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

2 21st European Conference on Operational Research Agenda PROBLEM DESCRIPTION DETERMINATION OF INITIAL SOLUTION -Constructive Algorithms -Polynomial Improvement Heuristics METAHEURISTIC ALGORITHMS COMPUTATIONAL RESULTS CONCLUSIONS

3 21st European Conference on Operational Research 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 }

4 21st European Conference on Operational Research STATEMENT OF THE PROBLEM Fixed standard processing time Fixed relative speed of machine processing time

5 21st European Conference on Operational Research PROBLEM DESCRIPTION Setup times − Sequence-dependent setup times − Machine-dependent setup times No preemption No precedence constraints

6 21st European Conference on Operational Research PROBLEM DESCRIPTION C max + (1- )  T OBJECTIVE: Minimization of a convex combination of makespan and number of tardy jobs:

7 21st European Conference on Operational Research PROBLEM DESCRIPTION OBJECTIVES: Formulation of a mathematical model Development of constructive and iterative algorithms

8 21st European Conference on Operational Research EXACT ALGORITHMS Formulation of a 0-1 mixed integer programming problem Use of the commercial software package (CPLEX and AMPL) Problems with up to five jobs can be solved in acceptable time

9 21st European Conference on Operational Research HEURISTIC ALGORITHMS DETERMINATION OF INITIAL SOLUTION − DISPATCHING RULES − FLOW SHOP MAKESPAN HEURISTCS − POLYNOMIAL IMPROVEMENT HEURISTICS METAHEURISTIC ALGORITHMS − SIMULATED ANNEALING − TABU SEARCH − GENETIC ALGORITHMS

10 21st European Conference on Operational Research DETERMINATION OF INITIAL SOLUTION particular sequencing rule 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

11 21st European Conference on Operational Research 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

12 21st European Conference on Operational Research 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

13 21st European Conference on Operational Research DETERMINATION OF INITIAL SOLUTION FLOW SHOP MAKESPAN HEURISTICS − PALMER (PAL) − CAMPBELL, DUDEK, SMITH (CDS) − GUPTA (GUP) − DANNENBRING (DAN) − NAWAZ, ENSCORE, HAM (NEH) CONSTRUCTIVE ALGORITHMS

14 21st European Conference on Operational Research 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

15 21st European Conference on Operational Research 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

16 21st European Conference on Operational Research 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

17 21st European Conference on Operational Research 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

18 21st European Conference on Operational Research DETERMINATION OF INITIAL SOLUTION Shift Neighborhood − (n-1) 2 neighbors NEIGHBORHOODS 12345

19 21st European Conference on Operational Research DETERMINATION OF INITIAL SOLUTION Pairwise Interchange Neighborhood − n  (n-1)/2 neighbors NEIGHBORHOODS

20 21st European Conference on Operational Research METAHEURISTIC ALGORITHMS Parameters − INITIAL TEMPERATURE , IN STEP OF , 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

21 21st European Conference on Operational Research METAHEURISTIC ALGORITHMS Parameters − NEIGHBORHOOD STRUCTURES Pairwise Interchange neighborhood Shift neighborhood − LENGTH OF TABU LIST 5, 10, 15, 20 − NUMBER OF NEIGHBORS , IN STEP OF 10 TABU SEARCH

22 21st European Conference on Operational Research 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

23 21st European Conference on Operational Research METAHEURISTIC ALGORITHMS − CROSSOVER RATE , IN STEPS OF 0.1 − MUTATION RATE , IN STEPS OF 0.1 GENETIC ALGORITHM

24 21st European Conference on Operational Research METAHEURISTIC ALGORITHMS PMX CROSSOVER Parent 1 Parent 2 Offspring 1 Offspring 2

25 21st European Conference on Operational Research METAHEURISTIC ALGORITHMS OX Based OPX CROSSOVER Parent 1 Parent 2 Offspring 1

26 21st European Conference on Operational Research METAHEURISTIC ALGORITHMS PBX based PMX CROSSOVER Parent 1 Parent 2 Offspring 1 Offspring 2

27 21st European Conference on Operational Research 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

28 21st European Conference on Operational Research COMPUTATIONAL RESULTS DISPATCHING RULES S/P

29 21st European Conference on Operational Research COMPUTATIONAL RESULTS FLOW SHOP HEURISTICS

30 21st European Conference on Operational Research COMPUTATIONAL RESULTS POLYNOMIAL IMPROVEMENT HEURISTICS

31 21st European Conference on Operational Research COMPUTATIONAL RESULTS SA PARAMETERS

32 21st European Conference on Operational Research COMPUTATIONAL RESULTS SA PARAMETERS

33 21st European Conference on Operational Research 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.

34 21st European Conference on Operational Research COMPUTATIONAL RESULTS TS PARAMETERS

35 21st European Conference on Operational Research COMPUTATIONAL RESULTS TS PARAMETERS

36 21st European Conference on Operational Research COMPUTATIONAL RESULTS TS PARAMETERS

37 21st European Conference on Operational Research COMPUTATIONAL RESULTS TS PARAMETERS: -NUMBER OF NEIGHBORS 20 -LENGTH OF TABU LIST 10 -PI IS BETTER THAN SM FOR =0, OTHERWISE SM.

38 21st European Conference on Operational Research COMPUTATIONAL RESULTS GA PARAMETERS

39 21st European Conference on Operational Research COMPUTATIONAL RESULTS GA PARAMETERS

40 21st European Conference on Operational Research COMPUTATIONAL RESULTS GA PARAMETERS

41 21st European Conference on Operational Research 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 21st European Conference on Operational Research COMPUTATIONAL RESULTS COMPARATIVE RESULTS

43 21st European Conference on Operational Research COMPUTATIONAL RESULTS COMPARATIVE RESULTS

44 21st European Conference on Operational Research COMPUTATIONAL RESULTS COMPARATIVE RESULTS

45 21st European Conference on Operational Research 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 21st European Conference on Operational Research 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%

47 21st European Conference on Operational Research 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