Parallel-machine group scheduling with inclusive processing set restrictions, outsourcing option and serial-batching under the effect of step-deterioration


This paper investigates a parallel-machine group scheduling problem where non-identical jobs with arbitrary sizes and inclusive processing set restrictions can be either processed on in-house parallel machines in the form of serial batch or outsourced with cost. The objective of our study is aimed at minimizing the weighted sum of the in-house makespan and the total outsourcing cost for a platform manufacturing enterprise. Some structural properties are identified for the optimal solution in some special cases of the studied problem, which contribute to the optimal solution for the studied problem. Further, based on these properties, a novel hybrid algorithm VNS–NKEA is proposed to solve the studied problem, which integrates neighborhood knowledge-based evolutionary algorithm (NKEA) and variable neighborhood search (VNS). To demonstrate the better performance including solution quality and the convergence speed of the proposed algorithm, computational experiments are conducted to evaluate its performance by comparing with other proposed algorithms. The experiment results show that the hybrid algorithm performs quite better than other compared algorithms for each instance, which reflect that the hybrid algorithm can solve the studied problem effectively.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16


  1. 1.

    Ikura, Y., Gimple, M.: Efficient scheduling algorithms for a single batch processing machine. Oper. Res. Lett. 5(2), 61–65 (1986)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Uzsoy, R.: Scheduling a single batch processing machine with non-identical job sizes. Int. J. Prod. Res. 32(7), 1615–1635 (1994)

    Article  Google Scholar 

  3. 3.

    Dupont, L., Ghazvini, F.J.: Minimizing makespan on a single batch processing machine with non-identical job sizes. J. Eur. Syst. Autom. 32(4), 431–440 (1998)

    Google Scholar 

  4. 4.

    Liu, P., Zhou, X., Tang, L.: Two-agent single-machine scheduling with position-dependent processing times. Int. J. Adv. Manuf. Technol. 48(1–4), 325–331 (2010)

    Article  Google Scholar 

  5. 5.

    Tang, L., Zhao, X., Liu, J., Leung, J.Y.T.: Competitive two-agent scheduling with deteriorating jobs on a single parallel-batching machine. Eur. J. Oper. Res. 263(2), 401–411 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ozturk, O., Espinousea, M.L., Mascolo, M.D., Gouin, A.: Makespan minimization on parallel batch processing machines with non-identical job sizes and release dates. Int. J. Prod. Res. 50(20), 6022–6035 (2012)

    Article  Google Scholar 

  7. 7.

    Wang, J.Q., Fan, G.Q., Zhang, Y., Zhang, C.W., Leung, J.Y.: Two-agent scheduling on a single parallel-batching machine with equal processing time and non-identical job sizes. Eur. J. Oper. Res. 258(2), 478–490 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cheng, B., Yang, S., Hu, X., Chen, B.: Minimizing makespan and total completion time for parallel batch processing machines with non-identical job sizes. Appl. Math. Model. 36(7), 3161–3167 (2012)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Abedi, M., Seidgar, H., Fazlollahtabar, H., Bijani, R.: Bi-objective optimization for scheduling the identical parallel batch-processing machines with arbitrary job sizes, unequal job release times and capacity limits. Int. J. Prod. Res. 53(6), 1680–1711 (2015)

    Article  Google Scholar 

  10. 10.

    Damodaran, P., Manjeshwar, P.K., Srihari, K.: Minimizing makespan on a batch-processing machine with non-identical job sizes using genetic algorithms. Int. J. Prod. Econ. 103(2), 882–891 (2006)

    Article  Google Scholar 

  11. 11.

    Melouk, S., Damodaran, P., Chang, P.Y.: Minimizing makespan for single machine batch processing with non-identical job sizes using simulated annealing. Int. J. Prod. Econ. 87(2), 141–147 (2004)

    Article  Google Scholar 

  12. 12.

