1. Opracowanie językowe dr inż. J. Jarnicki
Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Location and Placement Problems in Information and Communication Systems

2. PRESENTATION OUTLINE 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

3. LOCATION AND PLACEMENT PROBLEMS
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 …

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

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, meta2heuristics
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. SOLUTION METHODOLOGY. APPROXIMATE METHODS
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
electromagnetic search
* * * * *
METHODS RESISTANT
TO LOCAL EXTREMES

8. RADIO NETWORK DESIGN (RND) PROBLEM. CLASSICAL MATHEMATICAL MODELx
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
Ri(Pi)
Ci(Pi)
Ci(Pi)
Ci(Pi) Pi Pi Pi Pi

13. MORE REALISTIC RND PROBLEMS. COVERAGE
CHECKING POINT (pi, qi)
SOLUTION; ANTENNA LOCATED IN POINTS FROM K;
POWERS ARE Pi

14. THE OPTIMIZATION PROBLEM
GOAL FUNCTION VALUE
UNDER CONSTRAINTS

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

16. SOLUTION METHODS. DECOMPOSITION: MIDDLE LEVEL
GOAL FUNCTION VALUE
UNDER CONSTRAINTS

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