Algorithms for bicriteria minimization in the permutation flow shop scheduling problem

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January 2011, 7(1): 253-282. doi: 10.3934/jimo.2011.7.253

Algorithms for bicriteria minimization in the permutation flow shop scheduling problem

1.	Departamento de Fundamentos de Economía, Universidad de Alcalá, Alcalá de Henares, Madrid, 28802, Spain

Received August 2009 Revised November 2010 Published January 2011

Abstract
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This paper presents two bi-objective simulated annealing procedures to deal with the classical permutation flow shop scheduling problem considering the makespan and the total completion time as criteria. The proposed methods are based on multi-objective simulated annealing techniques combined with constructive and heuristic algorithms. A computational experiment has been carried out and different metrics have been computed to check various attributes of each method. For all the tested instances a net set of potentially efficient schedules has been obtained and compared with previously published results. Results indicate that the proposed algorithms provide efficient solutions with little computational effort which can serve as input for interactive procedures.

Keywords: Multi-Objective Optimization, Heuristics., Management Decision Making, Deterministic Scheduling Theory, Flow-shop.

Mathematics Subject Classification: Primary: 90B35; 90B50; 90C29; Secondary: 90C5.

Citation: Ethel Mokotoff. Algorithms for bicriteria minimization in the permutation flow shop scheduling problem. Journal of Industrial & Management Optimization, 2011, 7 (1) : 253-282. doi: 10.3934/jimo.2011.7.253

References:

