A GENETIC A LGORITHM FOR PARALLEL MACHINE TOTAL TARDINESS PROBLEM

EURO / INFORMS İstanbul 2003 July 06-10

A GENETIC ALGORITHM FOR PARALLEL MACHINE

TOTAL TARDINESS PROBLEMM. Furkan Kıraç

Ümit BilgeMüjde Kurtulan

Department of Industrial EngineeringBoğaziçi University


Objective Genetic Algorithms are rooted from a strong idea with a simple basic

mechanics that involves only the process of copying strings and swapping partial strings.

Implicit parallelism which traverse the search space climbing many hills in parallel.

However GAs are prone to premature convergence and impose numerous parameters to fine-tune.

In this study, a generic adaptive control mechanism to slow down or prevent this premature convergence and reduce the parameter dependence of a Basic Genetic Algorithm (GA) is developed and implemented over a hard to solve problem: The Parallel Machine Total Tardiness Problem (PMTT).

The fundamental elements of GA are investigated and the solution strategy developed is benchmarked with the literature for performance evaluation.


Outline

Problem definition and characteristics for PMTT Basic Genetic Algorithm (GA) approach to

PMTT and experimentation Adaptive Control Mechanism over Basic GA and

experimentation Results compared to literature Conclusions


Parallel Machine Total Tardiness Problem ‘n’ independent jobs to be scheduled on ‘m’ uniform parallel

machines

Each job has a distinct ready time ri a distinct due date di an integer processing time pi

Sequence dependent setup time sij

Objective is to minimize the total tardiness of all the jobs, ∑Ti, Ti is the respective tardiness of job i calculated as Ti = max{0, Ci - di} Ci is the completion time of job i


Problem Characteristics In most studies from the literature the general assumption is that

the machines are identical all jobs are available at time zero and setup times do not exist

These assumptions are far too simplistic when confronted with the real world situations

In this study, these features are also incorporated into the model to approach the problem with real world situations

Each machine in our problem set has a speed factor associated with it. Machines are not identical.


Chromosome Encoding for PMTT The chromosome representation used encodes each job

in the schedule as a gene on the chromosome

Machine sequences are separated by an asterisk (*) on the chromosome

Sequence of jobs on each machine

Machine 1: 1-2-3 Machine 2: 4-5-6-7

Machine 3: 8-9

1 2 3 * 4 5 6 7 * 8 9


Details of Basic GA Algorithm Initial population: Random population + solutions generated by

list scheduling heuristics such as EDD, SPT, SST, ERT Parent selection: Ranking Roulette Wheel

Less bias is introduced since the fitness values are based on a ranking of the total tardiness values

Crossover operator: Uniform order-based crossoverThe crossover operator generate a binary string where the number of “1”s and “0”s can be controlled. This binary string is used as a template to combine the genetic information and properties of the two parents.

Mutation operator: Swap operationConsists of swapping two randomly selected genes.


Crossover operator

Parent1 1 2 3 * 4 5 6 7 * 8 9

Binary 1 0 0 1 0 1 0 0 1 1 0

Child 1 - - * - 5 - - * 8 -

Parent2 9 8 * 7 6 1 * 2 3 4 5

Child’ 1 9 7 * 6 5 2 3 * 8 4


Transient Population Generation

The population generation method is Transient Creates a transient phase in the progress from one

generation to the next Transient population consists of the old population and the

new offspring, where N is population size Nc is number of children produced

To keep the population size constant, Nc individuals need to be eliminated

Gives a greater chance of survival to the old population members as long as they are fit enough


Transient Population Elimination

BasicElimination

150149148

::

