Transcript 21st European Conference on Operational Research

21st European Conference on Operational Research
Algorithms for flexible flow shop problems
with unrelated parallel machines,
setup times and dual criteria
Frank Werner
Otto-von-Guericke-University, Germany
Jitti Jungwattanakit
Manop Reodecha
Paveena Chaovalitwongse
Chulalongkorn University, Thailand
EURO XXI in Iceland July 2-5, 2006
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}
• mt unrelated parallel machines;
i  {1, 2, ..., mt}
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STATEMENT OF THE PROBLEM
• Fixed standard processing time
ps
• Fixed relative speed of machine
t
ij
t
j
v
processing time
p 
t
ij
ps
t
j
t
ij
v
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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
• OBJECTIVE: Minimization of a convex
combination of makespan and number of
tardy jobs:
Cmax + (1- ) T
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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
HEURISTIC SCHEDULE CONSTRUCTION
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.
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DETERMINATION OF INITIAL SOLUTION
CONSTRUCTIVE ALGORITHMS
• DISPATCHING RULES
− SPT
− LPT
− ERD
− EDD
− MST
− S/P
− HSE
: Shortest Processing Time rule
: Longest Processing Time rule
: Earliest Release Date rule
: Earliest Due Date rule
: Minimum Slack Time rule
: Slack time per Processing time
: Hybrid SPT and EDD rule
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DETERMINATION OF INITIAL SOLUTION
DISPATCHING RULES
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.
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DETERMINATION OF INITIAL SOLUTION
CONSTRUCTIVE ALGORITHMS
• FLOW SHOP MAKESPAN HEURISTICS
− PALMER (PAL)
− CAMPBELL, DUDEK, SMITH (CDS)
− GUPTA (GUP)
− DANNENBRING (DAN)
− NAWAZ, ENSCORE, HAM (NEH)
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DETERMINATION OF INITIAL SOLUTION
FLOW SHOP HEURISTCS
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.
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DETERMINATION OF INITIAL SOLUTION
NEH ALGORITHM
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.
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DETERMINATION OF INITIAL SOLUTION
POLYNOMIAL IMPROVEMENT HEURISTICS
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.
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DETERMINATION OF INITIAL SOLUTION
POLYNOMIAL IMPROVEMENT HEURISTICS
− 2-SHIFT MOVES
− ALL-SHIFT MOVES
− 2-PAIR INTERCHANGES
− ALL-PAIR INTERCHANGES
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:O (n)
:O (n2)
:O (n)
:O (n2)
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DETERMINATION OF INITIAL SOLUTION
NEIGHBORHOODS
• Shift Neighborhood
− (n-1)2 neighbors
1
2
3
4
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DETERMINATION OF INITIAL SOLUTION
NEIGHBORHOODS
• Pairwise Interchange Neighborhood
− n(n-1)/2 neighbors
1
2
3
4
5
1
2
3
4
5
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METAHEURISTIC ALGORITHMS
SIMULATED ANNEALING
• 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 : Tnew = Told
• Lundy&Mees
: Tnew = Told/(1+Told)
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METAHEURISTIC ALGORITHMS
TABU SEARCH
• 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
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METAHEURISTIC ALGORITHMS
GENETIC ALGORITHM
• 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
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METAHEURISTIC ALGORITHMS
GENETIC ALGORITHM
− CROSSOVER RATE
• 0.1 - 0.9, IN STEPS OF 0.1
− MUTATION RATE
• 0.1 - 0.9, IN STEPS OF 0.1
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METAHEURISTIC ALGORITHMS
PMX CROSSOVER
Parent 1
1
2
3
4
5
Offspring 1
1
2
4
5
3
Offspring 2
2
1
3
4
5
Parent 2
2
1
4
5
3
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METAHEURISTIC ALGORITHMS
OPX CROSSOVER
• OX Based
Parent 1
1
2
3
4
5
2
1
4
5
3
Offspring 1
Parent 2
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METAHEURISTIC ALGORITHMS
PMX CROSSOVER
• PBX based
Parent 1
1
2
3
4
5
Offspring 1
2
1
3
4
3
2
1
4
5
3
Offspring 2
Parent 2
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COMPUTATIONAL RESULTS
PROBLEM GENERATION
• 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
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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
(TNEW = 0.85 TOLD)
- 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(n2) 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.
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21st European Conference on Operational Research
THANK YOU FOR YOUR ATTENTION
-----------------------------QUESTIONS AND SUGGESTIONS