1. Internet Engineering Czesław Smutnicki Discrete Mathematics – Location and Placement Problems in Information and Communication Systems

2. location and placement problems, solution methodology, classical RND problems, more realistic RND problem, map topology, cell model, coverage, the optimization problem, solution methods, computer experiments, conclusions PRESENTATION OUTLINE

3. VLSI floorplanning, service or warehouse or facility location (known as QAP, Quadratic Assignment Problem), databases and network services migration and replication, antenna placement in mobile telecommunication, cell planning for cellular networks, distribution of access points in wireless networks, ad hoc networks, planning of distribution of wireless sensors … LOCATION AND PLACEMENT PROBLEMS

4.  Please wait. Calculations will last 3 289 years INSTANCE FROM PRACTICE ! ! ? NONLINEAR FUNCTION OF 2000 VARIABLES !!! CURSE OF DIMENSIONALITY SOLUTION METHODOLOGY. TIME OF CALCULATIONS/COST OF CALCULATION LAB INSTANCE 5..20 VARIABLES NP- HARDNESS

5. SOLUTION METHODOLOGY. CURRENT STATE IN DISCRETE OPTIMIZATION Theory of NP-completness Polynomial-time algorithms Exact methods (B&B, DP, ILP, BLP, MILP, SUB,…) Packages and solvers (LINDO, CPLEX, ILOG, …) Approximate methods (…): heuristics, metaheuristics, meta 2 heuristics Quality measures of approximation (absolute, relative, …) Analysis of quality measures (worst-case, probabilistic, experimental) Calculation cost (pessimistic, average, experimentally tested) Approximation schemes (AS, polynomial-time PTAS, fully polynomial-time FPTAS) Competitive analysis (no-line algorithms) Inapproximality theory Useful experimental methods (…) „No free lunch” theorem Public benchmarks Parallel and distributed methods: new class of algorithms Simulation

6. SOLUTION METHODOLOGY. CURRENT STATE IN DISCRETE OPTIMIZATION

7. constructive/improvement priority rules random search greedy randomized adaptive simulated annealing simulated jumping estimation of distribution tabu search adaptive memory search variable neighborhhod search evolutionary, genetic search differential evolution biochemistry methods immunological methods ant colony optimization particle swarm optimization neural networks threshold accepting bee search path search beam search scatter search harmony search path relinging adaptive search constraint satisfaction descending, hill climbing multi-agent memetic search intelligent wather drops harmony search electromagnetic search * * * * * METHODS RESISTANT TO LOCAL EXTREMES SOLUTION METHODOLOGY. APPROXIMATE METHODS

8. RADIO NETWORK DESIGN (RND) PROBLEM. CLASSICAL MATHEMATICAL MODEL x xx x x x CELL MODEL k n m

9. RADIO NETWORK DESIGN (RND) PROBLEM. CLASSICAL MATHEMATICAL MODEL PROBLEM DATA SOLUTION CONSTRAINTS GOAL FUNCTION Percentage of covered region,  =2

10. RADIO NETWORK DESIGN (RND) PROBLEM. CLASSICAL MATHEMATICAL MODEL cont. MULTIPLE CRITERIA CASE NP-hard problems Balance between criteria Scalarising Pareto set, Pareto frontier Approximate algorithms Approximation of Pareto frontier

11. MORE REALISTIC RND PROBLEMS. MAP TOPOLOGY

12. MORE REALISTIC RND PROBLEMS. CELL MODEL PiPi PiPi PiPi PiPi Ci(Pi)Ci(Pi)Ci(Pi)Ci(Pi)Ci(Pi)Ci(Pi)Ri(Pi)Ri(Pi)

13. MORE REALISTIC RND PROBLEMS. COVERAGE SOLUTION; ANTENNA LOCATED IN POINTS FROM K; POWERS ARE P i CHECKING POINT (p i, q i )

14. THE OPTIMIZATION PROBLEM UNDER CONSTRAINTS GOAL FUNCTION VALUE

15. SOLUTION METHODS. DECOMPOSITION: LOWER LEVEL UNDER CONSTRAINTS GOAL FUNCTION VALUE

16. SOLUTION METHODS. DECOMPOSITION: MID D LE LEVEL UNDER CONSTRAINTS GOAL FUNCTION VALUE

17. SOLUTION METHODS. DECOMPOSITION: UPPER LEVEL GOAL FUNCTION VALUE

18. SOLUTION METHODS LOWER LEVEL: EXACT SOLUTION MIDDLE LEVEL: KNAPSACK (APPROXIMATION) UPPER LEVEL: SIMULATED ANNEALING, AUTOTUNNIG VERSION WITH BOLTZMAN COOLING SCHEME AND SOME STEPS IN FIXED TEMPERATURE; SPECIFIC NEIGHBORHOOD BASED ON LOCAL VICINITY OF THE LOCATION POINT

19. COMPUTER EXPERIMENTS

20. CONCLUSIONS AND FURTHER RESEARCH the algorithm offers more realistic model of RND problem the model is smaller size and scalable new constraints can be embedded in the model model can be extended to multicriteria case further research are needed for evaluating the quality of the proposed methods on broader test of instances approximate solutions should be compared to exact solutions (CPLEX package) to evaluate their quality

21. Thank you for your attention LOCATION AND PLACEMENT PROBLEMS IN INFORMATION AND COMMUNICATION SYSTEMS Czesław Smutnicki