A multidimensional probabilistic model based evolutionary algorithm for the energy-efficient distributed flexible job-shop scheduling problem

With escalating environmental effects, the spotlight on low-carbon manufacturing has garnered significant attention. The rise of distributed production has emerged as a prominent trend in response to the imperatives of economic globalization. This article focuses on addressing the energy-efficient distributed flexible job-shop scheduling problem (EE_DFJSP), with the aim of minimizing both makespan and total energy consumption (TEC) simultaneously. The production process contains four pivotal phases: 1) job assignment in distributed factories; 2) machine selection within factories; 3) operation allocation on flexible machines; and 4) machine speed adjustment for processing. Given the problem's multi-phase and strong coupling characteristics, it is imperative to develop a promising evolutionary algorithm (EA) for EE_DFJSP. To tackle this challenge, we propose a multidimensional probabilistic model-based EA (MPMEA) paradigm. First, problem-specific encoding and decoding schemes are developed based on the solution features of EE_DFJSP. Second, a hybrid initialization strategy incorporating four heuristic rules is devised to yield an initial population with diversity. Third, an effective union probabilistic model (UPM) is formulated to learn promising patterns from superior solutions, and an efficient sampling strategy is designed to produce high-quality offspring individuals. To achieve a balance between global exploration and local exploitation, problem-specific multiple neighborhood operators are proposed to perform an in-depth local search. Furthermore, a two-stage energy-saving speed adjustment strategy is designed for the superior solutions obtained through local search. Finally, computational comparisons and simulation studies are conducted to validate the effectiveness and superiority of the MPMEA in effectively addressing EE_DFJSP. © 2024