[1]	J. Arroyo and V. Armentano, Genetic local search for multi-objective flowshop scheduling problems,, Eur. J. Oper. Res., 167 (2005), 717. doi: 10.1016/j.ejor.2004.07.017. Google Scholar
[2]	T. P. Bagchi, "Multiobjective Scheduling by Genetic Algorithms,", Kluwer Academic Publishers, (1999). Google Scholar
[3]	P. C. Chang, S. H. Chen and C. H. Liu, Sub-population genetic algorithm with mining gene structures for multiobjective flowshop scheduling problems,, Expert. Syst. Appl., 33 (2007), 762. doi: 10.1016/j.eswa.2006.06.019. Google Scholar
[4]	P. C. Chang, J. C. Hsieh and S. G. Lin, The development of gradual priority weighting approach for the multi-objective flowshop scheduling problem,, Int. J. Prod. Econ., 79 (2002), 171. doi: 10.1016/S0925-5273(02)00141-X. Google Scholar
[5]	P. Czyzak and A. Jaszkiewicz, Pareto Simulated Annealing-a metaheuristic technique for multiple objective combinatorial optimization,, J. Multicriteria Dec. Anal., 7 (1998), 34. doi: 10.1002/(SICI)1099-1360(199801)7:1<34::AID-MCDA161>3.0.CO;2-6. Google Scholar
[6]	R. L. Daniels and R. J. Chambers, Multiobjective flow-shop scheduling,, Nav. Res. Log., 37 (1990), 981. doi: 10.1002/1520-6750(199012)37:6<981::AID-NAV3220370617>3.0.CO;2-H. Google Scholar
[7]	J. Dorn, M. Girsch, G. Skele and W. Slany, Comparison of iterative improvement techniques for schedule optimization,, Eur. J. Oper. Res., 94 (1996), 349. doi: 10.1016/0377-2217(95)00162-X. Google Scholar
[8]	M. Ehrgott and X. Gandibleux, Multiobjective Combinatorial Optimization: Theory, Methodology, and Applications,, in, (2002), 369. Google Scholar
[9]	V. A. Emelichev and V. A. Perepelista, On cardinality of the set of alternatives in discrete many-criterion problems,, Discrete Math. Appl., 2 (1992), 461. doi: 10.1515/dma.1992.2.5.461. Google Scholar
[10]	J. M. Framinan, R. Leisten and R. Ruiz-Usano, Efficient heuristics for flowshop sequencing with the objectives of makespan and flowtime minimisation,, Eur. J. Oper. Res., 141 (2002), 559. doi: 10.1016/S0377-2217(01)00278-8. Google Scholar
[11]	M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness,", Freeman, (1979). Google Scholar
[12]	M. Geiger, On operators and search space topology in multi-objective flow shop scheduling,, Eur. J. Oper. Res., 181 (2007), 195. doi: 10.1016/j.ejor.2006.06.010. Google Scholar
[13]	R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey,, Ann. Discrete Math., 5 (1979), 287. doi: 10.1016/S0167-5060(08)70356-X. Google Scholar
[14]	J. N. D. Gupta, V. R. Neppalli and F. Werner, Minimizing total flow time in a two-machine flowshop problem with minimum makespan,, Int. J. Prod. Econ., 69 (2001), 323. doi: 10.1016/S0925-5273(00)00039-6. Google Scholar
[15]	J. C. Ho and Y. L. Chang, A new heuristic for the n-job, m-machine flowshop problem,, Eur. J. Oper. Res., 52 (1991), 194. doi: 10.1016/0377-2217(91)90080-F. Google Scholar
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[18]	E. Ignall and L. E. Schrage, Application of the branch-and-bound technique to some flow-shop scheduling problems,, Oper. Res., 13 (1965), 400. doi: 10.1287/opre.13.3.400. Google Scholar
[19]	H. Ishibuchi and T. Murata, A multi-objective genetic local search algorithm and its application to flowshop scheduling,, IEEE T. Syst. Man. Cy. C., 28 (1998), 392. doi: 10.1109/5326.704576. Google Scholar
[20]	A. Jaszkiewicz, A comparative study of multiple-objective metaheuristics on the bi-objective set covering problem and the pareto memetic algorithm,, Annals of Operations Research, 13 (2004), 135. doi: 10.1023/B:ANOR.0000039516.50069.5b. Google Scholar
[21]	S. M. Johnson, Optimal two- and three-stage production schedules with setup times included,, Nav. Res. Log., 1 (1954), 61. doi: 10.1002/nav.3800010110. Google Scholar
[22]	D. F. Jones, S. K. Mirrazavi and M. Tamiz, Multi-objective meta-heuristics: An overview of the current state of the art,, Eur. J. Oper. Res., 137 (2002), 1. doi: 10.1016/S0377-2217(01)00123-0. Google Scholar
[23]	S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671. doi: 10.1126/science.220.4598.671. Google Scholar
[24]	J. Knowles and D. Corne, On Metrics Comparing Nondominated Sets,, in, (2002), 711. Google Scholar
[25]	J. D. Landa-Silva, E. K. Burke and S. Petrovic, An introduction to multiobjective metaheuristics for scheduling and timetabling,, Lect. Notes Econ. Math., 535 (2004), 91. doi: 10.1007/978-3-642-17144-4_4. Google Scholar
[26]	C. J. Liao, W. C. Yu and C. B. Joe, Bicriterion scheduling in the two-machine flowshop,, J. Oper. Res. Soc., 48 (1997), 929. Google Scholar
[27]	G. B. McMahon, Optimal production schedules for flow shop,, Can. Oper. Res. Soc. J., 7 (1969), 141. Google Scholar
