A Hybrid Evolutionary Algorithm for Multi-Stage Workshop Sequencing in Car Production

Abstract

During the car production process, diverse production workshops have distinct prerequisites for car body sequencing. This results in the intricate nature of sequencing within multi-stage car workshops. In this study, an optimization method for car body sequencing is proposed that combines a hybrid evolutionary algorithm with heuristic rules. In the welding workshop, a genetic algorithm is employed to optimize the vehicle sequencing. Simultaneously, a differential evolution algorithm is used to optimize the inbound sequence of the buffer zone between the welding and painting workshops, as well as the inbound sequence of the buffer zone between the painting and assembly workshops. Heuristic rules are applied to optimize the outbound sequence of the buffer zone between the welding and painting workshops, as well as the outbound sequence of the buffer zone between the painting and assembly workshops. In addition, in order to improve the quality of the initial population, a heuristic method-based initial population construction method is proposed. The optimization objectives are the number of vehicle model changes in the welding workshop, the number of color changes in the painting workshop, and the total number of overloads in the assembly workshop. The experimental results show that the proposed method performs better than the five outstanding evolutionary algorithms.

1. Introduction

In the face of intensifying market competition, automotive manufacturers dealing with customized and diversified products are encountering substantial challenges. Implementing Toyota Production System (TPS) and Heijunka can help automotive manufacturers effectively address these challenges. The automobile manufacturing process is highly intricate, involving collaborative efforts from various workshops. These primarily encompass the stamping workshop, welding workshop, painting workshop, and assembly workshop [1]. Due to process differences in different workshops, the factors considered in the production sequencing for different workshops are inconsistent. For example, within a painting workshop, the optimal production sequencing should aim to minimize the number of color changes. However, in an assembly workshop, factors such as the total number of overloads or idle time become significant considerations [2]. The collaborative optimization of production sequences in different workshops contributes to the implementation of TPS and Heijunka, thereby improving overall production efficiency and reducing the overall production costs for the automotive manufacturers. As a result, the sequencing optimization problem has garnered significant attention from researchers.

Currently, many studies focus on the sequencing problem of a single workshop. However, the sequencing of the painting workshop is influenced by the production sequence of the welding workshop, while the sequencing of the assembly workshop is influenced by the production sequence of the painting workshop. Collaborative sequencing of multiple workshops can simultaneously optimize the optimization objectives of each workshop, such as the number of color changes, total idle time, and number of overloads. However, compared to the sequencing problem of a single workshop, the collaborative sequencing problem of multiple workshops is more difficult and has received less attention.

Aiming at the collaborative sequencing problem of the welding workshop, painting workshop, and assembly workshop, this paper studies the sequencing method with the optimization objectives of number of vehicle model changes, number of color changes, and total number of overloads. This study takes into account the presence of physical buffer zones between the welding workshop and the painting workshop, as well as between the painting workshop and the assembly workshop. In order to effectively address the sequencing problem of three-stage workshops, a hybrid multi-objective evolutionary algorithm has been proposed by combining two types of evolutionary algorithms with multiple heuristic rules. In the welding workshop, a genetic algorithm is utilized to optimize the car sequencing. Concurrently, a differential evolution algorithm is employed to optimize the inbound sequence of the buffer zone between the welding and painting workshops, as well as the inbound sequence of the buffer zone between the painting and assembly workshops. Heuristic rules are applied to optimize the outbound sequence of the buffer zone between the welding and painting workshops, as well as the outbound sequence of the buffer zone between the painting and assembly workshops. Furthermore, in order to enhance the quality of the initial population, a heuristic-based method for constructing the initial population is proposed. To validate the effectiveness of the introduced algorithm, a comparison is conducted between the proposed approach and five prominent multi-objective evolutionary algorithms. The experimental outcomes demonstrate that the suggested algorithm outperforms five outstanding algorithms.

The remaining content of this paper is organized according to the following structure: Section 2 provides a review of the related works; Section 3 provides a detailed description of the problem and makes some assumptions; Section 4 presents the proposed hybrid evolutionary algorithm, and describes in detail the process of the proposed hybrid evolutionary algorithm to solve the problems in this study; Section 5 conducted extensive experiments to verify the effectiveness of the proposed method; Section 6 summarizes this paper from multiple aspects and proposes future research directions.

2. Related Works

At present, sequencing problems in automobile manufacturing can be divided into virtual sequencing and physical sequencing according to the types of buffer zones. In environments where there are no buffer zones between upstream and downstream workshops, only virtual sequencing methods can be employed to optimize production sequences. In an environment where there is a buffer zone between upstream and downstream workshops, the buffer zone can be used to resequencing downstream production sequences. Kampker et al. studied the proactive sequencing problem involving buffers and constructed a hybrid mathematical model to resequencing the production sequence when it changes [3]. Manzini et al. examined the production line imbalance arising from a diverse range of products and introduced a predictive–reactive method to tackle assembly line sequencing problems involving time uncertainty [4]. Müllerklein et al. proposed the integration of feeding planning and production scheduling for automotive production lines, and proposed a mixed integer programming model to solve these two planning problems [5]. Denizhan et al. solved the problem of deviation in the automotive painting sequencing process in a just in time production environment by utilizing observations, expert opinions, and data analysis [6]. Leng et al. investigated the resequencing challenge within a hybrid buffer system in the painting workshop. They introduced a reinforcement learning approach to address the issue of color switching in the painting shop [7]. Winter et al. introduced a comprehensive neighborhood search approach to tackle the optimization of color switching in painting workshop. The experimental outcomes demonstrated the effectiveness and competitiveness of their proposed method [8]. Tabar et al. proposed a rule-based geometric quality-oriented method for solving the optimization problem of welding workshop sequencing [9]. Tabar et al. proposed a recognition and sequence optimization method for geometric spot welding in a digital twin environment, which optimized the production sequence of the welding workshop [10]. Tian et al. proposed a small-world optimization algorithm and applied it to the resequencing problem between the painting workshop and the assembly workshop [11]. Yilmazlar and Kurz proposed an adaptive local search algorithm for car sequencing problem, and they demonstrated that the algorithm should focus on randomly exploring more non-degenerate solutions rather than spending time accessing degenerate solutions [12]. Thiruvady et al. designed a large neighborhood search algorithm to solve the car sequencing problem, and combined Lagrangian relaxation and ant colony optimization to identify promising regions [13]. Hottenrott et al. studied a robust car sequencing method considering failure probabilities. They developed a branch and bound algorithm to solve small instances. For large instances, they designed a sampling-based adaptive large neighborhood search heuristic algorithm to solve them [14].