    Ng, C.T., Cheng, T.C.E., Yuan, J.J., Liu, Z.H.: On the single machine serial batching scheduling problem to minimize total completion time with precedence constraints, release dates and identical processing times. Oper. Res. Lett. 31(4), 323–326 (2003)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Yuan, J.J., Lin, Y.X., Cheng, T.C.E., Ng, C.T.: Single machine serial-batching scheduling problem with a common batch size to minimize total weighted completion time. Int. J. Prod. Econ. 105(2), 402–406 (2007)

    Article  Google Scholar 

  14. 14.

    Shen, L., Buscher, U.: Solving the serial batching problem in job shop manufacturing systems. Eur. J. Oper. Res. 221(1), 14–26 (2012)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Browne, S., Yechiali, U.: Scheduling deteriorating jobs on a single processor. Oper. Res. 38, 495–498 (1990)

    Article  Google Scholar 

  16. 16.

    Mosheiov, G.: Scheduling jobs under simple linear deterioration. Comput. Oper. Res. 21, 653–659 (1994)

    Article  Google Scholar 

  17. 17.

    Cheng, T.C.E., Ding, Q.: Single machine scheduling with deadlines and increasing rate of processing times. Acta Inf. 36(9–10), 673–692 (2000)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mosheiov, G.: Scheduling jobs with step-deterioration: minimizing makespan on a single and multi-machine. Comput. Ind. Eng. 28(4), 869–879 (1995)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Cheng, T.C.E., Ding, Q., Kovalyov, M.Y., Bachman, A., Janiak, A.: Scheduling jobs with piecewise linear decreasing processing times. Nav. Res. Logist. 50, 531–554 (2003)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Sundararaghavan, P.S., Kunnathur, A.S.: Single machine scheduling with start time dependent processing times: some solvable cases. Eur. J. Oper. Res. 79, 394–403 (1994)

    Article  Google Scholar 

  21. 21.

    Jeng, A.A.K., Lin, B.M.T.: Makespan minimization in single-machine scheduling with step-deterioration of processing times. J. Oper. Res. Soc. 55, 247–256 (2004)

    Article  Google Scholar 

  22. 22.

    Low, C., Hsu, C., Su, C.: Minimizing the makespan with an availability constraint on a single machine under simple linear deterioration. Comput. Math. Appl. 56, 257–265 (2008)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Layegh, J., Jolai, F., Amalnik, M.S.: A memetic algorithm for minimizing the total weighted completion time on a single machine under step-deterioration. Adv. Eng. Softw. 40, 1074–1077 (2009)

    Article  Google Scholar 

  24. 24.

    Li, S., Ng, C.T., Cheng, T.C.E., Yuan, J.J.: Parallel-batch scheduling of deteriorating jobs with release dates to minimize the makespan. Eur. J. Oper. Res. 210, 482–488 (2011)

    MathSciNet  Article  Google Scholar 

  25. 25.

    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)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Liao, B., Pei, J., Yang, S., Pardalos, P.M., Lu, S.: Single-machine and parallel-machine parallel batching scheduling considering deteriorating jobs, various group, and time-dependent setup time. Informatica 29(2), 281–301 (2018)

    Google Scholar 

  27. 27.

    Xu, Y.T., Zhang, Y., Huang, X.: Single-machine ready times scheduling with group technology and proportional linear deterioration. Appl. Math. Model. 38, 384–391 (2014)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Wu, C.C., Shiau, Y.R., Lee, W.C.L.: Single-machine group scheduling problems with deterioration consideration. Comput. Oper. Res. 35(5), 1652–1659 (2008)

    Article  Google Scholar 

  29. 29.

    Wang, J.B., Huang, X., Wu, Y.B., Ji, P.: Group scheduling with independent setup times, ready times, and deteriorating job processing times. Int. J. Adv. Manuf. Technol. 60, 643–649 (2012)

    Article  Google Scholar 

  30. 30.

    Wu, C.C., Lee, W.C.: Single-machine group-scheduling problems with deteriorating setup times and job-processing times. Int. J. Prod. Econ. 115, 128–133 (2008)

    Article  Google Scholar 

  31. 31.

    Wang, J.B., Gao, W.J., Wang, L.Y., Wang, D.: Single machine group scheduling with general linear deterioration to minimize the makespan. Int. J. Adv. Manuf. Technol. 43, 146–150 (2009)

    Article  Google Scholar 

  32. 32.

