Solving flow shop scheduling problems by quantum differential evolutionary algorithm

  • Tianmin Zheng
  • Mitsuo Yamashiro


This paper proposed a novel quantum differential evolutionary algorithm (QDEA) based on the basic quantum-inspired evolutionary algorithm (QEA) for permutation flow shop scheduling problem (PFSP). In this QDEA, the quantum chromosomes are encoded and decoded by using the quantum rotating angle and a simple strategy named largest rotating angle value rule to determine job sequence based on job’s quantum information is proposed for the representation of PFSP, firstly. Then, we merge the advantages of differential evolution strategy, variable neighborhood search and QEA by adopting the differential evolution to perform the updating of quantum gate and variable neighborhood search to raise the performance of the local search. We adopted QDEA to minimize the makespan, total flowtime and the maximum lateness of jobs and make the simulations. The results and comparisons with other algorithms based on famous benchmarks demonstrated the effectiveness of the proposed QDEA. Another contribution of this paper is to report new absolute values of total flowtime and maximum lateness for various benchmark problem sets.


Permutation flow shop scheduling Quantum-inspired evolutionary algorithm Differential evolution Variable neighborhood search 


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  1. 1.
    Garey M, Johnson D, Sethi R (1976) The complexity of flowshop and jobshop scheduling. Math Oper Res 24(1):117–129CrossRefMathSciNetGoogle Scholar
  2. 2.
    Nawaz M, Enscore E Jr, Ham I (1983) A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. OMEGA 11:91–95CrossRefGoogle Scholar
  3. 3.
    Kalczynski PJ, Kaznburowski J (2007) On the NEH heuristic for minimizing the makespan in permutation flow shops. OMEGA 35:53–60CrossRefGoogle Scholar
  4. 4.
    Murata T, Ishibuchi H, Tanaka H (1996) Genetic algorithms for flowshop scheduling problems. Comput Ind Eng 30(4):1061–1071CrossRefGoogle Scholar
  5. 5.
    Reeves CR, Yamada T (1998) Genetic algorithms, path relinking and the flowshop sequencing problem. Evol Comput 6:45–60CrossRefGoogle Scholar
  6. 6.
    Iyer SK, Saxena B (2004) Improved genetic algorithm for the permutation flowshop scheduling problem. Comput Oper Res 34(4):593–606CrossRefMathSciNetGoogle Scholar
  7. 7.
    Doyen A, Engin O, Ozkan C (2003) A new artificial immune system approach to solve permutation flow shop scheduling problems. Tukish Symposium on Artificial Immune System and Neural Networks TAINN’03.Google Scholar
  8. 8.
    Nowicki E, Smutnicki C (1996) A fast tabu search algorithm for the permutation flow-shop problem. Eur J Oper Res 91:160–175zbMATHCrossRefGoogle Scholar
  9. 9.
    Grabowski J, Wodecki M (2004) A very fast tabu search algorithm for the permutation flow shop problem with makespan criterion. Comput Oper Res 31:1891–1909zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Osman IH, Potts CN (1989) Simulated annealing for permutation flow-shop scheduling. OMEGA 17:551–557CrossRefGoogle Scholar
  11. 11.
    Ishibuchi H, Misaki S, Tanaka H (1995) Modified simulated annealing algorithms for the flow shop sequencing problem. Eur J Oper Res 81(2):388–398zbMATHCrossRefGoogle Scholar
  12. 12.
    Merkle D, Middendorf M (2001) A new approach to solve permutation scheduling problems with ant colony optimization. Lect Notes Comput Sci 2037:484–494CrossRefGoogle Scholar
  13. 13.
    Yinga K-C, Liao C-J (2004) An ant colony system for permutation flow-shop sequencing. Comput Oper Res 31(5):791–801CrossRefGoogle Scholar
  14. 14.
    Rameshkumar K, Suresh RK, Mohanasundaram KM (2005) Discrete particle swarm optimization (DPSO) algorithm for permutation flowshop scheduling to minimize makespan. Adv Nat Comp 3612:572–581CrossRefGoogle Scholar
  15. 15.
    Jarboui B, Ibrahim S, Siarry P, Rebai A (2008) A combinatorial particle swarm optimisation for solving permutation flowshop problems. Comput Ind Eng 54(3):526–538CrossRefGoogle Scholar
  16. 16.
    Stützle T (1998) Applying iterated local search to the permutation flow shop problem. Technical report, AIDA-98-04, FG Intellektik, TU DarmstadtGoogle Scholar
  17. 17.
