _{→ R such that}

Replacing continuity by increasing monotonicity:

Theorem 15 (Marichal-Mathonet, 2001)

Assume that E is open. The function M : En _{→ R is} increasing, idempotent, and comparison meaningful if and only if there exists a nonconstant set function c : 2N _{→ {0, 1}, with c(∅) = 0, such that}

M (x) = L_c(x) (x _{∈ E}n)

Connection with Choquet integral

Proposition 2 (Murofushi-Sugeno, 1993) If v _{∈ FN} is _{{0, 1}-valued then Cv}(x) = L_v(x) Conversely, we have L_c(x) = _Cc(x)

Proposition 3 (Radojevi´c, 1998)

A function M : En _{→ R is a Choquet integral if and} only if it is a weighted arithmetic mean of lattice polynomials Cv(x) = q X i=1 ω_i L_ci(x)

This decomposition is not unique !

0.2x₁ + 0.6x₂ + 0.2(x_{1 ∧ x2})

Proposition 4 (Marichal, 2001)

Any Choquet integral can be expressed as a lattice polynomial of weighted arithmetic means

Cv(x) = Lc(g1(x), . . . , gn(x)) Example (continued)

0.2x₁ + 0.6x₂ + 0.2(x_{1 ∧ x2})

= (0.4x₁ + 0.6x₂) _{∧ (0.2x1} + 0.8x₂)

The converse is not true ! The function

x₁ + x₂

2



∧ x3 is not a Choquet integral.