A Modified Binary Particle Swarm Optimization Algorithm for Permutation Flow Shop Problem LEI YUAN1, ZHENDONG ZHAO2 1School

of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, P.R. China 2Institute

of information and networking, Nanjing University of Posts and Telecommunications, P.R. China

ICMLC 2007 MBPSO for FSP Lei Yuan [email protected]

Outline z Introduction { Flow Shop Problem { Particle Swarm Optimizer & its Binary Version

z Previous Works { Traditional Methods { PSO methods

z Proposed Methods { Basic Modification: the Smallest Position Value Rule { Further Modification: new set of Update Formula { Other Modifications: Local Search, etc.

z Performance z Conclusions and Future Work

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Introduction Flow Shop Problem

zDescription of Job Shop Scheduling {A finite set of n jobs {Each job consists of a chain of operations {A finite set of m machines {Each machine can only handle one operation at a time {Each operation needs to be processed during an uninterrupted period of a given length on a given machine {Purpose is to find a schedule, that is, an allocation of the operations to time intervals to machines, that has minimal length

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Introduction Flow Shop Problem

z Formal Definition of JSS {Job set J = { j1 , j2 , ... jn } {Machine set M = {m1 , m2 , ... mm } {Operations Oi = {oi1 , oi 2 ,...oim } {Each operation has processing time {τ i1 ,τ i 2 ,...τ imi } i

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Introduction Particle Swarm Optimizer

z A stochastic, population-based computer problemsolving algorithm z Swarm Intelligence z Social influence and v t +1 = w ⋅ v t + C ⋅ r ⋅ ( p t − x t ) i i i i 1 1 social learning + C 2 ⋅ r2 ⋅ ( pgt − xit ) z Information Sharing

x

t +1 i

= x +v t i

t +1 i

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Previous Works zTraditional Methods {Mixed integer linear programming (1960) {Branch and Bound, etc.

zPSO methods {Two basic thinking: flying in the free space, then convert, or flying in the solution space {“Free space” approach: Smallest Position Rule {Discrete approach: special operators MBPSO for FSP Lei Yuan [email protected]

Proposed Methods Basic Modification

z Smallest Position Value Rule {Basic Purpose: to build up an easy connection between the positions and solutions {Method: using the sorting information

Data structure (Actual Vector)

1 0

0 1

0 0

1 1

1

1

1

1

Decoded Vector

5

3

1

7

Corresponding Sequence

2

3

4

1

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Proposed Methods Further Modification

z Motivation {Slow Convergence {Bad accuracy {Unstable Performance

z New Set of Updating Formula {Inspiration: the traditional thinking of the continuous version PSO

v id = w ⋅ v id + C1 ⋅ r1 ⋅ ( pid ⊕ xid ) + C 2 ⋅ r2 ⋅ ( pgd ⊕ xid ) if ( rand () < S ( v id )) then x id = x id else x id = x id MBPSO for FSP Lei Yuan [email protected]

Proposed Methods Further Modification

z Comparison between the formula { Continuous PSO v it +1 = w ⋅ v it + C1 ⋅ r1 ⋅ ( pit − xit ) + C 2 ⋅ r2 ⋅ ( pgt − xit )

x it +1 = x it + v it +1 { BPSO v it +1 = w ⋅ v it + C1 ⋅ r1 ⋅ ( pit − xit ) + C 2 ⋅ r2 ⋅ ( pgt − xit )

if ( rand () < S (v id )) then xid = 1 else xid = 0 { MBPSO v id = w ⋅ v id + C1 ⋅ r1 ⋅ ( pid ⊕ xid ) + C 2 ⋅ r2 ⋅ ( pgd ⊕ xid )

if ( rand () < S (v id )) then x id = x id else x = x d i

d i

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Proposed Methods Other Modifications

zLocal Search {Motivation: New set of formula brings fast convergence but high possibility of premature {Method: make the particles search a little “around” them, not only the lines between them

zPerturbation {Method: restart a particle once it got “stuck” somewhere MBPSO for FSP Lei Yuan [email protected]

Performance zCriteria {Relative Percent Deviation (RPD) Δ {average RPD Δ avg {maximum and minimum RPD Δ max , Δ min {standard deviation of RPD σ RPD

C min − C * Δ= × 100 * C MBPSO for FSP Lei Yuan [email protected]

Performance Problem Size n×m

Δ avg

Δ min

Δ max

σ RPD

20×5

0.1609

0

0.5384

0.0019

20×10

0.4862

0

1.0423

0.0028

20×20

0.4344

0

0.8522

0.0023

50×5

0.1837

0

0.3950

0.0012

50×10

1.0510

0

1.8401

0.0046

50×20

2.4296

0

3.3837

0.0049 MBPSO for FSP Lei Yuan [email protected]

Conclusions A modified version of Binary Particle Swarm Optimization Algorithm (MBPSO) is presented, and to avoid the stagnation in local optima, local search and perturbation are employed to improve the performance. Experiments fully demonstrate that MBPSO improve the Performance of the PFSPs solving. Several further works, such as the implementation of different local search mechanisms, different coding and decoding methods, or a new way to represent the solution, remain to be studied. MBPSO for FSP Lei Yuan [email protected]

Acknowledgements zAuthors acknowledge Yuxuan Wang, for he gave us a variety of comments which helped us a lot to improve the quality of our present work

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Thanks for your listening. Questions?

Lei Yuan [email protected]

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