A BRKGA-DE algorithm for parallel-batching scheduling with deterioration and learning effects on parallel machines under preventive maintenance consideration


This paper introduces a parallel-batching scheduling problem with deterioration and learning effects on parallel machines, where the actual processing time of a job is subject to the phenomena of deterioration and learning. All jobs are first divided into different parallel batches, and the processing time of the batches is equal to the largest processing time of their belonged jobs. Then, the generated batches are assigned to parallel machines to be processed. Motivated by the characteristics of machine maintenance activities in a semiconductor manufacturing process, we take the machine preventive maintenance into account, i.e., the machine should be maintained after a fixed number of batches have been completed. In order to solve the problem, we analyze several structural properties with respect to the batch formation and sequencing. Based on these properties, a hybrid BRKGA-DE algorithm combining biased random-key genetic algorithm (BRKGA) and Differential Evolution (DE) is proposed to solve the parallel-batching scheduling problem. A series of computational experiments is conducted to demonstrate the effectiveness and efficiency of the proposed algorithm.

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  1. 1.

    Kunnathur, A.S., Gupta, S.K.: Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem. Eur. J. Oper. Res. 47 (1), 56–64 (1990)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Alidaee, B., Womer, N.K.: Scheduling with time dependent processing times: Review and extensions. J. Oper. Res. Soc. 50, 711–720 (1999)

    MATH  Google Scholar 

  3. 3.

    Cheng, T.C.E., Ding, Q., Lin, B.M.T.: A concise survey of scheduling with time dependent processing times. Eur. J. Oper. Res. 152(1), 1–13 (2004)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Gawiejnowicz, S.: Time-Dependent Scheduling, monographs in theoretical computer science, an EATCS series. Springer, Berlin (2008)

    Google Scholar 

  5. 5.

    Biskup, D.: A state-of-the-art review on scheduling with learning effect. Eur. J. Oper. Res. 188(2), 315–329 (2008)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cheng, T.C.E., Wu, C.-C., Lee, W.-C.: Some scheduling problems with sum-of processing-times-based and job-position-based learning effects. Inf. Sci. 178(11), 2476–2487 (2008)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Li, M., Petruzzi, N.C.: Demand uncertainty reduction in decentralized supply chains. Prod. Oper. Manag. 26(12), 156–161 (2017)

    Google Scholar 

  8. 8.

    Tang, L., Zhao, X., Liu, J., Joseph, Y.-T.: Leung. Competitive two-agent scheduling with deteriorating jobs on a single parallel-batching machine, Eur. J. Oper. Res (2017). https://doi.org/10.1016/j.ejor.2017.05.019

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Tan, Y., Carrillo, J., Cheng, H.K.: The agency model for digital goods. Dec. Sci. 4, 628–660 (2016)

    Google Scholar 

  10. 10.

    Zhao, C., Hsu, C.: Scheduling deteriorating jobs with machine availability constraints to minimize the total completion time. J. Ind. Prod. Eng. 34 (5), 323–329 (2017). https://doi.org/10.1080/21681015.2017.1295404

    Google Scholar 

  11. 11.

    Pei, J., Cheng, B., Liu, X., Pardalos, P.M., Kong, M.: Single-machine and parallel-machine serial-batching scheduling problems with position-based learning effect and linear setup time, Ann. Oper Res. (2017). https://doi.org/10.1007/s10479-017-2481-8

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bensoussan, A., Cakanyildirim, M., Li, M., Suresh, P.: Sethi. Managing inventory with cash register information: Sales recorded but not demands. Prod. Oper. Manag. 25(1), 9–21 (2016)

    Google Scholar 

  13. 13.

    Wang, J.-B., Wang, J.-J.: Single machine scheduling with sum-of-logarithm-processing-times based and position based learning effects. Optim Lett. 8(3), 971–982 (2014)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Lee, W.-C.: A note on deteriorating jobs and learning in single-machine scheduling problems. Int. J. Bus. Econ. 3, 83–89 (2004)

    Google Scholar 

  15. 15.