In an environment where buffer zones are present within workshops, employing a physical resequencing method can effectively optimize the production sequence of downstream workshops. However, in situations where no buffer zone exists between upstream and downstream workshops, virtual resequencing technology can optimize the downstream production sequence to a certain extent. Virtual resequencing does not actually exchange the actual positions between cars, but simply exchanges the properties of two cars. For this reason, not all cars can be exchanged. For example, before the painting workshop, cars with the same type can be exchanged with each other, while before the assembly workshop, only cars with the same type and color can be exchanged with each other. Virtual resequencing can optimize production sequences even in the absence of buffer zones; therefore, virtual resequencing technology has garnered the attention of numerous researchers. Sun investigated the virtual resequencing problem within a painting workshop. They formulated a 0–1 programming model for the problem, focusing on minimizing the number of color changes, and introduced an ant colony optimization algorithm to address it [15]. Cao and Sun conducted research on a virtual resequencing challenge within mixed-model automobile assembly lines. This involved resequencing preassigned configurations to vehicles while maintaining their original sequence positions. They proposed three heuristic algorithms based on beam search to solve the problem [16]. Lübben et al. studied the resequencing problem between the painting workshop and the assembly workshop. They not only allowed the use of physical buffers to reorder the production sequence, but also allowed the use of virtual resequencing technology to reorder the production sequence [17]. Xu and Zhou studied the virtual resequencing problem in the painting workshop. On the premise of keeping the body sequence unchanged, they used a variety of heuristics to optimize the number of color changes. [18]. Sun et al. proposed a combination of virtual resequencing technology and physical resequencing technology for optimizing number of color changes in painting workshops [19]. Fournier and Agard investigated the resequencing problem in multiple workshops, with each workshop containing physical buffer zones. Their study allowed for virtual resequencing ahead of each physical buffer area [20].

Multi-workshop collaborative optimization allows for the simultaneous optimization of multiple objectives but also increases the complexity of implementation, which is why there is limited research in this area. Ribeiro et al. introduced a hybrid heuristic approach to tackle the multi-objective optimization problem of the painting workshop and assembly workshop, with the optimization objectives being the number of assembly constraint violations and the number of color changes [2]. Wu et al. established an integer programming model for the sequencing problem of multi-level workshops and designed corresponding heuristic algorithms to solve the problem [21]. Joly et al. studied the sequencing optimization problem of painting workshop and assembly workshop. They proposed a multi-objective heuristic algorithm to solve the multi-objective optimization problem of painting workshop and assembly workshop [22].

3. Problem Description and Assumptions

The manufacturing of a car usually involves various processes in the stamping workshop, welding workshop, painting workshop, and assembly workshop. Buffers are often set up between upstream and downstream workshops, which can provide conditions for resequencing production sequences in downstream workshops. Moreover, due to the inconsistent production speed in each workshop, the buffer zone can maintain a certain level of safety inventory, ensuring the smooth progress of production. Buffers can be divided into four types: automated storage and retrieval system (AS/RS), selectivity/mix bank, pull-off table, and insert buffer. Among them, the selectivity bank is widely used in practice, and the buffer considered in this study is selectivity bank. A selectivity bank typically contains multiple parallel lanes with the same capacity, and each vehicle can choose one of the lanes to enter the station, while also selecting one of the lanes to exit the station. The selectivity bank always maintains a stable inventory of work in process (WIP). A typical selectivity bank is shown in Figure 1. Due to the fact that the workpiece in the stamping workshop exists in the form of components, there is no need to set a buffer zone. Therefore, this paper studies the sequencing problem in an environment with painted body storage (PBS) and welding body storage (WBS). WBS refers to the buffer zone after the welding workshop and before the painting workshop, while PBS refers to the buffer zone after the painting.

Through the utilization of the selectivity bank, it becomes feasible to reorder the production sequences in the downstream workshops. For example, by employing the selectivity bank, it becomes possible to reorder the production sequence after the welding workshop. This allows for the optimization of the production sequence in the painting workshop, aiming to minimize the number of color changes within the painting workshop. Likewise, through the application of the selectivity bank, it is viable to reorder the production order following the painting workshop. This enables the optimization of the assembly workshop’s production sequence, aiming to minimize the total number of overloads. Employing the selectivity bank for reordering production sequences involves addressing two distinct problems: The first is the vehicle entry choice problem, which involves determining which lane each vehicle in the upstream production sequence selects to enter the station; the second is the vehicle exit decision problem, which involves determining how vehicles in each lane exit the station. The entry and exit strategies of vehicles affect the downstream production sequence and are the key to resequencing the downstream production sequence. It is assumed that the production sequence of the welding workshop is represented as $W=\left\{w_1,w_2,\dots ,w_n\right\}$, where n represents the total number of vehicles. The sequence of vehicles entering WBS through the welding workshop is $P^{in}=\left\{p_1^{in},p_2^{in},\dots ,p_n^{in}\right\}$, where $p_i^{in}$ represents the lane where $w_i$ chooses to enter the station. The outbound sequence of vehicles in WBS is represented as $P^{out}=\left\{p_1^{out},p_2^{out},\dots ,p_n^{out}\right\}$, where $p_i^{out}$ represents the lane where $p_i$ comes from in the production sequence $P=\left\{p_1,p_2,\dots ,p_n\right\}$ of the painting workshop. From the above, it can be concluded that $W,P^{in},P^{out}$, and $P$ have the relationship shown in Formula (1).

$$P=f^{\prime }\left(W,P^{in},P^{out}\right)$$

(1)

It is assumed that that the production sequence of the painting workshop entering the PBS is $Q^{in}=\left\{q_1^{in},q_2^{in},\dots ,q_n^{in}\right\}$, where $q_i^{in}$ represents the lane where $p_i$ chooses to enter the station. The outbound sequence of vehicles in PBS is represented as $Q^{out}=\left\{q_1^{out},q_2^{out},\dots ,q_n^{out}\right\}$, where $q_i^{out}$ represents the lane from which $s_i$ in the production sequence $S=\left\{s_1,s_2,\dots ,s_n\right\}$ of the assembly workshop comes from. $f^{\prime }$ represents the functional relationship between $P$ and $W$, $P^{in}$, and $P^{out}$. From the above, it can be concluded that $P, Q^{in}, Q^{out}$, and $S$ have the relationship shown in Formula (2).

$$S=f^{″}\left(P,Q^{in},Q^{out}\right)$$

(2)

where $f^{″}$ represents the functional relationship between $S$ and $P$, $Q^{in}$, and $Q^{out}$. By combining Formulas (1) and (2), the relationship between the production sequence of the assembly workshop and the welding workshop can be obtained as shown in Formula (3).

$$S=f^{″}\left(f^{\prime }\left(W,P^{in},P^{out}\right),Q^{in},Q^{out}\right)$$

(3)

