Soft Computing

, Volume 21, Issue 8, pp 2091–2103 | Cite as

A two-machine flowshop scheduling problem with precedence constraint on two jobs

  • Shuenn-Ren Cheng
  • Yunqiang Yin
  • Chih-Hou Wen
  • Win-Chin Lin
  • Chin-Chia Wu
  • Jun Liu
Methodologies and Application


Job precedence can be found in some real-life situations. For the application in the scheduling of patients from multiple waiting lines or different physicians, patients in the same waiting line for scarce resources such as organs, or with the same physician often need to be treated on the first-come, first-served basis to avoid ethical or legal issues, and precedence constraints can restrict their treatment sequence. In view of this observation, this paper considers a two-machine flowshop scheduling problem with precedence constraint on two jobs with the goal to find a sequence that minimizes the total tardiness criterion. In searching solutions to this problem, we build a branch-and-bound method incorporating several dominances and a lower bound to find an optimal solution. In addition, we also develop a genetic and larger-order-value method to find a near-optimal solution. Finally, we conduct the computational experiments to evaluate the performances of all the proposed algorithms.


Scheduling Two-machine flowshop Total tardiness Precedent constraint 


  1. Armentano VA, Ronconi DP (1999) Tabu search for total tardiness minimization in flow-shop scheduling problems. Comput Oper Res 26:219–235MathSciNetCrossRefMATHGoogle Scholar
  2. Baker KR (1974) Introduction to sequencing and scheduling. Wiley, New YorkGoogle Scholar
  3. Bank M, Fatemi Ghomi SMT, Jolai F, Behnamian J (2012) Two-machine flow shop total tardiness scheduling problem with deteriorating jobs. Appl Math Model 36(11):5418–5426MathSciNetCrossRefMATHGoogle Scholar
  4. Bean JC (1994) Genetic algorithms and random keys for sequencing and optimization. ORSA J Comput 6:154–160CrossRefMATHGoogle Scholar
  5. Chandra C, Liu Z, He J, Ruohonen T (2014) A binary branch and bound algorithm to minimize maximum scheduling cost. Omega 42:9–15CrossRefGoogle Scholar
  6. Chen ZY, Tsai CF, Eberle W, Lin WC, Ke W-C (2014) Instance selection by genetic-based biological algorithm. Soft Comput. doi:10.1007/s00500-014-1339-0
  7. Chen P, Wu C-C, Lee WC (2006) A bi-criteria two-machine flowshop scheduling problem with a learning effect. J Oper Res Soc 57:1113–1125CrossRefMATHGoogle Scholar
  8. Chung CS, Flynn J, Kirca O (2006) A branch and bound algorithm to minimize the total tardiness for m-machine permutation flowshop problems. Eur J Oper Res 174:1–10MathSciNetCrossRefMATHGoogle Scholar
  9. Della Croce F, Narayan V, Tadei R (1996) The two-machine total completion time flow shop problem. Eur J Oper Res 90:227–237CrossRefMATHGoogle Scholar
  10. Essafi I, Mati Y, Dauzere-Peres S (2008) A genetic local search algorithm for minimizing total weighted tardiness in the job-shop scheduling problem. Comput Oper Res 35:2599–2616MathSciNetCrossRefMATHGoogle Scholar
  11. Etiler O, Toklu B, Atak M, Wilson J (2004) A generic algorithm for flow shop scheduling problems. J Oper Res Soc 55(8):830–835CrossRefMATHGoogle Scholar
  12. Fisher ML (1971) A dual algorithm for the one-machine scheduling problem. Math Program 11:229–251MathSciNetCrossRefMATHGoogle Scholar
  13. French S (1982) Sequencing and scheduling: an introduction to the mathematics of the job shop. British Library Cataloguing in Publish Data, Ellis Horwood Limited, ChichesterMATHGoogle Scholar
  14. Gao KZ, Li H, Pan QK, Li JQ, Liang JJ (2010) Hybrid heuristics based on harmony search to minimize total flow time in no-wait flow shop. Control and decision conference (CCDC), 2010 Chinese. IEEE, 2010Google Scholar
  15. Garey MR, Johnson DS, Sethi PR (1979) The complexity of flowshop and jobshop scheduling. Math Oper Res 1:117–129MathSciNetCrossRefMATHGoogle Scholar
  16. Gelders LF, Sambandam N (1978) Four simple heuristics for scheduling a flowshop. Int J Prod Res 16:221–231CrossRefGoogle Scholar
  17. Gen M, Lin L (2012) Multiobjective genetic algorithm for scheduling problems in manufacturing systems. Ind Eng Manag Syst 11:310–330Google Scholar
  18. Gong D, Wang G, Sun X, Han Y (2014) A set-based genetic algorithm for solving the many-objective optimization problem. Soft Comput. doi:10.1007/s00500-014-1284-y
  19. Ishibuchi H, Murata T (1998) A multi-objective genetic local search algorithm and its application to flowshop scheduling. IEEE Trans Syst Man Cybern-Part C: Appl Rev 28(3):392–403CrossRefGoogle Scholar
