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Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion

A formal justification of a simple aggregation function based on criteria and rank weights

• Ch. Labreuche

Thales Research & Technology, Palaiseau, France

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion

Table of contents

1

Motivation & Background Motivation Background

2

Definition of a semantics on the two weight vectors p and w Aim Constraints on p Constraints on w Set of capacities consistent with p and w

3

Finding a parcimonous solution Optimization problem Analytical solution in a particular case Interpretation of the solution

4

Conclusion

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Context

Aggregation functions MCDA: Multi-Criteria Decision Analysis Aggregation function: H : RN → R on criteria N = {1, . . . , n}. Many existing models:

Weighted Sum (WS), based on criteria weights p = (p1, . . . , pn), Ordered Weighted Average (OWA), based on rank weights w = (w1, . . . , wn), Choquet integral,. . .

Choice of the aggregation model based on principles:

Ability to capture real-life, subtle decision strategies

YES for the Choquet integral (model interaction) Not really for WS and OWA

Ability to be interpretable (the simpler the better)

YES for WS and OWA Not really for the Choquet integral

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Motivation

Motivation Idea to reconciliate these two principles: combine criteria weights p and rank weights w,

in order to take advantage of the flexibility of both WS and OWA

capture relative importance of criteria and interaction among them

while using only a very small amount of information from the decision maker.

Existing proposals that combine criteria weights and rank weights

Weighted OWA (WOWA) operator [Torra’97] Hybrid Weighted Averaging (HWA) operator [Xu & Dai’03] Semi-Uninorm OWA (SUOWA) operator [ Llamazares’13], Ordered Weighted Averaging Weighted Average (OWAWA) [ Merigo’11]

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Motivation

Criticism of the existing proposals HWA has a simple expression but fails to fulfill basic important properties, such as idempotency. OWAWA is a simple linear combination of WS and OWA, and its interpretation is not so intuitive. WOWA and SUOWA operators have quite complex expressions, which are not intuitive for a decision maker. Moreover, what is the real contribution of p and w in the formula?

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Motivation

Aim of this work Define aggregation functions based on two weight vectors p and w, in which these two vectors have a clear semantics. How to do it?

Start with a class of very general aggregation functions able to capture both importance of criteria and interaction among them; Define clear semantics of p and w, in the form of constraints; Consider all capacities satisfying these constraints; Look for the simplest one.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

The Choquet integral

How to generalize the Weighted Sum? Starting from 0N, improve conjunctly

• n several criteria.

For A ⊆ N, we set µ(A) := H(1A, 0−A) Idempotent: µ(∅) = 0 as H(0, . . . , 0) = 0 µ(N) = 1 as H(1, . . . , 1) = 1 Monotony: If A ⊆ B then H(1A, 0−A) ≤ H(1B, 0−B) X1 X2

(bad)

1

(good) (bad)

1

(good)

µ(∅) µ({1}) µ({2}) µ({1, 2})

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

The Choquet integral

Capacity A capacity (fuzzy measure) on N is a set function µ : P(N) − → [0, 1], satisfying the following axioms. (i) µ(∅) = 0, µ(N) = 1. (ii) A ⊂ B implies µ(A) ≤ µ(B), for A, B ∈ P(N). MN: set of capacities Remarks A ⊂ N: coalition of criteria µ(A): importance or strength of the coalition A for evaluating products A capacity may be additive, i.e. µ(A) =

• i∈A

µ({i}) A capacity may by symmetric, i.e. µ(A) depends only on |A|

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Choquet integral

Choquet integral [Choquet’53] The Choquet integral of a ∈ Rn with respect to a capacity µ is defined by Cµ(a1, . . . , an) =

n

• i=1
• aσ(i) − aσ(i−1)
• × µ ({σ(i), σ(i + 1), . . . , σ(n)})

=

n

• i=1

aσ(i) ×

• µ ({σ(i), σ(i + 1), . . . , σ(n)})

−µ ({σ(i + 1), . . . , σ(n)})

• with aσ(0) := 0, and σ a permutation of indices so that aσ(1) ≤ · · · ≤ aσ(n).

Remark The Choquet integral represents a kind of average of a1, . . . , an, taking into account importance and interaction of criteria.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Capacities and Choquet integral

Particular cases Weighted Sum (WS): for criteria weights p = (p1, . . . , pn) CµWSp (a) = WSp(a) :=

• i∈N

pi ai where µWSp(S) =

i∈S pi.

Ordered Weighted Average (OWA): for rank weights w = (w1, . . . , wn) CµOWAw (a) = OWAw(a) :=

n

• j=1

wj aσ(n−j+1) where µOWAw (S) = |S|

j=1 wj

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Interpretation of a capacity

Mean importance of criteria: Shapley value [Shapley’53] φi(µ) =

• S⊆N\{i}

|S|!(n − |S| − 1)! n! (µ(S ∪ {i}) − µ(S)).

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Motivation Background

Interpretation of a capacity

General interaction: Intolerance index [Marichal’07] Counterpart of attitude of the Decision Maker towards risk in MCDA

• rness(Cµ) =

Cµ−min max−min = 1 n−1

n−1

t=1 1

(

n t)

• T⊆N

|T|=t

µ(T) The index of k-intolerance is the mean value of Cµ(a) over all a such that aσ(k) = 0: intolk(µ) = n − k + 1 (n − k) n

k

• K⊆N

|K|=k

Cµ(0K, ·) = 1 − 1 n − k

n−k

• t=0

1 n

t

• T⊆N

|T|=t

µ(T).

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Aim Constraints on p Constraints on w Set of capacities consistent with p and w

What do we wish to do?

We are given criteria weights p and rank weights w. We consider the class of capacities MN We wonder which capacity µ shall correspond to p and w In other words, which constraints shall µ satisfy?