From Formula (3), it can be seen that the production sequence of the assembly workshop is related to the production sequence $W$ of the welding workshop, the inbound sequence $P^{in}$ of WBS, the outbound sequence $P^{out}$ of WBS, the inbound sequence $Q^{in}$ of PBS, and the outbound sequence $Q^{out}$ of PBS. From formula (2), it can be seen that the production sequence of the painting workshop is related to the production sequence $W$ of the welding workshop, the inbound sequence $P^{in}$ of WBS, and the outbound sequence $P^{out}$ of WBS. From this, it can be seen that the optimization of the production sequence in the welding workshop, painting workshop, and final assembly workshop involves decision-making issues for $W$, $P^{in}$, $P^{out}$, $Q^{in}$, and $Q^{out}$. Figure 2 shows a floor plan of the location of the workshops and buffers. This study aims to optimize the production sequence of the welding workshop, painting workshop, and assembly workshop, in order to minimize the number of vehicle model changes (VMC) in the welding workshop, number of color changes (CC) in the painting workshop, and the total number of overloads (TO) in the assembly workshop. A vehicle model switching occurs when there is a change in the vehicle type between two consecutive vehicles produced in the welding workshop. The calculation method of VMC is shown in Formula (4). A color switch occurs when there is a change in the color between two consecutive vehicles produced in the painting workshop. The calculation method of CC is shown in Formula (5). In the assembly workshop, if an optional part has undergone k consecutive assembly operations and the k value exceeds a set value h, it is considered that the optional part has been overloaded once, and the overload condition of the optional part is represented by k/h. The calculation method of TO is shown in Formula (6).

$$\mathrm{V}\mathrm{M}\mathrm{C}={\sum }_{i=1}^{n-1}fv(w_{i+1},w_i)$$

(4)

$$\mathrm{C}\mathrm{C}={\sum }_{i=1}^{n-1}fc(p_{i+1},p_i)$$

(5)

$$\mathrm{T}\mathrm{O}={\sum }_{i=1}^{n-h}{\sum }_{j=1}^m{ft}_j(s_{i+h},s_i)$$

(6)

When the vehicle types of $w_{i+1}$ and $w_i$ are not consistent, $fv(w_{i+1},w_i)$ is equal to 1; otherwise, it is equal to 0. When the colors of $p_{i+1}$ and $p_i$ are not consistent, $fc(p_{i+1},p_i)$ is equal to 1; otherwise, it is equal to 0. If an optional part j has been assembled in cars $s_{i+h}$ to $s_i$, then ${ft}_j(s_{i+h},s_i)$ is equal to 1, otherwise it is equal to 0.

To effectively address the collaborative sequencing challenge across the three workshops, this study formulated the following assumptions:

(1)

Within a single lane of a selectivity bank, the positioning of neighboring vehicles cannot be interchanged, and all vehicles in the same lane adhere to a first-in-first-out principle;

(2)

This paper investigates the sequencing problem of production tasks within a short timeframe (such as a single day production cycle) and does not take into account delivery time. Here, it is assumed that all orders can be delivered punctually as planned;

(3)

All vehicles have the same priority;

(4)

All workshops follow the production sequence entered into the workshop while manufacturing vehicles, without any occurrences of queue jumping.

4. The Proposed Hybrid Evolutionary Algorithm

As can be seen from the previous text, the optimization of the production sequence in the welding workshop, painting workshop, and final assembly workshop involves decision-making issues for $W$, $P^{in}$, $P^{out}$, $Q^{in}$, and $Q^{out}$. $W$ is a sequencing optimization problem, and this paper employs a genetic algorithm to optimize $W$. For $P^{in}$ and $Q^{in}$, this paper employs a differential evolution approach for their optimization. For the optimization of $P^{out}$ and $Q^{out}$, this paper utilizes heuristic rule-based methods. Through the collaborative optimization of $W$, $P^{in}$, $P^{out}$, $Q^{in}$, and $Q^{out}$, the production sequence in the welding workshop, painting workshop, and assembly workshop is optimized, thereby achieving the objective of optimizing the number of vehicle model changes, number of color changes, and total number of overloads.

4.1. W. Optimization Method Based on Genetic Algorithm

The optimization of $W$ is a sequencing optimization problem, assuming that there is a set of vehicles $C=\left\{c_1,c,\dots ,c_n\right\}$, the optimization of $W$ requires finding the corresponding position of each vehicle in $W$. In the encoding scheme, the i-th individual is represented using the following encoding method: $X_i^w=\left\{x_1^p,x_2^p,\dots ,x_n^p\right\}$, each component represents a certain vehicle in the vehicle set $C$. $X_i^w=[\mathrm{2,1},\mathrm{5,3},4]$ indicates that vehicle $c_2$ is welded first, followed by vehicle $c_1$, and then by vehicle $c_5$, and so on. The two key operations in genetic algorithms are crossover and mutation operations. In the optimization problem of $W$, we use the order crossover operation to cross two individuals. For the given two individuals $X_i^w$ and $X_j^w$, the detailed steps of the crossover operation are shown in Algorithm 1. For a given individual $X_i^w$, we use a mutation operation such as Algorithm 2.

Algorithm 1. Pseudocode of the crossover operation

01. $\mathrm{Input}\text{:} \mathrm{two} \mathrm{individuals} X_i^w$$\mathrm{and} X_j^w$. 02. $\mathrm{Output}\text{:} \mathrm{two} \mathrm{new} \mathrm{individuals} O_i^w$$\mathrm{and} O_j^w$. 03. Randomly generate an integer D between [1, n]. 04. $\mathrm{Divide} X_i^w$ into two parts, mark the first D components of $X_i^w$$\mathrm{as} \dot{X_i^w}$, find the fragments of the remaining n-D components of $X_i^w$$\mathrm{in} X_j^w$ and keep the order unchanged, mark them as $\ddot{X_i^w}$. Similarly, divide $X_j^w$ into two parts, mark the first D components of $X_j^w$$\mathrm{as} \dot{X_j^w}$, find the fragments of the remaining n-D components of $X_j^w$$\mathrm{in} X_i^w$ and keep the order unchanged, mark them as $\ddot{X_j^w}$. 05. $\mathrm{Assign} \dot{X_i^w}$$\mathrm{to} \mathrm{the} \mathrm{first} D \mathrm{components} \mathrm{of} O_i^w$$\mathrm{and} \ddot{X_i^w}$$\mathrm{to} \mathrm{the} \mathrm{remaining} \mathrm{components} \mathrm{of} O_i^w$. 06. $\mathrm{Assign} \dot{X_j^w}$$\mathrm{to} \mathrm{the} \mathrm{first} D \mathrm{components} \mathrm{of} O_j^w$$\mathrm{and} \ddot{X_j^w}$$\mathrm{to} \mathrm{the} \mathrm{remaining} \mathrm{components} \mathrm{of} O_j^w$.

