Soft Computing

, Volume 23, Issue 19, pp 9617–9628 | Cite as

Dominance rule and opposition-based particle swarm optimization for two-stage assembly scheduling with time cumulated learning effect

  • Dujuan Wang
  • Huaxin QiuEmail author
  • Chin-Chia Wu
  • Win-Chin Lin
  • Kunjung Lai
  • Shuenn-Ren Cheng
Methodologies and Application


This paper introduces a two-stage assembly flowshop scheduling model with time cumulated learning effect, which exists in many realistic scheduling settings. By the time cumulated learning effect, we mean that the actual job processing time of a job depends on its scheduled position as well as the processing times of the jobs already processed. The first stage consists of two independently working machines where each machine produces its own component. The second stage consists of a single assembly machine. The objective is to identify a schedule that minimizes the total completion time of all jobs. With analysis on the discussed problem, some dominance rules are developed to optimize the solving procedure. Incorporating with the developed dominance rules, a dominance rule and opposition-based particle swarm optimization algorithm (DR-OPSO) and branch-and-bound are devised. Computational experiments have been conducted to compare the performances of the proposed DR-OPSO and branch-and-bound through comparing with the standard O-PSO and PSO. The results fully demonstrate the efficiency and effectiveness of the proposed DR-OPSO algorithm, providing references to the relevant decision-makers in practice.


Two-stage assembly Flowshop scheduling Time cumulated learning function Dominance rule Particle swarm optimization 



This paper was funded in part by the National Natural Science Foundation of China (Nos. 71501024, 71871148), by Taiwan’s Ministry of Science and Technology (No. MOST105-2221-E-035-053-MY3), by China Postdoctoral Science Foundation (Nos. 2018T110631, 2017M612099), and by Sichuan University (No. 2018hhs-47).

Compliance with ethical standards

Conflict of interest

All of the authors declare that they have no conflict of interest.

Human participants or animals

This paper does not contain any studies with human participants or animals performed by any of the authors.


