**2. Related Works**

ABC has been successfully applied to solve shop scheduling problems due to its advantages such as few control parameters and ease of implementation [20]. As there is no research related to ABC for solving FHFGSP, this section reviews the work related to the use of ABC for solving shop scheduling problems.

Li et al. [18] proposed a novel hybrid ABC and tabu search algorithm (TABC) to solve the HFSP finite buffers, employing a TS-based adaptive neighborhood strategy that gives the TABC algorithm the ability to learn and generate neighborhood solutions in different promising regions as a means to minimize makespan. Yue et al. [21] investigated the batching and hybrid model scheduling problem in a flexible parallel production line, considering the sequence-dependent setup time between hybrid model products with the aim of minimizing the manufacturing cycle time of the line while balancing the workload between lines and maximizing the net profit. In addition, a new material availability constraint is introduced to the problem. A novel Pareto guided ABC is designed to address the current problem. Gong et al. [22] considered the impact and potential of human factors on improving productivity and reducing production costs in real production systems and proposed a hybrid ABC to solve flexible job shop scheduling problems (FJSP) with worker flexibility. Zadeh et al. [23] proposed a heuristic model based on an ABC for the dynamic FJSP. Lei et al. [24] studied the distributed unrelated parallel machine scheduling problem with preventive maintenance (DUPMSP) and proposed an ABC with division to minimize MS. Xie et al. [25] proposed an improved ABC considering machining structure evaluation to solve the flexible integrated scheduling problem of networked equipment, which is an extension of job shop scheduling. Xuan et al. [20] proposed an improved DABC with the introduction of a genetic algorithm to solve FJSP for uncorrelated parallel machines with progressively deteriorating jobs and timing dependencies.

As flow shops are very common in practical production activities, the HFSP is of high research value. Wang et al. [19] proposed a new decoding method that simultaneously considers spindle speed optimization and scheduling scheme optimization and acts on the distribution estimation algorithm to simultaneously reduce energy consumption and makspan in the turning shop. Li et al. [26] proposed an improved ABC to solve the distributed flow shop problem (DFSP) with the objective of minimizing MS. Li et al. [27] proposed a hybrid ABC to solve the parallel batch DFSP with deteriorating jobs. In the proposed algorithms, two types of problem-specific heuristics are proposed, namely batch allocation and right-shift heuristics, which can significantly shorten makespan. Gong et al. [28] proposed a hybrid multi-objective DABC for solving the blocked batch flow process shop scheduling problem with two conflicting criteria of minimizing MS and lead time. With the objective of minimizing the total process time, Pan et al. [29] solved the distributed arrangemen<sup>t</sup> flow job scheduling problem based on a high-performance framework of DABC. Li et al. [30] proposed an improved ABC to solve a multi-objective optimization model with the objectives of minimizing MS and processing cost for the hybrid flow shop process planning and production scheduling independently of each other. Peng et al. [31] investigated the problem of flow shop rescheduling in the actual steelmaking process, considering interruptions caused by machine failures and controllable processing times in the final stages, and proposed an improved ABC to solve the problem.

However, in actual production, the processing time of jobs is often uncertain and there is very little research on ABC solutions to fuzzy HFSP. Zhong et al. [32] proposed a new artificial swarm algorithm, the improved artificial swarm algorithm, for the multi-objective fuzzy FJSP. The objectives are to minimize the maximum fuzzy MS, maximize the weighted consistency index and minimize the maximum fuzzy machine workload.

Most of the research on the use of ABC to solve shop scheduling problems is in the area of improving economic efficiency. Very little research has been done on saving energy and reducing pollution emissions. Li et al. [33] designed an improved ABC to solve a multi-objective low-carbon job shop scheduling problem with variable machining speed constraints. Zhang et al. [34] studied HFGSP with variable machine processing speed to minimize MS and TEC and proposed a multi-objective DABC (MDABC) to solve HFGSP. However, in HFGSP, the processing time of the job is set to an exact value, which is not fully compatible with the actual production environment. In real life, the processing time of the job often deviates due to the operator's business ability, machine aging, etc. Moreover, the neighborhood search adopted by MDABC in the employed bee phase and the binary race strategy adopted in the onlooker bee phase make the algorithm suffer from the problem that it cannot fully explore in the solution space, the convergence of the algorithm is not high, and it is easy to fall into local optimum.

For this reason, this paper studies the FHFGSP with uncertain job processing time and proposes SDABC to solve FHFGSP. In SDABC, the dominant individuals guide the poor individuals to update in the employed bee phase, which improves the convergence speed of the population, and the proposed ranking-based selection strategy and mutation strategy can prevent individuals from falling into local optimum in the onlooker bee phase. FHFGSP is consistent with the actual production environment and production requirements, but it is not common in previous studies.