Algorithm 2. Pseudocode of the mutation operation

01. $\mathrm{Input}\text{:} \mathrm{individual} X_i^w$. 02. $\mathrm{Output}\text{:} \mathrm{new} \mathrm{individual} O_i^w$. 03. $\mathrm{Randomly} \mathrm{generate} \mathrm{two} \mathrm{integers} k_1$$\mathrm{and} k_2$ between [1, n]. 04. $\mathrm{Assign} X_i^w$$\mathrm{to} O_i^w$. 05. $\mathrm{if} k_1<k_2$ 06.  $\mathrm{Insert} \mathrm{the} k_2$$\text{-}\mathrm{th} \mathrm{component} \mathrm{of} O_i^w$$\mathrm{before} \mathrm{the} k_1$-th component. 07. else 08.  $\mathrm{Insert} \mathrm{the} k_1$$\text{-}\mathrm{th} \mathrm{component} \mathrm{of} O_i^w$$\mathrm{before} \mathrm{the} k_2$-th component. 09. end

4.2. Optimization Method of $P^{in}$ and $Q^{in}$ Based on Differential Evolution

For the optimization problem of $P^{in}$ and $Q^{in}$, this study designs differential evolution to optimize them. Each vector of the differential evolution is represented as $X_i^p=\left\{x_1^p,x_2^p,\dots ,x_n^p\right\}$, and each component represents the lane selection of the corresponding vehicle. For example, for the production sequence $X_i^w=[\mathrm{2,1},\mathrm{5,3},4]$ of the welding workshop, the vector $X_i^p=[\mathrm{3,1},\mathrm{2,1},2]$ of a differential evolution indicates that vehicle 2 enters from lane 3, vehicle 1 enters from lane 1, vehicle 5 enters from lane 2, vehicle 3 enters from lane 1, and vehicle 4 enters from lane 2. There are numerous factors that influence the performance of a differential evolution, including parameter control methods, mutation operations, crossover operations, and selection operations. Among them, mutation operations and parameter control methods are research hotspots. So far, researchers have developed many mutation operations, each with unique benefits. For example, DE/rand/1 has good global search ability, while DE/current to best/1 has good local search ability. Usually, a single mutation operation is difficult to have both good global and local search capabilities. Many researchers pay attention to the combination of multiple mutation operations and the realization of the balance between the global search ability and the exploration ability of differential evolution by using the characteristics of different mutation operations. Hence, this paper employs a hybrid approach incorporating two mutation operations to enhance the performance of differential evolution. The two mutation operations are depicted in Formulas (7) and (8).

$$U_i=X_{r1}^p+F\times \left({X_{best}^p-X}_{r1}^p\right)+F\times \left({X_{r2}^p-X}_{r3}^p\right)$$

(7)

$$U_i=X_{r1}^p+F\times \left({X_i^p-X}_{r1}^p\right)+F\times \left({X_{r2}^p-X}_{r3}^p\right)$$

(8)

where $X_{r1}^p$, $X_{r2}^p$, and $X_{r3}^p$ are three different individuals randomly selected from the population ($r1\ne r2\ne r3\ne i$), and $U_i=\left\{u_1,u_2,\dots ,u_n\right\}$ represents the mutation vector. $X_{best}^p$ is the best individual in the current population, and F is the scaling factor. In the hybrid mutation strategy, the mutation operation of Formula (7) is used to enhance the local search ability of the algorithm, while the mutation operation of Formula (8) is used to enhance the global search ability of the algorithm. Following the mutation operation, the crossover operation is applied to the mutant vector $U_i$ and the target vector $X_i^p$, as illustrated in Formula (9), to obtain the trial vector $V_i$.

$$v_j=\{\begin{array}{cc}{u}_j\hfill & \hfill \mathrm{if} rand\le CR \mathrm{or} jrand=j\\ x_j^p\hfill & \hfill \mathrm{otherwise}\end{array}$$

(9)

where $v_j$ represents the j-th component of the trial vector $V_i$, rand is a random number between 0 and 1, and CR represents the crossover rate. $jrand$ represents an integer randomly generated between [1, n]. Each component of the trial vector is obtained through Formula (9), so the crossover rate CR determines the proportion of the target vector and the mutation vector in the trial vector. After obtaining the trial vector $V_i$, the selection operation of Formula (10) is applied to the trial vector $V_i$ and $X_i^p$ to obtain a new individual.

$$O_i^p=\{\begin{array}{cc}{V}_i\hfill & \hfill \mathrm{if} f\left(V_i\right)\le f\left(X_i^p\right)\\ X_i^p\hfill & \hfill \mathrm{otherwise}\end{array}$$

(10)

where $f\left(V_i\right)$ represents the fitness value of $V_i$, $f\left(X_i^p\right)$ represents the fitness value of $X_i^p$, and $O_i^p$ represents a new individual. Formula (10) represents a greedy selection operation, which involves selecting the best one between the trial vector and the target vector to survive to the next generation.

It is worth noting that since the vehicles of WBS and PBS are optimized by evolutionary algorithm when entering the station, according to the designed coding method, there may be multiple vehicles placed in the same lane at the same time, resulting in the problem of exceeding the capacity of the lane. Therefore, we design the optimization of $P^{in}$ and $Q^{in}$ as a constrained optimization problem, that is, when the vehicles placed in a certain lane exceeds the capacity of the lane, the solution is set as an infeasible solution, and the amount of excess is used as the degree of constraint violation.

4.3. Optimization Method for $P^{out}$ and $Q^{out}$ Based on Heuristic Rules

In this study, heuristic rules were used to solve the vehicle outbound optimization problem in WBS and PBS. For the painting workshop, the number of color changes is the optimization objective. Therefore, during the exit process of vehicles in the WBS buffer zone, it is advisable to continuously exit vehicles of the same color as much as possible to avoid color switching in the painting workshop. The following WBS exit rules are adopted in this study, and each outbound vehicle must be the first vehicle in the lane.

(1)

If there is no outbound vehicle, select the lane with the largest number of vehicles in all lanes for vehicle exit. If there are multiple lanes with the same maximum number of vehicles, randomly select one lane among these lanes for vehicle exit;

(2)

If there are outbound vehicles, select the vehicle with the same color as the previous vehicle to exit. If the current vehicle in more than one lane has the same color as the previous vehicle, select the lane with the largest number of vehicles in these lanes to exit the vehicle. If the current vehicle in multiple lanes has the same color as the previous vehicle, and there are multiple lanes with the same maximum number of vehicles in these lanes, the lane exit vehicle is randomly selected from the lanes with the maximum number of vehicles;

(3)

If there are outbound vehicles, but no vehicle with the same color as the previous vehicle can be found, select the outbound vehicle with the largest number of vehicles in all lanes. If there are multiple lanes with the same maximum number of vehicles, randomly select one lane from these lanes to exit the vehicle.

For the assembly workshop, the optimization objective is to minimize the total number of overloads. Therefore, in the exit strategy of PBS vehicles, the goal shall be to reduce the total number of overloads. For this purpose, the following PBS vehicle exit strategies are adopted in this study.

(1)

If there is no outbound vehicle, select the lane with the highest number of vehicles from all lanes for vehicle exit. If there are multiple lanes with the same maximum number of vehicles, randomly select one lane from these lanes for vehicle exit;

(2)

If there are already exiting outbound vehicles, please select the vehicle with the largest difference in optional accessories from the previous vehicle for exiting. If there are multiple lanes in which the current vehicle has the greatest difference from the optional accessories of the previous vehicle, select the lane with the maximum number of vehicles in these lanes to exit vehicle. If there are multiple lanes in which the current vehicle has the greatest difference from the optional accessories of the previous vehicle, and there are multiple lanes with the maximum number of vehicles, then randomly select one lane from these lanes with the maximum number of vehicles to exit vehicle.

