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

Optimization Letters volume 16, pages 59–78 (2022)Cite this article

Abstract

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.

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Fig. 1

Notes

  1. https://github.com/yotiti/Instances1machineTardiness.

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Acknowledgements

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

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Authors and Affiliations

  1. Departamento de Engenharia de Produção, Universidade Federal de Viçosa, Viçosa, Brazil

    Thiago Henrique Nogueira

  2. University Pompeu Fabra, Barcelona, Spain

    Helena Lourenço Ramalhinho

  3. Departamento de Engenharia de Produção, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Carlos R. V. de Carvalho

  4. Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Martín Gómez Ravetti

Authors
  1. Thiago Henrique Nogueira
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  2. Helena Lourenço Ramalhinho
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  3. Carlos R. V. de Carvalho
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  4. Martín Gómez Ravetti
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Correspondence to Thiago Henrique Nogueira.

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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 http://cran.r-project.org/web/packages/SPOT/index.html. 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
Full size table
Table 4 Parameter bounds for tuning the Volume Algorithm
Full size table

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
Full size table

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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). https://doi.org/10.1007/s11590-019-01525-7

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