Internet EngineeringInternet Engineering

Czesław SmutnickiCzesław Smutnicki

Discrete Mathematics Discrete Mathematics – – Location Location aand nd Placement Problems Placement Problems iin Information n Information aand nd Communication SystemsCommunication Systems

•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

•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

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

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

SOLUTION METHODOLOGY. CURRENT STATE IN DISCRETE OPTIMIZATION

• 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

RADIO NETWORK DESIGN (RND) PROBLEM.CLASSICAL MATHEMATICAL MODEL

x x x

x x x

CELL MODEL

k

n

m

RADIO NETWORK DESIGN (RND) PROBLEM. CLASSICAL MATHEMATICAL MODEL

PROBLEM DATA

SOLUTION

CONSTRAINTS

GOAL FUNCTION

Percentage of covered region, =2

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

MORE REALISTIC RND PROBLEMS. MAP TOPOLOGY

MORE REALISTIC RND PROBLEMS. CELL MODEL

Pi Pi Pi Pi

Ci(Pi)Ci(Pi)Ci(Pi)Ri(Pi)

MORE REALISTIC RND PROBLEMS. COVERAGE

SOLUTION; ANTENNA LOCATED IN POINTS FROM K; POWERS ARE Pi

CHECKING POINT (pi, qi)

THE OPTIMIZATION PROBLEM

UNDER CONSTRAINTS

GOAL FUNCTION VALUE

SOLUTION METHODS. DECOMPOSITION: LOWER LEVEL

UNDER CONSTRAINTS

GOAL FUNCTION VALUE

SOLUTION METHODS. DECOMPOSITION: MIDDLE LEVEL

UNDER CONSTRAINTS

GOAL FUNCTION VALUE

SOLUTION METHODS. DECOMPOSITION: UPPER LEVEL

GOAL FUNCTION VALUE

SOLUTION METHODS

• LOWER LEVEL: EXACT SOLUTION

• MIDDLE LEVEL: KNAPSACK (APPROXIMATION)

• UPPER LEVEL: SIMULATED ANNEALING, AUTOTUNNIG VERSIONWITH BOLTZMAN COOLING SCHEME AND SOME STEPS IN FIXED TEMPERATURE; SPECIFIC NEIGHBORHOOD BASED ON LOCAL

VICINITY OF THE LOCATION POINT

COMPUTER EXPERIMENTS

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

Thank you for your attention

LOCATION AND PLACEMENT PROBLEMS IN INFORMATION AND COMMUNICATION SYSTEMS

Czesław Smutnicki