[28]	G. Minella, R. Ruiz and M. Ciavotta, A review and evaluation of multi-objective algorithms for the flowshop scheduling problem,, Informs. Journal on Computing, 20 (2007), 451. doi: 10.1287/ijoc.1070.0258. Google Scholar
[29]	E. Mokotoff, Multi-objective simulated annealing for permutation flow shop problems,, in, (2009), 101. doi: 10.1007/978-3-642-02836-6_4. Google Scholar
[30]	T. Murata, H. Ishibuchi and H. Tanaka, Multi-objective genetic algorithm and its applications to flowshop scheduling,, Comp. Ind. Eng., 30 (1996), 957. doi: 10.1016/0360-8352(96)00045-9. Google Scholar
[31]	A. Nagar, S. S. Heragu and J. Haddock, A branch and bound approach for a two machine flowshop scheduling problem,, J. Oper. Res. Soc., 46 (1995), 721. Google Scholar
[32]	M. Nawaz, E. E. Jr Enscore and I. Ham, A heuristic algorithm for the m machine, n job flowshop sequencing problem,, OMEGA-Int J. Manage S., 11 (1983), 91. doi: 10.1016/0305-0483(83)90088-9. Google Scholar
[33]	V. L. Neppalli, C. L. Chen and J. N. D. Gupta, Genetic algorithms for the two-stage bicriteria flowshop problem,, Eur. J. Oper. Res., 95 (1996), 356. doi: 10.1016/0377-2217(95)00275-8. Google Scholar
[34]	S. Parthasarathy and C. Rajendran, An experimental evaluation of heuristics for scheduling in a real-life flowshop with sequence-dependent setup times of jobs,, Int. J. Prod. Econ., 49 (1997), 255. doi: 10.1016/S0925-5273(97)00017-0. Google Scholar
[35]	T. Pasupathy, C. Rajendran and R. K. Suresh, A multi-objective genetic algorithm for scheduling in flow shops to minimize the makespan and total flow time of jobs,, Int. J. Adv. Manuf. Technol., 27 (2006), 804. doi: 10.1007/s00170-004-2249-6. Google Scholar
[36]	C. Rajendran, Two-stage flowshop scheduling problem with bicriteria,, J. Oper. Res. Soc., 43 (1992), 879. Google Scholar
[37]	C. Rajendran, Heuristic algorithm for scheduling in a flowshop to minimize total flowtime,, Int. J. Prod. Econ., 29 (1993), 65. doi: 10.1016/0925-5273(93)90024-F. Google Scholar
[38]	C. Rajendran, Heuristics for scheduling in flowshop with multiple objectives,, Eur. J. Oper. Res., 82 (1995), 540. doi: 10.1016/0377-2217(93)E0212-G. Google Scholar
[39]	A. H. G. Rinnooy Kan, "Machine Scheduling problems: Classification, Complexity and Computations,", Martinus Nijhoff, (1976). Google Scholar
[40]	S. Sayin and S. Karabati, A bicriteria approach to the two-machine flow shop scheduling problem,, Eur. J. Oper. Res., 113 (1999), 435. doi: 10.1016/S0377-2217(98)00009-5. Google Scholar
[41]	W. J. Selen and D. D. Hott, A mixed-integer goal-programming formulation of the standard flow-shop scheduling problem,, J. Oper. Res. Soc., 12 (1986), 1121. Google Scholar
[42]	P. Serafini, Simulated annealing for multiple objective optimization problems,, in, (1992), 87. Google Scholar
[43]	F. S. Sivrikaya-Serifoglu and G. Ulusoy, A bicriteria two machine permutation flowshop problem,, Eur. J. Oper. Res., 107 (1998), 414. doi: 10.1016/S0377-2217(97)00338-X. Google Scholar
[44]	N. Srinivas and K. Deb, Multiobjective function optimization using nondominated sorting genetic algorithms,, Evol. Comp., 2 (1995), 221. doi: 10.1162/evco.1994.2.3.221. Google Scholar
[45]	V. T'kindt and J. C. Billaut, "Multicriteria scheduling: Theory, Models and Algorithms,", 2nd edition, (2006). Google Scholar
[46]	V. T'kindt, J. N. D. Gupta and J. C. Billaut, Two machine flowshop scheduling problem with a secondary criterion,, Comp. Oper. Res., 30 (2003), 505. doi: 10.1016/S0305-0548(02)00021-7. Google Scholar
[47]	V. T'kindt, N. Monmarche and F et al. Tercinet, An ant colony optimization algorithm to solve a 2-machine bicriteria flowshop scheduling problem,, Eur. J. Oper. Res., 142 (2002), 250. doi: 10.1016/S0377-2217(02)00265-5. Google Scholar
[48]	E. Taillard, Some efficient heuristic methods for the flow shop sequencing problem,, Eur. J. Oper. Res., 47 (1990), 67. doi: 10.1016/0377-2217(90)90090-X. Google Scholar
[49]	E. Taillard, Benchmark for basic scheduling problems,, Eur. J. Oper. Res., 64 (1993), 278. doi: 10.1016/0377-2217(93)90182-M. Google Scholar
[50]	T. K. Varadharajan and C. Rajendran, A multi-objective simulated-annealing algorithm for scheduling in flowshops to minimize the makespan and total flowtime of jobs,, Eur. J. Oper. Res., 167 (2005), 772. doi: 10.1016/j.ejor.2004.07.020. Google Scholar
[51]	J. M. Wilson, Alternative formulation of a flow shop scheduling problem,, J. Oper. Res. Soc., 40 (1989), 395. Google Scholar
[52]	B. Yagmahan and M. M. Yenisey, Ant colony optimization for multi-objective flow shop scheduling problem,, Comp. Ind. Eng., 54 (2008), 411. doi: 10.1016/j.cie.2007.08.003. Google Scholar
[53]	E. Zitzler, "Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications,", Ph.D thesis, (1999). Google Scholar