    Ruiz R, Stützle T (2007) A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. Eur J Oper Res 177(3):2033–2049zbMATHCrossRefGoogle Scholar
  18. 18.
    Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82CrossRefGoogle Scholar
  19. 19.
    Zheng D, Wang L (2003) An effective hybrid heuristic for flow shop scheduling. Int J Adv Manuf Technol 21(1):38–44CrossRefGoogle Scholar
  20. 20.
    Wang L, Zhang L, Zheng Da-Zhong (2006) An effective hybrid genetic algorithm for flow shop scheduling with limited buffers. Comput Oper Res 33(10):2960–2971zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Liu Z, Wang S (2006) Hybrid Particle Swarm Optimization for Permutation Flow Shop Scheduling. Intelligent Control and Automation, 2006. WCICA 2006. The Sixth World Congress on. 1, pp. 3245–3249Google Scholar
  22. 22.
    Chandrasekaran, S. Ponnambalam, S.G..Suresh, R.K. Vijayakumar (2006) A Hybrid Discrete Particle Swarm Optimization Algorithm to Solve Flow Shop Scheduling Problems. Cybernetics and Intelligent Systems, 2006 IEEE Conference on, pp.1–6.Google Scholar
  23. 23.
    Tavakkoli-Moghaddama R, Rahimi-Vaheda A, Mirzaei AH (2007) A hybrid multi-objective immune algorithm for a flow shop scheduling problem with bi-objectives: weighted mean completion time and weighted mean tardiness. Inf Sci 177(22):5072–5090CrossRefGoogle Scholar
  24. 24.
    Andreas AC, Nearchou C (2004) A novel metaheuristic approach for the flow shop scheduling problem. Eng Appl Artif Intell 17(3):289–300CrossRefGoogle Scholar
  25. 25.
    Talbi E-G, Rahoual M, Mabed MH, Dhaenens C (2001) A hybrid evolutionary approach for multicriteria optimization problems: application to the flow shop. Lect Notes Comput Sci 1993:416–428CrossRefGoogle Scholar
  26. 26.
    Ponnambalam SG, Jagannathan H, Kataria M, Gadicherla A (2009) A TSP-GA multi-objective algorithm for flow-shop scheduling. Int J Adv Manuf Technol 23(11–12):909–915Google Scholar
  27. 27.
    Han K-H (2000) Genetic quantum algorithm and its application to combinatorial optimization problem. In: IEEE Proc. Of the 2000 Congress on Evolutionary Computation, San Diego, USA IEEE PressGoogle Scholar
  28. 28.
    KH Han, KH Park, CH Lee, JH Kim Parallel quantum-inspired genetic algorithm for combinatorial optimization problem。Evolutionary Computation, 2001. Proceedings of the 2001.Google Scholar
  29. 29.
    KH Han, JH Kim, On setting the parameters of quantum-inspired evolutionary algorithm for practical application Evolutionary Computation, 2003. CEC’03. The 2003 Congress on 2003.Google Scholar
  30. 30.
    Han K-H, Kim J-H. Quantum-inspired Evolutionary Algorithm for a Class of Combinatorial Optimization. IEEE Trans on Evolutionary Computation, 2002.Google Scholar
  31. 31.
    Han K-H, Kim J-H. Quantum-inspired Evolutionary Algorithms with a New Termination Criterion H,Gate and Two-Phase Scheme. IEEE Trans on Evolutionary Computation 2004.Google Scholar
  32. 32.
    Wang Ling, Wu Hao, and Zheng Da-Zhong (2005) A quantum-inspired genetic algorithm for scheduling problems. Lecture Notes in Computer Science, v 3612, n PART III. Advances in Natural Computation: First International Conference, ICNC 2005. Proceedings, pp. 417–423.Google Scholar
  33. 33.
    Wang L, Wu H, Tang F, Zheng DZ (2005) A hybrid quantum-inspired genetic algorithm for flow shop scheduling. Lect Notes Comput Sci 3645:636–644CrossRefGoogle Scholar
  34. 34.
    Li Bin-Bin and Wang Ling (2006) A hybrid quantum-inspired genetic algorithm for multi-objective scheduling. Lecture Notes in Computer Science, v 4113 LNCS—I, International Conference on Intelligent Computing, ICIC 2006, Proceedings, pp. 511–522.Google Scholar
  35. 35.
    Bean JC (1994) Genetics and random keys for sequencing and optimization. ORSA J Comput 6(2):154–160zbMATHGoogle Scholar
  36. 36.
