job shop schedulingga modifi

8/13/2019 Job Shop SchedulingGA Modifi

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Genetic Algorithms ForJob Shop Scheduling


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Scheduling

Scheduling: Establishes the timing of theuse of equipment, facilities and humanactivities in an organization


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Scheduling

Effective scheduling can yield Cost savings

Increases in productivity

Improved customer satisfaction


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Measures to EvaluatePerformance of a Scheduling

Method Job Flow Time: Time elapsed from the

release of a job until it is completed.

Lateness: Difference between completiontime and due date of a job (may benegative).

Tardiness: The positive difference betweenthe completion time and the due date of a

job. Makespan: Flow time of the job completed

last.


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Job Shop Scheduling

Job : A piece of work that goes through series of

operations.

Shop : A place for manufacturing or repairing of

goods or machinery.

Scheduling : Decision process aiming to deduce the

order of processing.


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Job Shop Scheduling

J M JSSP: J jobs with M machines. Each job has M operations, each operation

requires a particular machine to run. A job j visits in a specified sequence a

subset of the machines

Precedence Constraint; each job has exactordering of operations to follow. Non-preemptible.


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A typical Job Shop Problem

ObjectivesMinimization of make span

Minimization of cost

Minimization of delays

Parameters Number of jobs

Number of operations within each job

Processing time of each operation within each job

Machining sequence of operations within each job


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Job Shop


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Job Shop

set of n jobs which is to be processed on a set of m machines


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Job Shop


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Disjunctive graph

The JSSP can be formally described by adisjunctive graph G = (VC UD ), where

V is a set of nodes representing operationsof the jobs together with two special nodes,a source (0) and a sink * , representing the

beginning and end of the schedule,respectively.


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Disjunctive graph

C is a set of conjunctive arcs representingtechnological sequences of the operations.

D is a set of disjunctive arcs representing pairs of operations that must be performedon the same machines.

The processing time for each operation isthe weighted value attached to thecorresponding nodes.


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Disjunctive graph

The processing of job Jj on machine Mr is called the operation Ojr .

Operation Ojr requires the exclusiveuse of Mr for an uninterrupted duration

pjr , its processing time.


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Disjunctive graph

Job-shop scheduling can also be viewed asdefining the ordering between all operations

that must be processed on the samemachine, i.e. to fix precedences betweenthese operations.

In the disjunctive graph model, this is done by turning all undirected (disjunctive) arcsinto directed ones.


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Disjunctive graph

A selection is a set of directed arcs selectedfrom disjunctive arcs.

A selection is complete if all thedisjunctions are selected.

It is consistent if the resulting directedgraph is acyclic.


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Disjunctive graph

path P is called a critical path and iscomposed of a sequence of critical

operations . A sequence of consecutive critical

operations on the same machine is called a critical block .


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Disjunctive graph

The distance between two schedules S and Tcan be measured by the number of

differences in the processing order ofoperations on each machine

It can be calculated by summing the

disjunctive arcs whose directions aredifferent between S and T . This distance thedisjunctive graph (DG) distance


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Active schedules

An optimal schedule is clearly active so it issafe and efficient to limit the search space to

the set of all active schedules. An active schedule is generatedby the GT

algorithm proposed by Giffler and

Thompson which is described in Algorithm


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Disjunctive graph

In the algorithm, the earliest starting time ES (O) and earliest completion time EC (O)

of an operation O denote its starting andcompletion times when processed with thehighest priority among all currently

schedulable operations on the samemachine.


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Disjunctive graph

An active schedule is obtained by repeatingthe algorithm until all operations are

processed.


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Disjunctive graph


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A 3 job 3 machine problem

J1 M1 M2 M3

J2 M1 M3 M2

J3 M3 M2 M1

Machining Sequence

Jobs Operations

Processing time

Jobs Operations

J1 6 5 3

J2 4 3 4

J3 10 5 2


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Why Nature Inspired Algorithms

Evaluation on a set of points. Better search.

Better chances for globaloptimal solution.

Suitable for Multi-objective optimisation.

Flexibility, becauseconstraints can be taken

care of.

Evaluation on a pointeach time.

Often terminate intolocal optima.

Not suited for multi-objective optimisation.

Not flexible, driven byheuristics, constraints

not handled easily.

GAs vs Other methods


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Issues in Genetic Algorithms when

applied to job shop problemsRepresentation of schedule(phenotype) by suitablegenotype.

