Jitti Jungwattanakit Manop Reodecha Paveena Chaovalitwongse Chulalongkorn University, Thailand

1. 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

2. Agenda • PROBLEM DESCRIPTION • DETERMINATION OF INITIAL SOLUTION • Constructive Algorithms • Polynomial Improvement Heuristics • METAHEURISTIC ALGORITHMS • COMPUTATIONAL RESULTS • CONCLUSIONS

3. PROBLEM DESCRIPTION Flexible flow shop scheduling (FFS): • n independent jobs; j {1, 2, ..., n} • k stages; t {1, 2, ..., k} • mtunrelated parallel machines; i {1, 2, ..., mt}

4. STATEMENT OF THE PROBLEM • Fixed standard processing time • Fixed relative speed of machine processing time

5. PROBLEM DESCRIPTION • Setup times • Sequence-dependent setup times • Machine-dependent setup times • No preemption • No precedence constraints

6. PROBLEM DESCRIPTION • OBJECTIVE: Minimization of a convex combination of makespan and number of tardy jobs: Cmax+ (1- )T

7. PROBLEM DESCRIPTION OBJECTIVES: • Formulation of a mathematical model • Development of constructive and iterative algorithms

8. 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

9. HEURISTIC ALGORITHMS • DETERMINATION OF INITIAL SOLUTION • DISPATCHING RULES • FLOW SHOP MAKESPAN HEURISTCS • POLYNOMIAL IMPROVEMENT HEURISTICS • METAHEURISTIC ALGORITHMS • SIMULATED ANNEALING • TABU SEARCH • GENETIC ALGORITHMS

10. 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.

11. DETERMINATION OF INITIAL SOLUTION CONSTRUCTIVE ALGORITHMS • 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

12. 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.

13. DETERMINATION OF INITIAL SOLUTION CONSTRUCTIVE ALGORITHMS • FLOW SHOP MAKESPAN HEURISTICS • PALMER (PAL) • CAMPBELL, DUDEK, SMITH (CDS) • GUPTA (GUP) • DANNENBRING (DAN) • NAWAZ, ENSCORE, HAM (NEH)

14. 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.

15. 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.

16. 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.

17. DETERMINATION OF INITIAL SOLUTION POLYNOMIAL IMPROVEMENT HEURISTICS • 2-SHIFT MOVES :O (n) • ALL-SHIFT MOVES :O (n2) • 2-PAIR INTERCHANGES :O (n) • ALL-PAIR INTERCHANGES :O (n2)

18. DETERMINATION OF INITIAL SOLUTION NEIGHBORHOODS • Shift Neighborhood • (n-1)2 neighbors 1 2 3 4 5

19. DETERMINATION OF INITIAL SOLUTION NEIGHBORHOODS • Pairwise Interchange Neighborhood • n(n-1)/2 neighbors 1 2 2 3 4 4 5 1 2 3 4 5

20. 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)

21. 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

22. 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

23. 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

24. 1 2 4 5 3 1 1 2 2 3 3 4 4 5 5 2 1 3 4 5 2 2 1 1 4 4 5 5 3 3 METAHEURISTIC ALGORITHMS PMX CROSSOVER 1 2 3 3 4 4 4 5 Parent 1 3 Offspring 1 5 Offspring 2 2 2 1 1 4 4 5 5 3 3 Parent 2

25. 2 1 2 1 3 4 4 5 3 5 METAHEURISTIC ALGORITHMS OPX CROSSOVER • OX Based 1 2 3 4 5 Parent 1 Offspring 1 2 1 4 5 3 Parent 2

26. 1 2 2 1 1 2 4 3 4 5 5 4 3 3 5 METAHEURISTIC ALGORITHMS PMX CROSSOVER • PBX based 1 2 3 4 5 Parent 1 2 1 3 4 3 Offspring 1 Offspring 2 2 1 4 5 3 Parent 2

27. 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

28. S/P COMPUTATIONAL RESULTS DISPATCHING RULES

29. COMPUTATIONAL RESULTS FLOW SHOP HEURISTICS

30. COMPUTATIONAL RESULTS POLYNOMIAL IMPROVEMENT HEURISTICS

31. COMPUTATIONAL RESULTS SA PARAMETERS

32. COMPUTATIONAL RESULTS SA PARAMETERS

33. COMPUTATIONAL RESULTS • SA PARAMETERS: • INITIAL TEMPERATURE T=10 • GEOMETRIC COOLING SCHEME (TNEW = 0.85 TOLD) - PI IS BETTER THAN SM FOR  =0, OTHERWISE SM.

34. COMPUTATIONAL RESULTS TS PARAMETERS

35. COMPUTATIONAL RESULTS TS PARAMETERS

36. COMPUTATIONAL RESULTS TS PARAMETERS

37. COMPUTATIONAL RESULTS • TS PARAMETERS: • NUMBER OF NEIGHBORS 20 • LENGTH OF TABU LIST 10 • PI IS BETTER THAN SM FOR  =0, OTHERWISE SM.

38. COMPUTATIONAL RESULTS GA PARAMETERS

39. COMPUTATIONAL RESULTS GA PARAMETERS

40. COMPUTATIONAL RESULTS GA PARAMETERS

41. 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

42. COMPUTATIONAL RESULTS COMPARATIVE RESULTS

43. COMPUTATIONAL RESULTS COMPARATIVE RESULTS

44. COMPUTATIONAL RESULTS COMPARATIVE RESULTS

45. 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.

46. 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%

47. 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.

48. THANK YOU FOR YOUR ATTENTION ------------------------------ QUESTIONS AND SUGGESTIONS