4.4. Overall Implementation of the Proposed Hybrid Evolutionary Algorithm

In this study, a hybrid method is proposed to simultaneously optimize the production sequence of three workshops. The production sequence of welding workshop is optimized by a genetic algorithm, the vehicle inbound sequences of WBS and PBS are optimized by a differential evolution algorithm, and the vehicle outbound sequences of WBS and PBS are optimized by heuristic rules. The hybrid method is referred to as HDEGA. HDEGA uses a vector with a length of $3n$ to represent an individual, where the first $n$ components represent the production sequence of the welding workshop, the components between $n+1~2n$ represent the inbound sequence of WBS, and the components between $2n+1~3n$ represent the inbound sequence of PBS. In order to further improve the performance of HDEGA, a heuristic rule-based population initialization method is proposed, as shown in Algorithm 3.

Algorithm 3. Population initialization method based on heuristic rules

01. $\mathrm{Input}\text{:} \mathrm{Population} \mathrm{size} pop\text{\_}num$. 02. $\mathrm{Output}\text{:} \mathrm{Initial} \mathrm{population} {Pop}^0$. 03. Sort according to the model of each vehicle, and then sort according to the color of the vehicles within the same model to obtain a production sequence for the welding workshop ${heu}_1^w$. 04. Sort the vehicles based on their color, and then sort them by vehicle model within vehicles of the same color to obtain a production sequence for the welding workshop ${heu}_2^w$. 05. for $i=1:pop\text{\_}num$ 06.  $\mathrm{if} i=1$ 07.   $\mathrm{Assign} {heu}_1^w$ to the first n components of the i-th individual in the population. 08.   If two consecutive vehicles in ${heu}_1^w$ are of the same color, find the corresponding vehicles in the $n+1~2n$ components of the i-th individual, and assign the same value (the same lane) to these vehicles. 09.   $\mathrm{Initialize} \mathrm{each} \mathrm{component} \mathrm{of} 2n+1~3n$ of the i-th individual of the population (select a lane at random in PBS). 10.  end 11.  $\mathrm{if} i=2$ 12.   $\mathrm{Assign} {heu}_2^w$ to the first n components of the i-th individual of the population. 13.   $\mathrm{Initialize} \mathrm{each} \mathrm{component} \mathrm{of} n+1~2n$ of the i-th individual of the population (randomly select a lane in WBS). 14.   $\mathrm{Initialize} \mathrm{each} \mathrm{component} \mathrm{of} 2n+1~3n$ of the i-th individual of the population (select a lane at random in PBS). 15.  end 16.  $\mathrm{if} i\ge 3$ 17.   $\mathrm{Initialize} \mathrm{the} \mathrm{first} n \mathrm{components} \mathrm{of} \mathrm{the} i\text{-}\mathrm{th} \mathrm{individual} \mathrm{of} \mathrm{the} \mathrm{population} (\mathrm{randomly} \mathrm{generate} \mathrm{a} \mathrm{sequence} \mathrm{of} 1~n$). 18.   $\mathrm{Initialize} \mathrm{each} \mathrm{component} \mathrm{of} n+1~2n$ of the i-th individual of the population (randomly select a lane in WBS). 19.   $\mathrm{Initialize} \mathrm{each} \mathrm{component} \mathrm{of} 2n+1~3n$ of the i-th individual of the population (select a lane at random in PBS). 20.  end 21. end

Algorithm 4 shows the pseudo code of HDEGA. HDEGA integrates genetic algorithm, differential evolution algorithm and heuristic rules. Genetic algorithm is used to optimize the sequence of welding workshops. In the inbound operation of PBS and WBS, differential evolution algorithm is used to optimize them. Finally, in the outbound operations of PBS and WBS, the outbound selection strategy is optimized based on heuristic rules. The three methods are carried out cooperatively. The production sequence of the welding workshop obtained by genetic algorithm affects the optimization of the production sequence of the painting workshop, while the number of color changes in the painting workshop and the total number of overloads in the assembly workshop will also affect the selection of individuals in the genetic algorithm, thus affecting the optimization of the production sequence of the welding workshop. The production sequence of the painting workshop affects the optimization of the production sequence of the welding workshop, and the total number of overloads of the assembly workshop will also affect the optimization of the production sequence of the painting workshop in turn. To this end, HDEGA utilized three algorithms to synergistically optimize the production sequences of the welding workshop, painting workshop, and assembly workshop, and optimizing the number of vehicle model changes, number of color changes, and total number of overloadings.

Algorithm 4. Pseudocode of HDEGA

01. Set the population size to $pop\text{\_}num$; initialize population ${Pop}^0$ using Algorithm 3; calculate the fitness value ${fit}_i=[f_1,f_2,f_3]$ for each individual and the degree of constraint violation ${Con}_i$ for each individual, where $f_1$ represents the number of vehicle model changes, $f_2$ represents the number of color changes, and $f_3$ represents the total number of overloads. 02. Set the maximum number of function evaluations to $Maxfes$, set the current number of function evaluation $fes=0$, set the offspring population $Pof={Pop}^0$, and set $k=1$. 03. $\mathrm{while} fes\le Maxfes$ 04.  $t=0$. 05.  $\mathrm{for} i=1:pop\text{\_}num/2$ 06.   Based on the values of $f_1$$, f_2$, and $f_3$ (first $f_1$, then $f_2$, and finally $f_3$), use the tournament selection to select two individuals. 07.   Use Algorithm 1 to perform a crossover operation on the first n components of the selected two individuals, and then use Algorithm 2 to perform a mutation operation. Assign two new individuals to the first n components of the t-th individual ${Pof}_t$ and the $t+1\text{-}\mathrm{th} \mathrm{individual} {Pof}_{t+1}$ of the offspring respectively. 08.   $t=t+2$. 09.  end 10.  $\mathrm{for} i=1:pop\text{\_}num$ 11.  Randomly select a value from [0.6, 0.8, 1.0] to assign to F, and randomly select a value from [0.1, 0.2, 1.0] to assign to CR. 12.   $\mathrm{if} rand<0.5$ 13.    Using Formula (7) to perform mutation operation on the components between n + 1 and 3n of the current individual to obtain a mutation vector. 14.   else 15.    Using Formula (8) to perform mutation operation on the components between n + 1 and 3n of the current individual to obtain a mutation vector. 16.   end 17.   Using Formula (9) to perform crossover operation on the components between n + 1 and 3n of the mutation vector to obtain a trial vector. 18.   Use Formula (10) to select a survivor and assign it to the $n+1~3n$$\mathrm{component} \mathrm{of} \mathrm{the} t\text{-}\mathrm{th} \mathrm{individual} {Pof}_t$ in the offspring population. 19.  end 20.  $\mathrm{Calculate} \mathrm{the} \mathrm{fitness} \mathrm{value} \mathrm{of} \mathrm{each} \mathrm{individual} \mathrm{in} \mathrm{the} \mathrm{offspring} \mathrm{population} Pof$. 21.  Merge the offspring population $Pof$ and the parent population ${Pop}^k$, and sort the merged population based on constraint domination principle and crowding distance. Select the first n individuals to form a new generation population ${Pop}^{k+1}$. 22.  $k=k+1$. 23. end

5. Experiment

In order to verify the performance of the proposed HDEGA, HDEGA is compared with five outstanding evolutionary algorithms, which are NSGA-III [23], AR-MOEA [24], TiGE-2 [25], DCNSGA-III [26], and NSGA-II [27]. The experiment is divided into two parts. In the first part, the performance of HDEGA was analyzed and compared with NSGA-III, ARMOEA, DCNSGA-III, NSGA-II, and TiGE2. The second part validates the proposed initial population construction method through an ablation experiment. All algorithms were implemented using MATLAB 2018b. The implementations of NSGA-III, ARMOEA, NSGA-II, DCNSGA-III, and TiGE2 were based on the PlatEMO [28].