    Storn R, Price K (1999) Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95-012, ICSI.Google Scholar
  37. 37.
    Storn R, Price K (1997) Differential evolution—a simple evolution strategy for fast optimization. Dr. Dobb’s J 78:18–24Google Scholar
  38. 38.
    Pan QK, Tasgetiren MF, Liang YC (2008) A discrete differential evolution algorithm for the permutation flowshop scheduling problem. Comput Ind Eng 55(4):795–816CrossRefGoogle Scholar
  39. 39.
    Qian B, Wang L, Rong Hu, Wang W-L, Huang De-Xian, Wang X (2008) A hybrid differential evolution method for permutation flow-shop scheduling. Int J Adv Manuf Technol 38(5–6):757–777CrossRefGoogle Scholar
  40. 40.
    Hansen P, Mladenovic N (2003) A tutorial on variable neighborhood search. GERAD and Mathematical Institute, SANUGoogle Scholar
  41. 41.
    Ong YS, Lim MH, Zhu N (2006) Classification of adaptive memetic algorithms: a comparative study. IEEE Transact Sys, Man and Cyb-B: Cyb 36:141–152CrossRefGoogle Scholar
  42. 42.
    Hart W E, Krasnogor N, Smith J E. ( 2004) Recent advances in memetic algorithms. Springer: Heidelberg.Google Scholar
  43. 43.
    Carlier J (1978) Ordonnancements a contraintes disjonctives. Rech Opér 12(4):333–350zbMATHMathSciNetGoogle Scholar
  44. 44.
    Reeves CR (1995) A genetic algorithm for flowshop sequencing. Comp Ope Res 22(1):5–13zbMATHCrossRefGoogle Scholar
  45. 45.
    Heller J (1960) Some numerical experiments for an MxJ flow shop and its decision-theoretical aspects. Oper Res 8:178–184zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Taillard E (1993) Benchmarks for basic scheduling problems. Eur J Oper Res 64(2):278–285zbMATHCrossRefGoogle Scholar
  47. 47.
    Demirkol E, Mehta S, Uzsoy R (1998) Benchmarks for shop scheduling problems. Eur J Oper Res 109:137–141zbMATHCrossRefGoogle Scholar
  48. 48.
    Tasgetiren MF, Liang YC, Sevkli M, Gencyilmaz G. (2004) Particle swarm optimization algorithm for makespan and maximum lateness minimization in permutation flowshop sequencing problem. In: Proceedings of the fourth international symposium on intelligent manufacturing systems. Turkey: Sakarya. pp. 431–41.Google Scholar
  49. 49.
    Liao C-J, Tseng C-T, Luarn P (2007) A discrete version of particle swarm optimization for flowshop scheduling problems. Comput Oper Res 34(10):3099–3111zbMATHCrossRefGoogle Scholar
  50. 50.
    Liu J, Reeves CR (2001) Constructive and composite heuristic solutions to the scheduling P//ΣCi problem. Eur J Oper Res 132:439–452zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Rajendran C, Ziegler H (2004) Ant-colony algorithms for permutation flowshop scheduling to minimize makespan/total flowtime of jobs. Eur J Oper Res 155:426–438zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Rajendran C, Ziegler H (2005) Two ant-colony algorithms for minimizing total flowtime in permutation flowshops. Comput Ind Eng 48(4):789–797CrossRefGoogle Scholar
  53. 53.
    Rajendran C, Chaudhuri D (1991) A flowshop scheduling algorithm to minimize maximum lateness. J Oper Res Soc Jpn 34:28–45zbMATHGoogle Scholar
  54. 54.
    Laha D, Sarin SC (2009) A heuristic to minimize total flow time in permutation flow shop. Omega 37(3):734–739CrossRefGoogle Scholar
  55. 55.
    Framinan JM, Leisten R (2003) An efficient constructive heuristic for flowtime minimisation in permutation flow shops. Omega 31(4):311–317CrossRefGoogle Scholar
  56. 56.
    Rad SF, Ruiz R, Boroojerdian N (2009) New high performing heuristics for minimizing makespan in permutation flowshops. Omega 37(2):331–345CrossRefGoogle Scholar
  57. 57.
    Zhang C, Sun J (2009) An alternate two phases particle swarm optimization algorithm for flow shop scheduling problem. Expert Systems Appl 36(3):5162–5167CrossRefGoogle Scholar
  58. 58.
