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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 Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Finding a parcimonous solution Conclusion Table of contents Motivation & Background 1 Motivation Background Definition of a semantics on the two weight vectors p and w 2 Aim Constraints on p Constraints on w Set of capacities consistent with p and w Finding a parcimonous solution 3 Optimization problem Analytical solution in a particular case Interpretation of the solution Conclusion 4 Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion Context Aggregation functions MCDA: Multi-Criteria Decision Analysis Aggregation function: H : R N → R on criteria N = { 1 , . . . , n } . Many existing models: Weighted Sum (WS), based on criteria weights p = ( p 1 , . . . , p n ), Ordered Weighted Average (OWA), based on rank weights w = ( w 1 , . . . , w n ), 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 Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion 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] Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion 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? Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion 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. Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion The Choquet integral How to generalize the Weighted Sum? Starting from 0 N , improve conjunctly X 2 on several criteria. µ ( { 2 } ) µ ( { 1 , 2 } ) 1 For A ⊆ N , we set (good) µ ( A ) := H (1 A , 0 − A ) Idempotent: µ ( ∅ ) µ ( { 1 } ) (bad) 0 X 1 µ ( ∅ ) = 0 as H (0 , . . . , 0) = 0 0 1 µ ( N ) = 1 as H (1 , . . . , 1) = 1 (bad) (good) Monotony: If A ⊆ B then H (1 A , 0 − A ) ≤ H (1 B , 0 − B ) Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion 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 ). M N : 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 ) = µ ( { i } ) i ∈ A A capacity may by symmetric , i.e. µ ( A ) depends only on | A | Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion Choquet integral Choquet integral [ Choquet’53 ] The Choquet integral of a ∈ R n with respect to a capacity µ is defined by n � � � C µ ( a 1 , . . . , a n ) = a σ ( i ) − a σ ( i − 1) × µ ( { σ ( i ) , σ ( i + 1) , . . . , σ ( n ) } ) i =1 n � � = a σ ( i ) × µ ( { σ ( i ) , σ ( i + 1) , . . . , σ ( n ) } ) i =1 � − µ ( { σ ( i + 1) , . . . , σ ( n ) } ) with a σ (0) := 0, and σ a permutation of indices so that a σ (1) ≤ · · · ≤ a σ ( n ) . Remark The Choquet integral represents a kind of average of a 1 , . . . , a n , taking into account importance and interaction of criteria. Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion Capacities and Choquet integral Particular cases Weighted Sum (WS): for criteria weights p = ( p 1 , . . . , p n ) � C µ WS p ( a ) = WS p ( a ) := p i a i i ∈ N where µ WS p ( S ) = � i ∈ S p i . Ordered Weighted Average (OWA): for rank weights w = ( w 1 , . . . , w n ) n � C µ OWA w ( a ) = OWA w ( a ) := w j a σ ( n − j +1) j =1 where µ OWA w ( S ) = � | S | j =1 w j Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion Interpretation of a capacity Mean importance of criteria: Shapley value [Shapley’53] | S | !( n − | S | − 1)! � φ i ( µ ) = ( µ ( S ∪ { i } ) − µ ( S )) . n ! S ⊆ N \{ i } Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Definition of a semantics on the two weight vectors p and w Motivation Finding a parcimonous solution Background Conclusion Interpretation of a capacity General interaction: Intolerance index [Marichal’07] Counterpart of attitude of the Decision Maker towards risk in MCDA C µ − min � n − 1 1 1 orness ( C µ ) = max − min = � µ ( T ) T ⊆ N t =1 n − 1 ( n t ) | T | = t The index of k -intolerance is the mean value of C µ ( a ) over all a such that a σ ( k ) = 0: intol k ( µ ) = n − k + 1 � C µ (0 K , · ) � n � ( n − k ) k K ⊆ N | K | = k n − k 1 1 � � = 1 − µ ( T ) . � n � n − k t t =0 T ⊆ N | T | = t Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Aim Definition of a semantics on the two weight vectors p and w Constraints on p Finding a parcimonous solution Constraints on w Conclusion 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 M N 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 Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights

Motivation & Background Aim Definition of a semantics on the two weight vectors p and w Constraints on p Finding a parcimonous solution Constraints on w Conclusion Set of capacities consistent with p and w Constraints on p p has a clear interpretation in WS p : p i = WS p (1 i , 0 − i ) − WS p (0 , . . . , 0) . When the DM provides as inputs the importance p i of criterion i , the mean importance of criteria i should be precisely p i . Interpretation of p ∀ i ∈ N φ i ( µ ) = p i . Thales Ch. Labreuche Simple aggregation function based on criteria and rank weights