Genetic Algorithms for

multiple resource constraints

Production Scheduling with multiple levels of product structure

By : Pupong Pongcharoen (Ph.D. Research Student)

Supervisors : Prof. Paul Braiden

Dr. Chris Hicks

26 April 1999

Dept. of MMME, University of Newcastle upon Tyne

Overview of this presentation

Background and literature review Characteristics of production scheduling

problem Optimisation algorithms Genetic Algorithms(GAs) applied to production

scheduling Experimental Program Results Discussions and conclusions

What is scheduling ?

“ The allocation of resources over time to perform a collection of tasks ”

“ Scheduling problems in their simple static and deterministic forms are extremely simple to describe and formulate but difficult to solve ”

Baker(1974)

King and Spackis(1980)

Scheduling problems

n jobs & m machines = (n!)m possible solutions

e.g. 20 x 10 problem => 7.2651x10183 solutions

feasiblesolutions

infeasiblesolutions

Type of scheduling problems in literature

Job shop problem (JSP)

different routing of jobs machines Flow shop problem (FSP)

same routing of jobs machines Permutation scheduling problem (PSP)

same job sequence machinesKing and Spackis

(1980)

Literature review

JSPFSPPSP All jobs are independent

Single-level scheduling

No assembly operations

Nagar(1995)

Assumption

Products have no structureIn reality

Most products require some type of assembly

"Strongly relationship between product structure & sequencing"

Fry et.al.(1989)

Optimisation algorithms

Conventional optimisation algorithms

Example Branch & Bound, Integer Linear Programming and Dynamic Programming.

works well with small problems

slow can’t solve “big” problems

Approximation optimisation algorithms

Example Dispatching rules, Simulated Annealing, Taboo Search and Genetic Algorithms.

fast can be applied with big or

small problems approximate “optimal”

solutions.

Jain et.al. (1999)

Product structure from company

Type of scheduling environment

Machine environment

Single or or Multiple machines Product environment

Single or or Multiple products Capacity planning

Infinite or or Finite resources constraints Research methodology

Analytical or or Simulation methodology

The objectives of this research

Apply Genetic Algorithms to complex capital goods production scheduling problems

Minimising penalty cost due to earliness and tardiness

Assume finite capacity Using simulation methodology for testing

plans

Production Scheduling

Final product

Part no. 1

Operation no. 3

Part no. 3

Operation no. 2

Operation no. 2

Operation no. 1

Operation no. 1

Part no. 2

Operation no. 1

Part no. 5

Operation no. 5

Operation no. 2

Operation no. 1

Operation no. 4

Operation no. 3

Part no. 4

Operation no. 4

Operation no. 2

Operation no. 1

Operation no. 3

with multiple levels of product structure

Example of Gantt Chart

Fitness function

Minimise : Pe(Ec+Ep) + Pt(Tp)

Where Ec = max (0, Dc - Fc)

Ep = man (0, Dp - Fp)

Tp = max (0, Fp - Dp)

Genetic Algorithms

Chromosome

Chromosome

Chromosome

:: Parent 1

Parent 2

+ ==>

Offspring 1

Offspring 2

Parent 3 ==> Offspring 3

Mutation Operation

Crossover Operation

Genetic OperationPopulation

SolutionSpace

Fitness Measure

Offspring 1

Offspring 2

Offspring 3

FitnessTesting

Selection

NewPopulation

random

encoding

decoding

Crossover Operation

1 2 3 4 5 6 7 8

8 7 6 5 4 3 2 1

Parent 1

Parent 2

Child 1

Child 2

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

Mutation Operation

1 2 3 4 5 6 7 8Parent

1 2 6 5 4 3 7 8Child

Demonstration of Genetic Algorithm Program

Genetic Algorithms for scheduling problems was written by using Tcl/Tk programming language.

The program was runs on Unix system V release 4.0 on a Sun workstation.

Case study (data from Parsons)

Characteristics of the problemProblem sizes Products Operations Resources Levels

Small 2 34 13 4Medium 2 67 16 4Large 2 135 27 4

Experimental program

Factors No. oflevels

Levels

Problem size 3 small, medium, largeNumber of populations 2 20, 40Number of generations 2 20, 40Probability of crossover 4 0.6, 0.7, 0.8, 0.9Probability of mutation 5 0.02, 0.04, 0.06, 0.08, 0.1

Full factorial experimental design was performed.

Total number of runs = 3 x 2 x 2 x 4 x 5 = 240 (per replication)

Results from 240 runs on each problem sizes

Problem sizes Penalty costAverage Variance The best

Small 1.358 0.140 1.161Medium 2.085 0.166 1.692Large 3.881 0.676 2.509

Analysis of Variance

Source DF SS MS F P

Size 2 269.850 134.925 1264.24 0.000Population 1 2.099 2.099 19.66 0.000Generation 1 2.419 2.419 22.66 0.000Mutation 4 3.450 0.862 8.08 0.000Size*Population 2 1.671 0.836 7.83 0.001Size*Generation 2 1.655 0.827 7.75 0.001Size*Mutation 8 5.158 0.645 6.04 0.000Error 219 23.373 0.107Total 239 309.674

The best performance of GAs on the problems

Population average RunningProblem sizes Generation Improve Standard deviation Decreasing Time

first last (%) first last (%) (Min.)Small 2.215 1.242 43.9 0.279 0.033 88.2 2.4

Medium 3.124 1.955 37.4 0.425 0.157 63.1 8.7Large 6.232 2.709 56.5 0.659 0.134 79.7 25.6

Mean and standard deviation for each population

Effect of mean and standard deviation of population on GA performance

0

0.5

1

1.5

2

2.5

3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

No. of generations

Ave

rag

e p

enal

ty

Discussions

When the problem size increases the execution times increase exponentially.

Next step is to break “large” problems down into smaller independent problems that can be solved in a “reasonable” amount of time.

The solutions to the small problems will be integrated to give an overall solution.

Conclusions

Genetic algorithms represents a powerful technique for solving scheduling problems.

Practical software produced for solving scheduling problems.

Solutions far better than original schedules obtained from Company

Appropriate levels for Genetic Algorithm parameters identified.

Further Research

Bicriteria scheduling problems.

Multiple criteria scheduling problems.

Any questions please ?