Michael Heusch - IntCP 2006
30.0
1.0 6
Modeling and solving of a radio antennasdeployment support application with discreteand interval constraints
Michael Heusch - IntCP 20062
30.0
1.0 6
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 20063
30.0
1.0 6
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 20064
30.0
1.0 6
Minimal and maximal distances between antennas
Presentation of the LocRLFAP
Informal description of the de radio antennas deployment problem :
Difficulties : Hybrid combinatorial
optimisation problem non-linear continuous
constraints
Constraints involved : Distance between
frequencies depends on distance between antennas
Michael Heusch - IntCP 20065
30.0
1.0 6
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 P i to Pj
Optimisation problem: Minimise the maximal frequency used
Michael Heusch - IntCP 20066
30.0
1.0 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 20067
30.0
1.0 6
Comparing the RLFAP/LocRLFAP with 5 sites
RLFAP LocRLFAP
Michael Heusch - IntCP 20068
30.0
1.0 6
Comparing the RLFAP/LocRLFAP with 5 sites
RLFAP LocRLFAP
dist² (Si,Sj) = Σi (Xi - Xj)²
Michael Heusch - IntCP 20069
30.0
1.0 6
Comparing the RLFAP/LocRLFAP
Michael Heusch - IntCP 200610
30.0
1.0 6
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 200611
30.0
1.0 6
Discrete constraints
Co-site transmitter-receiver interference constraints:
Duplex distance constraints for each bidirectional link
Forbidden portions in the frequency range
Michael Heusch - IntCP 200612
30.0
1.0 6
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
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
d1
d2
Michael Heusch - IntCP 200613
30.0
1.0 6
Test set
Full deployment of networks with 5 to 10 sites
RLFAP
LocRLFAP
Michael Heusch - IntCP 200614
30.0
1.0 6
Progressive deployment of networks with 9 and 10 sites
P
P
P
P
P
P
P
P P
P
Michael Heusch - IntCP 200615
30.0
1.0 6
Solving with elementary constraints
Full deployment in both models
Michael Heusch - IntCP 200616
30.0
1.0 6
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 200617
30.0
1.0 6
Results with minDomain+maxConstraints
9 sites 10 sites
99% of the backtracks are performed on the continuous part of the search tree
A bit less backtracks on the hybrid model
Hybrid solving is 1 to 3 times slower
Progressive deployment in both models
Michael Heusch - IntCP 200618
30.0
1.0 6
Introducing the distn global constraint
distn ([P1, … , Pn], V)
Pi = Xi x Yi : Cartesian coordinates of the point pi V 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 200619
30.0
1.0 6
Applications of the constraint
Molecular conformation
Robotics
Antennas deployment
Michael Heusch - IntCP 200620
30.0
1.0 6
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| > Δ 1 }, (i,j,k), i≠j, i≠k, j≠k
{ Vij<d 2 } v { |fik-fjl| > Δ 2 }, (i,j,k), i≠j, i≠k, j≠k
Michael Heusch - IntCP 200621
30.0
1.0 6
Results using distn (9 sites)
Simple heuristics Advanced heuristics
hybrid model / 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 200622
30.0
1.0 6
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 200623
30.0
1.0 6
Quality of solutions
9 sites 10 sites
Michael Heusch - IntCP 200624
30.0
1.0 6
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 200625
30.0
1.0 6
Perspectives on the application
Validation on instances of greater size Take forbidden zone constraints into account Provide deployment zones using polygons
Michael Heusch - IntCP 200626
30.0
1.0 6
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 200627
30.0
1.0 6
Sketch of distn’s filtering algorithm
Michael Heusch - IntCP 200628
30.0
1.0 6
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
pi2
pik-1
pik
pi1
Pi
Pj
Michael Heusch - IntCP 200629
30.0
1.0 6
Filtering algorithm on polygons
P1
P2
Michael Heusch - IntCP 200630
30.0
1.0 6
Filtering algorithm on polygons
-
-
++
+
+
P1
P2
Michael Heusch - IntCP 200631
30.0
1.0 6
Filtering algorithm on polygons
P1
P2
Michael Heusch - IntCP 200632
30.0
1.0 6
Interval extension of the algorithm
P1
P2
Michael Heusch - IntCP 200633
30.0
1.0 6
Filtering algorithm of distn
P1
P2
Adjusting bounds of the distance variables
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