Scatter Search

(Heuristic Algorithm)

Authors: Manuel Laguna

Speaker: B.Y. Huang Adviser: Dr. Peitsang Wu

2007/04/10

Outline• Introduction

• Scatter Search Template

• A Scatter Search Illustration

• Conclusions

Introduction

• This development was launched in job shop scheduling problems (Glover, 1963).

• Fundamental part of Tabu Search (Glover and Laguna, 1977). Scatter Search is intimately related to the Tabu Search meta-heuristic, and derive additional advantages by making use of adaptive memory and associated memory

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Introduction

• Fred W. Glover (1998) first discussed the “Scatter Search” in “A Template for Scatter Search and Path Relinking” .

• Scatter Search is also called evolutionary method or population-based algorithm.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Introduction

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Path Relinking: Original path shown by heavy line and one possible relinked path shown by dotted line

Introduction

• It derives its foundations from strategies originally proposed for combining decision rules and constraints.

• The goal is to enable the implementation of solution procedures that can derive new solutions from combined elements.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Scatter Search Template

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

• Diversification Generation Method

• Improvement Method

• Reference Set Update Method

• Subset Generation Method

• Solution Combination Method

Scatter Search Template

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

• A Diversification Generation Method:

to generate a collection of diverse trial solutions, using an arbitrary trial solution (or seed solution) as an input

• An Improvement Method:

to transform a trial solution into one or more enhanced trial solutions

Scatter Search Template

• A Reference Set Update Method:

to build and maintain a consisting of “best” solutions found, organized to provide efficient accessing by other parts of the method.

Solutions gain membership to the reference set according to their quality or their diversity.

b

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

setreference

Scatter Search Template

• A Subset Generation Method:

to operate on the reference set, to produce a subset of its solutions as a basis for creating combined solutions.

• An Solution Combination Method:

to transform a given subset of solutions produced by the Subset Generation Method into one or more combined solution vectors.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

• Consider the following 0-1 knapsack problem:

A Scatter Search Illustration

• Diversification Generation Method:

( where is the number of variables in the problem). Let us choose .

......(the initial seed)

Type 1 ( )

Type 2 ( )

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration• For example:

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration• For example:

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

The 10 solutions can be used to create the reference set by applying the following Improvement Method.

A Scatter Search Illustration• Improvement Method:1.If the trial solution is infeasible: To start with the profit-to-weight ratio which is the smallest one, and change the variable values from one to zero.

2.If the trial solution is feasible: To start with the profit-to-weight ratio which is the largest one, and change the variable values from zero to one.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

• Solution 8:

→New Solution 8:

A Scatter Search Illustration

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

: infeasible solution

: the optimal solution

A Scatter Search Illustration

• Reference Set Update Method:

• We set a reference set ,where and .

• The subset of high-quality solutions:1, 2, 8.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

5b 31b22 b

A Scatter Search Illustration

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

• We define the distance between two solutions:

• Distance measures:

A Scatter Search Illustration

• Solution 3 should be added to the of the reference set, and so as Solution 7(by the maximum minimum distance)

7,3

8 ,2 ,1

2

1

b

b

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

• Subset Generation Method:

1) All 2-element subsets.

2) 3-element subsets. (derived from the 2-element subsets by augmenting each 2-element subset to include the best solution

not in this subset)

3) 4-element subsets. (derived from the 3-element subsets by augmenting each 3-element subset to include the best solution

not in this subset)

4) The subsets consisting of the best elements, for .

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

• Subset generation: (1, 2, 8, 3, 7)

add “8”

add “1”

add “2”

A Scatter Search Illustration• Solution Combination Method:

We use a combination method that creates only

one solution from each subset and calculates a

score for each variable.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

:variable :the subset

:the objective value of solution :is the value of the th variable in solution

A Scatter Search Illustration• The trial solution is constructed by rounding

the score for each variable to the nearest integer, i.e.,

• Consider the subset type 2 given by solutions 3, 7 and 8. The objective values are ,

and .

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

• For example:

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

A Scatter Search Illustration

• Score calculation for subset combination (3, 7, 8)

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Improvement

Method

Conclusions

• A solution may become a member if its objective value is better than the objective value of any of the solutions in the high-quality subset.

• Alternatively, if a new solution improve the diversity of the reference set, this solution can replace one that is currently in the diverse set.

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Conclusions

• If the reference set is modified, then the Subset Generation Method, the Solution Combination Method and the Improve Method are applied in sequence.

• The application of the methods continues until the reference set converges (i.e., no elements in the set are replaced).

Introduction→ Scatter Search Template → A Scatter Search Illustration→ Conclusions

Q & A

Thank you

for

your attention !