    Wang, J.-B.: Single-machine scheduling problems with the effects of learning and deterioration. Omega 35(4), 397–402 (2007)

    Google Scholar 

  16. 16.

    Wang, X., Cheng, T.C.E.: Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan. Eur. J. Oper Res. 178(1), 57–70 (2007)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Wang, J.-B., Guo, Q.: A due-date assignment problem with learning effect and deteriorating jobs. Appl. Math. Model 34(2), 309–313 (2010)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Lu, Y., Jin, J., Ji, P., Wang, J.-B.: Resource-dependent scheduling with deteriorating jobs and learning effects on unrelated parallel machine. Neural Comp. Appl. 27(7), 1993–2000 (2016)

    Google Scholar 

  19. 19.

    Rostami, M., Pilerood, A.E., Mazdeh, M.M.: Multi-objective parallel machine scheduling problem with job deterioration learning effect under fuzzy environment. Comp. & Ind. Eng. 85, 206–215 (2015)

    Google Scholar 

  20. 20.

    Wang, X.-Y., Wang, J.-J.: Scheduling deteriorating jobs with a learning effect on unrelated parallel machines. Appl. Math. Model. 38(21), 5231–5238 (2014)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Lee, C.Y., Uzsoy, R., Martin-Vega, L.A.: Efficient algorithms for scheduling semiconductor burn-in operations. Oper. Res. 40, 764–775 (1992)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Li, C., Lee, C.Y.: Scheduling with agreeable release times and due dates on a batch processing machine. Eur. J. Oper Res. 96, 564–569 (1997)

    MATH  Google Scholar 

  23. 23.

    Melouk, S., Damodaran, P., Chang, P.-Y.: Minimizing makespan for single machine batch processing with nonidentical job sizes using simulated annealing. Int. J. Prod. Econ. 87, 141–147 (2004)

    Google Scholar 

  24. 24.

    Malapert, A., Guéret, C., Rousseau, L.M.: A constraint programming approach for a batch processing problem with non-identical job sizes. Eur. J. Oper. Res. 3, 533–545 (2012)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Zhang, G., Cai, X., Lee, C.Y., Wong, C.K.: Minimizing makespan on a single batch processing machine with nonidentical job sizes. Nav. Res. Log. 3, 226–240 (2001)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Dupont, L., Dhaenens-Flipo, C.: Minimizing the makespan on a batch machine with non-identical job sizes: An exact procedure. Comput. & Oper. Res. 7, 807–819 (2001)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Qi, X., Zhou, S., Yuan, J.: Single machine parallel-batch scheduling with deteriorating jobs. Theor. Comput. Sci. 10, 830–836 (2009)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Li, S., Ng, C.T., Cheng, T.C.E., Yuan, J.: Parallel-batch scheduling of deteriorating jobs with release dates to minimize the makespan. Eur. J. Oper. Res. 10, 482–488 (2011)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Miao, C., Zhang, Y., Cao, Z.: Bounded parallel-batch scheduling on single and multi -machines for deteriorating jobs. Inf. Proc. Lett. 111, 798–803 (2011)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Yang, S.-J.: Single-machine scheduling problems with both start-time dependent learning and position dependent aging effects under deteriorating maintenance consideration. Appl. Math. Comput. 217(7), 3321–3329 (2010)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Yang, S.-J.: Group scheduling problems with simultaneous considerations of learning and deterioration effects on a single-machine. Appl. Math. Model. 35(8), 4008–4016 (2011)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Yang, S.-J.: Unrelated parallel-machine scheduling with deterioration effects and deteriorating multi-maintenance activities for minimizing the total completion time. Appl. Math Model. 37(5), 2995–3005 (2013)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Yang, S.-J., Yang, D.-L.: Minimizing the total completion time in single-machine scheduling with aging/deteriorating effects and deteriorating maintenance activities. Comput. & Math. Appl. 60(7), 2161–2169 (2010)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Pei, J., Liu, X., Fan, W., Pardalos, P.M., Lu, S.: A hybrid BA-VNS algorithm for coordinated serial-batching scheduling with deteriorating jobs, financial budget, and resource constraint in multiple manufacturers. Omega. https://doi.org/10.1016/j.omega.2017.12.003 (2017a)

    Google Scholar 

  35. 35.

