Parallel-batching scheduling with nonlinear processing times on a single and unrelated parallel machines

  • Min KongEmail author
  • Xinbao Liu
  • Jun PeiEmail author
  • Panos M. Pardalos
  • Nenad Mladenovic


Parallel-batching processing and job deterioration are universal in the real industry. Scholars have deeply investigated the problem of parallel-batching scheduling and the problem of scheduling with deteriorating jobs separately. However, the situations where both parallel-batching processing and job deterioration exist simultaneously were seldom considered. This paper studies the parallel-batching scheduling problem with nonlinear processing times on a single machine, and proposes several structural properties and an optimal algorithm to solve it. Based on the above properties and optimal algorithm for the single machine setting, we further study the problem of parallel-batching scheduling with nonlinear processing times under the unrelated parallel machine setting. Since the unrelated parallel machines scheduling problem is NP-hard, a hybrid SFLA-VNS algorithm combining Shuffle Frog Leap Algorithm (SFLA) with Variable Neighborhood Search Algorithm (VNS) is proposed. Computational experiments and comparison are finally conducted to demonstrate the effectiveness of the proposed algorithm.


Parallel-batching Scheduling Nonlinear processing times Meta-heuristic algorithm 



This work is supported by the National Natural Science Foundation of China (Nos. 71231004, 71871080, 71601065, 71690235, 71501058, 71601060, 71801071), 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), the Project of Key Research Institute of Humanities and Social Science in University of Anhui Province (No. SK2017A0055), the Philosophy and Social Science Cultivation Project of Hefei University of Technology (No. JS2017AJRW0031), the Fundamental Research Funds for the Central Universities (JZ2018HGBZ0129), 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).