  20. Kharbeche M, Haouari M (2013) MIP models for minimizing total tardiness in a two-machine flow shop. J Oper Res Soc 64:690–707CrossRefGoogle Scholar
  21. Kim Y-D (1993) A new branch and bound algorithm for minimizing mean tardiness in two-machine flowshops. Comput Oper Res 20:391–401MathSciNetCrossRefMATHGoogle Scholar
  22. Kim Y-D (1995) Minimizing total tardiness in permutation flowshops. Eur J Oper Res 85(3):541–555CrossRefMATHGoogle Scholar
  23. Koulamas C (1994) The total tardiness problem: review and extensions. Oper Res 42:1025–1041MathSciNetCrossRefMATHGoogle Scholar
  24. Lenstra JK, Rinnooy AHG, Brucker P (1977) Complexity of machine scheduling problems. Ann Discrete Math 1:1016–1019MathSciNetMATHGoogle Scholar
  25. Li J, Song Y (2013) Community detection in complex networks using extended compact genetic algorithm. Soft Comput 17(6):925–937CrossRefGoogle Scholar
  26. Mati Y, Xie X (2008) A genetic-search-guided greedy algorithm for multi-resource shop scheduling with resource flexibility. IIE Trans 40(12):1228–1240CrossRefGoogle Scholar
  27. Onwubolu GC, Mutingi M (1999) Genetic algorithm for minimizing tardiness in flow-shop scheduling. Prod Plan Control 10:462–471CrossRefGoogle Scholar
  28. Ow PS (1985) Focused scheduling in proportionate flowshops. Manag Sci 31:852–869CrossRefMATHGoogle Scholar
  29. Pan JCH, Chen J-S, Chao C-M (2002) Minimizing tardiness in a two-machine flow-shop. Comput Oper Res 29:869–885MathSciNetCrossRefMATHGoogle Scholar
  30. Pan JCH, Fan E-T (1997) Two-machine flowshop scheduling to minimize total tardiness. Int J Syst Sci 28(4):405–414CrossRefMATHGoogle Scholar
  31. Panwalker SS, Iskander W (1977) A survey of scheduling rules. Oper Res 25:45–61MathSciNetCrossRefMATHGoogle Scholar
  32. Parthasarathy S, Rajendran C (1997) A simulated annealing heuristic for scheduling to minimize mean weighted tardiness in a flowshop with sequence dependent setup times of jobs-a case study. Prod Plan Control 8:475–483 Google Scholar
  33. Pinedo M (2008) Scheduling: theory, algorithms and systems. Upper Saddle River, Prentice-HallMATHGoogle Scholar
  34. Qian B, Wang L, Huang D-X, Wang W-L, Wang X (2009) An effective hybrid DE-based algorithm for multi-objective flow shop scheduling with limited buffers. Comput Oper Res 36(1):209–233Google Scholar
  35. Reeves C (1995) Heuristics for scheduling a single machine subject to unequal job release times. Eur J Oper Res 80:397–403CrossRefGoogle Scholar
  36. Schaller J (2005) Note on minimizing total tardiness in a two-machine flowshop. Comput Oper Res 32(12):3273–3281MathSciNetCrossRefMATHGoogle Scholar
  37. Sen T, Dileepan P, Gupta JND (1989) The two-machine flowshop scheduling problem with total tardiness. Comput Oper Res 16:333–340MathSciNetCrossRefMATHGoogle Scholar
  38. Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359MathSciNetCrossRefMATHGoogle Scholar
  39. Ta QC, Billaut J-C, Bouquard J- L (2013) An hybrid metaheuristic, an hybrid lower bound and a Tabu search for the two-machine flowshop total tardiness problem. 2013 IEEE RIVF international conference on computing & communication technologies—research, innovation, and vision for the future (RIVF), pp 198–202Google Scholar
  40. Tasgetiren MF, Liang YC, Sevkli M, Gencyilmaz G (2004) Differential evolution algorithm for permutation flowshop sequencing problem with makespan criterion. In: Proceedings of 4th international symposium on intelligent manufacturing systems, Sakarya, Turkey, 2004Google Scholar
  41. Vallada E, Ruiz R, Minella G (2008) Minimising total tardiness in the \(m\)-machine flowshop problem: a review and evaluation of heuristics and metaheuristics. Comput Oper Res 35(4):1350–1373CrossRefMATHGoogle Scholar
  42. Wang L, Qian, B (2012) Hybrid differential evolution and scheduling algorithms. Beijing Tsinghua University Press. ISBN 978-7-302-28367-6 (in Chinese)Google Scholar
  43. Wu C-C, Lee W-C, Wang W-C (2007) A two-machine flowshop maximum tardiness scheduling problem with a learning effect. Int J Adv Manuf Technol 31(7–8):743–750Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Shuenn-Ren Cheng
    • 1
  • Yunqiang Yin
    • 2
  • Chih-Hou Wen
    • 3
  • Win-Chin Lin
    • 3
  • Chin-Chia Wu
    • 3
  • Jun Liu
    • 4
  1. 1.Graduate Institute of Business AdministrationCheng Shiu UniversityKaohsiungTaiwan
  2. 2.Faculty of ScienceKunming University of Science and TechnologyKunmingChina
  3. 3.Department of StatisticsFeng Chia UniversityTaichungTaiwan
  4. 4.School of Nuclear Engineering and GeophysicsEast China Institute of TechnologyNanchangChina

Personalised recommendations