200 8 ACADEMIC TOUR : University of Paderborn , Germany

1. 2008 ACADEMICTOUR: University of Paderborn, Germany ADVANCED METAHEURISTICS Prof. Jacek ZAK Poznan University of Technology Poznan University of Technology, Poland

2. ADVANCED METAHEURISTICSCONTENTS • INTRODUCTION TO METAHEURISTICS • BASIC NOTIONS, CONCEPTS AND FEATURES • REVIEW: LS, SA, GA, TS • SOLVING MULTIPLE OBJECTIVE OPTIMIZATION PROBLEMS • SPECIALIZED SINGLE OBJECTIVE METAHEURISTICS • ANT COLONIES (SWARM – BASED METAHEURISTIC) • SPECIALIZED METAHEURISTICS FOR VEHICLE ROUTING PROBLEM • CASE STUDIES, COMPUTATIONAL RESULTS • COMMERCIAL SOFWARE (EVOLVER – GA; OPTQUEST – TS) • PRESENTATION AND APPLICATION • MULTIPLE OBJECTIVE METAHEURISTICS • PARETO SIMULATED ANNEALING (  Crew Assigmnent + Scheduling) • MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH • HYBRID GENETIC ALGORITHM • SIGNLE OBJECTIVE GLS • MULTIPLE OBJECTIVE GLS • PARETO MEMETIC ALGORITHM • CASE STUDIES, COMPUTATIONAL RESULTS • CONLUSIONS Vehicle Assignment Problem Slide 2

3. ADVANCED METAHEURISTICS INTRODUCTION TO METAHEURISTICS

4. ADVANCED METAHEURISTICS Motivation & Need for Metaheuristics • Growing complexity of the real life problems; mathematical sophistication of their discription; • Many real life problems are NP-complete problems (Traveling Salesman Problem, Set Covering Problem) • Computational time increases exponentially with the increase of the size of instances; • Non-linear; non-proportional increase; NP – non polynomial computational time • Real need for efficient methods/ algorithms that would be able to solve NP-complete problems • The algorithm is efficient if the cost (measured by the time of its development and the size of used memory) of its application does not grow too fast with the growing size of the problem. Slide 4

5. ADVANCED METAHEURISTICS Computational Complexity • Computational complexity – theory of „how difficult” is to answer a decision problem DP, where a DP is a question that has either a „yes” or „no” answer • The difficulty is measured by the number of operations an algorithm needs to perform to find the correct answer to the DP in the worst case • A decision problem belongs to the class P of problems if there exists a deterministic algorithm that answers the decision problem and needs O(p(n)) operations; p is a polynomial in n; n is the size of the instance Slide 5

6. ADVANCED METAHEURISTICS Computational Complexity • A decision problem belongs to the class NP if there is a nondeterministic polynomial time algorithm that solves the decision problem • For optimization problems it is possible to check whether x belongs to X and f(x) < b; b is a constant in polynomial time • A decision problem DPis NP – complete if DP belongs to NP and DP’ transforms into DP in a polynomial time for all DP’ that belong to NP. • NP-Complete is a subset of NP Slide 6

7. ADVANCED METAHEURISTICSCOMPLETE PROBLEMS (NP – C) Formal definiton of NP-C problem A decision problem DP is NP- complete if: • DP is in NP 2. Each problem in NP is reducable/ transformable to DP (in a polynomial time) Slide 7

8. ADVANCED METAHEURISTICSTHE VIENNE DIAGRAM • The Vienne Diagram of complexity clases shows that P is not equal NP. • It shows also existance of problems outside P and NP-C. NP NP-C P Slide 8

9. Well konwn NP-complete problems ADVANCED METAHEURISTICSCOMPLETE PROBLEMS (NP – C) NP-C Hamiltonian path problem Vertex cover problem Traveling salesman problem Knapsack problem Graph coloring problem Boolean satisfiability problem Slide 9

10. ADVANCED METAHEURISTICSCOMPLETE PROBLEMS (NP – C) How to solve NP-Complete Problems? • Approximation • Randomization • Parametrisation • Restricion (in the sensce of restriction for input) • Heuristics Slide 10

11. ADVANCED METAHEURISTICSCOMPLETE PROBLEMS (NP – C) Approximation Instead of searching for an optimal solution, search for an "almost" optimal one. Many approximation algorithms emerge from the linear programming relaxation of the integer program It applies only to optimization problems and not to "pure" decision problems like satisfiability (although it's often possible to conceive optimization versions of such problems, such as the maximum satisfiability problem). Slide 11

