Integrated Offline and Online Learning to Solve a Large Class of Scheduling Problems

TL;DR

This work tackles single-machine scheduling with non-decreasing min-sum objectives by proposing a unified ML framework that operates over a time-indexed formulation. The core is the Unified Machine Scheduling Neural Network (UMSNN), which ingests raw inputs derived from $c_{jt}=z_j(t+p_j)$, processing times $p_j$, and release dates $r_j$, and outputs per-job windowed start-time probabilities via a CNN-augmented transformer. To overcome the lack of optimal labels, the authors introduce offline training on specially constructed large instances whose optimal solutions scale from small ones, along with data augmentation, and they further enable instance-specific refinement through online single-instance learning using a differentiable feasibility surrogate. Empirical results on problems with up to $n=1000$ jobs show that offline learning yields fast, high-quality solutions (gaps often under 5%), and the integrated offline+online approach achieves near-optimality (1–2% gaps) with modest additional time, highlighting strong practical potential for large-scale scheduling in manufacturing and related domains.

Abstract

In this paper, we develop a unified machine learning (ML) approach to predict high-quality solutions for single-machine scheduling problems with a non-decreasing min-sum objective function with or without release times. Our ML approach is novel in three major aspects. First, our approach is developed for the entire class of the aforementioned problems. To achieve this, we exploit the fact that the entire class of the problems considered can be formulated as a time-indexed formulation in a unified manner. We develop a deep neural network (DNN) which uses the cost parameters in the time-indexed formulation as the inputs to effectively predict a continuous solution to this formulation, based on which a feasible discrete solution is easily constructed. The second novel aspect of our approach lies in how the DNN model is trained. In view of the NP-hard nature of the problems, labels (i.e., optimal solutions) are hard to generate for training. To overcome this difficulty, we generate and utilize a set of special instances, for which optimal solutions can be found with little computational effort, to train the ML model offline. The third novel idea we employ in our approach is that we develop an online single-instance learning approach to fine tune the parameters in the DNN for a given online instance, with the goal of generating an improved solution for the given instance. To this end, we develop a feasibility surrogate that approximates the objective value of a given instance as a continuous function of the outputs of the DNN, which then enables us to derive gradients and update the learnable parameters in the DNN. Numerical results show that our approach can efficiently generate high-quality solutions for a variety of single-machine scheduling min-sum problems with up to 1000 jobs.

Figures (5)

• Figure 1: Architecture of the Unified Machine Scheduling Neural Network
• Figure 2: Step function and sigmoid function
• Figure 3: Offline supervised learning for subclasses M-9 to M-12
• Figure 4: Computation flows in MLPs
• Figure 5: Online single-instance learning for an instance of $1|r_j|\sum w_jT_j$ with 700 jobs

Theorems & Definitions (2)

• Theorem 1
• Theorem 2