  1. 1.
    Lee, C.Y., Uzsoy, R., Martin-Vega, L.A.: Efficient algorithms for scheduling semiconductor burn-in operations. Oper. Res. Int. Journal 40, 764–775 (1992)MathSciNetzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang, G., Cai, X., Lee, C.Y., Wong, C.K.: Minimizing makespan on a single batch processing machine with nonidentical job sizes. Nav. Res. Logist. 3, 226–240 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)CrossRefzbMATHGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Kumar, A., Tan, Y.: Demand effects of joint product advertising in online videos. Manage. Sci. 61, 1921–1937 (2015)CrossRefGoogle Scholar
  8. 8.
    Paul, A., Tan, Y., Vakharia, A.: Inventory planning for a modular product family. Prod. Oper. Manag. 24, 1033–1053 (2015)CrossRefGoogle Scholar
  9. 9.
    Tan, Y., Carrillo, J., Cheng, H.K.: The agency model for digital goods. Decis. Sci. 4, 628–660 (2016)CrossRefGoogle Scholar
  10. 10.
    Tan, Y., Carrillo, J.: Strategic analysis of the agency model for digital goods. Prod. Oper. Manag. (2017). Google Scholar
  11. 11.
    Gupta, J.N.D., Gupta, S.K.: Single facility scheduling with nonlinear processing times. Comput. Ind. Eng. 14, 387–393 (1988)CrossRefGoogle Scholar
  12. 12.
    Browne, S., Yechiali, U.: Scheduling deteriorating jobs on a single processor. Oper. Res. 3, 495–498 (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mosheiov, G.: Scheduling deteriorating jobs under simple linear deterioration. Comput. Oper. Res. 21, 653–659 (1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    Cheng, T.C.E., Lee, W.C., Wu, C.C.: Single-machine scheduling with deteriorating jobs and past-sequence-dependent setup times. Appl. Math. Model. 35, 1861–1867 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lai, P., Lee, W.C.: Single-machine scheduling with a nonlinear deterioration function. Inf. Process. Lett. 110, 455–459 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, J., Wang, M.: Single-machine scheduling with nonlinear deterioration. Optimization Letters 6, 87–98 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pei, J., Liu, X., Fan, W., Pardalos, P.M., Shaojun, L.: A hybrid BA-VNS algorithm for coordinated serial-batching scheduling with deteriorating jobs, financial budget, and resource constraint in multiple manufacturers. Omega (2017). Google Scholar
  19. 19.
    Fan, W., Pei, J., Liu, X., Pardalos, P.M., Kong, M.: Serial-batching group scheduling with release times and the combined effects of deterioration and truncated job-dependent learning. J. Global Optim. (2017). zbMATHGoogle Scholar
  20. 20.
    Pei, J., Pardalos, P.M., Liu, X., Fan, W., Yang, S.: Serial batching scheduling of deteriorating jobs in a two-stage supply chain to minimize the makespan. Eur. J. Oper. Res. 244(1), 13–25 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Alidaee, B., Womer, N.K.: Scheduling with time dependent processing times: review and extensions. J. Oper. Res. Soc. 50, 711–720 (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Cheng, T.C.E., Kang, L., Ng, C.T.: Due-date assignment and single machine scheduling with deteriorating jobs. J. Oper. Res. Soc. 55, 198–203 (2004)CrossRefzbMATHGoogle Scholar
  23. 23.
    Wu, C.C., Lee, W.C.: Scheduling linear deteriorating jobs to minimize makespan with an availability constraint on a single machine. Inf. Process. Lett. 87, 89–93 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ji, M., He, Y., Cheng, T.C.E.: Scheduling linear deteriorating jobs with an availability constraint on a single machine. Theoret. Comput. Sci. 362, 115–126 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, J.B.: Single-machine scheduling problems with the effects of learning and deterioration. Omega 35, 397–402 (2007)CrossRefGoogle Scholar
  26. 26.
    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, 1–13 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Toksarı, M.D., Güner, E.: Minimizing the earliness/tardiness costs on parallel machine with learning effects and deteriorating jobs: a mixed nonlinear integer programming approach. Int. J. Adv. Manuf. Technol. 38, 801–808 (2008)CrossRefGoogle Scholar
  28. 28.
    Ji, M., Cheng, T.C.E.: Parallel-machine scheduling of simple linear deteriorating jobs. Theoret. Comput. Sci. 410, 3761–3768 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mazdeh, M.M., Zaerpour, F., Zareei, A., Hajinezhad, A.: Parallel machines scheduling to minimize job tardiness and machine deteriorating cost with deteriorating jobs. Appl. Math. Model. 34, 1498–1510 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, S., Yuan, J.: Parallel-machine scheduling with deteriorating jobs and rejection. Theoret. Comput. Sci. 411, 3642–3650 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wang, J., Wang, L., Wang, D., Wang, X.: Single-machine scheduling with a time-dependent deterioration. Int. J. Adv. Manuf. Technol. 43, 805–809 (2009)CrossRefGoogle Scholar
  32. 32.
    Qi, X., Zhou, S., Yuan, J.: Single machine parallel-batch scheduling with deteriorating jobs. Theoret. Comput. Sci. 410, 830–836 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Miao, C., Zhang, Y., Cao, Z.: Bounded parallel-batch scheduling on single and multi-machines for deteriorating jobs. Inf. Process. Lett. 111, 798–803 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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. 210, 482–488 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wu, Y., Wang, M., Wang, J.: Some single-machine scheduling with both learning and deterioration effects. Appl. Math. Model. 35, 3731–3736 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lenstra, J.K., Rinnooy, A.H.G., Brucker, P.: Complexity of machine scheduling problems. J. Sched. 1, 343–362 (1977)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Eusuff, M.M., Lansey, K.E.: Optimization of water distribution network design using the shuffled frog leaping algorithm. J. Water Resour. Plan. Manag. 129, 210–225 (2003)CrossRefGoogle Scholar
  39. 39.
    Hansen, P., Mladenović, N.: Variable neighborhood search. Comput. Oper. Res. 24, 1097–1100 (1977)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hansen, P., Mladenović, N., Pérez, J.A.M.: Variable neighbourhood search: methods and applications. 4OR 175, 367–407 (2008)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Zhou, S., Liu, M., Chen, H., Li, X.: An effective discrete differential evolution algorithm for scheduling uniform parallel batch processing machines with non-identical capacities and arbitrary job sizes. Int. J. Prod. Econ. 179, 1–11 (2016)CrossRefGoogle Scholar
  42. 42.
    Jiang, L., Pei, J., Liu, X., Pardalos, P.M., Yang, Y., Qian, X.: Uniform parallel batch machines scheduling considering transportation using a hybrid DPSO-GA algorithm. Int. J. Adv. Manuf. Technol. (2016). Google Scholar
  43. 43.
    Bean, J.C.: Genetic algorithms and random keys for sequencing and optimization. ORSA J. Comput. 2, 154–160 (1994)CrossRefzbMATHGoogle Scholar
  44. 44.
    Borges, P., Eid, T., Bergseng, E.: Applying simulated annealing using different methods for the neighborhood search in forest planning problems. Eur. J. Oper. Res. 233, 700–710 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Liang, X., Li, W., Zhang, Y., Zhou, M.C.: An adaptive particle swarm optimization method based on clustering. Soft. Comput. 19, 431–448 (2015)CrossRefGoogle Scholar
  46. 46.
    Lei, D., Guo, X.: A shuffled frog-leaping algorithm for job shop scheduling with outsourcing options. Int. J. Prod. Res. 54, 1–12 (2016)CrossRefGoogle Scholar
  47. 47.
    Huang, X., Wang, J., Wang, L., Gao, W., Wang, X.: Single machine scheduling with time-dependent deterioration and exponential learning effect. Comput. Ind. Eng. 58, 58–63 (2010)CrossRefGoogle Scholar
  48. 48.
    Lai, P.J., Wu, C.C., Lee, W.C.: Single-machine scheduling with logarithm deterioration. Optimization Letters 6, 1719–1730 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Rudek, R.: Some single-machine scheduling problems with the extended sum-of-processing-time-based aging effect. Int. J. Adv. Manuf. Technol. 59, 299–309 (2012)CrossRefGoogle Scholar
  50. 50.
    Cheng, T.C.E., Tseng, S.C., Lai, P.J., Lee, W.C.: Single-machine scheduling with accelerating deterioration effects. Optimization Letters 8, 543–554 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ManagementHefei University of TechnologyHefeiChina
  2. 2.Key Laboratory of Process Optimization and Intelligent Decision-Making of Ministry of EducationHefeiChina
  3. 3.Center for Applied Optimization, Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  4. 4.Mathematical Institute, Serbian Academy of Sciences and ArtsBeogradSerbia

Personalised recommendations