Nuno Abreu, Zafeiris Kokkinogenis, Behdad Bozorg, Muhammad Ajmal 1

OUTLINE

               Introduction to TS Parameters of TS Basic concepts of TS TS vs. other Meta-heuristics Memory in TS Use of Memory in TS Tabus Aspiration criteria Stop condition TS basic algorithm Search Process – 1 Search Process – 2 Flowchart of a Standard TS Algorithm Example Pros and Cons 2

INTRODUCTION TO TS

 TS is an iterative procedure designed for the solution of optimization problems  Invented by Glover (1986)  Uses a neighborhood search procedure to iteratively move from a solution x to a solution x* in the neighborhood of x  Uses memory structures so that the algorithm does not visit a given solution repeatedly  Tabu Search Benefits  Cycle avoidance which also saves time  Guide search to promising regions of the search space 3

PARAMETERS OF TABU SEARCH

 Space search procedure  Neighborhood structure  Short-term memory: Tabu list  Types of moves  Addition of a Tabu move  Maximum size of Tabu list  Aspiration conditions  Stopping rule 4

BASIC CONCEPTS OF TABU SEARCH



Saves information

according to the exploration process  It will be used to

limit the moves

through the neighborhood  Structure of the neighborhood of the solutions varies from iteration to iteration  Infeasible solutions can be accepted and evaluated to

escape local minimum

.

 To prevent from cycling,

recent moves are forbidden

.

 A

tabu list

records forbidden moves, which are referred to as

tabu moves

 A tabu move can be accepted using

aspiration criteria

.

 Allows

exploitation

of good solution and

exploration

unvisited region of the search space of 5

TS VS OTHER META-HEURISTICS

 Traditional descent methods cannot allow

non-improving moves

, TS can.

 SA and GA does not have

memory

, TS has.

 SA uses

randomness forbidden moves

.

to escape local minimum, TS uses  TS claims that a bad strategic choice can yield more information than a random choice.

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MEMORY IN TS

 TS uses mainly two types of memory: 

Short-term

memory   Recent solutions Structure were tabu moves are stored  Avoids cycling 

Long-term

memory (frequency-based)  Number of iterations that “solution

components

” have been present in the current solution 7

USE OF MEMORY IN TS

 Use of memory leads to

learning

 Prevent the search from

repeating

moves 

Explore

the unvisited area of the solution space  By using memory to avoid certain moves, TS can be seen as

global optimizer

rather than local.

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TABUS

 Tabus are one of the distinctive elements of TS when compared to LS  Prevent cycling when moving away from local minimum through non-improving moves  Stored in the

short-term memory –

Tabu list 

Tabu tenure

is the number of iteration the move is tabu  Tabu list can be of fixed-length or dynamically varying 9

ASPIRATION

 If a move which is tabu

can lead to better solution

, the tabu status should be

overruled

 This will be performed using

aspiration

level conditions 

Aspiration criteria

: accepting an improving solution even if generated by a tabu move  A tabu move becomes admissible if it yields a solution that is better than an aspiration value  A tabu move becomes admissible if the direction of the search (improving or non-improving) does not change 10

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 solutions in N(i,k) such that the Tabu conditions are not violated or the aspiration conditions hold.

 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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STOPPING CONDITIONS

 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 12

SEARCH PROCESS - 1



Intensification

:     searches solutions

similar

wants to space

intensively

to the current solution explore known promising areas of the search creates solutions using the more attractive components solutions in memory (recency memory) of the best Another technique consists in changing the neighborhood’s structure by allowing different moves 

Diversification

:   Examines

unvisited

regions of the search space Generates different solutions   Using rarely components present in the current solution Biasing the evaluation of a move by modifying the objective function adding a term related to component frequencies  Intensification and diversification phases

alternate

search during the 13

SEARCH PROCESS - 2

  

Allowing infeasible solution

 Constraints defining the search space can lead the process to mediocre solutions  Induces diversification  By dropping some constraints (

relaxation

) a larger space can be explored  Penalize objective for the violation  A well known technique:

Strategic oscillation Surrogate objective

 Evaluates neighbors using a simpler function than the objective in order to spot promising candidates. (Intensification)

Auxiliary objective

 Objective function can’t drive the search to more interesting areas  Orient the search by measuring desirable attributes of solutions 14

FLOWCHART OF A STANDARD TABU SEARCH ALGORITHM Heuristic procedure Tabu restrictions Candidate lists Aspiration criteria Elite solutions Modified choice rules for diversification or intensification Generate initial solution and initialize memory structures Construct modified neighborhood Yes Select best neighbor Stop No More iteration?

Update memory structures Short and long term memory Restarting Strategic oscillation Execute specializes procedures Update best solution 15

EXAMPLE

• • Minimum spanning tree problem with constraints.

Objective: Connects all nodes with minimum costs

Costs A B 20 15 10 25 C D 5 40 30 E A B 20 15 10 C 25 D 5 40 30 E 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.)

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EXAMPLE

Iteration 1 Cost=50+200 (constraint penalties)   New cost = 75 (iteration 2) ( local optimum) B 20 30 A Delete 15 10 C 25 D 5 E 40 Add Add BE BE BE CD CD DE DE DE Delete CE AC AB AD AC CE AC AD Cost 75+200=275 70+200=270 60+100=160 60+100=160 65+300=365 85+100=185 80+100=180

75+0=75

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.)