show all references

References:

[1]	J. Arroyo and V. Armentano, Genetic local search for multi-objective flowshop scheduling problems,, Eur. J. Oper. Res., 167 (2005), 717. doi: 10.1016/j.ejor.2004.07.017. Google Scholar
[2]	T. P. Bagchi, "Multiobjective Scheduling by Genetic Algorithms,", Kluwer Academic Publishers, (1999). Google Scholar
[3]	P. C. Chang, S. H. Chen and C. H. Liu, Sub-population genetic algorithm with mining gene structures for multiobjective flowshop scheduling problems,, Expert. Syst. Appl., 33 (2007), 762. doi: 10.1016/j.eswa.2006.06.019. Google Scholar
[4]	P. C. Chang, J. C. Hsieh and S. G. Lin, The development of gradual priority weighting approach for the multi-objective flowshop scheduling problem,, Int. J. Prod. Econ., 79 (2002), 171. doi: 10.1016/S0925-5273(02)00141-X. Google Scholar
[5]	P. Czyzak and A. Jaszkiewicz, Pareto Simulated Annealing-a metaheuristic technique for multiple objective combinatorial optimization,, J. Multicriteria Dec. Anal., 7 (1998), 34. doi: 10.1002/(SICI)1099-1360(199801)7:1<34::AID-MCDA161>3.0.CO;2-6. Google Scholar
[6]	R. L. Daniels and R. J. Chambers, Multiobjective flow-shop scheduling,, Nav. Res. Log., 37 (1990), 981. doi: 10.1002/1520-6750(199012)37:6<981::AID-NAV3220370617>3.0.CO;2-H. Google Scholar
[7]	J. Dorn, M. Girsch, G. Skele and W. Slany, Comparison of iterative improvement techniques for schedule optimization,, Eur. J. Oper. Res., 94 (1996), 349. doi: 10.1016/0377-2217(95)00162-X. Google Scholar
[8]	M. Ehrgott and X. Gandibleux, Multiobjective Combinatorial Optimization: Theory, Methodology, and Applications,, in, (2002), 369. Google Scholar
[9]	V. A. Emelichev and V. A. Perepelista, On cardinality of the set of alternatives in discrete many-criterion problems,, Discrete Math. Appl., 2 (1992), 461. doi: 10.1515/dma.1992.2.5.461. Google Scholar
[10]	J. M. Framinan, R. Leisten and R. Ruiz-Usano, Efficient heuristics for flowshop sequencing with the objectives of makespan and flowtime minimisation,, Eur. J. Oper. Res., 141 (2002), 559. doi: 10.1016/S0377-2217(01)00278-8. Google Scholar
[11]	M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness,", Freeman, (1979). Google Scholar
[12]	M. Geiger, On operators and search space topology in multi-objective flow shop scheduling,, Eur. J. Oper. Res., 181 (2007), 195. doi: 10.1016/j.ejor.2006.06.010. Google Scholar
[13]	R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey,, Ann. Discrete Math., 5 (1979), 287. doi: 10.1016/S0167-5060(08)70356-X. Google Scholar
[14]	J. N. D. Gupta, V. R. Neppalli and F. Werner, Minimizing total flow time in a two-machine flowshop problem with minimum makespan,, Int. J. Prod. Econ., 69 (2001), 323. doi: 10.1016/S0925-5273(00)00039-6. Google Scholar
[15]	J. C. Ho and Y. L. Chang, A new heuristic for the n-job, m-machine flowshop problem,, Eur. J. Oper. Res., 52 (1991), 194. doi: 10.1016/0377-2217(91)90080-F. Google Scholar
[16]	J. A. Hoogeveen, "Single-Machine Bicriteria Scheduling,", Ph.D thesis, (1992). Google Scholar
[17]	G. Huang and A. Lim, Fragmental optimization on the 2-machine bicriteria flowshop scheduling problem,, in, (1992), 33. Google Scholar
[18]	E. Ignall and L. E. Schrage, Application of the branch-and-bound technique to some flow-shop scheduling problems,, Oper. Res., 13 (1965), 400. doi: 10.1287/opre.13.3.400. Google Scholar