12. ADVANCED METAHEURISTICSCOMPLETE PROBLEMS (NP – C) Randomization Randomized algorithm=probabilistic algorithm • The algorithm typically uses the random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" • Use randomness to get a faster average running time, and allow the algorithm to fail with some small probability. Slide 12

13. ADVANCED METAHEURISTICSCOMPLETE PROBLEMS (NP – C) Restricion (in the sensce of restriction for input) By restricting the structure of the input (e.g.to graphs), faster algorithms are usually possible. Parametrisation • The theory of parameterized complexity was developed in the 1990s by Rod Downey and Michael Fellows • Often there are fast algorithms if certain parameters of the input are fixed Slide 13

14. ADVANCED METAHEURISTICSHEURISTICS • Heuresis (gr.) = dicovering – the way of organizing the learning process based on self-dependent search; discovering the truth and solving the problem; Heuristic – the way of learning without a well organized hypothesis; „Trial-by error” • Heurisko; Heuriskein (gr.) = find, discover, finding (they find) – the art of discussing focused on discovering the truth, new facts and relationships • Heuristic – practical, experience – based, „intelligent” rule of conduct and behavior • Heuristic (algorithmic meaning) – „not fully valuable” procedure that allows to find a „sufficiently good”, approximate solution in the acceptable / reasonable time • Resigning from obtaining an optimal solution; trade – offs analysis; searching for a satisfactory, high quality solution Slide 14

15. ADVANCED METAHEURISTICSHEURISTICS • Heuristic algorithms should to be efficient to generate „reasonalbe” solutions in a „resonable time” • Heuristics are typically used to solve complex (large, nonlinear, nonconvex - containing many local minima) multivariate combinatorial optimization problems that are difficult to solve to optimality. • Heuristics are good at dealing with local optima without getting stuck in them while searching for the global optimum. Slide 15

16. ADVANCED METAHEURISTICSMETAEURISTICS Greek: meta = megas = large, great, huge, universal • METAHEURISTICS – Heuristic procedures; Provide general schemes for solving similar categories of problems; need customization METAHEURISTICS = mega algorithms = universal algorithms that help us to solve independently in the approximate way a certain decision problem Slide 16

17. ADVANCED METAHEURISTICSMETAEURISTICS – general idea • The goal of optimization is to find a discrete solution (vector of bits, array or another structure) • The solution optimizes (maximizes or minimizes) a function created by the user (goal function) • Solutions are called states and the whole set of all states (candidate solutions) is called search space • The nature of search space and states are different for particular problems • Metaheuristics are very often based on probabilistic procedures Slide 17

18. ADVANCED METAHEURISTICSMETAEURISTICS - inspirations Genetics Metalurgy Behaviour of animals Slide 18

19. ADVANCED METAHEURISTICSTYPICAL IDEAS OF METAHEURISTICS • Metaheuristics are based often on probabilistic procedures • Neihgbourhood relation • User specifies time budget (number of iterations or time bounds) Slide 19

20. ADVANCED METAHEURISTICSMETAEURISTICS - CLASSIFICATION METAHEURISTICS CLASSIC HEURISTICS LOCAL SEARCH BASED POPULATION BASED LS SA GA TS Slide 20

21. ADVANCED METAHEURISTICSMETAEURISTICS - history 1965: I. Rechenberg - Evolution strategies 1991: M. Dorigo –Ant Colonies Algorithms 1975: J. Holland – Genetic Algorithms 1983: W.K. Hastings, S. Kirkpatrick, C.D.Gelatt and M.P. Vecchi – Simulated Annealing 1986: F. Glover – Tabu Search (first mentioned the term meta-heuristic) Slide 21

22. ADVANCED METAHEURISTICSMETAEURISTICS - examples Local search Genetic algorithm Hill-climbing Tabu search Memetic algorithm Slide 22

23. ADVANCED METAHEURISTICS REVIEW OF METAHEURISTICS Slide 23

24. ADVANCED METAHEURISTICSLOCAL SEARCH LOCAL SEARCH One of the simplest and most popular metaheuristics; often used as a basic algorithm (component) of more advanced procedures Metaheuristic usually applied for solving hard optimization problems General idea is moving from one solution to another in the space of candidate solutions, only in the neighbourhood of a current solution Slide 24

