Lecture 06 Tabu Search

Transcription

1 Lecture 06 Tabu Search s.l. dr. ing. Ciprian-Bogdan Chirila Heuristic Methods

2 Outline Introduction The TS framework Example Notation and problem description Neighborhood search TS characteristics TS Memory Recency based tabu memory functions Aspiration criteria Frequency based memory

3 Outline Broader aspect of intensification and diversification Diversification vs. randomization Reinforcement by restriction Extrapolated relinking Solutions evaluated but not visited Interval specific penalties and incentives Candidate list procedures Compound neighborhoods Creating new attributes Strategic oscillation TS Applications Connections and conclusions

4 Introduction Antecedents in methods designed to cross boundaries of feasibility or local optimality Barriers to impose and release constraints to permit exploration of forbidden regions Modern form of TS derives from Glover Seminal ideas developed also by Hansen Steepest ascent/mildest descent The word tabu or taboo Charged with dangerous supernatural power Forbidden to profane use Banned on grounds of morality or taste

5 Introduction TS concerned with imposing restrictions to guide a search process to negotiate difficult regions Restrictions operate in several forms Direct exclusion of certain search alternatives classified as forbidden Translation into modified evaluations and probabilities of solutions selection Goals To present fundamental ways of TS views To exemplify the operations To point out directions for new applications and research Comparisons with SA, GA, NN

6 TS philosophy To derive and exploit principles of intelligent problem solving To use flexible memory Creating and exploiting structures for taking advantage of the history Combining the activities of acquiring and profiting from the information Four principal dimensions Recency Frequency Quality influence

7 TS example Permutation problems TSP Quadratic assignment problem Production sequencing Design problems To design a material Consists in insulating modules The arrangement order determines the overall insulating property The problem To find the ordering of modules that maximizes the overall insulation of the composite material

8 TS example assumptions 7 modules for a particular material The evaluation of the overall insulation is computationally expensive requirements To find an optimal or near-optimal solution To examine only a small subset of the total number of possible permutations (5040)

9 Related problems serial filtering pattern recognition signal processing to find the best filtering sequence job sequencing best sequences for processing a set of jobs

10 Insulation problem Initial solution can be constructed in some way To use problem specific structure modules module 2 is on the first position module 5 is on the second position etc. the resulting insulating property is 10

11 Insulation problem Neighborhood can be constructed to identify adjacent solutions that can be reached from any solution Pairwise exchanges (swaps) Used in permutation problems Lead from one solution to the next Exchanges the positions of two modules Complete neighborhood solution => 21 adjacent solutions Move value = change in the objective function

12 Insulation problem To classify a set of moves in neighborhood to be forbidden or tabu Classification depends on the history of search Recency Frequency attributes certain move or solution components that participated in generating past solutions Swap of module 5 and 6

13 Insulation problem To prevent swaps in the recent past Could revert the effects of previous moves Might return to previous positions We classify the swaps as tabu E.g the 3 most recent pairs A module pair will be kept tabu for a duration (tenure) of 3 iterations Swap 2,5 is the same like swap 5,2 Both may be represented by (2,5)

14 Tabu data structure Remaining tabu tenure for module pair (2,5)

15 Tabu data structure If cell (3,5)=0 then Modules 3 and 5 are free to be exchanged If cell (2,4)=2 then Modules 2 and 4 can not exchange positions for the next 2 iterations Move attributes for tabu restrictions can be defined differently Reference may be made to separate modules rather than pair Positions of modules Links between predecessors and successors etc

16 Tabu restrictions Involve an important exception Are not inviolable under all circumstances When a tabu move leads to a better solution The tabu classification may be overridden It is called aspiration criterion Next example Basic tabu procedure with paired module tabu restriction Best solution aspiration criterion

17 Iteration 0 Current solution Tabu structure Top 5 candidates Swap Value ,4 6 * 2 7,4 4 Insulation Value = , ,3 0 All entries zero 5 4,1-1 6

18 Iteration 0 Initial insulation value is 10 Tabu data structure is filled with zeroes No moves are classified as tabu Top five moves are evaluated We use only the best To find local maximum we swap modules 5 and 4 Total gain is 6 units

19 Iteration 1 Current solution Tabu structure Top 5 candidates Swap Value ,1 2 * 2 2,3 1 Insulation Value = , , ,1-4 6

20 Iteration 1 New current solution has insulating value 16 Swapping positions 4 and 5 is forbidden for 3 iterations The best move is to swap 3 and 1 The gain is 2

21 Iteration 2 Current solution Tabu structure Top 5 candidates Swap Value ,3-2 T 2 2,4-4 * Insulation Value = , ,5-7 T 5 5,3-9 6

22 Iteration 2 Current solution has value of 18 Two swaps are tabu 4,5 decreased from 3 to 2 2 remaining iterations 1,3 has tenure of 3 no candidates have positive values Most attracting non-improving move Is a move performed in previous iteration Is not selected since it is classified as tabu Instead move 2,4 is chosen indicated by the star symbol

23 Iteration 3 Current solution Tabu structure Top 5 candidates Swap Value ,5 6 T* 2 3 5,3 2 Insulation Value = , ,3-3 T 5 2,6-6 6

24 Iteration 3 Current solution has inferior insulation value As result to a move with negative value 3 moves are tabu Different remaining tabu tenures The top move is the swap of 4,5 Is tabu But the objective function is superior We use aspiration criteria to override tabu classification We select this move

25 Iteration 4 Current solution Tabu structure Top 5 candidates Swap Value ,1 0 * 2 2 4,3-3 Insulation Value = , ,4-6 T 5 2,6-8 6

26 Iteration 4 The current solution is the new best one We have 3 out of 21 possible swaps Tenures of 1 drops to 0 When a tenure of 3 is introduced Sometimes Desirable to increase the percentage of available tabu moves Increasing the tabu tenure Changing the tabu restriction E.g. to imagine a restriction related to one member of a module will prevent larger number of moves to be executed 15/21

27 Complementary Tabu Memory Structures Recency based memories Frequency based memories We assume that 25 TS iterations were performed The number of exchanges is stored in a expanded tabu data structure Under the lower diagonal we have frequency counts

28 Iteration 26 Current solution Tabu structure (Recency) Top 5 candidates Swap Value Penalized value ,4 3 2 T 2 2, Insulation Value = , * , , (Frequency)

29 Iteration 26 Recency memory Indicates the last 3 swaps (1,4), (3,6), (4,7) Frequency counts The distribution of moves through the first 25 iterations Used to diversify the search To drive into new regions Operates only on particular occasions In this case When no admissible improving moves exist Will penalize non-improving moves Will assign larger penalty for swaps with greater frequency

30 Iteration 26 The most improving move is 1,4 Has tabu tenure 3 It is not taken Move 2,4 has value -1 The next preferred It was the most frequently used Heavily penalized Losses attractiveness Move 3,7 Is selected as the best move in the iteration

31 Bibliography Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest - Introduction to Algorithms, ISBN , The MIT Press, Z. Michaelwitz, D.B. Fogel - How to solve it: Modern Heuristics, ISBN , Springer Verlag, Berlin, Heidelberg, Colin R. Reeves - Modern Heuristic Techniques for Combinatorial Problems, Blackwell Scientific Publications, Oxford, 1993.