    Nomden, G., van der Zee, D.J.: Virtual cellular manufacturing: configuring routing flexibility. Int. J. Prod. Econ. 112, 439–451 (2008)

    Article  Google Scholar 

  33. 33.

    Rustogi, K., Strusevich, V.A.: Simple matching vs linear assignment in scheduling models with positional effects: a critical review. Eur. J. Oper. Res. 222, 393–407 (2012)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Hwang, H.C., Chang, S.Y., Lee, K.: Parallel machine scheduling under a grade of service provision. Comput. Oper. Res. 31(12), 2055–2061 (2004)

    Article  Google Scholar 

  35. 35.

    Li, C., Li, Q.: Scheduling jobs with release dates, equal processing times, and inclusive processing set restrictions. J. Oper. Res. Soc. 66(3), 516–523 (2015)

    Article  Google Scholar 

  36. 36.

    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: algorithms and complexity. In: Graves, S.C., Rinnooy Kan, A.H.G., Zipkin, P.H. (eds.) Logistics of Production and Inventory, pp. 445–522. North Holland, Amsterdam (1993)

    Google Scholar 

  37. 37.

    Li, C.L., Wang, X.: Scheduling parallel machines with inclusive processing set restrictions and job release times. Eur. J. Oper. Res. 200(3), 702–710 (2010)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Wu, J.Z., Chien, C.F., Gen, M.: Coordinating strategic outsourcing decisions for semiconductor assembly using a bi-objective genetic algorithm. Int. J. Prod. Res. 50(1), 235–260 (2012)

    Article  Google Scholar 

  39. 39.

    Yadav, V., Gupta, R.K.: A paradigmatic and methodological review of research in outsourcing. Inf. Resour. Manag. J. 21(1), 27–43 (2008)

    Article  Google Scholar 

  40. 40.

    Gonzalez, R., Gasco, J., Llopis, J.: Information systems outsourcing: a literature analysis. Inf. Manag. 43(7), 821–834 (2006)

    Article  Google Scholar 

  41. 41.

    Qi, X.: Coordinated logistics scheduling for in-house production and outsourcing. IEEE Trans. Autom. Sci. Eng. 5, 188–192 (2008)

    Article  Google Scholar 

  42. 42.

    Ruiz-Torres, A.J., Ho, J.C., López, F.J.: Generating Pareto schedules with outsource and internal parallel resources. Int. J. Prod. Econ. 103(2), 810–825 (2006)

    Article  Google Scholar 

  43. 43.

    Qi, X.: Two-stage production scheduling with an option of outsourcing from a remote supplier. J. Syst. Sci. Syst. Eng. 18(1), 1–15 (2009)

    Article  Google Scholar 

  44. 44.

    Neto, R.F.T., Filho, M.G., Silva, F.M.: An ant colony optimization approach for the parallel machine scheduling problem with outsourcing allowed. J. Intell. Manuf. 26(3), 527–538 (2015)

    Article  Google Scholar 

  45. 45.

    Pei, J., Liu, X., Pardalos, P.M., Athanasios, M., Shanlin, Y.: Serial-batching scheduling with time-dependent setup time and effects of deterioration and learning on a single-machine. J. Global Optim. 67(1–2), 251–262 (2017)

    MathSciNet  Article  Google Scholar 

  46. 46.

    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. 71(1), 147–163 (2018)

    MathSciNet  Article  Google Scholar 

  47. 47.

    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).

    Article  Google Scholar 

  48. 48.

    Pei, J., Liu, X., Liao, B., Pardalos, P.M., Kong, M.: Single-machine scheduling with learning effect and resource-dependent processing times in the serial-batching production. Appl. Math. Model. 58, 245–253 (2017)

    MathSciNet  Article  Google Scholar 

  49. 49.

    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 (2017).

    Article  Google Scholar 

  50. 50.

    Liu, X., Lu, S., Pei, J., Pardalos, P.M.: A hybrid VNS–HS algorithm for a supply chain scheduling problem with deteriorating jobs. Int. J. Prod. Res. (2017).

    Article  Google Scholar 

  51. 51.