25. ADVANCED METAHEURISTICSLOCAL SEARCH LOCAL SEARCH • Major features of LS • Iterative modification of the current solution a • Defining the rule for generating the neighborhoodV(a)of the current solution a, which is a set of solutions similar to a (LS is based on the neighbourhood relation) • In each iteration one solution bfrom the neighborhood of the current solution a is selected – usually bgives a better value of the objective function) • Solution bbecomes a new current solution and a new neighborhood V(b)is generated • The cycle is repeated until the stop condition is reached Slide 25

26. ADVANCED METAHEURISTICSLOCAL SEARCH LOCAL SEARCH • Termination conditions: • When the best solution is found • Predefined time bound or number of steps • Impossibility of improving the solution for a given number of steps • The family of LS is wide, e.g. „Hill climbing” algorithm • Major versions • Greedy – finishes search when any solution giving the improvement of the objective function is found; next the neighborhood of a new solution is analyzed • Steepest descent – systematically reviews the neighborhood and selects the solution that gives the largest inprovement of the objective function Slide 26

27. ADVANCED METAHEURISTICSLOCAL SEARCH PSEUDOCODE N:=number of repetitions s:=0; fori=1 to Ndo s:=initial solution; while there is a neigbor of s with better quality do s:=one arbitrary neighbor of s with better quality; end while if s is better than sthen s:=s; end if end for return s; ~ ~ ~ ~ Slide 27

28. ADVANCED METAHEURISTICSLOCAL SEARCH HILL CLIMBING „Like climbing Everest in thick fog with amnesia” At each step, move to a neighbor of higher valuein hopes of getting to an optimal solution (highest possible value) Can easily modify this for problems where optimal means least possible value Slide 28

29. ADVANCED METAHEURISTICSTABU SEARCH TABU SEARCH • Local (neighborhood) search based metaheuristic; proved to be efficient and flexible optimization technique; Some of the first TS algorithms did not yield impressive results, but subsequent implementations were much more successful (20 years of experience) • The idea is using memory structures to remember potential solutions to avoid cycling • To improve efficiency of the exploration process one needs to keep track not only of local information (current value of the objective function) but also information on the exploration process • Inspiration - sociology Slide 29

30. ADVANCED METAHEURISTICSTABU SEARCH INSPIRATION – TABU = TABOO • TABU is a strong social prohibition against words, objects or actions, that are considered undesirable or offensive by a group, society or community • Breaking TABU is usually considered objectionable • Word TABU comes from Fijan and means „forbidden” or „not allowed” • Examples of TABU: • Gestures; subjects • Drags • Religion Slide 30

31. ADVANCED METAHEURISTICSTABU SEARCH In Tabu Search, sequences of solutions are examinedand the next move is made to the best neighbor of the current solutiona; non – improving moves are acceptable (escaping from local minima) To avoid cycling, solutions that were recently examined are forbidden, or tabu, for a number of iterations; the use of memory is helpful to forbid moves which might lead to currently visited solutions The structure of the neighborhood V(a)depends upon the itinerary and hence iterationk --- V (a, k) To alleviate time and memory requirements, it is customary to record an attribute of tabu solutions rather than the solutions themselves. Slide 31

32. ADVANCED METAHEURISTICSTABU SEARCH TABU list: X2 X3 X2 X1 X3 X4 Slide 32

33. ADVANCED METAHEURISTICSTABU SEARCH • Instead ofrecording solutions (impractical)– Tabu list of T solutions we keep track of the last T moves • For efficiency purposes several list Tr can be used at a time; constituents are given a tabu status • Relaxation of the tabu status – aspiration level conditions • Short and Long Term Memory – changing the goal function; intensification and diversification • Intensification – giving high priority to the solutions which have common features with the current solution • Diversification – spreading the exploration over different regions of the solution space Slide 33

34. ADVANCED METAHEURISTICSTABU SEARCH PSEUDOCODE Slide 34

35. ADVANCED METAHEURISTICSSIMULATED ANNEALING SIMULATED ANNEALING • Statistical Mechanics: • The behavior of systems with many degrees of freedom inthermal equilibrium at a finite temperature. • Combinatorial Optimization: • Finding the minimum of a given function depending on many variables. • Analogy: • If a liquid material cools and anneals too quickly, then the material will solidify into a sub-optimal configuration. If the liquid material cools slowly, the crystals within the material will solidify optimally into a state of minimum energy (i.e. ground state). • This ground state corresponds to the minimum of the cost function in an optimization problem. Slide 35