Conversion of genotype to phenotype

Choice of Schedule Builder

Type of Crossover and Mutation to be used

Avoiding Premature convergence.


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Schedule Builder!! Whats That

J1 M1 M2 M3

J2 M1 M3 M2

J3 M3 M2 M1

J1 6 5 3

J2 4 3 4

J3 10 5 2


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Representations of JSSP

Direct Chromosome represents a schedule.

e.g. list of starting times. Indirect

A chromosome represents a scheduling

guideline. e.g. list of priority rules, job permutation with

repetition. Risk of false competition


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Priority Rules

First come, first served (FCFS) Last come, first served (LCFS) Earliest due date (EDD) Shortest processing time (SPT) Longest processing time (LPT) Critical ratio (CR):

(Time until due date)/(processing time) Slack per remaining Operations (S/RO)

Slack /(number of remaining operations)


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Indirect Representations

Binary representation Each bit represent orientation of a

disjunctive arc in the disjunctive graph. Job permutation with repetition

Indicate priority of a job to break conflictin GT.

3 3 2 2 1 2 3 1 1


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Indirect Representations (cont.)

Job sequence matrix Similar to permutation, but separated to each

machine.


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JSSP-specific Crossover

Operators Chromosome level

Work directly on chromosomes.

Schedule level Work on decoded schedules, not directly

on chromosomes. Not representation-sensitive


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Chromosome-level Crossover

Subsequence Exchange Crossover (SXX) Job sequence matrix representation. Search for exchangeable subsequences, then

switch ordering. Job-based Ordered Crossover (JOX)

Job sequence matrix representation. Separate jobs into two set, derive ordering

between one set of job from a parent


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Chromosome-level Crossover

(cont.) Precedence Preservative Crossover (PPX)

Permutation with repetition representation.

Use template.

Order-based Giffler and Thompson (OBGT) Uses order-based crossover and mutation

operators, then use GT to repair the offspring


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Schedule-level Crossover

GT Crossover Use GT algorithm.

Randomly select a parent to derive prioritywhen breaking conflict. Time Horizon Exchange (THX)

Select a point in time, offspring retain orderingof operations starting before that point from one

parent, the rest from another.


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1. Binary Representation

Genotype is binary matrix ofRows = Number of job pairs

Columns = Number of machines

InterpretationM ij = 0 / 1 depending on whether job1is executed later or prior to job2.


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Crossover

The crossover applied is simple one point

crossover.

DemeritsRedundancy in representation.

2mj(j-1)/2 bits are required for (!j) m schedules.

Forcing techniques required for replacement of

illegal schedules.

Forcing means the replacement of the original stringwith a feasible one.


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2. Job Based RepresentationTypical chromosome [ J i J j Jk ]

For [ J 2 J1 J3 ]

All operations of job 2 folllowed by 1 and then by 3are scheduled in the available processing times .


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Demerits

Approach is very constrained Not many possibilities are explored

Merits

Scheduling is very easyAlways yields a feasible schedule, hence forcing

not required.


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3. Permutation Representation(Partitioned)

Chromosome is set of permutation of jobs on

each machine.M1 M2 M3

1 3 2 3 2 1 3 1 2

Cross Over (SXX)Subsequence Exchange Crossover

Searches for exchangeable subsequence pairs

in parents, and swaps them.

Job sequence matrix for 3 X 3 problem


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Subsequence Exchange Crossover

M1 M2 M3

P0

1 2 3 6 4 5 3 2 1 5 6 4 2 3 5 6 1 4

P 1 6 2 1 3 4 5 3 2 6 4 5 1 6 3 5 4 2 1

C 0 2 1 3 4 6 5 3 2 5 1 6 4 2 6 3 5 1 4

C 1 6 1 2 3 4 5 3 2 6 4 1 5 3 5 6 4 2 1


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Merits

Demerits

GA operators used for TSP can be applied here

Simple representation

Does not always give active schedules

Robust Schedule builder is requiredSXX does not always guarantee a crossover


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4. Permutation Representation

(Repetitive)Also known as operation based representation

Typical genotype is a unpartitioned permutation

with m repetitions for each job.

1 2 2 3 1 3 2 1 3

M1 1 3 2

M2 2 3 1

M3 2 1 3


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MeritsVery simple representation

All decoding leads to active schedules

Schedule building is straightforward

Crossover results in passing of characteristics from

both parents in most cases.

DemeritsProblem of Premature convergence

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