1. I T & S A e r o s p a c eD e f e n c e Integration of a Multicriteria Decision Model in Constraint Programming F. Le Huédé M. Grabisch C. Labreuche P. Savéant

2. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 2Objective Integrate in in order to solve Multicriteria Combinatorial Optimization Problems. Preference Modeling with fuzzy measures and the Choquet integral Constraint Programming

3. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 3 Why These Approaches? Why CP? For the software engineering aspect of CP. modelling capacities For the modelling capacities of CP languages and the models flexibility. For the efficiency of the combination between search and constraints propagation. Why a multicriteria model based on an aggregation function? Automated Decision In an Automated Decision context, decisions are taken by the software or an embedded system. unique solutiongood quality  The algorithm must return a unique solution of good quality. a priori  Preferences must be modelled « a priori » precisely.

4. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 4 Why the Choquet Integral? C1 C2 A B C D Intolerance Complementary criteria C1 C2 A B C D Independence C1 C2 A B C D Tolerance Redundant criteria The Choquet integral allows to model: the importance of a criterion, interactions between criteria. Many different behaviors. Criterion 1 Criterion 2 A B C D 4 solutions to compare:

5. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 5 Main Issue To find a good multicriteria solution within a satisfying time.  How can we model and propagate the multicriteria model in CP?  How can we guide the search towards good solutions?

6. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 6 The Multicriteria Examination Timetabling Problem Data: A set of exams to plan. A set of rooms of given capacities. A set of students, each enrolled to a set of exams. A set of periods. Objective: To give each examination a period number and a room. Constraints: Disjunction constraints: two exams followed by a same student can’t be planned on the same period. Room capacity: the number of students having an exam in a room on a given period can’t exceed the capacity of this room. Criteria: Examination duration. Number of used rooms. Spreading of the exams.

7. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 7 Outline I.The MCDM Model II.Modelling a Multicriteria Model in CP III.The Search of Solutions for Multicriteria Optimisation Problems

8. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 8 Components of the Multicriteria Model Formalizing criteria Modeling interactions between criteria Aggregating Utility functions Fuzzy measure Choquet Integral

9. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 9 MCDM Methodlogy: Criteria Formalization 0 1 Overall Evaluation C1 C2 C3 Definition of reference levels Construction of an interval scale over each criterion Construction of utility functions 0 1 a1a1 0 1 a2a2 0 1 a3a3 u1u1 u2u2 u3u3 Hierarchical organization of criteria Aggregation Criterion Attribute Utility

10. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 10 Utility function 0 1 LB  1.LB  2.LB  3.LB max Utility = Degree of satisfaction of the criteria: Minimise the duration Attribute: Duration of the exams

11. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 11 Fuzzy measures Define a weight, not only for each criterion but also for each criteria coalition. Definition: A fuzzy measure µ on N is a function µ: P(N)  [0,1], such that: µ(  )=0, µ(N)=1 A  B  µ(A)  µ(B). Definition: A fuzzy measure µ on N is a function µ: P(N)  [0,1], such that: µ(  )=0, µ(N)=1 A  B  µ(A)  µ(B). N : criteria set A  N : criteria coalition µ(A) : weight of coalition A

12. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 12 The Choquet integral u = (u 1, u 2, u 3 ) : satisfaction degrees of a solution a. C µ (u)=  u (i)  [ µ ( { (i),...,(n) } ) - µ ( { (i+1),...,(n) } ) ] i=1 n Choquet integral of u with respect to µ : µ({(2),(3)}) µ({(1),(2),(3)}) µ({(3)}) u (1) u (2) u (3) where (.) indicate that the indices have been permuted so that: u (1)  u (2)  u (3) and µ({(n+1)}) = 0 C µ ( u 1, u 2, u 3 )

13. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 13 Graphical representations of the Choquet integral Complementary criteria Redundant criteria u1u1 u2u2 u 1 =u 2 C µ (u 1,u 2 )=z 1 u1u1 u2u2 C µ (u 1,u 2 )=z 2 u 1 =u 2

14. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 14 Outline I.The MCDM Model II.Modelling a Multicriteria Model in CP III.The Search of Solutions for Multicriteria Optimisation Problems

15. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 15 Integration in a CP solver Choquet Constraint : y = C µ (u 1, u 2, u 3 ) y Overall evaluation u 1 =U 1 (a 1 ), u 2 =U 2 (a 2 ) u 3 =U 3 (a 3 ) Affine Constraint u1u1 u2u2 u3u3 Utilities / scores x 1, x 2,....................., x m Problem variables and constraints [0,1] FD a1a1 a2a2 a3a3 Attributes Objective functions a 1 =f 1 (x 1,…,x m ), a 2 =f 2 (x 1,…,x m ), a 3 =f 3 (x 1,…,x m )

16. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 16 The Choquet constraint Objective : Establish the relation : y = C µ (u 1,..., u n ) During the search, propagate domain reductions between the variables y, u 1,..., u n.

17. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 17 Constraint Propagation The CP resolution process alternates : construction of a search tree, constraint propagation at each node of the tree. Propagation = enforcing a consistency level for each constraint: Arc consistency : A constraint propagation technique enforces arc- consistency on a constraint c(x 1... x m ), if and only if when propagation stops the following statement holds: for every variable x i for every value val i in the domain of x i there are values val 1... val i-1 val i+1... val m in the domains of x 1... x i-1 x i+1... x m such that val 1... val i-1 val i val i+1... val m satisfy c Bound consistency techniques: consistency checks are done only on the bounds of the domains.

18. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 18 Propagation of the Choquet constraint Consider a constraint that maintains the relation y = C µ (u 1,...,u n ), it is bound-consistent iff conditions 1,2,3 and 4 hold: How can we deduce some new bounds for u k ? 1 2 3 4

19. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 19 Computing a new lower bound for u k Notation :  Find the smallest value,, such that: Determine on which piece of the function is located 1 Compute 2

20. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 20 The Choquet constraint propagation: Example µ 0 = 0 µ 1 = 0.1 µ 2 = 0.4 µ 3 = 0.1 µ 12 = 0.5 µ 13 = 0.2 µ 23 = 0.6 µ 123 = 1 y  [0.4,1] y  [0.4,0.44] u 1  [0,0.2] u 1  [0.1,0.2] u 2  [0,0.8] u 2  [0.7,0.8] u 3  [0,0.2] u 3  [0.12,0.2] y = C µ (u 1,u 2,u 3 )

21. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 21 The Choquet constraint propagation: Results Results for a model with redundant criteria Without PropagationWith Propagation Nb. Btkmsms msms Sc. 66025013640 Sc. 1540741492091061570 Sc. 2037494355460379318820 Sc. 23308855182569 s12266735790 s Results for a model with independant criteria Without PropagationWith Propagation Nb. Btkmsms msms Sc. 6150113041970 Sc. 152380128203908840 Sc. 207388691061005178616120 Sc. 23298168271234 s247009901179 s

22. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 22 Outline I.The MCDM Model II.Modelling a Multicriteria Model in CP III.The Search of Solutions for Multicriteria Optimisation Problems

23. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 23 Main Issue: Guiding the Search Objective: To define a search strategy able to guide the search quickly towards good solutions. Difficult due to: Conflicting criteria Compensation phenomena u2u2 u1u1 u 1 =u 2 C µ (u 1,u 2 )=z 1 A strategy that is good for optimising a criteria is often inefficient for another Good solutions offer a trade-off between the criteria

24. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 24 Guiding the Search in Multicriteria Optimisation: Related Papers Focacci-Godard CP-AI-OR’02 : Definition of a partial search algorithm for a three objectives job shop aggregated by a weighted sum. At each solution, bounding constraints are set on the objectives of the problem to speed-up the finding of good solutions. The best algorithm alternates several searches with different configurations of bounding constraints.  A practical and dedicated approach.  Incomplete search. Junker CP-AI-OR’02 : « Preference Based Search » A simple formalism allows to express some preferences between the criteria. The PBS algorithm performs successive mono-objective optimisations to find the set of non-dominated solutions. A suitable labelling strategy is selected for each mono-objective optimisation depending on the criterion to optimise.  The model returns a set of solutions.  The whole set of non-dominated solutions is hard to find in large size optimisation problems.

25. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 25 So What Do We Need? A general approach for the search of solutions to multicriteria combinatorial optimisation problems in an automated decision context.

26. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 26 Suggested Approach: The MCS Algorithm (1) General principle: to alternate searches on the problem criteria in order to find solutions of increasing quality. mono-criterion strategy A search on a criterion uses a dedicated mono-criterion strategy that allows to find quickly solutions of good quality on this criterion. criterion choice heuristic After a search on a criterion, the criterion choice heuristic dynamically chooses the criterion that will guide the next search. stopping condition The algorithm ends when a user-defined stopping condition is verified. local constraints. Before a mono-criterion search, an improvement on a criterion can be explicitly imposed thanks to local constraints.

27. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 27 Suggested Approach: The MCS Algorithm (2) Criterion choice heuristic s*) (s*) (s*,c) Search(u c,y,(c)) Search(u c,y,getStrategy(c)) (s*,b) c c,b s* false true Local constraints addition Solutions search Stopping Condition MCS( y, list (u1,…,un), list (S1,…,Sn),, )

28. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 28 Instantiating MCS: An Example  Search for a first solution on criterion 1; old_c := 1  If ( c = old_c) set local constraint ; local := true  Set constraint propagate bounds and  Select the criterion c that maximises:  Terminate if (local  true) and (s = nil) else old_c := c  Search for a solution on criterion c or nil

29. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 29 s*) (s*) (s*,c) Search(u c,y,(c)) Search(u c,y,getStrategy(c)) (s*,b) c c,b s* false true MCS Components: n Mono-Criterion Search Strategies

30. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 30 Many kinds of search strategies can be defined easily with dedicated search pattern libraries such as ToOLS© : Definition of the search tree: Variable choice heuristic ; Labelling strategy. Specification of the maximum number of solutions per search. Specification of complete or partial search strategies. A library of hybrid methods that combine tree search and local search. Definition of an overall time limit and of a time limit per search. MCS(y,(u1,…,un),(S1,…,Sn),,, ) MCS Components: n Mono-Criterion Search Strategies

31. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 31 (s*) (s*,c) Search(uc,y,) Search(uc,y,getStrategy(c)) (s*,b) c c,b s* false true MCS Components: The Criterion Choice Heuristic

32. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 32 MCS(y,(u1,…,un),(S1,…,Sn),,, )  Indicates the criterion on which it is the most interesting to get an improvement with respect to: The last found solution (we denote its performances on the problem criteria). The aggregation function coefficients. The upper bounds of utility variables (denoted ).  Some criterion choice heuristics: maximum improvement indicator Choose the criterion that maximises the maximum improvement indicator : average improvement indicator Choose the criterion that maximises the average improvement indicator : MCS Components: The Criterion Choice Heuristic

33. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 33 (s*) (s*,c) Search(uc,y,) Search(uc,y,getStrategy(c)) (s*,b) c c,b s* false true MCS components: Local Constraints

34. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 34 MCS(y,(u1,…,un),(S1,…,Sn),,, )  (s*,c) allows to add some constraints on the criteria (u c > u* c )  A local constraint is valid during only one search.  When a function posts a local constraint, it notifies the whole algorithm by returning true.  Advantages: makes the search more efficient for the search of a solution of better quality on a selected criterion (if such a solution exists).  Drawbacks: can make the search incomplete and take time for the search of a solution that does not exist.  Examples : systematicImprovement(s*,c): systematically add the constraint u c > u* c. consecutiveImprovement(s*,c): add the constraint u c > u* c when criterion c had been selected for the guiding of the previous search. MCS Components: Local Constraints

35. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 35 (s*) (s*,c) Search(uc,y,) Search(uc,y,getStrategy(c)) (s*,b) c c,b s* false true MCS components: The algorithm Stopping Condition

36. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 36 MCS(y,(u1,…,un),(S1,…,Sn),,, ) realises two kinds of operations: The termination of the algorithm (e.g., when a search can’t find any solution). The use of information returned by a search to reinforce the model. optimalityCondition(local,s*,s,u,c) if (getOverallValue(s*) = y.sup) return true else if (s =  ) \\ the last search did not find any solution if (local) \\ A local constraint was used setConstraint(u[c]  getValue(s*,c)), return false else return true else return false end MCS Components: The Algorithm Stopping Condition

37. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 37 Some Instances of MCS for Complete Search Results on the examination timetabling problem MCS Instances

38. This document and any data included are the property of THALES. They cannot be reproduced, disclosed or used without THALES' prior written approval. ©THALES 2003. Template trtco V 6.0.0 38 Conclusion and Future Work Contributions: Integration of an MCDM model in CP. The MCS algorithm, dedicated to the search of solutions to multicriteria optimisation problems. The criterion choice heuristic. The definition of MCS components to achieve complete search. Implementation of these results in the Eclair© solver. On going work: Instantiation of the MCS algorithm for partial search. Large combinatorial optimisation problems.