Optimization Methods vs Spatial Stochastic Models in a Hierarchical Network problem

1. Optimization Methods vs Spatial Stochastic Models in a Hierarchical Network problem Christian Destré, France Telecom R&D christian.destre@francetelecom.com The present document contains information that remains the property of France Telecom. The recipient’s acceptance of this document implies his or her acknowledgement of the confidential nature of its contents and his or her obligation not to reproduce, transmit to a third party, disclose or use for commercial purposes any of its contents whatsoever without France Telecom’s prior written agreement.

2. Introduction • Hierarchical network problem: • Location of network equipments • Compare 2 different ways to solve this problem: • Spatial Stochastic Models • Operational Research • Aim of this talk: • Present and discuss the approaches

3. Outline • 3-level hierarchical network • Model and cost function • Use of Spatial Stochastic models • Poisson-Voronoi Spanning Trees • Another approach: Operational Research • Location Problem / Weber Problem • Discussion

4. 2 1 0 3-level hierarchical network model • 3 types of equipment • 3-level hierarchical network • Connections are done according to shortest paths • Subscriber line model (PSTN)

5. Costs • For each link: • Capacity cost: between a point of level-i and the closest point of level-i+1 (medium used): • Infrastructure cost (civil engineering): (r: distance, A,B,, non negative parameters) • Constant equipment installation cost

6. Cost function

7. 2 1 0 Cost function

8. Level-1 • Level-0 and level-2 are given • Number and location of equipments on level-1 in order to minimize the total cost ?

9. Use of spatial geometry

10. π2 π1 π0 Poisson-Voronoi Spanning Trees • [Bacelli, Zuyev 96] • 3 independent homogeneous Poisson point processes π0 π1 π2 • Level-i is obtained from realizations of πi • λi is the intensity of πi

11. λ1 intensity • Given the intensities λ0 and λ2 and the installation cost • Find λ1 that minimizes the average total cost • Bacelli&Zuyev have proved that

12. Discussion • Average method: no exact result for specific instance • Need realizations • Analytical formula: direct result • No computation time (without considering the realizations of the point processes)

13. Operational Research

14. OR: Location problem • Given a set of facility locations and a set of customers: • Which facilities should be opened ? • Which customers served by which open facility ? • Minimize the total cost of serving all the customers • Facility locations are known • Simple Plant Location Problem

15. OR: Weber problem • Location in the plane (continuous) • Locate a given number m of new facilities to serve a set of n customers • To minimize the total service (transportation + installation) cost • NP-hard problem (depending on the number of customers and equipments)

16. OR: Formulation • Aj = coordinates of customer j • Xi = coordinates of new facility i • wij = decision variable : connecting customer i to facilty j • d(X,Y) = distance function

17. Brimberg et al. approach 04 • Fix m (unknown) • SPLP with customer = potential facility • Improve the solution in continuous space • Multi-phase heuristic approach • Work to do: • Adapt this approach to our problem • Distance function • Use the best heuristics • Time consumption / instance size (number of points)

18. OR: Discussion • Exact or approximated solution for concrete instances • Time consuming • Good for small instances • What about big instances ? (to define) • For example: Brimberg et al. solve instances with 1060 customers, 121 facilities (CPU time = 1370s on a Sun Spare Station 10)

19. Comparison • Which optimization methods is interesting ? • According to the instance size & time consumption • What about the quality of the solutions ? • Differences between the two approaches • Can we validate the stochastic approach by OR ? • Are OR solutions Poisson ?

20. Thank you