5.1. Experimental Data

Experimental data. In order to thoroughly validate the effectiveness of the proposed method, this study employed simulation techniques to generate 21 test cases for verifying the performance of the algorithms. This study designed some basic parameters for generating all test cases, as shown in

Table 1. Basic parameters.

Parameter	Value
Number of WBS lanes and capacity of each lane	The number of lanes is 6, and each lane has a capacity of 10
Number of PBS lanes and capacity of each lane	The number of lanes is 6, and each lane has a capacity of 10
Number of vehicle types in the welding workshop	10
Number of vehicle colors	10
Number of optional accessories	5

. In this study, there can be up to five optional accessories during the vehicle assembly process, and the k/h values of each optional accessory are shown in

Table 2. The k/h values for optional accessories.

Optional Accessory	k/h Value
1	4/3
2	5/3
3	5/4
4	6/4
5	6/5

. Based on the basic parameters in Table 1, 21 cases were constructed using simulation methods. The number of vehicles included in 21 cases varies from 50 to 250. The effectiveness of the algorithm in small-scale and large-scale problems was verified by using cases with different vehicle numbers.

5.2. Parameter Setting and Performance Evaluation Indicators

The population size of all algorithms is set to 100. In order to fairly compare all algorithms, the maximum number of function evaluations was used as the criterion for stopping the algorithm, with a maximum number of function evaluations equal to $2000\times \mathrm{V}\mathrm{e}\text{\_}\mathrm{N}\mathrm{u}\mathrm{m}$, where $\mathrm{V}\mathrm{e}\text{\_}\mathrm{N}\mathrm{u}\mathrm{m}$ represents the number of vehicles. Except for the above parameter settings, HDEGA has no other parameters to set, and the detailed parameter settings for NSGA-III, AR-MOEA, DCNSGA-III, NSGA-II, and Ti-GE2 can refer to the PlatEMO platform. The parameters of each algorithm were set using the same settings as the original paper. Due to the randomness of evolutionary algorithms, each algorithm runs independently 25 times on each case in order to reliably analyze the performance of the algorithms. As the problem being solved is a multi-objective optimization problem, this paper adopts the most used evaluation metric-hypervolume (HV) to evaluate the quality of the solutions obtained by each algorithm. The larger the value of HV, the better the solution obtained by the algorithm, and vice versa. The calculation method of HV is shown in Formula (11). In addition, to verify the significance of the differences between algorithms, a Wilcoxon rank-sum test was used to test the significance of the differences between the two algorithms, with a significance level set at 0.05.

$$\mathrm{H}\mathrm{V}(ps,\mathbf{z})=VOL({\bigcup }_{\mathbf{x}\in ps}\left[f_1\left(\mathbf{x}\right),z_1\right]\times \dots \times [f_m\left(\mathbf{x}\right),z_m])$$

(11)

where $ps$ is the non-dominated solution set and $\mathbf{z}=\{z_1,\dots ,z_m\}$ is the reference point (m is the number of objectives). VOL(*) is the Lebesgue measure. $\{f_1\left(\mathbf{x}\right),\dots ,f_m\left(\mathbf{x}\right)\}$ represents the objective values of x.

5.3. Analysis of Comparison Results

Table 3. HV results of HDEGA and compared algorithms.

Problem	NSGA-II	DCNSGA-III	NSGA-III	AR-MOEA	TiGE-2	HDEGA
J50	4.73 × 10−1	1.17 × 10−1	4.66 × 10−1	4.34 × 10−1	4.26 × 10−1	5.38 × 10−1
	2.04 × 10−2 (+)	2.03 × 10−2 (+)	2.75 × 10−2 (+)	2.88 × 10−2 (+)	3.16 × 10−2 (+)	6.73 × 10−3
J60	4.25 × 10−1	1.04 × 10−1	4.10 × 10−1	3.74 × 10−1	3.77 × 10−1	5.55 × 10−1
	2.49 × 10−2 (+)	1.92 × 10−2 (+)	2.56 × 10−2 (+)	2.77 × 10−2 (+)	2.47 × 10−2 (+)	6.47 × 10−3
J70	3.71 × 10−1	6.32 × 10−2	3.56 × 10−1	2.85 × 10−1	3.22 × 10−1	5.56 × 10−1
	3.23 × 10−2 (+)	1.20 × 10−2 (+)	2.94 × 10−2 (+)	3.68 × 10−2 (+)	2.77 × 10−2 (+)	8.31 × 10−3
J80	3.65 × 10−1	6.57 × 10−2	3.50 × 10−1	3.07 × 10−1	3.09 × 10−1	5.59 × 10−1
	1.79 × 10−2 (+)	1.23 × 10−2 (+)	2.88 × 10−2 (+)	3.25 × 10−2 (+)	2.26 × 10−2 (+)	1.14 × 10−2
J90	3.16 × 10−1	6.81 × 10−2	3.03 × 10−1	2.82 × 10−1	2.63 × 10−1	4.99 × 10−1
	2.49 × 10−2 (+)	1.39 × 10−2 (+)	2.40 × 10−2 (+)	3.20 × 10−2 (+)	2.65 × 10−2 (+)	6.28 × 10−3
J100	3.16 × 10−1	5.58 × 10−2	3.04 × 10−1	2.49 × 10−1	2.44 × 10−1	5.23 × 10−1
	2.34 × 10−2 (+)	7.21 × 10−3 (+)	2.72 × 10−2 (+)	2.45 × 10−2 (+)	2.22 × 10−2 (+)	1.65 × 10−2
J110	3.64 × 10−1	6.31 × 10−2	3.46 × 10−1	3.20 × 10−1	2.92 × 10−1	6.69 × 10−1
	2.04 × 10−2 (+)	1.05 × 10−2 (+)	2.58 × 10−2 (+)	2.99 × 10−2 (+)	2.23 × 10−2 (+)	6.87 × 10−3
J120	3.39 × 10−1	5.91 × 10−2	3.38 × 10−1	2.96 × 10−1	2.75 × 10−1	6.37 × 10−1
	2.11 × 10−2 (+)	1.28 × 10−2 (+)	2.43 × 10−2 (+)	2.07 × 10−2 (+)	2.17 × 10−2 (+)	6.45 × 10−3
J130	2.99 × 10−1	4.76 × 10−2	3.01 × 10−1	2.42 × 10−1	2.46 × 10−1	5.61 × 10−1
	2.08 × 10−2 (+)	6.43 × 10−3 (+)	2.35 × 10−2 (+)	2.14 × 10−2 (+)	1.67 × 10−2 (+)	9.24 × 10−3
J140	2.87 × 10−1	5.83 × 10−2	2.81 × 10−1	2.44 × 10−1	2.33 × 10−1	5.59 × 10−1
	2.04 × 10−2 (+)	1.25 × 10−2 (+)	1.86 × 10−2 (+)	2.29 × 10−2 (+)	1.78 × 10−2 (+)	4.03 × 10−2
J150	2.58 × 10−1	3.94 × 10−2	2.56 × 10−1	2.17 × 10−1	2.13 × 10−1	5.81 × 10−1
	2.31 × 10−2 (+)	6.68 × 10−3 (+)	1.45 × 10−2 (+)	2.51 × 10−2 (+)	1.77 × 10−2 (+)	7.85 × 10−3
J160	2.81 × 10−1	4.43 × 10−2	2.78 × 10−1	2.23 × 10−1	2.20 × 10−1	6.12 × 10−1
	1.74 × 10−2 (+)	6.45 × 10−3 (+)	2.69 × 10−2 (+)	2.15 × 10−2 (+)	1.81 × 10−2 (+)	7.61 × 10−3
J170	2.45 × 10−1	3.12 × 10−2	2.32 × 10−1	1.62 × 10−1	1.85 × 10−1	5.56 × 10−1
	1.30 × 10−2 (+)	2.63 × 10−3 (+)	1.82 × 10−2 (+)	1.49 × 10−2 (+)	1.66 × 10−2 (+)	1.09 × 10−2
J180	2.50 × 10−1	3.55 × 10−2	2.45 × 10−1	1.66 × 10−1	1.90 × 10−1	5.89 × 10−1
	1.70 × 10−2 (+)	3.19 × 10−3 (+)	1.99 × 10−2 (+)	2.35 × 10−2 (+)	1.10 × 10−2 (+)	3.29 × 10−2
J190	2.37 × 10−1	3.81 × 10−2	2.27 × 10−1	1.83 × 10−1	1.82 × 10−1	5.43 × 10−1
	1.67 × 10−2 (+)	7.76 × 10−3 (+)	1.56 × 10−2 (+)	2.44 × 10−2 (+)	1.59 × 10−2 (+)	2.02 × 10−2
J200	2.11 × 10−2	1.28 × 10−2	2.43 × 10−2	2.07 × 10−2	2.17 × 10−2	6.45 × 10−3
	2.11 × 10−2 (+)	1.28 × 10−2 (+)	2.43 × 10−2 (+)	2.07 × 10−2 (+)	2.17 × 10−2 (+)	6.45 × 10−3
J210	2.34 × 10−1	3.59 × 10−2	2.30 × 10−1	1.89 × 10−1	1.72 × 10−1	5.76 × 10−1
	1.92 × 10−2 (+)	5.11 × 10−3 (+)	1.74 × 10−2 (+)	2.96 × 10−2 (+)	1.32 × 10−2 (+)	1.12 × 10−2
J220	2.37 × 10−1	4.19 × 10−2	2.37 × 10−1	1.97 × 10−1	1.77 × 10−1	6.35 × 10−1
	1.29 × 10−2 (+)	9.97 × 10−3 (+)	2.07 × 10−2 (+)	2.76 × 10−2 (+)	1.88 × 10−2 (+)	1.03 × 10−2
J230	2.14 × 10−1	3.35 × 10−2	2.19 × 10−1	1.84 × 10−1	1.67 × 10−1	5.55 × 10−1
	1.39 × 10−2 (+)	6.42 × 10−3 (+)	1.19 × 10−2 (+)	2.18 × 10−2 (+)	1.52 × 10−2 (+)	1.00 × 10−2
J240	2.24 × 10−1	4.07 × 10−2	2.24 × 10−1	1.97 × 10−1	1.69 × 10−1	5.83 × 10−1
	1.46 × 10−2 (+)	1.04 × 10−2 (+)	1.79 × 10−2 (+)	2.35 × 10−2 (+)	1.57 × 10−2 (+)	4.77 × 10−2
J250	2.10 × 10−1	3.01 × 10−2	2.12 × 10−1	1.73 × 10−1	1.55 × 10−1	6.03 × 10−1
	1.68 × 10−2 (+)	6.06 × 10−3 (+)	1.50 × 10−2 (+)	2.19 × 10−2 (+)	1.40 × 10−2 (+)	1.07 × 10−2

Note: “+” indicates that the corresponding algorithm performs worse than HDEGA.

lists the results of HDEGA and the compared algorithms in 21 cases. In Table 3, “+” was used to indicate that the result of HDEGA is significantly better than that of the corresponding compared algorithm, “−” was used to indicate that the result of HDEGA is significantly worse than that of the corresponding compared algorithm, and “=” was used to indicate that the result of HDEGA is not significantly different from that of the corresponding compared algorithm. From the HV results in Table 3, it can be seen that HDEGA obtained larger HV values than the other five algorithms, and the Wilcoxon rank-sum test results showed that HDEGA’s HV results were significantly better in each case than compared algorithms. In addition, HDEGA did not perform worse than any other algorithm in any case. Similarly, Figure 3 shows that the HV results obtained by HDEGA are better than those obtained by other algorithms. DCNSGA-III performs the worst among all algorithms, with smaller HV values in each case compared to other algorithms. From the results in Table 3, it can be seen that the performance of NSGA-II is second only to HDEGA, and NSGA-III ranks third. It can be seen that in the problems studied in this study, NSGA-III did not perform as well as NSGA-II, indicating that there are differences in algorithm performance among different optimization problems.

In order to prove that in addition to the proposed population initialization method, other improvements are also effective, we next conducted a comparison between HDEGA and the other five methods after excluding the proposed population initialization approach (HDEGA-I).

Table 4. HV results of HDEGA-I and compared algorithms.