10099989796959493::654321

Best 53 individuals preserved

48 48 individuals individuals eliminatedeliminated

Worst 2 Worst 2 individuals individuals eliminatedeliminated

SORTED SORTED

TRANSIENT TRANSIENT

POPULATIONPOPULATION

consisting of 150 consisting of 150

individualsindividuals

N = 100N = 100

Nc = 50Nc = 50


Analysis of Basic GA GA has a high number of parameters that can be regulated for higher

performance, but this introduces the difficulty of fine-tuning the parameters Population Size New Generation Creation Method Fitness Evaluation Method Parent Selection method Crossover Probability & Operator Mutation Probability & Operator Mutation Strength …

GA is prone to the risk of premature convergence i.e. the population converges to a set of good performing and highly similar

members or to an individual without having much chance of generating representatives of

diverse hyperplanes of the solution space


Unstable ? Why not control it ? The weakness of GAs can be attributed to the high

sensitivity of the GA parameters strong parameter dependence affects the robustness

Therefore, the GA can be termed as unstable from the control theory point of view

When a system is defined as unstable, the natural attitude is to try to control it

Classical control theory proposes closed-loop systems for robust control of a system


Closed-loop Control Systems A closed-loop system is one that considers the output of the

previous state as a feedback input for the successive state In this study, a control mechanism consisting of two

complementary subcomponents is devised

CONTROLLERCONTROLLER

KK

errorreference

+ -sensor

output


Adaptive Control over Basic GA Preliminary experiments performed with Basic GA

indicate that the problem under study favors rather high mutation rates high diversity within the GA search

Therefore, the population diversity is the first performance indicator to be controlled for higher performance aims to overcome the risk of premature convergence due

to the dominance of some fit individuals

Additionally, a training mechanism is developed designed to operate on the weak offspring in the population

to bring them to a level of maturity


Diversity Control An adaptive mechanism to control the population diversity

whenever it deviates from a threshold value is developed

The operating principle is simple in that whenever the population diversity falls below a

given percentage, the control mechanism is triggered

A set of diversifying operations are performed on the population

At the end of these moves the population diversity increases and the Basic GA is resumed until diversity falls below the threshold level


Control for Population Diversity

7504

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6950

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6150

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:

k = 4 Non-mutants

Mutated

Mutated

1st cluster of equal fitness chromosomes

2nd cluster of equal fitness chromosomes

3rd cluster of equal fitness chromosomes

nth cluster of equal fitness chromosomes

Mutated

Mutated

NNOOTT MMUUTTAATTEEDD





Effect of Adaptive Diversity ControlBar charts showing the population distribution

BEFOREThe instant when the diversity threshold is reached and the control mechanism is triggered

AFTERBy the operation of diversity control, the peak consisting of converged individuals is suppressed and the population distribution is smoothed

# of individuals

66

tardiness160000

# of individuals

22

tardiness800000


Training In order to further exploit the recombining strength of

the crossover operator, an adaptation from real life occurrences is introduced at this stage

This is called “training” based on the argument that a newborn child is not capable of surviving in the environment without first going through training

This concept is extended to encompass the entire set of unfit individuals in the population instead of just the offspring


Training Parameters The trigger of training is a performance measure of the

system that stimulates steepest descent when the search stagnates for a proportion of the entire search duration

This proportion is set to be 1.0%,i.e. 100 non-improving generations

the duration of the training session applied over each of the individuals(number of iterations for which steepest descent will take over )

the number of individuals to be educated


Effect of Training ControlBEFORE The function of the training phase is to improve the fitness of the worst population members so that the population distribution curve is smoothed out

AFTER

In other words, the function of training can be defined as decreasing the skewness in the population distribution.

# of individuals

25

tardiness500000

# of individuals

26

tardiness180000


Effect of Diversity andTraining Control 1

2 3 4 5 6 7 8 9 : : :

79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Systematic mutation applied over the best individuals in the

population

Worst TP% individuals in the population APPLY

TEACHING PHASE

FF II TT

NNEE

SS SS

k non-mutants


Experimentation The problem set used for experimentation consists of parallel

machine scheduling problems of 40, and 60 jobs, developed and tested by Sivrikaya-Şerifoğlu, F. and G.Ulusoy to study a GA

The same problem set is addressed by Bilge,Ü., F.Kıraç, M. Kurtulan and P. Pekgün in a deterministic TS approach

These problem sets are as follows: Instances with n = 40, and n = 60 were randomly generated (n:

number of jobs) Number of machines, m, is 2 or 4 20 distinct instances generated for each group.