[19]	H. Ishibuchi and T. Murata, A multi-objective genetic local search algorithm and its application to flowshop scheduling,, IEEE T. Syst. Man. Cy. C., 28 (1998), 392. doi: 10.1109/5326.704576. Google Scholar
[20]	A. Jaszkiewicz, A comparative study of multiple-objective metaheuristics on the bi-objective set covering problem and the pareto memetic algorithm,, Annals of Operations Research, 13 (2004), 135. doi: 10.1023/B:ANOR.0000039516.50069.5b. Google Scholar
[21]	S. M. Johnson, Optimal two- and three-stage production schedules with setup times included,, Nav. Res. Log., 1 (1954), 61. doi: 10.1002/nav.3800010110. Google Scholar
[22]	D. F. Jones, S. K. Mirrazavi and M. Tamiz, Multi-objective meta-heuristics: An overview of the current state of the art,, Eur. J. Oper. Res., 137 (2002), 1. doi: 10.1016/S0377-2217(01)00123-0. Google Scholar
[23]	S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671. doi: 10.1126/science.220.4598.671. Google Scholar
[24]	J. Knowles and D. Corne, On Metrics Comparing Nondominated Sets,, in, (2002), 711. Google Scholar
[25]	J. D. Landa-Silva, E. K. Burke and S. Petrovic, An introduction to multiobjective metaheuristics for scheduling and timetabling,, Lect. Notes Econ. Math., 535 (2004), 91. doi: 10.1007/978-3-642-17144-4_4. Google Scholar
[26]	C. J. Liao, W. C. Yu and C. B. Joe, Bicriterion scheduling in the two-machine flowshop,, J. Oper. Res. Soc., 48 (1997), 929. Google Scholar
[27]	G. B. McMahon, Optimal production schedules for flow shop,, Can. Oper. Res. Soc. J., 7 (1969), 141. Google Scholar
[28]	G. Minella, R. Ruiz and M. Ciavotta, A review and evaluation of multi-objective algorithms for the flowshop scheduling problem,, Informs. Journal on Computing, 20 (2007), 451. doi: 10.1287/ijoc.1070.0258. Google Scholar
[29]	E. Mokotoff, Multi-objective simulated annealing for permutation flow shop problems,, in, (2009), 101. doi: 10.1007/978-3-642-02836-6_4. Google Scholar
[30]	T. Murata, H. Ishibuchi and H. Tanaka, Multi-objective genetic algorithm and its applications to flowshop scheduling,, Comp. Ind. Eng., 30 (1996), 957. doi: 10.1016/0360-8352(96)00045-9. Google Scholar
[31]	A. Nagar, S. S. Heragu and J. Haddock, A branch and bound approach for a two machine flowshop scheduling problem,, J. Oper. Res. Soc., 46 (1995), 721. Google Scholar
[32]	M. Nawaz, E. E. Jr Enscore and I. Ham, A heuristic algorithm for the m machine, n job flowshop sequencing problem,, OMEGA-Int J. Manage S., 11 (1983), 91. doi: 10.1016/0305-0483(83)90088-9. Google Scholar
[33]	V. L. Neppalli, C. L. Chen and J. N. D. Gupta, Genetic algorithms for the two-stage bicriteria flowshop problem,, Eur. J. Oper. Res., 95 (1996), 356. doi: 10.1016/0377-2217(95)00275-8. Google Scholar
[34]	S. Parthasarathy and C. Rajendran, An experimental evaluation of heuristics for scheduling in a real-life flowshop with sequence-dependent setup times of jobs,, Int. J. Prod. Econ., 49 (1997), 255. doi: 10.1016/S0925-5273(97)00017-0. Google Scholar
[35]	T. Pasupathy, C. Rajendran and R. K. Suresh, A multi-objective genetic algorithm for scheduling in flow shops to minimize the makespan and total flow time of jobs,, Int. J. Adv. Manuf. Technol., 27 (2006), 804. doi: 10.1007/s00170-004-2249-6. Google Scholar
