Skip to main content

A hybrid VNS-Lagrangean heuristic framework applied on single machine scheduling problem with sequence-dependent setup times, release dates and due dates


In this work, we propose a hybrid VNS-Lagrangean heuristic applied on the single machine scheduling problem with sequence-dependent setup times, release dates, and due dates. The objective function is the minimization of the total tardiness. The proposed hybrid heuristic is a Lagrangean relaxation integrated with the variable neighborhood search (VNS). The methodology can generate strong bounds, using the information of the Lagrangean multipliers to construct and perturb feasible solutions within the VNS framework. We compare its performance with previous hybrid approaches and find that the upper bounds obtained are optimal for several cases and tight for others. The methodology presents competitive results when compared with previous related works.

This is a preview of subscription content, access via your institution.

Fig. 1




  1. Barahona, F., Anbil, R.: The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87(3), 385–399 (2000)

    Article  MathSciNet  Google Scholar 

  2. Bartz-Beielstein, T.: Spot: An R package for automatic and interactive tuning of optimization algorithms by sequential parameter optimization (2010). arXiv preprint arXiv:1006.4645

  3. Blum, C., Puchinger, J., Raidl, G.R., Roli, A.: Hybrid metaheuristics in combinatorial optimization: a survey. Appl. Soft Comput. 11(6), 4135–4151 (2011)

    Article  Google Scholar 

  4. Boschetti, M., Maniezzo, V.: Benders decomposition, Lagrangean relaxation and metaheuristic design. J. Heuristics 15(3), 283–312 (2009)

    Article  Google Scholar 

  5. Boschetti, M., Maniezzo, V., Roffilli, M.: Decomposition techniques as metaheuristic frameworks. In: Matheuristics, pp. 135–158. Springer (2010)

  6. Brimberg, J., Mladenović, N., Todosijević, R., Urošević, D.: A basic variable neighborhood search heuristic for the uncapacitated multiple allocation p-hub center problem. Optim. Lett. 11(2), 313–327 (2017)

    Article  MathSciNet  Google Scholar 

  7. Brimberg, J., Mladenović, N., Todosijević, R., Urošević, D.: General variable neighborhood search for the uncapacitated single allocation p-hub center problem. Optim. Lett. 11(2), 377–388 (2017)

    Article  MathSciNet  Google Scholar 

  8. de Paula, M.R., Mateus, G.R., Ravetti, M.G.: A non-delayed relax-and-cut algorithm for scheduling problems with parallel machines, due dates and sequence-dependent setup times. Comput. Oper. Res. 37(5), 938–949 (2010)

    Article  MathSciNet  Google Scholar 

  9. de Paula, M.R., Ravetti, M.G., Mateus, G.R.: Solving parallel machines scheduling problems with sequence-dependent setup times using variable neighbourhood search. IMA J. Manag. Math. 18, 101–115 (2007)

    Article  MathSciNet  Google Scholar 

  10. Fisher, M.L.: Optimal solution of scheduling problems using lagrange multipliers: Part I. Oper. Res. 21, 1114–1127 (1973)

    Article  Google Scholar 

  11. Fonseca, G., Nogueira, T., Ravetti, M.: A hybrid lagrangian metaheuristic for the cross-docking flow shop scheduling problem. Eur. J. Oper. Res. 275, 139–154 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  12. Gagné, C., Gravel, M., Price, W.: Using metaheuristic compromise programming for the solution of multiple-objective scheduling problems. J. Oper. Res. Soc. 56(6), 687–698 (2005)

    Article  Google Scholar 

  13. Gagné, C., Price, W., Gravel, M.: Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times. J. Oper. Res. Soc. pp. 895–906 (2002)

  14. Hansen, P., Mladenović, N.: An introduction to variable neighborhood search. In: Meta-heuristics, pp. 433–458. Springer (1999)

  15. Hansen, P., Mladenović, N.: Variable neighborhood search. In: Search methodologies, pp. 313–337. Springer (2014)

  16. Kirlik, G., Oguz, C.: A variable neighborhood search for minimizing total weighted tardiness with sequence dependent setup times on a single machine. Comput. Oper. Res. 39(7), 1506–1520 (2012)

    Article  MathSciNet  Google Scholar 

  17. Lemaréchal, C.: An extension of davidon methods to non differentiable problems. In: Nondifferentiable Optimization, pp. 95–109. Springer (1975)

  18. Lemaréchal, C.: Nondifferentiable optimization. In: Nemhauser, G.L., Rinnooy Kan, A.H.G., Todd, M.J. (eds.) Hanbooks in Operations Research, pp. 529–572. North Holland, New York (1989)

  19. Lucena, A.: Non delayed relax-and-cut algorithms. Ann. Oper. Res. 140(1), 375–410 (2005)

    Article  MathSciNet  Google Scholar 

  20. Marinakis, Y., Migdalas, A., Sifaleras, A.: A hybrid particle swarm optimization-variable neighborhood search algorithm for constrained shortest path problems. Eur. J. Oper. Res. 261(3), 819–834 (2017)

    Article  MathSciNet  Google Scholar 

  21. Nawaz, M., Enscore Jr., E.E., Ham, I.: A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega 11(1), 91–95 (1983)

    Article  Google Scholar 

  22. Nogueira, T.H., Carvalho, C., Ravetti, M.G., Souza, M.: Analysis of mixed integer programming formulations for single machine scheduling problems with sequence dependent setup times and release dates. Pesquisa Oper. 39, 109–154 (2019)

