Algorithms for flexible flow shop problems with unrelated parallel machines, setup times and dual criteria

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Oct 23, 2013 (3 years and 7 months ago)

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21st European Conference on Operational Research


Algorithms for flexible flow shop problems
with unrelated parallel machines,

setup times and dual criteria

Jitti Jungwattanakit

Manop Reodecha

Paveena Chaovalitwongse

Chulalongkorn University, Thailand

Frank Werner

Otto
-
von
-
Guericke
-
University, Germany

EURO XXI in Iceland July 2
-
5, 2006

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Agenda


PROBLEM DESCRIPTION


DETERMINATION OF INITIAL SOLUTION

-
Constructive Algorithms

-
Polynomial Improvement Heuristics


METAHEURISTIC ALGORITHMS


COMPUTATIONAL RESULTS


CONCLUSIONS


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PROBLEM DESCRIPTION

Flexible flow shop scheduling (FFS):


n

independent jobs;
j



{1, 2, ...,
n
}


k
stages;
t



{1, 2, ...,
k
}


m
t

unrelated parallel machines
;


i



{1, 2, ...,
m
t
}

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STATEMENT OF THE PROBLEM


Fixed standard processing time


Fixed relative speed of machine


processing time


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PROBLEM DESCRIPTION


Setup times


Sequence
-
dependent setup times


Machine
-
dependent setup times


No preemption


No precedence constraints


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PROBLEM DESCRIPTION


C
max

+ (1
-


)


T



OBJECTIVE: Minimization of a convex
combination of makespan and number of
tardy jobs:

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PROBLEM DESCRIPTION

OBJECTIVES:


Formulation of a mathematical model


Development of constructive and iterative
algorithms


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EXACT ALGORITHMS


Formulation of a 0
-
1 mixed integer
programming problem


Use of the commercial software package
(CPLEX 8.0.0 and AMPL)


Problems with up to five jobs can be
solved in acceptable time


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HEURISTIC ALGORITHMS


DETERMINATION OF INITIAL SOLUTION


DISPATCHING RULES


FLOW SHOP MAKESPAN HEURISTCS


POLYNOMIAL IMPROVEMENT HEURISTICS


METAHEURISTIC ALGORITHMS


SIMULATED ANNEALING


TABU SEARCH


GENETIC ALGORITHMS

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DETERMINATION OF INITIAL SOLUTION

Step 1:

Sequence the jobs by using a

particular sequencing rule

(first
-
stage
sequence.

Step 2:

Assign the jobs to the machines at
every stage using the job sequence from
either the First
-
In
-
First
-
Out (FIFO) rule or
the Permutation rule.

Step 3:

Return the best solution.

HEURISTIC SCHEDULE CONSTRUCTION

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DETERMINATION OF INITIAL SOLUTION


DISPATCHING RULES


SPT : Shortest Processing Time rule


LPT : Longest Processing Time rule


ERD : Earliest Release Date rule


EDD : Earliest Due Date rule


MST : Minimum Slack Time rule


S/P : Slack time per Processing time


HSE : Hybrid SPT and EDD rule

CONSTRUCTIVE ALGORITHMS

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DETERMINATION OF INITIAL SOLUTION

Step 1:

Select the representatives of relative
speeds and setup times for every job and every
stage by using the combinations of the min, max
and average data values.

Step 2:

Use
the dispatching rule

to find the
first
-
stage sequence.

Step 3:

Apply the Heuristic Schedule Construction

Step 4:

Return the best solution.

DISPATCHING RULES

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DETERMINATION OF INITIAL SOLUTION


FLOW SHOP MAKESPAN HEURISTICS


PALMER (PAL)


CAMPBELL, DUDEK, SMITH (CDS)


GUPTA (GUP)


DANNENBRING (DAN)


NAWAZ, ENSCORE, HAM (NEH)

CONSTRUCTIVE ALGORITHMS

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DETERMINATION OF INITIAL SOLUTION

Step 1:

Select the representatives of relative
speeds and setup times for every job and every
stage by using the nine combinations.