    Pei, J., Wang, X., Fan, W., Pardalos, P.M.: Scheduling step-deteriorating jobs on bounded parallel-batching machines to maximise the total net revenue. J. Oper. Res. Soc. (2018).

    Article  Google Scholar 

  52. 52.

    Ma, C., Kong, M., Pei, J., Pardalos, P.M.: BRKGA–VNS for parallel-batching scheduling on a single machine with step-deteriorating jobs and release times. In: International Workshop on Machine Learning, Optimization, and Big Data, pp. 414–425. Springer, Cham (2017)

  53. 53.

    Pei, J., Liu, X., Pardalos, P.M., Fan, W., Wang, L., Yang, S.: Solving a supply chain scheduling problem with non-identical job sizes and release times by applying a novel effective heuristic algorithm. Int. J. Syst. Sci. 47(4), 765–776 (2016)

    MathSciNet  Article  Google Scholar 

  54. 54.

    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. Discrete Math. 5, 287–326 (1979)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Lee, I.S., Sung, C.S.: Single machine scheduling with outsourcing allowed. Int. J. Prod. Econ. 111, 623–634 (2008)

    Article  Google Scholar 

  56. 56.

    Dupont, L., Dhaenens-Flipo, C.: Minimizing the makespan on a batch machine with non-identical job sizes: an exact procedure. Comput. Oper. Res. 29, 807–819 (2002)

    MathSciNet  Article  Google Scholar 

  57. 57.

    Wen, Y., Xu, H., Yang, J.: A heuristic-based hybrid genetic-variable neighborhood search algorithm for task scheduling in heterogeneous multiprocessor system. Inf. Sci. 181(3), 567–581 (2011)

    Article  Google Scholar 

  58. 58.

    Duarte, A., Pantrigo, J.J., Pardo, E.G., Mladenović, N.: Multi-objective variable neighborhood search: an application to combinatorial optimization problems. J. Global Optim. 63(3), 515–536 (2015)

    MathSciNet  Article  Google Scholar 

  59. 59.

    Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)

    MathSciNet  Article  Google Scholar 

  60. 60.

    Hansen, P., Mladenović, N., Pérez, J.A.M.: Variable neighbourhood search: methods and applications. 4OR Q. J Oper. Res. 6(4), 319–360 (2008)

    MathSciNet  Article  Google Scholar 

  61. 61.

    Mladenović, N., Urošević, D., Ilić, A.: A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem. Eur. J. Oper. Res. 220(1), 270–285 (2012)

    MathSciNet  Article  Google Scholar 

  62. 62.

    Jarboui, B., Derbel, H., Hanafi, S., Mladenović, N.: Variable neighborhood search for location routing. Comput. Oper. Res. 40(1), 47–57 (2013)

    MathSciNet  Article  Google Scholar 

  63. 63.

    Yu, Z., Wong, H., Wang, D., Wei, M.: Neighborhood knowledge-based evolutionary algorithm for multiobjective optimization problems. IEEE Trans. Evol. Comput. 15(6), 812–830 (2011)

    Article  Google Scholar 

  64. 64.

    Lei, D.: Variable neighborhood search for two-agent flow shop scheduling problem. Comput. Ind. Eng. 80(C), 125–131 (2015)

    Article  Google Scholar 

Download references


This work is supported by the National Natural Science Foundation of China (Nos. 71871080, 71601065, 71501058, 71690235, 71231004), 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), and Base of Introducing Talents of Discipline to Universities for Optimization and Decision-making in the Manufacturing Process of Complex Product (111 Project). Panos M. Pardalos is partially supported by the Project of “Distinguished International Professor by the Chinese Ministry of Education” (MS2014HFGY026).

Author information



Corresponding author

Correspondence to Jun Pei.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liao, B., Song, Q., Pei, J. et al. Parallel-machine group scheduling with inclusive processing set restrictions, outsourcing option and serial-batching under the effect of step-deterioration. J Glob Optim (2018).

Download citation


  • Serial-batching
  • Group scheduling
  • Step-deterioration
  • Parallel machines
  • Outsourcing
  • Inclusive processing set restrictions
  • Platform enterprise