    Changsheng Zhang, Jiaxu Ning and Dantong Ouyang (2009) A hybrid alternate two phases particle swarm optimization algorithm for flow shop scheduling problem. Comp Ind Eng (in press)Google Scholar
  59. 59.
    Rajkumar R, Shahabudeen P (2009) Scheduling jobs on flowshop environment applying simulated annealing algorithm. Int Jour Serv Op Info 4(3):212–231Google Scholar
  60. 60.
    Rajkumar R, Shahabudeen P (2009) An improved genetic algorithm for the flowshop scheduling problem. Int J Prod Res 47(1):233–249zbMATHCrossRefGoogle Scholar
  61. 61.
    Zobolas GI, Tarantilisa CD, Ioannoua G (2009) Minimizing makespan in permutation flow shop scheduling problems using a hybrid metaheuristic algorithm. Comput Oper Res 36(4):1249–1267zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Qian B, Wang L, Huang DX, Wang X (2009) An effective hybrid DE-based algorithm for flow shop scheduling with limited buffers. Int J Prod Res 47(1):1–24CrossRefMathSciNetGoogle Scholar
  63. 63.
    Tseng L-Y, Lin Y-T (2009) A hybrid genetic local search algorithm for the permutation flowshop scheduling problem. Eur J Oper Res 198(1):84–92zbMATHCrossRefGoogle Scholar
  64. 64.
    Laha D, Chakraborty UK (2009) An efficient hybrid heuristic for makespan minimization in permutation flow shop scheduling. Int J Adv Manuf Technol 44(5–6):559–569CrossRefGoogle Scholar
  65. 65.
    Kuo I-Hong, Horng S-J, Kao T-W, Lin T-L, Lee C-L, Terano T, Pan Y (2009) An efficient flow-shop scheduling algorithm based on a hybrid particle swarm optimization model. Expert Systems Appl 36(2):7027–7032CrossRefGoogle Scholar
  66. 66.
    Lian Z, Gu X, Jiao B (2008) A novel particle swarm optimization algorithm for permutation flow-shop scheduling to minimize makespan. Chaos, Solitons Fractals 35:851–861zbMATHCrossRefGoogle Scholar
  67. 67.
    Zhang Y, Li X, Wang Q (2009) Hybrid genetic algorithm for permutation flowshop scheduling problems with total flowtime minimization. Eur J Oper Res 196(3):869–876zbMATHCrossRefGoogle Scholar
  68. 68.
    PC Chang, WH Huang, CJ Ting, LC Wu, CM Lai (2009) A Hybrid Genetic-Immune Algorithm with Improved Offsprings and Elitist Antigen for Flow-shop Scheduling Problems. 11th IEEE International Conference on High Performance Computing and Communications.Google Scholar
  69. 69.
    Bożejko W (2009) Solving the flow shop problem by parallel programming. J Parallel Distrib Comput 69(5):470–481CrossRefGoogle Scholar
  70. 70.
    Vallada E, Ruiz R (2009) Cooperative metaheuristics for the permutation flowshop scheduling problem. Eur J Oper Res 193(2):365–376zbMATHCrossRefGoogle Scholar
  71. 71.
    Bai D, Tang L (2009) New heuristics for flow shop problem to minimize makespan. J Oper Res Soc (in press)Google Scholar
  72. 72.
    Naderi B, Tavakkoli-Moghaddam R, Khalili M (2009) Electromagnetism-like mechanism and simulated annealing algorithms for flowshop scheduling problems minimizing the total weighted tardiness and makespan. Knowledge-Based Systems (in press).Google Scholar
  73. 73.
    Li X, Wang Q, Cheng Wu (2009) Efficient composite heuristics for total flowtime minimization in permutation flow shops. Omega 37(1):155–164CrossRefGoogle Scholar
  74. 74.
    Sheibani K (2009) A fuzzy greedy heuristic for permutation flow-shop scheduling. J Oper Res Soc (in press)Google Scholar
  75. 75.
    Chin-Chia Wu and Wen-Chiung Lee (2009) A note on the total completion time problem in a permutation flowshop with a learning effect. Eur J Oper Res 192(1):343–347CrossRefGoogle Scholar
  76. 76.
    Rahimi-Vahed A, Dangchi M, Rafiei H, Salimi E (2009) A novel hybrid multi-objective shuffled frog-leaping algorithm for a bi-criteria permutation flow shop scheduling problem. Int J Adv Manuf Technol 41(11–12):1227–1239CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Department of Industrial and Information Systems EngineeringAshikaga Institute of TechnologyAshikagaJapan326-8558

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