    Pei, J., Liu, X., Pardalos, P.M., Fan, W., Yang, S.: Scheduling deteriorating jobs on a single serial-batching machine with multiple job types and sequence-dependent setup times. Ann. Oper. Res. 249, 175–195 (2017b)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Yang, D.-L., Kuo, W.-H.: Some scheduling problems with deteriorating jobs and learning effects. Comput. & Ind. Eng. 58(1), 25–28 (2010)

    Google Scholar 

  37. 37.

    Valdez-Flores, C., Feldman, R.M.: A survey of preventive maintenance models for stochastically deteriorating single - unit systems. Nav. Res. Log. 36(4), 419–446 (1989)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Yao, X., Fernandez-Gaucherand, E., Fu, M.C., Marcus, S.I.: Optimal preventive maintenance scheduling in semiconductor manufacturing. IEEE Trans. on Sem. Manuf. 17(3), 345–356 (2004)

    Google Scholar 

  39. 39.

    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Ann. Discr. Math. 5, 287–326 (1979)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, London (1967)

    Google Scholar 

  41. 41.

    Gonçalves, J.F., Almeida, J.R.D.: A hybrid genetic algorithm for assembly line balancing. J. Heur. 8(6), 629–642 (2002)

    Google Scholar 

  42. 42.

    Zhang, J., Sanderson, A.C.: JADE: Adaptive Differential evolution with optional external archive. IEEE Trans. Evolut. Comput. 13(5), 945–958 (2009)

    Google Scholar 

  43. 43.

    Jia, Z., Leung, Y.-T.: A meta-heuristic to minimize makespan for parallel batch machines with arbitrary job sizes. Eur. J. Oper. Res. 240(3), 649–665 (2015)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Nickabadi, A., Ebadzadeh, M., Safabakhsh, R.: A novel particle swarm optimization algorithm with adaptive inertia weight. Appl. Soft Comput. 11(4), 3658–3670 (2011)

    Google Scholar 

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This work is supported by the National Natural Science Foundation of China (Nos. 71231004, 71871080, 71601065, 71690235, 71501058, 71601060), and Innovative Research Groups of the National Natural Science Foundation of China (71521001), the Humanities and Social Sciences Foundation of the Chinese Ministry of Education (No. 15YJC630097), Anhui Province Natural Science Foundation (No. 1608085QG167), Base of Introducing Talents of Discipline to Universities for Optimization and Decision-making in the Manufacturing Process of Complex Product (111 project), the Project of Key Research Institute of Humanities and Social Science in University of Anhui Province, Open Research Fund Program of Key Laboratory of Process Optimization and Intelligent Decision-making(Hefei University of Technology), Ministry of Education. Panos M. Pardalos is partially supported by the project of Distinguished International Professor by the Chinese Ministry of Education (MS2014HFGY026).

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Correspondence to Jun Pei.



Table 5 Results for instances with n = 50 and C = 3
Table 6 Results for instances with n = 50 and C = 5
Table 7 Results for instances with n = 75 and C = 3
Table 8 Results for instances with n = 75 and C = 5
Table 9 Results for instances with n = 100 and C = 3
Table 10 Results for instances with n = 100 and C = 5

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Kong, M., Liu, X., Pei, J. et al. A BRKGA-DE algorithm for parallel-batching scheduling with deterioration and learning effects on parallel machines under preventive maintenance consideration. Ann Math Artif Intell 88, 237–267 (2020). https://doi.org/10.1007/s10472-018-9602-1

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  • Scheduling
  • Deterioration
  • Learning
  • Parallel-batching

Mathematics Subject Classification (2010)

  • 90B35