A GENETIC ALGORITHM FOR PARALLEL MACHINE TOTAL TARDINESS PROBLEM

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A GENETIC ALGORITHM FOR PARALLEL MACHINE TOTAL
TARDINESS PROBLEM

• M. Furkan Kiraç
• Ümit Bilge
• Müjde Kurtulan
• Department of Industrial Engineering
• Bogaziçi University

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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.

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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

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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 max0, Ci - di
• Ci is the completion time of job i

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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.

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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

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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 1s and 0s
  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.

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Crossover operator
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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

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Transient Population Elimination
Best 53 individuals preserved

• Basic
• Elimination

48 individuals eliminated
Worst 2 individuals eliminated
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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

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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

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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

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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

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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

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Control for Population Diversity
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Effect of Adaptive Diversity Control
Bar charts showing the population distribution
BEFORE The 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
of individuals 22
tardiness 800000
tardiness 160000
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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

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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

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Effect of Training Control
AFTER In other words, the function of training
can be defined as decreasing the skewness in the
population distribution.
BEFORE 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
of individuals 25
of individuals 26
tardiness 500000
tardiness 180000
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Effect of Diversity andTraining Control
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Experimentation

• The problem set used for experimentation consists
  of parallel machine scheduling problems of 40,
  and 60 jobs, developed and tested by
  Sivrikaya-Serifoglu, F. and G.Ulusoy to study a
  GA
• The same problem set is addressed by Bilge,Ü.,
  F.Kiraç, 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.

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Performance Measure

• 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
  basis
• where,
• i 1, 2, 3, 4, 5 denotes different replications
• j 1, 2, , 20 denotes the instance number in a
  given problem set

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Performance Ratio (PR)
Best Known Result
?TS
?GA

• 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 possible
• A ratio greater than 1.0 means that the GAs
  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 .

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Performance of Adaptive GA
Problem Set Basic GA Performance Ratio Adaptive GA Performance Ratio Times Better
40 Job 2 Machine 11.329 0.809 14.00
60 Job 2 Machine 6.534 0.591 11.06

40 Job 4 Machine 7.065 1.121 6.30
60 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)
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Improved Best Known Results
Those values marked with a () are contributed by
the adaptive GA algorithm devised in this study
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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.

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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.