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Tabu Search. INTRODUCTION. Background of TS. TS was first proposed by Glover (1986) and was also developed by Hansen (1986 ) TS has its roots in methods that cross boundaries of feasibility and local optimality
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Background of TS • TS was first proposed by Glover (1986) and was also developed by Hansen (1986) • TS has its roots in methods that cross boundaries of feasibility and local optimality • Examples of such methods include use of surrogate constraints and cutting plane approaches • Tabu search (TS) is a neighborhood search method which employs "intelligent" search and flexible memory technique to avoid being trapped at local optimum. • Tabu Search (TS) is a metaheuristic that guides a local heuristic search procedure to explore the solution space beyond local optimality
Basic notions of TS • The word tabu (or taboo) comes from Tongan, a language of Polynesia, where it indicates things that cannot be touched because they are scared • Now it also means “a prohibition imposed by social custom” • In TS, tabu status of forbidden elements shift according to time and circumstance, based on an evolving memory
Basic notions of TS (cont.) • Tabu status can be overruled for a preferrable alternative • Hence TS uses adaptive (flexible) memory • TS also uses responsive exploration, i.e. exploitation of good solutions and exploration of new promising regions
PRINCIPAL TABU SEARCH FEATURES • Tabu search is based on the premise that problem solving, in order to qualify as intelligent, must incorporate adaptive memory and responsive exploration. The use of adaptive memory contrasts with "memoryless" designs, such as those inspired by metaphors of physics and biology, and with "rigid memory" designs, such as those exemplified by branch and bound and its AI-related cousins. • The emphasis on responsive exploration (and hence purpose) in tabu search, whether in a deterministic or probabilistic implementation, derives from the supposition that a bad strategic choice can yield more information than a good random choice.
Main Features • TS emulates the human problem solving process. • It takes advantage of search history. • The historical record is usually maintained for the characteristics of the moves applied, rather than the solutions visited. • Recent moved are classified as tabus to restrict the search space. • TS is a variable neighborhood method. • Tabu restrictions are not inviolable under all circumstances. • Several types of memories are used, both short term and long term, in order to improve the exploration quality.
PRINCIPAL TABU SEARCH FEATURES (Contd…) Adaptive Memory • Selectivity (including strategic forgetting) • Abstraction and decomposition (through explicit and attributive memory) • Timing: • recency of events • frequency of events • differentiation between short term and long term • Quality and impact: • relative attractiveness of alternative choices • magnitude of changes in structure or constraining • relationships • Context: • regional interdependence • structural interdependence • sequential interdependence
PRINCIPAL TABU SEARCH FEATURES (Contd…) • Responsive Exploration • Strategically imposed restraints and inducements • (tabu conditions and aspiration levels) • Concentrated focus on good regions and good solution features • (intensification processes) • Characterizing and exploring promising new regions • (diversification processes) • Non-montonic search patterns • (strategic oscillation) • Integrating and extending solutions • (path relinking)
Parameters of Tabu Search: • Local search procedure • Neighborhood structure • Aspiration conditions • Form of tabu moves • Addition of a tabu move • Maximum size of tabu list • Stopping rule
Basic Ingredients of Tabu Search • A chief way to exploit memory in tabu search is to classify a subset of the moves in a neighborhood as forbidden (or tabu). • A neighborhood is constructed to identify adjacent solutions that can be reached from current solution. • The classification depends on the history of the search, and particularly on the recency or frequency that certain move or solution components, called attributes, have participated in generating past solutions. • A tabu list records forbidden moves, which are referred to as tabu moves. • Tabu restrictions are subject to an important exception. When a tabu move has a sufficiently attractive evaluation where it would result in a solution better than any visited so far, then its tabu classification may be overridden. A condition that allows such an override to occur is called an aspiration criterion.
Four dimensions of TS memory • Recency based (short term) memory • Frequency based (long term) memory • Quality: ability to differenciate the merit of solutions • Use memory to identify elements common to good solutions • Reinforce (discourage) actions that lead to good (bad) solutions • Influence: impact of the choices made in search on both quality and structure of solutions
Use of memory in TS • Use of memory leads to learning • Memory in TS is explicit and attributive • Explicit:- Records complete solution… • Attribute: - For guiding purpose • Solution attributes (or elements or components) that change in moving from one solution to another are recorded for guiding the search • Attributes can be nodes or arcs repositioned in a graph or job indeces in scheduling • In addition to recency or frequency of solution attributes, elite solutions and their attractive neighbors are explicitely recorded
Intensification and diversification • Intensification: a form of exploitation • Based on modifying choice rules to encourage good move combinations and solution attributes • May lead to return to attractive regions • Examines neighbors of prerecorded elite solutions • Diversification: a form of exploration • Examines unvisited regions, generates different solutions
Applications of TS • Applications of Tabu Search are as follows: • 1. Scheduling • Flow Time Cell Manufacturing • Workforce Planning • 2. Design • Computer Aided Design • Diagram Coherency • Irregular cutting Problems • 3. Location and Allocation • Lay-out Planning • Multilevel Generalized Assignment • Multimodity Allocation • 4. Technology • Seismic Inversion • Electrical Power Distribution • Engineering Structural Design • 5. Graph Optimization • Graph Partitioning • Graph Coloring
Basic Tabu Search Algorithm • Step 1: Choose an initial solution i in S. Set i* = i and k=0. • Step 2: Set k=k+1 and generate a subset V* of solution in N(i,k) such that either one of the Tabu conditions is violated or at least one of the aspiration conditions holds. • Step 3: Choose a best j in V* and set i=j. • Step 4: If f(i) < f(i*) then set i* = i. • Step 5: Update Tabu and aspiration conditions. • Step 6: If a stopping condition is met then stop. Else go to Step 2.
