2. 2
History Of Tabu Search
• A very simple memory mechanism is described in
Glover (1977) to implement the oscillating
assignment heuristic
• Hansen . 1986. put The steepest ascent heuristic for
coalition programming , Congress on numerical
methods in coalition optimization .
• Glover (1986) introduces tabu search as a “meta-
heuristic” To impose on another heuristic and Glover
(1989) provide a full description of the method .
3. Introduction
3
The word tabu (or taboo) comes from Tongan, a
language of Polynesia, where it indicates things be
touched because they are sacred.
Now it also means “a prohibition imposed by social
custom”.
In TS, when finding the next solution to visit , some
solution elements (or moves) are regarded as tabu ,
they cannot be used in constructing the next solution
.
4. Overview of Tabu search
▪ Tabu search is based on introducing flexible
memory structures in conjunction with strategic
restrictions and aspiration levels as a means for
exploiting search spaces.
▪ Meta-heuristic that guides a local heuristic search
procedure to explore the solution space beyond
local optimum by use of a Tabu list.
▪ Originated from surrogate constraint methods and
cutting plane approaches .
4
5. Overview Of Tabu Search
▪ Cutting plane: A cutting plane (or cut) for any IP
is a new functional constraint that reduces the feasible
region for the relaxation without eliminating any
feasible solution for the IP problem.
▪ Used to solve fusion (finite solution set) optimization
problems
▪ A dynamic neighborhood search method use of a
flexible memory to restrict the next solution choice to
some subset of neighborhood of current solution
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6. Tabu Search Strategy
▪ There are three main strategies :
▪ Forbidding strategy: control what enters
to the tabu list.
▪ Freeing strategy: control what exits the
tabu list and when.
▪ Short-term strategy: manage interplay
between the forbidding strategy and
freeing strategy to select trial solutions.
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7. 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
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8. Basic components 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 specially 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 .
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9. 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.
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10. Tabu Search Stopping Conditions
Some immediate stopping conditions could be the
following :
1. N(i, K+1) = 0. (no feasible solution in the
neighborhood of solution i)
2. K is larger than the maximum number of iterations
allowed.
3. The number of iterations since the last improvement
of i* is larger than a specified number.
4. Evidence can be given than an optimum solution has
been obtained.
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11. Flowchart Of Tabu Search Algorithm
11
Initial solution
(i in S)
Create a candidate
list of solutions
Evaluate solutions
Choose the best
admissible solution
Stopping conditions
satisfied ?
Update Tabu &
Aspiration
Conditions
Final solution
No
Yes
12. Minimum Spanning Tree
▪ Given a connected , undirected graph , a spanning tree
of that graph is a subgraph which is a tree and connect
all the Crests together .
▪ A single graph can have many different spanning trees.
we can also assign a weight to each edge, which is a
number representing how unfavorable it is, and use this
to assign a weight to a spanning tree by computing the
sum of the weights of the edges in that spanning tree. A
minimum spanning tree or minimum weight
spanning tree is then a spanning tree with weight less
than or equal to the weight of every other spanning tree.
12
13. ▪ One example would be a cable TV company laying
cable to a new neighborhood. If it is constrained to
bury the cable only along certain paths, then there
would be a graph representing which points are
connected by those paths.
▪ Some of those paths might be more expensive,
because they are longer, or require the cable to be
buried deeper; these paths would be represented by
edges with larger weights.
▪ A spanning tree for that graph would be a subset of
those paths that has no cycles but still connects .
There might be several spanning trees possible. A
minimum spanning tree would be one with the lowest
total cost.
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14. EXAMPLE
▪ Minimum spanning tree problem with constraints.
▪ Objective: Connects all nodes with minimum costs
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50
19. Procedure
▪ Allows non-improving solution to be accepted in
order to escape from a local optimum
▪ 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.
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20. 20
▪ Too many parameters to be
determined.
▪ Number of iterations could be very
large.
▪ Global optimum may not be found,
depends on parameter settings .
Conclusions
21. Advanced Topics
▪ Intensification: Intensification strategies are
based on modifying choice rules to courage move
combinations and solution features historically
found good.
▪ Diversification: penalize solutions close to the
current solution
▪ The TS notions of intensification and
diversification are beginning to find their way
into other meta-heuristics.
▪ It is important to keep in mind that these ideas are
somewhat different than the old control theory.
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22. Some Convergence Results
▪ Memory Tabu Search converges to the
global optimum with probability one if
randomly generated vectors (x) follows
Gaussian or uniform distribution .
▪ Convergent Tabu Search converges to the
global optimum with probability one .
22
23. References
[1] Glover, F., Kelly, J. P., and Laguna, M. 1995. Genetic Algorithms and Tabu
Search: Hybrids for Optimization. Computers and Operations Research. Vol. 22,
No. 1, pp. 111 – 134.
[2] Glover, F. and Laguna, M. 1997. Tabu Search. Norwell, MA: Kluwer Academic
Publishers.
[3] Hanafi, S. 2001. On the Convergence of Tabu Search. Journal of Heuristics. Vol.
7, pp. 47 – 58.
[4] Hertz, A., Taillard, E. and Werra, D. A Tutorial on Tabu Search. Accessed on
April 14, 2005: http://www.cs.colostate.edu/~whitley/CS640/hertz92tutorial.pdf
[5] Hillier, F.S. and Lieberman, G.J. 2005. Introduction to Operations Research. New
York, NY: McGraw-Hill. 8th
Ed.
[6] Ji, M. and Tang, H. 2004. Global Optimizations and Tabu Search Based on
Mamory. Applied Mathematics and Computation. Vol. 159, pp. 449 – 457.
[7] Pham, D.T. and Karaboga, D. 2000. Intelligent Optimisation Techniques –
Genetic Algorithms, Tabu Search, Simulated Annealing and Neural Networks.
London: Springer-Verlag.
[8] Reeves, C.R. 1993. Modern Heuristic Techniques for Combinatorial Problems.
John Wiley & Sons, Inc.23
Cutting plane: A cutting plane (or Cut) for any IP problem is a new functional constraint that reduces the feasible region for the LP relaxation without eliminating any feasible solution for the IP problem.