36. ADVANCED METAHEURISTICSSIMULATED ANNEALING Fast cooling scheme Slow cooling scheme Example illustrating the effect of cooling scheme on the structure of the material (cristalic structure of the metal) Slide 36

37. ADVANCED METAHEURISTICSSIMULATED ANNEALING TERMINOLOGY • X (or R orG) = Design Vector (i.e. Design, Architecture, Configuration) • E = System Energy (i.e. Objective Function Value) • T = System Temperature • D = Difference in System Energy Between Two Design Vectors Slide 37

38. ADVANCED METAHEURISTICSSIMULATED ANNEALING THE SIMULATED ANNEALING ALGORITHM • Choose a random Xi, select the initial system temperature, and specify thecooling (i.e. annealing) scheme • Evaluate E(Xi) using a simulation model • PerturbXito obtain a neighboring Design Vector (Xi+1) • Evaluate E(Xi+1) using a simulation model • If E(Xi+1)< E(Xi), Xi+1is the new current solution • If E(Xi+1)> E(Xi), then accept Xi+1as the new current solution with aprobability e(-D/T) where D = E(Xi+1) -E(Xi). • Reduce the system temperature according to the cooling scheme • Terminate the algorithm. Slide 38

39. ADVANCED METAHEURISTICSSIMULATED ANNEALING Scheme of SA Slide 39

40. ADVANCED METAHEURISTICSSIMULATED ANNEALING Comparision of SA and LS value value Local minimum Global minimum Local minimum Global minimum solution solution Local and global extremes in SA Local and global extremes in LS In SA algorithm not only the best solutions are evaluated, so the algorithm may escape from local minimum region Slide 40

41. ADVANCED METAHEURISTICSSIMULATED ANNEALING Pseudocode Slide 41

42. ADVANCED METAHEURISTICSGENETIC ALGORITHM GENETIC ALGORITHMS • PARTICULAR CLASS OF EVOLUTION - BASED ALGORITHMS • ALGORITHM INSPIARED BY EVOLUTIONERY BIOLOGY • TYPICAL FOR GA ARE: • Crossover • Selection • Population • Chromosome • Goal Function called Fitness Function Slide 42

43. ADVANCED METAHEURISTICSGENETIC ALGORITHM INSPIRATION IN EVOLUTION POPULAION IN ENVIRONMENT SURVIVING POPULAION AFTER SOME TIME: REPRODUCTION POPULAION INDIVIDUALS STONES PREDATORS Slide 43

44. ADVANCED METAHEURISTICSGENETIC ALGORITHM SCHEME OF THE GENETIC ALGORITHMS Initialization New population Evolution Reproduction YES Continue? NO Final polulation Slide 44

45. ADVANCED METAHEURISTICSGENETIC ALGORITHM TYPICAL GENETIC ALGORITHM Genetic algorithm Genetic representation of the solution domain Fitness function A typical representation of the solution is a vector or an array of bits (but also of integers). The fitness function measures the quality of the solution and depends always on the problem. Slide 45

46. ADVANCED METAHEURISTICSGENETIC ALGORITHM REPRESENTATION The representation of solution is called chromosome Chromosome can be a vector or array of bools, another data type or a tree data structure Represetation has huge influence on efficiency of algorithm Slide 46

47. ADVANCED METAHEURISTICSGENETIC ALGORITHM REPRESENTATION Chromosome as a vector of bits Chromosome as an array of bits Crossing-over Slide 47

48. ADVANCED METAHEURISTICSGENETIC ALGORITHM FITNESS FUNCTION Every chromosome is ranked by fitness function Best chromosomes are allowed to crossover and produce a new generation Fitness function should be very fast because of many iterations of the algorithm The main problem is to create a proper fitness function Slide 48

49. ADVANCED METAHEURISTICSGENETIC ALGORITHM FITNESS FUNCTION Every chromosome is ranked by fitness function Best chromosomes are allowed to crossover and produce a new generation Fitness function should be easily computed because of many iterations of the algorithm The main problem is to create a proper fitness function Slide 49

50. ADVANCED METAHEURISTICSGENETIC ALGORITHM GENETIC PROCES Slide 50