Michael Heusch - IntCP 2006 3 0 . 0 1 . 0 6 Modeling and solving of a radio antennas deployment support application with discrete and interval constraints
Dec 21, 2015
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
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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 Pi 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
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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
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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
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Test set
Full deployment of networks with 5 to 10 sites
RLFAP
LocRLFAP
Michael Heusch - IntCP 200614
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Progressive deployment of networks with 9 and 10 sites
P
P
P
P
P
P
P
P P
P
Michael Heusch - IntCP 200615
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Solving with elementary constraints
Full deployment in both models
Michael Heusch - IntCP 200616
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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
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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
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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
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Applications of the constraint
Molecular conformation
Robotics
Antennas deployment
Michael Heusch - IntCP 200620
30.0
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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
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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
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Quality of solutions
9 sites 10 sites
Michael Heusch - IntCP 200624
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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
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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
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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
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Sketch of distn’s filtering algorithm
Michael Heusch - IntCP 200628
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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
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Filtering algorithm on polygons
P1
P2
Michael Heusch - IntCP 200630
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Filtering algorithm on polygons
-
-
+
+
+
+
P1
P2
Michael Heusch - IntCP 200631
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Filtering algorithm on polygons
P1
P2
Michael Heusch - IntCP 200632
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Interval extension of the algorithm
P1
P2
Michael Heusch - IntCP 200633
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Filtering algorithm of distn
P1
P2
Adjusting bounds of the distance variables