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Deep learning-driven scheduling algorithm for a single machine problem minimizing the total tardiness

Michal Bouška, Přemysl Šůcha, Antonín Novák, Zdeněk Hanzálek

TL;DR

The paper tackles the NP-hard problem of minimizing total tardiness on a single machine, defined as $1||\sum T_j$, by introducing a deep-learning-guided decomposition (dhs) that augments Lawler's edd-based and Della Croce's spt-based decompositions with a regressor to estimate optimal subproblem tardiness. The regressor, implemented as a two-layer recurrent network (preferably LSTM, capacity ~256) operating on normalized job features, guides the selection of splitting positions, enabling fast, single-pass exploration of the solution space while preserving problem structure. Training data is efficiently generated via Generate & Solve and a Subproblem Generator, with the latter yielding millions of informative subproblems and enabling the model to generalize to much larger instances (up to 800 jobs) with average gaps around $0.26\%$, outperforming state-of-the-art heuristics and tight time-bounded exact methods. The results demonstrate that integrating machine learning with classical decomposition can realize substantial practical gains in scheduling, and the authors outline promising directions for extending this approach to other NP-hard problems and alternative objective functions.

Abstract

In this paper, we investigate the use of the deep learning method for solving a well-known NP-hard single machine scheduling problem with the objective of minimizing the total tardiness. We propose a deep neural network that acts as a polynomial-time estimator of the criterion value used in a single-pass scheduling algorithm based on Lawler's decomposition and symmetric decomposition proposed by Della Croce et al. Essentially, the neural network guides the algorithm by estimating the best splitting of the problem into subproblems. The paper also describes a new method for generating the training data set, which speeds up the training dataset generation and reduces the average optimality gap of solutions. The experimental results show that our machine learning-driven approach can efficiently generalize information from the training phase to significantly larger instances. Even though the instances used in the training phase have from 75 to 100 jobs, the average optimality gap on instances with up to 800 jobs is 0.26%, which is almost five times less than the gap of the state-of-the-art heuristic.

Deep learning-driven scheduling algorithm for a single machine problem minimizing the total tardiness

TL;DR

The paper tackles the NP-hard problem of minimizing total tardiness on a single machine, defined as , by introducing a deep-learning-guided decomposition (dhs) that augments Lawler's edd-based and Della Croce's spt-based decompositions with a regressor to estimate optimal subproblem tardiness. The regressor, implemented as a two-layer recurrent network (preferably LSTM, capacity ~256) operating on normalized job features, guides the selection of splitting positions, enabling fast, single-pass exploration of the solution space while preserving problem structure. Training data is efficiently generated via Generate & Solve and a Subproblem Generator, with the latter yielding millions of informative subproblems and enabling the model to generalize to much larger instances (up to 800 jobs) with average gaps around , outperforming state-of-the-art heuristics and tight time-bounded exact methods. The results demonstrate that integrating machine learning with classical decomposition can realize substantial practical gains in scheduling, and the authors outline promising directions for extending this approach to other NP-hard problems and alternative objective functions.

Abstract

In this paper, we investigate the use of the deep learning method for solving a well-known NP-hard single machine scheduling problem with the objective of minimizing the total tardiness. We propose a deep neural network that acts as a polynomial-time estimator of the criterion value used in a single-pass scheduling algorithm based on Lawler's decomposition and symmetric decomposition proposed by Della Croce et al. Essentially, the neural network guides the algorithm by estimating the best splitting of the problem into subproblems. The paper also describes a new method for generating the training data set, which speeds up the training dataset generation and reduces the average optimality gap of solutions. The experimental results show that our machine learning-driven approach can efficiently generalize information from the training phase to significantly larger instances. Even though the instances used in the training phase have from 75 to 100 jobs, the average optimality gap on instances with up to 800 jobs is 0.26%, which is almost five times less than the gap of the state-of-the-art heuristic.
Paper Structure (20 sections, 2 theorems, 3 equations, 12 figures, 5 tables, 1 algorithm)

This paper contains 20 sections, 2 theorems, 3 equations, 12 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Single Machine Total Tardiness Problems
  4. Use of Machine Learning in Algorithms for Combinatorial Optimization Problem
  5. Problem Statement
  6. Proposed Decomposition Heuristic Algorithm
  7. Problem Decompositions
  8. Scheduling Algorithm
  9. Regressor
  10. Normalization of the Input Data
  11. Neural Network
  12. Training Data Set Generation
  13. Time Complexity of the Scheduling Algorithm
  14. Experimental Results
  15. Experimental setup
  16. ...and 5 more sections

Key Result

Theorem 4.1

(Lawler, 1977) Suppose jobs $J$ are ordered in edd order and the splitting job is $l^{EDD}(J\xspace)\xspace = \arg \max_{i \in J\xspace}{p_i}$. Then, there is some integer $k$, $l^{EDD}(J\xspace)\xspace \le k\xspace \le n\xspace\xspace$, such that there exists an optimal sequence $\pi^*{}\xspace$ in

Figures (12)

  • Figure 1: Regressor architecture.
  • Figure 2: Distribution of training sample size for Subproblem generator method with different range of instances and Generate $\&$ Solve.
  • Figure 3: Distribution of $\mathit{rdd}$ and $\mathit{tf}$ over $n$ in the data set generated by Subproblem generator.
  • Figure 4: Optimality gap for instances with $p_{max}\xspace = 100.$
  • Figure 5: Optimality gap for instances with $p_{max}\xspace = 5000.$
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 4.2