21st european conference on operational research algorithms for flexible flow shop problems with...

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 ChaovalitwongseChulalongkorn University, Thailand

Frank Werner Otto-von-Guericke-University,

Germany

EURO XXI in Iceland July 2-5, 2006 EURO XXI in Iceland July 2-5, 2006

221st European Conference on Operational Research

Agenda

• PROBLEM DESCRIPTION

• DETERMINATION OF INITIAL SOLUTION

- Constructive Algorithms

- Polynomial Improvement Heuristics

• METAHEURISTIC ALGORITHMS

• COMPUTATIONAL RESULTS

• CONCLUSIONS


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}


STATEMENT OF THE PROBLEM

• Fixed standard processing time

• Fixed relative speed of machine

processing time

tjpstijv

tij

tjt

ij v

psp


PROBLEM DESCRIPTION

• Setup times−Sequence-dependent setup times−Machine-dependent setup times

• No preemption

• No precedence constraints


PROBLEM DESCRIPTION

Cmax + (1- ) T

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


PROBLEM DESCRIPTION

OBJECTIVES:

• Formulation of a mathematical model

• Development of constructive and iterative algorithms


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


HEURISTIC ALGORITHMS

• DETERMINATION OF INITIAL SOLUTION−DISPATCHING RULES−FLOW SHOP MAKESPAN HEURISTCS−POLYNOMIAL IMPROVEMENT HEURISTICS

• METAHEURISTIC ALGORITHMS−SIMULATED ANNEALING−TABU SEARCH−GENETIC ALGORITHMS


DETERMINATION OF INITIAL SOLUTION

Step 1: Sequence the jobs by using a particular sequencing ruleparticular 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



• 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



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


DISPATCHING RULES



• FLOW SHOP MAKESPAN HEURISTICS −PALMER (PAL)−CAMPBELL, DUDEK, SMITH (CDS)−GUPTA (GUP)−DANNENBRING (DAN)−NAWAZ, ENSCORE, HAM (NEH)

CONSTRUCTIVE ALGORITHMS



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


FLOW SHOP HEURISTCS



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



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



−2-SHIFT MOVES :O (n)−ALL-SHIFT MOVES :O (n2)−2-PAIR INTERCHANGES :O (n)−ALL-PAIR INTERCHANGES :O (n2)




• Shift Neighborhood− (n-1)2 neighbors

NEIGHBORHOODS

1 2 3 4 5



• Pairwise Interchange Neighborhood− n(n-1)/2 neighbors

NEIGHBORHOODS

1 3 5

1 2 3 4 5

2 442


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 : Tnew = Told

• Lundy&Mees : Tnew = Told/(1+Told)

SIMULATED ANNEALING



• 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



• 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



−CROSSOVER RATE• 0.1 - 0.9, IN STEPS OF 0.1

−MUTATION RATE• 0.1 - 0.9, IN STEPS OF 0.1

GENETIC ALGORITHM



PMX CROSSOVER

1 2 3 54

2 1 4 5 3

3

1 2 3 4 5

2 1 4 5 34 5

3 44

312

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



• OX Based

OPX CROSSOVER

1 2 3 54

2 1 4 5 3

1 2 3 54

2 1 4 35

Parent 1

Parent 2

Offspring 1



• PBX based

PMX CROSSOVER

1 2 3 54

2 1 4 5 3

3 42 1 3

1 2 3 54

2 1 4 352 1 4 35

Parent 1

Parent 2

Offspring 1

Offspring 2


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



DISPATCHING RULES

S/P



FLOW SHOP HEURISTICS



SA PARAMETERS



• SA PARAMETERS:

- INITIAL TEMPERATURE T=10

- GEOMETRIC COOLING SCHEME

(TNEW = 0.85 TOLD)

- PI IS BETTER THAN SM FOR =0, OTHERWISE SM.



TS PARAMETERS



• TS PARAMETERS:

- NUMBER OF NEIGHBORS 20

- LENGTH OF TABU LIST 10

- PI IS BETTER THAN SM FOR =0, OTHERWISE SM.



GA PARAMETERS



• 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



COMPARATIVE RESULTS


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.


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%


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.


THANK YOU FOR YOUR ATTENTIONTHANK YOU FOR YOUR ATTENTION

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QUESTIONS AND SUGGESTIONSQUESTIONS AND SUGGESTIONS

21st european conference on operational research algorithms for flexible flow shop problems with...

Documents

european conference

operational research

acceptable time slide

fifo rule

permutation rule

iterative algorithms

processing time hse

date rule mst