    Article  Google Scholar 

  23. Pessoa, A., Uchoa, E., Aragão, M., Rodrigues, R.: Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems. Math. Program. Comput. 2, 259–290 (2010)

    Article  MathSciNet  Google Scholar 

  24. Pinedo, M.L.: Scheduling: Theory, Algorithms, and Systems. Springer, Berlin (2008)

    MATH  Google Scholar 

  25. Pirkwieser, S., Raidl, G.R.: Boosting a variable neighborhood search for the periodic vehicle routing problem with time windows by ilp techniques. In: Proceedings of the 8th Metaheuristic International Conference, Hamburg, Germany, pp. 33–35 (2009)

  26. Pirkwieser, S., Raidl, G.R., Gottlieb, J.: Improved packing and routing of vehicles with compartments. In: Computer Aided Systems Theory–EUROCAST 2011, pp. 392–399. Springer (2011)

  27. Pirkwieser, S., Raidl, G.R., Puchinger, J.: Combining lagrangian decomposition with an evolutionary algorithm for the knapsack constrained maximum spanning tree problem. In: Evolutionary Computation in Combinatorial Optimization, pp. 176–187. Springer (2007)

  28. Rubin, P.A., Ragatz, G.L.: Scheduling in a sequence dependent setup environment with genetic search. Comput. Oper. Res. 22(1), 85–99 (1995)

    Article  Google Scholar 

  29. Sousa, J.P., Wolsey, L.A.: A time indexed formulation of non-preemptive single machine scheduling problems. Math. Program. 54, 353–367 (1992)

    Article  Google Scholar 

  30. Subramanian, A., Battarra, M., Potts, C.N.: An iterated local search heuristic for the single machine total weighted tardiness scheduling problem with sequence-dependent setup times. Int. J. Prod. Res. 52(9), 2729–2742 (2014)

    Article  Google Scholar 

  31. Todosijević, R., Urošević, D., Mladenović, N., Hanafi, S.: A general variable neighborhood search for solving the uncapacitated \(r\)-allocation \(p\)-hub median problem. Optim. Lett. 11(6), 1109–1121 (2017)

    Article  MathSciNet  Google Scholar 

  32. Xu, H., Lu, Z., Yin, A., Shen, L., Buscher, U.: A study of hybrid evolutionary algorithms for single machine scheduling problem with sequence-dependent setup times. Comput. Oper. Res. 50, 47–60 (2014)

    Article  MathSciNet  Google Scholar 

Download references


This research was partially supported by FAPEMIG, CAPES and CNPq (Brazil). MGR acknowledges partial support from FUNDEP.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Thiago Henrique Nogueira.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A. Parameters tuning

A. Parameters tuning

The Sequential Parameter Optimization Toolbox (SPOT) is the used method to determine the parameters vector of the Hybrid algorithms. This parameters vector comprises different values of the parameters of the algorithm. The SPOT is an implementation of the Sequential Parameter Optimization (SPO), which is an iterative model-based method of tuning algorithms (see [2]). The tuning process is based on the parameters data, and their utility delivered by the performance of the hybrid proposed algorithm for generated viable solutions. SPO performs a multi-stage procedure where the model is updated at each iteration with a set of new vectors and new predictions of utility to improve the algorithm’s convergence. The SPOT is implemented with R package, and is available in the R archive network at See [2] for more details.

1.1 A. 1 Experimental setup

Therefore, the Sequential Parameter Optimization Toolbox (SPOT), consists of two optimization problems: problem-solving and parameter tuning. The problem-solving aims to find an optimal solution for the problem. The parameters tuning uses a tuning method to find the best parameter values for the algorithm. Thereby, the experimental setup consists of a set of eight instances with different sizes (four instances of each author—[13, 28]) of the single-machine scheduling problem, showed on Table 3. We summarize in Table 4 the types, the lower and upper bounds of the hybrid algorithm’s parameters used to define the region of interest (ROI) of the tuning algorithm.

Table 3 Sampling of instances bounds
Table 4 Parameter bounds for tuning the Volume Algorithm

1.2 A.2 Results of the experiment

The SPOT algorithm, implemented in the R package, is connected with the hybrid algorithm implemented in C++. The computer used in the tests is a Linux Maya with a single thread of 2.4 GHz and 4GB of RAM. Table 5 presents the best result found by the SPOT method. The first column indicates the iteration of SPOT, the second one presents the indicated parameters vector, and the last column indicates the solution value obtained by the tested instances. The best parameters vector indicated for the volume algorithm is Tolerance with 0.01, \(\alpha \) with 0.11, \(Y_{limit}\) with 9, \(R_{limit}\) with 28, and, \(\pi \) with n / 2.16, \(VNS_{max}\) with n, \(\phi \) with 3, where n is the number of jobs.

Table 5 Four best parameters settings for the volume algorithm

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nogueira, T.H., Ramalhinho, H.L., de Carvalho, C.R.V. et al. A hybrid VNS-Lagrangean heuristic framework applied on single machine scheduling problem with sequence-dependent setup times, release dates and due dates. Optim Lett 16, 59–78 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: