## 1 Introduction  
The flexible job shop scheduling problem (FJSP) aims to address the challenges of determining the processing sequence and machine allocation for operations within production systems [1]. It has been proven to be a complex combinatorial optimization problem [2]. Moreover, manufacturing enterprises often have multiple factories, each of which can be considered a flexible job shop [3]. As a result, it is essential to consider the allocation of jobs across these factories, which is the so-called distributed flexible job shop scheduling problem (DFJSP). The DFJSP is an extension of the classical FJSP, and it is characterized by its enhanced applicability to real-world production environments [4]. Specifically, it addresses substantial challenges regarding the management of distributed resources and coordination across factories. The development of efficient algorithms for the DFJSP is a critical topic in both academic and engineering research. In real production environments, processing times are often uncertain due to factors such as process damage, the equipment status, and operator proficiency [5]. These dynamic events can directly disrupt the original scheduling plan, necessitating real-time adjustments to job sequencing and resource allocation. As a result, the traditional DFJSP model needs to be extended to a dynamic DFJSP with variable processing time (DDFJSP-VPT) to address the impact of processing time variability, as a dynamic event, on scheduling decisions. In the DDFJSP-VPT, the scheduling problem requires not only the design of an efficient solution in the initial stage but also the ability to adapt to the deviations from estimated times. It is important to adjust the scheduling strategy in real time to maintain the system's performance and stability. This imposes higher requirements on scheduling algorithms, demanding both robustness and flexibility to respond to such dynamic fluctuations [6]. The methods for solving the DFJSP can be mainly divided into exact algorithms, heuristic algorithms, and metaheuristic algorithms. The exact methods are not able to find feasible solutions for comparatively complex models in acceptable timeframes. Moreover, although heuristic algorithms are easy to implement, it is difficult to obtain high-quality solutions. In contrast, metaheuristic algorithms have been widely used since they can maintain a balance between the solution quality and operational efficiency. As a relatively new member of this group of methods, the grey wolf algorithm (GWO) is inspired by the social hierarchy and predatory behavior of grey wolves; it has the advantages of simplicity, flexibility, and ease of implementation. So far, GWOs have been applied to a variety of scheduling problems, such as the job shop scheduling problem [7], the flexible job shop scheduling problem [8; 9; 10], the flow job shop scheduling problem [11], and the distributed flexible job shop scheduling problem [12; 13]. However, to the best of our knowledge, there is no reported work about using the GWO to solve the DDFJSP-VPT. Few existing studies have fully considered the effects of variable processing times on the DFJSP since they assume that the processing time is predetermined and unchangeable, overlooking the potential for dynamic changes during practical production. Moreover, the existing scheduling methods struggle to address this issue, quickly respond to similar events, and make reasonable decisions. Thus, it remains essential to model and examine the changes in the processing time of the DFJSP. To address the above challenges, this paper introduces a novel approach that focuses on handling dynamic changes in processing times within the framework of the DDFJSP-VPT. Specifically, we make the following contributions:  
1. A multi-objective DDFJSP-VPT mathematical model is established with the objectives of simultaneously minimizing the makespan, total factory load, and tardiness, in which the processing time fluctuation follows a normal distribution. 2.