Modeling and solving of a radio antennas deployment support application with discrete and interval constraints

1. Modeling and solving of a radio antennasdeployment support application with discreteand interval constraints Michael Heusch - IntCP 2006

2. Outline of the talk • Presentation of the application • Modeling with discrete and interval constraints • Defining search heuristics • Modeling the problem with the distn constraint • Experimental results on solving the progressive deployment problem Michael Heusch - IntCP 2006

3. Presentation of the LocRLFAP • Informal description of the de radio antennas deployment problem : • Constraints involved : • Distance between frequencies depends on distance between antennas Michael Heusch - IntCP 2006

4. Minimal and maximal distances between antennas Presentation of the LocRLFAP • Informal description of the de radio antennas deployment problem : • Constraints involved : • Distance between frequencies depends on distance between antennas • Difficulties : • Hybrid combinatorial optimisation problem • non-linear continuous constraints Michael Heusch - IntCP 2006

5. Specification of the problem • Formulation as a constrained optimisation problem: • Data • Fixed set of antennas (transmitter-receiver) • Dispatched on n sites {P1, … , Pn} • The links to establish is known in advance • Variables of the problem: • A solution associates one frequency to each antenna and a position to each site • Pi = (Xi,Yi): Position of a site • fi,j : frequency allocated to the link from Pi to Pj • Optimisation problem: • Minimise the maximal frequency used Michael Heusch - IntCP 2006

6. Constraints of the problem • Constraints of the problem • discrete constraints: • Compatibility between antennas • Forbidden frequencies • continuous constraints • Maximum distance between antennas (range) • Minimum distance between the antennas (security, interference) • mixed constraints • Compatibility between the allocation and the deployment Michael Heusch - IntCP 2006

7. Comparing the RLFAP/LocRLFAP with 5 sites • LocRLFAP • RLFAP Michael Heusch - IntCP 2006

8. RLFAP LocRLFAP dist² (Si,Sj) = Σi (Xi - Xj)² Comparing the RLFAP/LocRLFAP with 5 sites Michael Heusch - IntCP 2006

9. Comparing the RLFAP/LocRLFAP Michael Heusch - IntCP 2006

10. Hybrid solving with collaborating solvers • Original approach • Modeling with the finite domain constraint solver Eclair • Full discretization of the problem • Modeling three types of constraints • Discrete constraints • Continuous constraints • Mixed constraints Michael Heusch - IntCP 2006

11. Discrete constraints • Co-site transmitter-receiver interference constraints: • Duplex distance constraints for each bidirectional link • Forbidden portions in the frequency range Michael Heusch - IntCP 2006

12. Mixed constraints: • Compatibility constraints • If dist(Pi,Pj)< d1, great interference • If d1 <= dist(Pi,Pj)< d2, limited interference • Expression with elementary constraints • { dist(Pi,Pj)< d1 } v { |fik-fjl| > Δ1 }, (i,j,k), i≠j, i≠k, j≠k • { dist(Pi,Pj)< d2 } v { |fik-fjl| > Δ2 }, (i,j,k), i≠j, i≠k, j≠k d2 d1 Continuous and mixed constraints • Elementary continuous constraints: • dist²(Pi,Pj) > mij² , for all i<j • dist²(Pi,Pj) < Mij² , if there exists a radio link between Pi and Pj Michael Heusch - IntCP 2006

13. LocRLFAP Test set • Full deployment of networks with 5 to 10 sites RLFAP Michael Heusch - IntCP 2006

14. P P P P P P P P P P Progressive deployment of networks with 9 and 10 sites Michael Heusch - IntCP 2006

15. Solving with elementary constraints • Full deployment in both models Michael Heusch - IntCP 2006

16. Improvements to the search algorithm • Usage of a naïve Branch & Bound with: • Distinction of the type of variables • The problem is under-constrained on positions • Branch on disjunctions? • Branch first on constraints entailing a strong interdistance? • Variable selection heuristics • minDomain • min(dom/deg) • minDomain+maxConstraints Michael Heusch - IntCP 2006

