Scientific Research Group in Egypt (SRGE) Meta-heuristics techniques (I) Tabu search Scientific Research Group in Egypt (SRGE) Dr. Ahmed Fouad Ali Suez Canal University, Dept. of Computer Science, Faculty of Computers and informatics Member of the Scientific Research Group in Egypt

Scientific Research Group in Egypt www.egyptscience.net

Meta-heuristics techniques

Outline 1. Motivation 2. Tabu search (TS)(Background) 3. TS (main concepts) 4. TS algorithm 5. TS examples 6. TS applications

Motivation ? ? barrier to local search starting point descend direction local minima global minima

Tabu search (TS)(Background) Tabu search (TS) algorithm was proposed by Glover (1986). In the 1990s, the tabu search algorithm became very popular in solving optimization problems. Nowadays, it is one of the most wide spread (single ) S-metaheuristics. The use of memory represents the particular feature of tabu search. TS behaves like a steepest LS algorithm, but it accepts nonimproving solutions to escape from local optima.

Tabu search (main concepts) The key feature of TS method is the use of memory, which records information related of the search process. TS generates a neighborhood solution from the current solution and accepts the best solution even if is not improving the current solution. This strategy may lead to cycles i.e the previous visited solutions could be selected again.

Tabu search (main concepts) In order to avoid cycles, TS discards the solution that have been previously visited by using memory which is called tabu list. The length of the memory (tabu list) control the search process. A high length of the tabu list is high the search will explore larger regions and forbids revisiting high number of solution. A low length of the tabu list concentrates the search on a small area of the search space.

Tabu search (main concepts) At each iteration the tabu list is updated (first in – first out queue). The tabu list contains a constant number of tabu moves called tabu tenure, which is the length of time for which a move is forbidden. If a move is good and can improve the search process but it is in tabu list, there is no need to be prohibited and the solution is accepted in a process called aspiration criteria.

Tabu search algorithm

Tabu search examples N(x0) X0 Neighborhood trail solutions N(x1) x0 x1

1||SwjTj TS examples Jobs 1 2 3 4 wj 5 pj 12 8 15 9 dj 16 26 25 27 Determine Sc by the best schedule in the neighborhood that is not tabu Use tabu-list length = 2 The tabu list is denoted by L

Iteration 1 TS examples Step 1: S0=S1=(1,3,2,4). G(S1)=136. Set L={}. Step 2. N(S1)= {(3,1,2,4), (1,2,3,4), (1,3,4,2)} with respective cost = {174, 115, 141} => Sc=S0=S2=(1,2,3,4). Set L={(3,2)}, i.e., swapping 3 and 2 is not allowed (Tabu) Step 3: Let k=2

Iteration 2 TS examples Step 2. Step 3: Let k=3 X N(S2)= {(2,1,3,4), (1,3,2,4), (1,2,4,3)} with respective costs = {131, - , 67} => Sc=S3=(1,2,4,3) Set S0=Sc Set L={(3,4),(3,2)} Step 3: Let k=3 X

Iteration 3 TS examples Step 2 Step 3: Let k=4 X N(S3)= {(2,1,4,3), (1,4,2,3), (1,2,3,4)} with respective costs = {83, 72, -} => Sc=S4=(1,4,2,3) Set L={(2,4),(3,4)} Step 3: Let k=4 X

Iteration 4 TS examples Step 2 Step 3: Let k=5 X N(S4)= {(4,1,2,3), (1,2,4,3), (1,4,3,2)} with respective costs = {92, -, 123} => Sc=S5=(4,1,2,3) Set L={(1,4),(2,4)} Step 3: Let k=5 X

Iteration 5 TS examples Step 2 Step 3: Let k=6 X N(S5)= {(1,4,2,3), (4,2,1,3), (4,1,3,2)} with respective costs = {-, 109, 143} => Sc=S6=(4,2,1,3) Set L={(2,1),(4,1)} Step 3: Let k=6 X

TS Applications Scheduling Quadratic assignment Frequency assignment Car pooling Capacitated p-median, Resource constrained project scheduling (RCPSP) Vehicle routing problems Graph coloring Retrieval Layout Problem Maximum Clique Problem, Traveling Salesman Problems Database systems Nurse Rostering Problem Neural Nets Grammatical inference, Knapsack problems SAT Constraint Satisfaction Problems Network design Telecomunication Network Global Optimization

References Metaheuristics From design to implementation, El-Ghazali Talbi, University of Lille – CNRS – INRIA. F. Glover, Future paths for integer programming and links to artificial intelligence, Computers and Operations Research 13 (1986), 533-549.

Thank you Ahmed_fouad@ci.suez.edu.eg http://www.egyptscience.net