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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 Delete B 20 30 Escape local optimum Add A 10 C 5 E Add AD AD AD Delete DE* CE AC Cost Tabu move 85+100=185 80+100=180 15 25 D 40 Tabu BE BE BE CE AC AB 100+0=100 95+0=95

85+0=85

CD CD DE* CE 60+100=160 95+100=195

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.)

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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 Add Delete Cost 20 B 30 Tabu AB AB AB BE* CE AC Tabu move 100+0=100 95+0=95 10 A C 5 E 15 25 D Add 40 Tabu Delete AD AD AD DE* CE AC 60+100=160 95+0=95 90+0=90 CD CD DE* CE

70+0=70

105+0=105

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.)

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EXAMPLE

  Optimal Solution Cost = 70   Additional iterations only find inferior solutions A B 20 15 10 25 C D 5 40 30 E 20

PROS AND CONS

 Pros:  Allows non-improving solution to be accepted in order to

escape from a local optimum

 The use of

Tabu list

 For larger and more difficult problems tabu search can beat other approaches  Cons:  

Too many parameters

to be determined Number of iterations could be very large  Global optimum may not be found, depends on parameter settings 

Too complex

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REFERENCES

 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.

 Glover, F. and Laguna, M. 1997. Tabu Search. Norwell, MA: Kluwer Academic Publishers.

 Hanafi, S. 2001. On the Convergence of Tabu Search. Journal of Heuristics. Vol. 7, pp. 47 – 58.

 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

 Gendreau, M. 2002. An Introduction To Tabu Search, Centre de Recherche sur les Transports et Département d´informatique et de Recherche opérationnelle, Université de Montréal.  Hillier, F.S. and Lieberman, G.J. 2005. Introduction to Operations Research. New York, NY: McGraw-Hill. 8th Ed.

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OUTLINE

 Introduction    Tabu Search Basic Approach – 1 Basic Approach – 2  The TS framework for multi-dimensional bin packing  Search methods  Computational test - Dataset  Computational test  Sample of results  Conclusions 24

INTRODUCTION

 The paper deals with

multi-dimensional

algorithms cutting and packing  Assumes that the item will be packed with

fixed orientation

(no rotation)  Lodi, Martello and Vigo present an effective

BP Tabu Search framework

 The main characteristic is the adoption of a search scheme and a neighborhood independent of the packing problem to be solved 25

TABU SEARCH

 The goal is

empty

a specified target bin  Given a current solution, the moves modify it by changing the packing of a set S of items   The target bin is the one

minimizing

a filling function φ(.) φ(.) measures the

easiness of emptying the bin

 The idea is to favor target bins packing a

small area

and a relatively

large

number of items 26

BASIC APPROACH - 1

 Select

target

bin  Include

one item

, j, from the

target

other bins in a set S bin, and contents of

k

 Execute an greedy-type heuristic A on S  The size of the neighborhood

k

is automatically updated  If the items of S are packed into k (or less) bins, item j is

removed

selecting a different set of k bins, or a different item from the target bin from the target bin. Otherwise, S is changed by  Then, a new item is selected, a new set S is defined and a new move is performed  The execution is

halted

as soon as a proven optimal solution is found, or a time limit is reached 27

BASIC APPROACH - 2

 If the algorithm gets stuck  The target bin is not emptied, the neighborhood is enlarged by increasing the value of up to a prefixed upper limit  Tabu list stores,for each forbidden move, the sum of the filling function values of the k+1 involved bins  Variable Neighborhood Search strategy    Small values of k = Small neighborhoods, fast to explore Accepting moves dealing with increased k = Enlarged neighborhood, more chances to improve solution By changing the size of the neighborhood, the algorithm plays “Intensification” and “Diversification”  The execution is

halted

as soon as a proven optimal solution is found, or a time limit is reached 28

THE TS FRAMEWORK FOR MULTI-DIMENSIONAL BIN PACKING 29

SEARCH METHODS



Intensification

procedure explores the neighborhood of the current solution (Inner loop)  Two types of

diversification

are defined:  “

soft

” diversification - select as target bin the one having the second smallest filling function value  “

hard

” diversification - re-pack into separate bins the items currently packed in the z/2 bins ( z being the number of bins in the current solution) with smallest filling function value  The

tabu list

stores, for each forbidden move:  the sum of the

filling function values

 the

last moves

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COMPUTATIONAL TESTS - DATASET

 The benchmark consists of 500 random instances with n = {20,40,60,80,100}  Ten different classes of instances were used  wi (width), hi (height) generated from uniform distribution of varying intervals 31

COMPUTATIONAL TESTS

 Measures the

improvement

to three

greedy-type

heuristics: brought by TS when comparing  Hybrid Best-Fit algorithm (HBP)   Knapsack Packing (KP) Alternate Directions (AD)  Results show that the TS allows a

good improvement

in the quality of the solution obtained by the greedy-type heuristics 32

SAMPLE

OF RESULTS

33

CONCLUSIONS

 The main idea of the framework:  

Isolate the information

concerning the problem Let a

greedy-type feasible solutions

heuristic take care of the structure and construct  Tabu Search is then used to

drive the search

through the solution space by:  

re-combining

the packed items,

exploring a neighborhood

by alternating between “

intensification

” and “

diversification

”  This results in a very general algorithm for

multidimensional bin packing

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