Tabu Search Stopping Conditions Some immediate stopping conditions could be the following : • N(i, K+1) = 0. (no feasible solution in the neighborhood of solution i) • K is larger than the maximum number of iterations allowed. • The number of iterations since the last improvement of i* is larger than a specified number. • Evidence can be given than an optimum solution has been obtained. Hillier and Lieberman outlined the tabu search stopping criterion by, for example, using a fixed number of iterations, a fixed amount of CPU time, or a fixed number of consecutive iterations without an improvement in the best objective function value. Also stop at any iteration where there are no feasible moves into the local neighborhood of the current trial solution.
Flowchart of a Standard Tabu Search Algorithm Initial solution (i in S) Create a candidate list of solutions Evaluate solutions Stopping conditions satisfied ? Update Tabu & Aspiration Conditions No Choose the best admissible solution Yes Final solution
A E A B D C E C D B Example • Minimum spanning tree problem with constraints. • Objective: Connects all nodes with minimum costs Costs 20 30 20 30 10 10 5 5 25 25 15 40 15 40 An optimal solution without considering constraints Constraints 1: Link AD can be included only if link DE also is included. (penalty:100) Constraints 2: At most one of the three links – AD, CD, and AB – can be included. (Penalty of 100 if selected two of the three, 200 if all three are selected.)
A B D C E Example Iteration 1 Cost=50+200 (constraint penalties) New cost = 75 (iteration 2) ( local optimum) 20 30 10 5 25 15 40 Delete Add Constraints 1: Link AD can be included only if link DE also is included. (penalty:100) Constraints 2: At most one of the three links – AD, CD, and AB – can be included. (Penalty of 100 if selected two of the three, 200 if all three are selected.)
A B D C E Example Tabu list: DE Iteration 2 Cost=75 * A tabu move will be considered only if it would result in a better solution than the best trial solution found previously (Aspiration Condition) Iteration 3 new cost = 85 Escape local optimum Delete Add 20 30 10 5 25 Tabu 15 40 Constraints 1: Link AD can be included only if link DE also is included. (penalty:100) Constraints 2: At most one of the three links – AD, CD, and AB – can be included. (Penalty of 100 if selected two of the three, 200 if all three are selected.)
A B D C E Example Tabu list: DE & BE Iteration 3 Cost=85 * A tabu move will be considered only if it would result in a better solution than the best trial solution found previously (Aspiration Condition) Iteration 4 new cost = 70 Override tabu status Tabu 20 30 10 5 Add 25 Tabu 15 40 Delete Constraints 1: Link AD can be included only if link DE also is included. (penalty:100) Constraints 2: At most one of the three links – AD, CD, and AB – can be included. (Penalty of 100 if selected two of the three, 200 if all three are selected.)
Optimal Solution Cost = 70 Additional iterations only find inferior solutions A B D C E Example 20 30 10 5 25 15 40
Travelling salesman problem using tabu search
Travelling salesman problem A Classical Traveling Salesman Problem (TSP) can be defined as a problem where starting from a node it is required to visit every other node only once in a way that the total distance covered is minimized. City 2 City 1 City 3 City 5 City 4 Distance Matrix
Tabu Search for TSP 1. Solution Representation: A feasible solution is represented as a sequence of nodes, each node appearing only once and in the order it is visited. The first and the last visited nodes are fixed to 1. The starting node is not specified in the solution representation and is always understood to be node 1. 2. Initial Solution: A good feasible, yet not-optimal, solution to the TSP can be found quickly using a greedy approach. Starting with the first node in the tour, find the nearest node. Each time find the nearest unvisited node from the current node until all the nodes are visited.