17. Results with minDomain+maxConstraints • Progressive deployment in both models • 10 sites • 9 sites A bit less backtracks on the hybrid model Hybrid solving is 1 to 3 times slower 99% of the backtracks are performed on the continuous part of the search tree Michael Heusch - IntCP 2006

18. Introducing the distn global constraint • distn ([P1, … , Pn], V)Pi =Xi x Yi : Cartesian coordinates of the point piV i,j : distance to maintain between Pi and Pj • distn(p1, … , pn], v) satisfied if and only if dist(pi,pj) = vi,j • Filtering algorithm uses geometric approximation techniques Michael Heusch - IntCP 2006

19. Applications of the constraint • Molecular conformation • Robotics • Antennas deployment Michael Heusch - IntCP 2006

20. Using distn in the model • Second formulation of the problem with the global constraint: • Simple continuous constraints • Introduction of a matrix {Vi,j} of distance variables: • Domain(Vi,j)=[mi,j , Mi,j] • Expression of the set of min and max distance constraints: • distn([P1, … , Pn], V) • Expression of the mixed « distant compatibility » disjunctions • distn([P1, … , Pn], V) • { Vij<d 1} v { |fik-fjl| > Δ1 }, (i,j,k), i≠j, i≠k, j≠k • { Vij<d 2} v { |fik-fjl| > Δ2 }, (i,j,k), i≠j, i≠k, j≠k Michael Heusch - IntCP 2006

21. Simple heuristics Advanced heuristics Results using distn (9 sites) hybridmodel / discrete model comparison: 1.8 times slower 1.5 times more backtracks Similar performance of both models wrt. simple model, distn divides by 2 the nb. of backtracks Michael Heusch - IntCP 2006

22. Results using distn (10 sites) • Simple heuristics • Advanced heuristics hybrid model / discrete model comparison: 4 additional instances are solved • Performance on the solved instances: • 63% less backtracks • All instances are solved Michael Heusch - IntCP 2006

23. Quality of solutions • 9 sites • 10 sites Michael Heusch - IntCP 2006

24. Conclusion and perspectives • We showed the relevance of coupling discrete and continuous constraints • Obtain solution of greater quality • Better performance when solving • Independence w.r.t. the discretization step • Validation on one industrial application • Key points • Definition of appropriate search heuristics • Usage of the distn global constraint Michael Heusch - IntCP 2006

25. Perspectives on the application • Validation on instances of greater size • Take forbidden zone constraints into account • Provide deployment zones using polygons Michael Heusch - IntCP 2006

26. Other approaches for solving the RLFAP • Other approaches for solving the classical RLFAP • Graph coloring • Branch & Cut • CP • LDS [Walser – CP96] • Russian Doll Search [Schiex et. al - CP97] • Heuristics • Tabou [Vasquez – ROADEF 2001] • Simulated annealing, evolutionary algorithms… • Motivations for an approach using CP • Robustness wrt modification of the constraints of the problem Michael Heusch - IntCP 2006

27. Sketch of distn’s filtering algorithm Michael Heusch - IntCP 2006

28. Filtering algorithm on polygons Method using polygons for representing domains • Theorem by K. Nurmela et P. Östergård (1999) • M. Markót et T. Csendes: A New Verified Optimization Technique for the ``Packing Circles in a Unit Square'' Problems. SIAM Journal of Optimization, 2005 pi1 pi2 Pj Pi pik-1 pik Michael Heusch - IntCP 2006

29. Filtering algorithm on polygons P2 P1 Michael Heusch - IntCP 2006

30. + - + - + + Filtering algorithm on polygons P2 P1 Michael Heusch - IntCP 2006

31. Filtering algorithm on polygons P2 P1 Michael Heusch - IntCP 2006

32. Interval extension of the algorithm P2 P1 Michael Heusch - IntCP 2006

33. Filtering algorithm of distn • Adjusting bounds of the distance variables P2 P1 Michael Heusch - IntCP 2006