Step 2:

Use a
flow shop makespan heuristic
(e.g. NEH)

to find the first
-
stage sequence.

Step 3:

Apply the Heuristic Schedule Construction

Step 4:

Return the best solution.

FLOW SHOP HEURISTCS

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DETERMINATION OF INITIAL SOLUTION


Step 1:

Sort the jobs according to non
-
increasing
total operating times (setup + processing times)


Step 2:

Insert the next job according to the
above list in an existing partial job sequence and
take in any step the partial sequence with the
best function value for further extension.




NEH ALGORITHM

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DETERMINATION OF INITIAL SOLUTION

Step 1:

Select the first tardy job in the original job
sequence not yet considered.

Step 2:

Interchange or shift the chosen job
(considering one or more possibilities) and
evaluate the objective function values.

Step 3:

Update the current best job sequence.

Step 4:

Go to Step 1 until all tardy jobs have been
considered.

Step 5:

Return the best job sequence.

POLYNOMIAL IMPROVEMENT HEURISTICS

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DETERMINATION OF INITIAL SOLUTION



2
-
SHIFT MOVES


:O (
n
)


ALL
-
SHIFT MOVES



:O (
n
2
)


2
-
PAIR INTERCHANGES


:O (
n
)


ALL
-
PAIR INTERCHANGES

:O (
n
2
)

POLYNOMIAL IMPROVEMENT HEURISTICS

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DETERMINATION OF INITIAL SOLUTION


Shift Neighborhood



(
n
-
1)
2
neighbors



NEIGHBORHOODS

1

2

3

4

5

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DETERMINATION OF INITIAL SOLUTION


Pairwise Interchange Neighborhood



n

(
n
-
1)/2 neighbors


NEIGHBORHOODS

1

3

5

1

2

3

4

5

2

4

4

2

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METAHEURISTIC ALGORITHMS


Parameters


INITIAL TEMPERATURE


10
-
100, IN STEP OF 10


100
-

1000, IN STEP OF 100


NEIGHBORHOOD STRUCTURES



Pairwise Interchange




Shift neighborhood





COOLING SCHEME


Geometric scheme : T
new

=

T
old


Lundy&Mees


: T
new

= T
old
/(1+

T
old
)

SIMULATED ANNEALING

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METAHEURISTIC ALGORITHMS


Parameters


NEIGHBORHOOD STRUCTURES



Pairwise Interchange neighborhood




Shift neighborhood





LENGTH OF TABU LIST


5, 10, 15, 20


NUMBER OF NEIGHBORS



10
-
50, IN STEP OF 10


TABU SEARCH

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METAHEURISTIC ALGORITHMS


Parameters


POPULATION SIZES



30, 50, 70


CROSSOVER TYPE


PMX :Partially mapped crossover


OPX :Combined order and position
-
based crossover


MUTATION TYPE



Pairwise Interchange Neighborhood



Shift Neighborhood

GENETIC ALGORITHM

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METAHEURISTIC ALGORITHMS


CROSSOVER RATE


0.1
-

0.9, IN STEPS OF 0.1


MUTATION RATE


0.1
-

0.9, IN STEPS OF 0.1

GENETIC ALGORITHM

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METAHEURISTIC ALGORITHMS



PMX CROSSOVER

1

2

3

5

4

2

1

4

5

3

3

1

2

3

4

5

2

1

4

5

3

4

5

3

4

4

3

1

2

5

1

2

3

4

5

2

1

4

3

2

1

4

5

3

4

5

3

1

2

5

Parent 1

Parent 2

Offspring 1

Offspring 2

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METAHEURISTIC ALGORITHMS


OX Based

OPX CROSSOVER

1

2

3

5

4

2

1

4

5

3

1

2

3

5

4

2

1

4

3

5

Parent 1

Parent 2

Offspring 1

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METAHEURISTIC ALGORITHMS


PBX based

PMX CROSSOVER

1

2

3

5

4

2

1

4

5

3

3

4

2

1

3

1

2

3

5

4

2

1

4

3

5

2

1

4

3

5

Parent 1

Parent 2

Offspring 1

Offspring 2

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COMPUTATIONAL RESULTS


STD PROCESSING TIMES: [10, 100]


RELATIVE SPEED: [0.7, 1.3]


SETUP TIMES: [0, 50]


DUE DATES: similar to Rajendran et.al.