[36]	C. Rajendran, Two-stage flowshop scheduling problem with bicriteria,, J. Oper. Res. Soc., 43 (1992), 879. Google Scholar
[37]	C. Rajendran, Heuristic algorithm for scheduling in a flowshop to minimize total flowtime,, Int. J. Prod. Econ., 29 (1993), 65. doi: 10.1016/0925-5273(93)90024-F. Google Scholar
[38]	C. Rajendran, Heuristics for scheduling in flowshop with multiple objectives,, Eur. J. Oper. Res., 82 (1995), 540. doi: 10.1016/0377-2217(93)E0212-G. Google Scholar
[39]	A. H. G. Rinnooy Kan, "Machine Scheduling problems: Classification, Complexity and Computations,", Martinus Nijhoff, (1976). Google Scholar
[40]	S. Sayin and S. Karabati, A bicriteria approach to the two-machine flow shop scheduling problem,, Eur. J. Oper. Res., 113 (1999), 435. doi: 10.1016/S0377-2217(98)00009-5. Google Scholar
[41]	W. J. Selen and D. D. Hott, A mixed-integer goal-programming formulation of the standard flow-shop scheduling problem,, J. Oper. Res. Soc., 12 (1986), 1121. Google Scholar
[42]	P. Serafini, Simulated annealing for multiple objective optimization problems,, in, (1992), 87. Google Scholar
[43]	F. S. Sivrikaya-Serifoglu and G. Ulusoy, A bicriteria two machine permutation flowshop problem,, Eur. J. Oper. Res., 107 (1998), 414. doi: 10.1016/S0377-2217(97)00338-X. Google Scholar
[44]	N. Srinivas and K. Deb, Multiobjective function optimization using nondominated sorting genetic algorithms,, Evol. Comp., 2 (1995), 221. doi: 10.1162/evco.1994.2.3.221. Google Scholar
[45]	V. T'kindt and J. C. Billaut, "Multicriteria scheduling: Theory, Models and Algorithms,", 2nd edition, (2006). Google Scholar
[46]	V. T'kindt, J. N. D. Gupta and J. C. Billaut, Two machine flowshop scheduling problem with a secondary criterion,, Comp. Oper. Res., 30 (2003), 505. doi: 10.1016/S0305-0548(02)00021-7. Google Scholar
[47]	V. T'kindt, N. Monmarche and F et al. Tercinet, An ant colony optimization algorithm to solve a 2-machine bicriteria flowshop scheduling problem,, Eur. J. Oper. Res., 142 (2002), 250. doi: 10.1016/S0377-2217(02)00265-5. Google Scholar
[48]	E. Taillard, Some efficient heuristic methods for the flow shop sequencing problem,, Eur. J. Oper. Res., 47 (1990), 67. doi: 10.1016/0377-2217(90)90090-X. Google Scholar
[49]	E. Taillard, Benchmark for basic scheduling problems,, Eur. J. Oper. Res., 64 (1993), 278. doi: 10.1016/0377-2217(93)90182-M. Google Scholar
[50]	T. K. Varadharajan and C. Rajendran, A multi-objective simulated-annealing algorithm for scheduling in flowshops to minimize the makespan and total flowtime of jobs,, Eur. J. Oper. Res., 167 (2005), 772. doi: 10.1016/j.ejor.2004.07.020. Google Scholar
[51]	J. M. Wilson, Alternative formulation of a flow shop scheduling problem,, J. Oper. Res. Soc., 40 (1989), 395. Google Scholar
[52]	B. Yagmahan and M. M. Yenisey, Ant colony optimization for multi-objective flow shop scheduling problem,, Comp. Ind. Eng., 54 (2008), 411. doi: 10.1016/j.cie.2007.08.003. Google Scholar
[53]	E. Zitzler, "Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications,", Ph.D thesis, (1999). Google Scholar

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2018 Impact Factor: 1.025

American Institute of Mathematical Sciences