3. Neighborhood: A neighborhood to a given solution is defined as any other solution that is obtained by a pair wise exchange of any two nodes in the solution. This always guarantees that any neighborhood to a feasible solution is always a feasible solution (i.e., does not form any sub-tour). If we fix node 1 as the start and the end node, for a problem of N nodes, there are such neighborhoods to a given solution. At each iteration, the neighborhood with the best objective value (minimum distance) is selected. 4. Neighborhood solution: Obtained by swapping the order of visit of cities 5 and 3.
5. Tabu List: To prevent the process from cycling in a small set of solutions, some attribute of recently visited solutions is stored in a Tabu List, which prevents their occurrence for a limited period. For our problem, the attribute used is a pair of nodes that have been exchanged recently. A Tabu structure stores the number of iterations for which a given pair of nodes is prohibited from exchange as illustrated in figure. 6. Aspiration Criterion: Tabus may sometimes be too powerful: they may prohibit attractive moves, even when there is no danger of cycling, or they may lead to an overall stagnation of the searching process. It may, therefore, become necessary to revoke tabus at times. The criterion used for this to happen in the present problem of TSP is to allow a move, even if it is tabu, if it results in a solution with an objective value better than that of the current best-known solution.
7. Diversification: Quite often, the process may get trapped in a space of local optimum. To allow the process to search other parts of the solution space (to look for the global optimum), it is required to diversify the search process, driving it into new regions. This is implemented in the current problem using “frequency based memory”. The use of frequency information is used to penalize non-improving moves by assigning a larger penalty to swaps with greater frequency counts. This diversifying influence is allowed to operate only on occasions when no improving moves exist. Additionally, if there is no improvement in the solution for a pre-defined number of iterations, frequency information can be used for a pair wise exchange of nodes that have been explored for the least number of times in the search space, thus driving the search process to areas that are largely unexplored so far. 8. Termination Criteria: The algorithm terminates if a pre-specified number of iterations is reached.
Results and discussion: The algorithms for Tabu Search consisting of the strategies described are implemented using MATLAB. The results obtained for the implementation of Tabu Search for TSP provide solution for the small sized 5-city problem tested. Further improvements can be obtained by fine tuning the various parameters & by better implementation of diversification and intensification in Tabu Search. City 2 City 1 City 3 Distance Covered = 9 units City 5 City 4
Some Conclusions • One basic element of the tabu search technique is based on the interplay between tabu restrictions and aspiration criteria. • Tabu restrictions are used to prevent cyclic behavior. • The basic idea of aspiration criteria is to allow improving moves to escape a tabu classification under certain conditions. These moves are usually treated as inferior improving moves. • While tabus are defined by properties of the moves, aspiration criteria are usually based on the outcome of the moves. • Another interaction which plays an important role in TS is that between diversification and intensification.
Pros & Cons of TS • Pros: • Allows non-improving solution to be accepted in order to escape from a local optimum • The use of Tabu list • Can be applied to both discrete and continuous solution spaces • For larger and more difficult problems (scheduling, quadratic assignment and vehicle routing), tabu search obtains solutions that rival and often surpass the best solutions previously found by other approaches [1]. • Cons: • Too many parameters to be determined • Number of iterations could be very large • Global optimum may not be found, depends on parameter settings
Main Weakness of TS • No theory has yet been formulated to support TS and its convergence behavior. • Good understanding of the problem structure isrequired. Domain specific knowledge is needed for selection for tabus and aspiration criteria. • Experiment might have to be done using different tabu classification schemes and aspiration criteria. • It needs considerable memory resources. • Efficient data structure must be used for tabu list manipulation. • TS is still a relatively young technique and has many unexplored issues.
REFERENCES Glover, F. (1977), “Heuristics for Integer Programming Using Surrogate Constraints”, Decision Sciences 8, 156-166. Glover, F. (1986), “Future Paths for Integer Programming and Links to Artificial Intelligence”, Computers and Operations Research 13, 533-549. Glover, F. (1989), “Tabu Search – Part I”, ORSA Journal on Computing 1, 190-206. Glover, F. (1990), “Tabu Search – Part II”, ORSA Journal on Computing 2, 4-32. Glover, F. (1992), “Ejection chains, Reference Structures and Alternating Path Methods for Traveling Salesman Problems”, University of Colorado. Shortened version published in Discrete AppliedMathematics 65, 223-253, 1996. Glover, F. and M. Laguna (1993), “Tabu Search”, in C.R Reeves (ed.), in Modern Heuristic Techniques for Combinatorial Problems, C.R. Reeves (ed.), Blackwell, pp. 70-150. Glover, F. and M. Laguna (1997), Tabu Search, Kluwer Academic Publishers, Norwell, MA. Glover, F., M. Laguna, E. Taillard and D. de Werra (eds.) (1993), “Tabu Search”, Annals of Operations Research 41, J.C. Baltzer Science Publishers, Basel, Switzerland.