10 JOBS 5 STAGES, 30 JOBS 10 STAGES,


50 JOBS 20 STAGES




= 0.00, 0.05, 0.10, 0.50, 1.00

PROBLEM GENERATION

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COMPUTATIONAL RESULTS

DISPATCHING RULES

S/P

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COMPUTATIONAL RESULTS

FLOW SHOP HEURISTICS

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COMPUTATIONAL RESULTS

POLYNOMIAL IMPROVEMENT HEURISTICS

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COMPUTATIONAL RESULTS

SA PARAMETERS

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COMPUTATIONAL RESULTS

SA PARAMETERS

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COMPUTATIONAL RESULTS


SA PARAMETERS:

-
INITIAL TEMPERATURE T=10

-
GEOMETRIC COOLING SCHEME


(T
NEW

= 0.85

T
OLD
)

-

PI

IS BETTER THAN
SM

FOR


=0,
OTHERWISE
SM.

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COMPUTATIONAL RESULTS

TS PARAMETERS

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COMPUTATIONAL RESULTS

TS PARAMETERS

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COMPUTATIONAL RESULTS

TS PARAMETERS

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COMPUTATIONAL RESULTS


TS PARAMETERS:

-
NUMBER OF NEIGHBORS 20

-
LENGTH OF TABU LIST 10

-
PI

IS BETTER THAN
SM

FOR


=0,
OTHERWISE
SM.

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COMPUTATIONAL RESULTS

GA PARAMETERS

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COMPUTATIONAL RESULTS

GA PARAMETERS

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COMPUTATIONAL RESULTS

GA PARAMETERS

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COMPUTATIONAL RESULTS


GA PARAMETERS:

-
POPULATION SIZE 30

-
CROSSOVER:

OPX

IS BETTER THAN
PMX

-
CROSSOVER RATE 0.8

-
MUTATION:
PI

IS BETTER THAN
SM

FOR


=0,
OTHERWISE
SM.

-
MUTATION RATE 0.5


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COMPUTATIONAL RESULTS

COMPARATIVE RESULTS

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COMPUTATIONAL RESULTS

COMPARATIVE RESULTS

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COMPUTATIONAL RESULTS

COMPARATIVE RESULTS

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CONCLUSIONS


CONSTRUCTIVE ALGORITHMS: THE
NEH

RULE OUTPERFORMS THE OTHER
ALGORITHMS


DISPATCHING RULES: THE
HSE

RULE
OUTPERFORMS THE OTHERS FOR


= 0,
OTHERWISE THE
LPT

RULE IS BEST.

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CONCLUSIONS


POLYNOMIAL IMPROVEMENT HEURISTICS:


--

O(n) ALGORITHMS:

2
-
PI

OUTPERFORMS
2
-
SM

FOR


= 0, BUT
2
-
SM

BECOMES
BETTER THAN
2
-
PI

FOR


> 0,

THE APD IS REDUCED BY ABOUT 50 %


--

O(n
2
) ALGORITHMS:

A
-
PI

OUTPERFORMS
A
-
SM
.

THE APD IS REDUCED BY ABOUT 70%

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CONCLUSIONS


COMPARATIVE TESTS:
:


-

RSA

IS BETTER THAN
RTS

AND
RGA

-

C
-
SA

IS BETTER THAN
C
-
TS

AND
C
-
GA
,

-

MIF
-
GA

IS BETTER THAN THE OTHERS FOR THE
50
-
JOB PROBLEMS.

21st European Conference on Operational Research


THANK YOU FOR YOUR ATTENTION

------------------------------

QUESTIONS AND SUGGESTIONS