Performance Measure

ji

5

1i

K

1jj GA - literaturein reported Best value

K1

51GA

j

K

1jj TS - literaturein reported valueBest

K1TS

K is the number of problem instances over which the values are evaluated (20 in this case)Performance measure used in this study is a comparative relative measure which takes the best-known TS values for the problem instances reported in the literature [Bilge et al.] as a basiswhere,i = 1, 2, 3, 4, 5 denotes different replications

j = 1, 2, …, 20 denotes the instance number in a given problem set


Performance Ratio (PR)

TSGAPR

This ratio is used for a comparison of the relative achievements obtained via each metaheuristic The aim in this study is to obtain a ratio as low as possibleA ratio greater than 1.0 means that the GA’s performance is worse than the TS presented in [Bilge et al.] on the average.A ratio of 1.0 means that the average behavior of the GA is comparable to the average behavior of the TS presented in [Bilge et al.] A ratio less than 1.0 means that the average results obtained by the GA is better than the TS presented in [Bilge et al.]A ratio less than 0.0 means that the best known values in the literature are improved by the GA .

Best KnownResult TS GA


Performance of Adaptive GA

Problem SetBasic GA

PerformanceRatio

Adaptive GAPerformance

RatioTimesBetter

40 Job 2 Machine 11.329 0.809 14.0060 Job 2 Machine 6.534 0.591 11.06

40 Job 4 Machine 7.065 1.121 6.3060 Job 4 Machine 6.099 5.414 1.12

Diversity Non-Mutants = 10 out of 100 (Best fit individuals)Number of Trainees = 20 out of 100 (Worst fit individuals)Training Duration = 15 (Steepest Descent Steps)


Improved Best Known Results60 JOBS 60 JOBS 40 JOBS 40 JOBS

2 MACHINES 4 MACHINES 2 MACHINES 4 MACHINES1 14205 0 14079 02 6528 2737 3946 03 17296 155 3335 04 72330* (72406) 0 10095 05 34578* (34640) 2591 19671* (19695) 06 50138* (50492) 339 26372 07 26660 4744 18565 9148 8030* (8042) 0 37513 489 16790 0 1055 0

10 20899* (20943) 4626 1038 011 11204 4423 1726 012 14080 0 8199 013 12806 0 8382 280714 6834* (6874) 0 5860 270415 20017 0 21562* (21712) 138816 23883 58 43395* (43502) 017 12222 0 15816 018 38948 0 5866 019 164 0 27258 020 23514 0 2887* (2934) 0

Problem

UPDATED BEST KNOWN RESULTS

Those values marked with a (*)

are contributed by the adaptive GA

algorithm devised in this study


Conclusion The major enhancement brought to the GA concept in this study is the

generic adaptive control mechanism which aims to better exploit its strengths by diminishing its high parameter dependence

Population diversity is selected as the system output upon which the adaptive GA approach is based

In order to achieve a closed-loop form for the controller over the Basic GA, two complementary control strategies that operate upon different triggers are implemented

They complement each other such that whenever one of them is triggered, the result causes the other strategy to be triggered.


Conclusion Our usage of steepest descent algorithm as the base of the

training control mechanism is somewhat different from its proposed applications in the literature. Most studies propose climbing heuristics after the GA has converged to various local optima. This strategy can still be implemented over our approach.

Different control mechanisms and triggers can be developed for faster and more effective traversal of the search space. We only provided a certain way of forming a valid closed loop control system.

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A GENETIC A LGORITHM FOR PARALLEL MACHINE TOTAL TARDINESS PROBLEM