**1. Introduction**

The growth of manufacturing has brought economic and social prosperity. Shop scheduling, as a key part of manufacturing, plays an important role in economic development. Hybrid flow shop (HFS) is a common manufacturing environment [1] that combines the features of process shop and parallel machine scheduling and is widely used in container handling [2], electronics manufacturing, chemical production, and steel production [3–5], in addition to applications in internet service architecture [6], civil engineering [7], and production planting [8]. The hybrid flow shop scheduling problem (HFSP) refers to multiple jobs to be processed in multiple stages with one or more machines in each stage, and a specific optimization objective is achieved by determining the order in which the jobs are processed and the allocation of machines to each job in each stage [1]. It is worth noting that there are two other cases of HFSP in real life [9,10]: (1) the processing time

**Citation:** Li, M.; Wang, G.-G.; Yu, H. Sorting-Based Discrete Artificial Bee Colony Algorithm for Solving Fuzzy Hybrid Flow Shop Green Scheduling Problem. *Mathematics* **2021**, *9*, 2250. https://doi.org/10.3390/ math9182250

Academic Editor: Frank Werner

Received: 16 August 2021 Accepted: 8 September 2021 Published: 14 September 2021

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of a job is often not fixed but fluctuates within a limited range due to worker proficiency, newness of the machine. (2) The same machine processing different jobs requires a certain setup of the machine before processing, and due to the differences between jobs, the setup time required by the machine varies from job to job. Therefore, it is more meaningful and practical to study HFSP with setup time and fuzzy job processing time.

While the world is experiencing unprecedented economic and social prosperity, environmental pollution and energy scarcity are becoming a serious problem that seriously affects the future development of humanity. In particular, the manufacturing industry takes up most of the world's energy and produces a large amount of pollutant emissions [11]. Therefore, in order to solve the energy and environmental problems, green shop scheduling, as a key aspect of manufacturing, has become a hot spot for research [12]. The purpose of green shop scheduling is to reduce energy consumption, reduce environmental pressure, and achieve sustainable development without losing economic benefits. Therefore, the widely used hybrid flow green shop scheduling problem (HFGSP) has a high research value.

However, HFGSPs that consider fuzzy job processing time are not common at present. Fu et al. [13] developed a hybrid multi-objective optimization algorithm to solve HFSP with fuzzy processing time but did not consider the energy problem. Wang et al. [14] investigated the HFGSP of job processing time variation caused by the dynamic reconfiguration process of the device to minimize the energy consumption of makespan and the whole device and proposed an improved multi-objective whale optimization algorithm to solve it.

As HFSP has a wide range of application scenarios, the uncertain job processing time meets the actual production needs and the energy saving is in line with the future direction of manufacturing. In this paper, we study the fuzzy hybrid flow green shop scheduling problem (FHFGSP) which meets the above three scenarios and is less studied currently. FHFGSP considers fuzzy job processing time and machine setup time with the objective of minimizing both makespan (MS) and total energy consumption (TEC). Uncertain completion time is denoted by triangular fuzzy numbers (TFN) and TEC is divided into three parts: machine working time, machine setup time, and machine idle time. At present, there are not many HFGSPs that consider both fuzzy processing time and work sequence-related setup time, but FHFGSP is more in line with actual production scenarios and has higher research value.

Artificial bee colony (ABC) [15] is one of the swarm intelligence algorithms, which is divided into employed bees, onlooker bees, and scout bees according to the foraging behavior of the swarm, with good global exploration and local development. ABC has been shown to be superior or close to other classical swarm intelligence algorithms [16,17]. ABC is widely used to solve shop scheduling problems [18]. To solve FHFGSP, this paper proposed a sorting-based discrete artificial bee colony algorithm (SDABC). Individuals in the population are ranked according to non-dominated solutions and similarity to the ideal solution and adopt different search and follow strategies according to the location to achieve full exploration of the solution space and discover better solutions. It is worth mentioning that SDABC can be used not only to solve FHFGSP problems such as turning shop [19]. It can also be used to solve the expansion of FHFGSP described in the first paragraph.

The main contributions of this paper are as follows:


The paper is organized as follows: Section 2 gives the relevant works, Section 3 describes what the FHFGSP is, gives a symbolic representation and builds a mathematical model of the FHFGSP. Section 4 details the SDABC for solving the FHFGSP. Experimental validation is presented in Section 5 and the last section contains conclusions and outlook.
