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Modeling and Solving Radio Antennas Deployment Support

Explore the application with discrete and interval constraints, including modeling, search heuristics, and experimental results. Solve the progressive deployment problem with hybrid optimization techniques.

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Modeling and Solving Radio Antennas Deployment Support

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  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

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