Constraints for the semantics of p Constraints for the semantics of w

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Aim Constraints on p Constraints on w Set of capacities consistent with p and w

Constraints on p

p has a clear interpretation in WSp: pi = WSp(1i, 0−i) − WSp(0, . . . , 0). When the DM provides as inputs the importance pi of criterion i, the mean importance of criteria i should be precisely pi. Interpretation of p ∀i ∈ N φi(µ) = pi.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Aim Constraints on p Constraints on w Set of capacities consistent with p and w

Constraints on w

Case of OWAw all criteria are symmetric w describes only the way criteria are interacting together k-intolerant indices are good candidates to interpret w Computation for OWAw intolk(µOWAw ) = 1 − 1 n − k

n−k

• t=1

1 n

t

• T⊆N

|T|=t

µOWAw (T) = 1 − 1 n − k

n−k

• t=1

t

• j=1

wj = 1 − 1 n − k

n−k

• t=1

t wt. However, this relation is not so trivial, and it will not be convenient as a constraint.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Aim Constraints on p Constraints on w Set of capacities consistent with p and w

Constraints on w

Case of OWAw Essential property of the weights w: OWAw(1S, 0N\S) = w1 + w2 + · · · + ws for any S ⊆ N with |S| = s. Note that OWAw(1S, 0N\S) = µOWAw (S) Generalization for non-symmetric µ Replace OWAw(1S, 0N\S) by the average value of µ(S) over all subsets of cardinality s As(µ) := 1 n

s

• T⊆N

|T|=s

µ(T) Note that At(µOWAw ) = t

j=1 wj

There is a simple linear relation between the ∗-intolerant indices and the A∗ indices: intolk(µ) = 1 − 1 n − k

n−k

• t=1

At(µ)

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Aim Constraints on p Constraints on w Set of capacities consistent with p and w

Constraints on w

Interpretation of w ∀s ∈ {1, . . . , n − 1} 1 n

s

• S⊆N : |S|=s

µ(S) =

s

• k=1

wk. Remark: The relation with s = n has been removed as it is trivially satisfied by every capacity

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Aim Constraints on p Constraints on w Set of capacities consistent with p and w

Set of capacities consistent with the semantics of p and w

Set of capacities consistent with the semantics of p and w MN(p, w) :=

• µ ∈ MN :

∀i ∈ N

• S⊆N\{i}

|S|!(n − |S| − 1)! n! (µ(S ∪ {i}) − µ(S)) = pi , ∀s ∈ {1, . . . , n − 1} 1 n

s

• S⊆N : |S|=s

µ(S) =

s

• k=1

wk

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Optimization problem Analytical solution in a particular case Interpretation of the solution

Optimization problem

Finding a parcimonous solution Optimization problem P: Minimize

• i∈N
• S⊆N\{i}

|S|!(n − |S| − 1)! n! E(µ(S ∪ {i}) − µ(S)) under µ ∈ MN(p, w) where E can be

• pposite of Shanon entropy: E(u) = u ln u [Marichal 2002]

based on R´ enyi entropy: E(u) = uα variance: E(u) =

• u − 1

n

2 [Kojadinovic 2006]

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Optimization problem Analytical solution in a particular case Interpretation of the solution

Optimization problem

In general, optimization problem P can be very complex due to the numerous constraints.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Optimization problem Analytical solution in a particular case Interpretation of the solution

Optimization problem

Theorem Consider for E the variance, and assume that the inequality constraints are not saturated in P: Minimize

• i∈N
• S⊆N\{i}

|S|!(n − |S| − 1)! n! E(µ(S ∪ {i}) − µ(S)) under ∀i ∈ N ∀S ⊆ N \ {i} µ(S ∪ {i}) > µ(S) ∀i ∈ N

• S⊆N\{i}

|S|!(n−|S|−1)! n!

(µ(S ∪ {i}) − µ(S)) = pi ∀s ∈ {1, . . . , n − 1}

1

(

n s)

• S⊆N : |S|=s µ(S) = s

k=1 wk

Then there is a unique solution to this problem, which corresponds to Hp,w(a) := WSp(a) + OWAw(a) − AM(a).

• i.e. µ(S) = µWSp (S) + µOWAw (S) − µAM(S) =

i∈S pi + |S| j=1 wj − |S| n

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Optimization problem Analytical solution in a particular case Interpretation of the solution

Interpretation of the solution

Monotonicty conditions Hp,w is monotone iff min

i∈N pi + min k∈N wk ≥ 1

n.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Optimization problem Analytical solution in a particular case Interpretation of the solution

Interpretation of the solution

Difficulty in the interpretation of Hp,w: negative term AM. Definition Av p,v w (a) :=

• i∈N

v p

i ai +

• i∈N

v w

i aτ(n−i+1).

where v p = (v p

1 , . . . , v p n ) and v w = (v w 1 , . . . , v w n ) such that

∀i ∈ N v p

i ≥ 0 and v w i

≥ 0,

• k∈N

v p

k +

• k∈N

v w

k = 1.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Optimization problem Analytical solution in a particular case Interpretation of the solution

Interpretation of the solution

Proposition Models Hp,w and Av p,v w are equivalent. Example For p = (0.1, 0.1, 0.1, 0.1, 0.6) and w = (0.1, 0.1, 0.1, 0.1, 0.6) Hp,w(a) = 0.5 a5 + 0.5 aτ(1)

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights

Thales Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion

Conclusion

Contributions of the talk Provide a clear interpretation of the two weight vectors p and w

Interpretation of p through the Shapley value Interpretation of w through a variant of the intolerance indices

The most parcimonous solution (when monotonicity constraints are not saturated) is very simple and can be easily be interpreted.

• Ch. Labreuche

Simple aggregation function based on criteria and rank weights