  1. Al-Anzi FS, Allahverdi A (2013) An artificial immune system heuristic for two-stage multi-machine assembly scheduling problem to minimize total completion time. J Manuf Syst 32(4):825–830CrossRefGoogle Scholar
  2. Biskup D (1999) Single-machine scheduling with learning considerations. Eur J Oper Res 115:173–178CrossRefzbMATHGoogle Scholar
  3. Biskup D (2008) A state-of-the-art review on scheduling with learning effect. Eur J Oper Res 188:315–329MathSciNetCrossRefzbMATHGoogle Scholar
  4. Fattahi P, Hosseini SMH, Jolai F, Reza TM (2014) A branch and bound algorithm for hybrid flow shop scheduling problem with setup time and assembly operations. Appl Math Model 38(1):119–134MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ignall E, Schrage LE (1965) Application of the branch and bound technique to some flowshop scheduling problems. Oper Res 13:400–412CrossRefGoogle Scholar
  6. Janiak A, Krysiak T, Trela R (2011) Scheduling problems with learning and ageing effects: a survey. Decis Mak Manuf 5(1–2):19–36MathSciNetzbMATHGoogle Scholar
  7. Ji M, Tang X, Zhang X, Cheng TCE (2016) Machine scheduling with deteriorating jobs and DeJong’s learning effect. Comput Ind Eng 91(C):42–47CrossRefGoogle Scholar
  8. Jung S, Woo YB, Kim BS (2017) Two-stage assembly scheduling problem for processing products with dynamic component-sizes and a setup time. Pergamon Press, Inc, New YorkCrossRefGoogle Scholar
  9. Komaki GM, Kayvanfar V (2015) Grey Wolf Optimizer algorithm for the two-stage assembly flow shop scheduling problem with release time. Journal of Computational Science 8:109–120CrossRefGoogle Scholar
  10. Koulamas C, Kyparisis GJ (2007) Single-machine and two-machine flowshop scheduling with general learning functions. Eur J Oper Res 178:402–407MathSciNetCrossRefzbMATHGoogle Scholar
  11. Lee CY, Cheng TCE, Lin BMT (1993) Minimizing the makespan in the 3-machine assembly-type flowshop scheduling problem. Manag Sci 39:616–625CrossRefzbMATHGoogle Scholar
  12. Liu Y, Feng Z (2014) Two-machine no-wait flowshop scheduling with learning effect and convex resource-dependent processing times. Comput Ind Eng 75:170–175CrossRefGoogle Scholar
  13. Lu YY, Wei CM, Wang JB (2012) Several single-machine scheduling problems with general learning effects. Appl Math Model 36(11):5650–5656MathSciNetCrossRefzbMATHGoogle Scholar
  14. Mirjalili S, Jangir P, Mirjalili SZ, Saremi S, Trivedi IN (2017) Optimization of problems with multiple objectives using the multi-verse optimization algorithm. Knowl-Based Syst 134:50–71CrossRefGoogle Scholar
  15. Niknamfar AH, Niaki STA, Niaki SAA (2017) Opposition-based learning for competitive hub location: a bi-objective biogeography-based optimization algorithm. Knowl-Based Syst 128:1–19CrossRefGoogle Scholar
  16. Torabzadeh E (2010) Cloud theory-based simulated annealing approach for scheduling in the two-stage assembly flowshop. Adv Eng Softw 41(10):1238–1243CrossRefzbMATHGoogle Scholar
  17. Wang JB, Wang JJ (2015) Research on scheduling with job-dependent learning effect and convex resource-dependent processing times. Int J Prod Res 53(19):1–11Google Scholar
  18. Wang XY, Zhou Z, Zhang X, Ji P, Wang JB (2013) Several flow shop scheduling problems with truncated position-based learning effect. Comput Oper Res 40(12):2906–2929MathSciNetCrossRefzbMATHGoogle Scholar
  19. Wang XR, Wang JB, Jin J, Ji P (2014) Single machine scheduling with truncated job-dependent learning effect. Optim Lett 8(2):669–677MathSciNetCrossRefzbMATHGoogle Scholar
  20. Wong TC, Ngan SCA (2013) comparison of hybrid genetic algorithm and hybrid particle swarm optimization to minimize makespan for assembly job shop. Appl Soft Comput 13(3):1391–1399CrossRefGoogle Scholar
  21. Wu CC, Cheng SR (2013) Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions. J Oper Res Soc 64(1):147–156CrossRefGoogle Scholar
  22. Xiong F, Xing K (2014) Meta-heuristics for the distributed two-stage assembly scheduling problem with bi-criteria of makespan and mean completion time. Int J Prod Res 2(9):2743–2766CrossRefGoogle Scholar
  23. Xu J, Wu CC, Yin Y, Zhao C, Chiou YT, Lin WC (2016) An order scheduling problem with position-based learning effect. Comput Oper Res 74(C):175–186MathSciNetCrossRefzbMATHGoogle Scholar
  24. Yin Y, Xu D, Sun K, Li H (2009) Some scheduling problems with general position-dependent and time-dependent learning effects. Inf Sci 179:2416–2425MathSciNetCrossRefzbMATHGoogle Scholar
  25. Yin Y, Xu D, Wang J (2010a) Some single-machine scheduling problems past-sequence-dependent setup times and a general learning effect. Int J Adv Manuf Technol 48:1123–1132CrossRefGoogle Scholar
  26. Yin Y, Xu D, Wang J (2010b) Single-machine scheduling with a general sum-of-actual-processing-times-based and job-position-based learning effect. Appl Math Model 34:3623–3630MathSciNetCrossRefzbMATHGoogle Scholar
  27. Yin Y, Xu D, Huang X (2011) Notes on “Some single-machine scheduling problems with general position-dependent and time-dependent learning effects”. Inf Sci 181:2209–2217MathSciNetCrossRefzbMATHGoogle Scholar
  28. Yin Y, Liu M, Hao J, Zhou M (2012a) Single machine scheduling with job position-dependent learning and time-dependent deterioration. IEEE Trans Syst Man Cybern A Syst Hum 42:192–200CrossRefGoogle Scholar
  29. Yin Y, Wu C-C, Wu W-H, Cheng S-R (2012b) The single-machine total weighted tardiness scheduling problem with position-based learning effects. Comput Oper Res 39:1109–1116MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Dujuan Wang
    • 1
  • Huaxin Qiu
    • 2
    Email author
  • Chin-Chia Wu
    • 3
  • Win-Chin Lin
    • 3
  • Kunjung Lai
    • 3
  • Shuenn-Ren Cheng
    • 4
  1. 1.China Business Executives AcademyDalianChina
  2. 2.School of Management Science and EngineeringDalian University of TechnologyDalianChina
  3. 3.Department of StatisticsFeng Chia UniversityTaichungTaiwan
  4. 4.Shandong Yingcai UniversityJinanChina

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