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An Expectation-Maximization Algorithm-based Autoregressive Model for the Fuzzy Job Shop Scheduling Problem

Yijian Wang, Tongxian Guo, Zhaoqiang Liu

TL;DR

This work tackles the fuzzy job shop scheduling problem (FJSSP) by reframing it as a generative task and introducing EMARM, an expectation–maximization–based autoregressive model. EMARM uses a hand-crafted fuzzy prior, a Graph Attention Network encoder, and a two-part decoder to generate scheduling schemes and a policy $p_ heta(\bm{\pi} \mid I)$, trained with an EM-based strategy to handle latent optimal schedules. The approach demonstrates faster solution times and superior fuzzy makespans compared with established baselines (NSODE, HADE, GA, PSO) on large synthetic datasets, indicating strong potential for practical fuzzy scheduling. The work also outlines avenues for extending the framework with diffusion, GAN, or flow-based generative models and for applying it to other scheduling domains.

Abstract

The fuzzy job shop scheduling problem (FJSSP) emerges as an innovative extension to the job shop scheduling problem (JSSP), incorporating a layer of uncertainty that aligns the problem more closely with the complexities of real-world manufacturing environments. This improvement increases the computational complexity of deriving the solution while improving its applicability. In the domain of deterministic scheduling, neural combinatorial optimization (NCO) has recently demonstrated remarkable efficacy. However, its application to the realm of fuzzy scheduling has been relatively unexplored. This paper aims to bridge this gap by investigating the feasibility of employing neural networks to assimilate and process fuzzy information for the resolution of FJSSP, thereby leveraging the advancements in NCO to enhance fuzzy scheduling methodologies. To achieve this, we approach the FJSSP as a generative task and introduce an expectation-maximization algorithm-based autoregressive model (EMARM) to address it. During training, our model alternates between generating scheduling schemes from given instances (E-step) and adjusting the autoregressive model weights based on these generated schemes (M-step). This novel methodology effectively navigates around the substantial hurdle of obtaining ground-truth labels, which is a prevalent issue in NCO frameworks. In testing, the experimental results demonstrate the superior capability of EMARM in addressing the FJSSP, showcasing its effectiveness and potential for practical applications in fuzzy scheduling.

An Expectation-Maximization Algorithm-based Autoregressive Model for the Fuzzy Job Shop Scheduling Problem

TL;DR

This work tackles the fuzzy job shop scheduling problem (FJSSP) by reframing it as a generative task and introducing EMARM, an expectation–maximization–based autoregressive model. EMARM uses a hand-crafted fuzzy prior, a Graph Attention Network encoder, and a two-part decoder to generate scheduling schemes and a policy , trained with an EM-based strategy to handle latent optimal schedules. The approach demonstrates faster solution times and superior fuzzy makespans compared with established baselines (NSODE, HADE, GA, PSO) on large synthetic datasets, indicating strong potential for practical fuzzy scheduling. The work also outlines avenues for extending the framework with diffusion, GAN, or flow-based generative models and for applying it to other scheduling domains.

Abstract

The fuzzy job shop scheduling problem (FJSSP) emerges as an innovative extension to the job shop scheduling problem (JSSP), incorporating a layer of uncertainty that aligns the problem more closely with the complexities of real-world manufacturing environments. This improvement increases the computational complexity of deriving the solution while improving its applicability. In the domain of deterministic scheduling, neural combinatorial optimization (NCO) has recently demonstrated remarkable efficacy. However, its application to the realm of fuzzy scheduling has been relatively unexplored. This paper aims to bridge this gap by investigating the feasibility of employing neural networks to assimilate and process fuzzy information for the resolution of FJSSP, thereby leveraging the advancements in NCO to enhance fuzzy scheduling methodologies. To achieve this, we approach the FJSSP as a generative task and introduce an expectation-maximization algorithm-based autoregressive model (EMARM) to address it. During training, our model alternates between generating scheduling schemes from given instances (E-step) and adjusting the autoregressive model weights based on these generated schemes (M-step). This novel methodology effectively navigates around the substantial hurdle of obtaining ground-truth labels, which is a prevalent issue in NCO frameworks. In testing, the experimental results demonstrate the superior capability of EMARM in addressing the FJSSP, showcasing its effectiveness and potential for practical applications in fuzzy scheduling.

Paper Structure

This paper contains 15 sections, 31 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fuzzy number
  4. FJSSP
  5. EMARM
  6. Autoregressive model for FJSSP
  7. Hand-crafted fuzzy prior
  8. Encoder
  9. Decoder
  10. Training strategy based on EM algorithm
  11. Numerical Result and Comparisons
  12. Dataset and test instances
  13. Architecture, hyperparameters, and training
  14. Comparisons to other algorithms
  15. Conclusion and Future Work

Figures (1)

  • Figure 1: Disjunctive graph model. (a) illustrates a 3 × 3 FJSSP instance. The black solid lines delineate set A, while the colored dashed lines enclose set E. Operations with the same frame color must be executed on the same machine. (b) presents a solution for (a), where all undirected edges have been directed.