Problem	NSGA-II	DCNSGA-III	NSGA-III	AR-MOEA	TiGE-2	HDEGA-I
J50	4.73 × 10−1	1.17 × 10−1	4.66 × 10−1	4.34 × 10−1	4.26 × 10−1	5.38 × 10−1
	2.04 × 10−2 (+)	2.03 × 10−2 (+)	2.75 × 10−2 (+)	2.88 × 10−2 (+)	3.16 × 10−2 (+)	6.73 × 10−3
J60	4.21 × 10−1	9.82 × 10−2	4.07 × 10−1	3.71 × 10−1	3.72 × 10−1	5.47 × 10−1
	2.51 × 10−2 (+)	1.82 × 10−2 (+)	2.55 × 10−2 (+)	2.80 × 10−2 (+)	2.50 × 10−2 (+)	6.98 × 10−3
J70	3.71 × 10−1	6.32 × 10−2	3.56 × 10−1	2.85 × 10−1	3.22 × 10−1	5.56 × 10−1
	3.23 × 10−2 (+)	1.20 × 10−2 (+)	2.94 × 10−2 (+)	3.68 × 10−2 (+)	2.77 × 10−2 (+)	8.31 × 10−3
J80	3.65 × 10−1	6.57 × 10−2	3.50 × 10−1	3.07 × 10−1	3.09 × 10−1	5.59 × 10−1
	1.79 × 10−2 (+)	1.23 × 10−2 (+)	2.88 × 10−2 (+)	3.25 × 10−2 (+)	2.26 × 10−2 (+)	1.14 × 10−2
J90	3.18 × 10−1	6.91 × 10−2	3.05 × 10−1	2.84 × 10−1	2.65 × 10−1	5.04 × 10−1
	2.48 × 10−2 (+)	1.41 × 10−2 (+)	2.41 × 10−2 (+)	3.20 × 10−2 (+)	2.66 × 10−2 (+)	6.12 × 10−3
J100	3.11 × 10−1	5.36 × 10−2	2.99 × 10−1	2.45 × 10−1	2.40 × 10−1	5.13 × 10−1
	2.34 × 10−2 (+)	7.04 × 10−3 (+)	2.73 × 10−2 (+)	2.44 × 10−2 (+)	2.22 × 10−2 (+)	1.65 × 10−2
J110	3.70 × 10−1	6.51 × 10−2	3.52 × 10−1	3.25 × 10−1	2.97 × 10−1	6.73 × 10−1
	2.03 × 10−2 (+)	1.07 × 10−2 (+)	2.55 × 10−2 (+)	2.99 × 10−2 (+)	2.23 × 10−2 (+)	6.86 × 10−3
J120	3.35 × 10−1	5.67 × 10−2	3.34 × 10−1	2.92 × 10−1	2.71 × 10−1	6.27 × 10−1
	2.12 × 10−2 (+)	1.25 × 10−2 (+)	2.41 × 10−2 (+)	2.08 × 10−2 (+)	2.16 × 10−2 (+)	7.23 × 10−3
J130	2.84 × 10−1	4.24 × 10−2	2.86 × 10−1	2.30 × 10−1	2.34 × 10−1	5.37 × 10−1
	2.09 × 10−2 (+)	6.11 × 10−3 (+)	2.37 × 10−2 (+)	2.13 × 10−2 (+)	1.66 × 10−2 (+)	1.04 × 10−2
J140	2.71 × 10−1	4.86 × 10−2	2.65 × 10−1	2.31 × 10−1	2.18 × 10−1	5.17 × 10−1
	2.01 × 10−2 (+)	1.14 × 10−2 (+)	1.84 × 10−2 (+)	2.33 × 10−2 (+)	1.73 × 10−2 (+)	3.90 × 10−2
J150	2.72 × 10−1	4.35 × 10−2	2.70 × 10−1	2.29 × 10−1	2.24 × 10−1	5.89 × 10−1
	2.31 × 10−2 (+)	7.22 × 10−3 (+)	1.46 × 10−2 (+)	2.52 × 10−2 (+)	1.77 × 10−2 (+)	7.92 × 10−3
J160	2.78 × 10−1	4.28 × 10−2	2.75 × 10−1	2.20 × 10−1	2.17 × 10−1	6.01 × 10−1
	1.74 × 10−2 (+)	6.37 × 10−3 (+)	2.68 × 10−2 (+)	2.15 × 10−2 (+)	1.79 × 10−2 (+)	8.12 × 10−3
J170	2.45 × 10−1	3.12 × 10−2	2.32 × 10−1	1.62 × 10−1	1.85 × 10−1	5.56 × 10−1
	1.30 × 10−2 (+)	2.63 × 10−3 (+)	1.82 × 10−2 (+)	1.49 × 10−2 (+)	1.66 × 10−2 (+)	1.09 × 10−2
J180	2.47 × 10−1	3.42 × 10−2	2.42 × 10−1	1.64 × 10−1	1.88 × 10−1	5.75 × 10−1
	1.69 × 10−2 (+)	3.22 × 10−3 (+)	1.96 × 10−2 (+)	2.34 × 10−2 (+)	1.10 × 10−2 (+)	3.17 × 10−2
J190	2.32 × 10−1	3.65 × 10−2	2.22 × 10−1	1.80 × 10−1	1.78 × 10−1	5.30 × 10−1
	1.67 × 10−2 (+)	7.57 × 10−3 (+)	1.55 × 10−2 (+)	2.43 × 10−2 (+)	1.59 × 10−2 (+)	1.99 × 10−2
J200	2.58 × 10−1	4.42 × 10−2	2.56 × 10−1	2.28 × 10−1	2.14 × 10−1	6.46 × 10−1
	1.84 × 10−2 (+)	8.86 × 10−3 (+)	1.71 × 10−2 (+)	3.47 × 10−2 (+)	1.61 × 10−2 (+)	1.18 × 10−2
J210	2.35 × 10−1	3.61 × 10−2	2.31 × 10−1	1.89 × 10−1	1.72 × 10−1	5.72 × 10−1
	1.91 × 10−2 (+)	5.14 × 10−3 (+)	1.73 × 10−2 (+)	2.98 × 10−2 (+)	1.32 × 10−2 (+)	1.14 × 10−2
J220	2.46 × 10−1	4.42 × 10−2	2.45 × 10−1	2.04 × 10−1	1.82 × 10−1	6.23 × 10−1
	1.31 × 10−2 (+)	1.06 × 10−2 (+)	2.06 × 10−2 (+)	2.82 × 10−2 (+)	1.89 × 10−2 (+)	1.13 × 10−2
J230	2.12 × 10−1	3.28 × 10−2	2.17 × 10−1	1.83 × 10−1	1.65 × 10−1	5.48 × 10−1
	1.39 × 10−2 (+)	6.34 × 10−3 (+)	1.18 × 10−2 (+)	2.18 × 10−2 (+)	1.52 × 10−2 (+)	1.03 × 10−2
J240	2.28 × 10−1	4.17 × 10−2	2.27 × 10−1	2.01 × 10−1	1.72 × 10−1	5.74 × 10−1
	1.47 × 10−2 (+)	1.05 × 10−2 (+)	1.79 × 10−2 (+)	2.36 × 10−2 (+)	1.59 × 10−2 (+)	4.66 × 10−2
J250	2.05 × 10−1	2.85 × 10−2	2.07 × 10−1	1.69 × 10−1	1.51 × 10−1	5.85 × 10−1
	1.69 × 10−2 (+)	5.80 × 10−3 (+)	1.49 × 10−2 (+)	2.19 × 10−2 (+)	1.38 × 10−2 (+)	1.16 × 10−2

Note: “+” indicates that the corresponding algorithm performs worse than HDEGA-I.

lists the HV results obtained by all algorithms in 21 cases. It can be seen that HDEGA-I performs better than other algorithms in any case. Similarly, Figure 4 shows that the HV results obtained by HDEGA-I are better than those obtained by other algorithms. The HV values obtained by DCNSGA-III are very small; therefore, the standard deviations of the results obtained by DCNSGA-III are better than that of HDEGA-I. In addition, the standard deviations of the results obtained by HDEGA-I are better than all the algorithms, which shows that the performance of HDEGA-I is very stable. This shows that the performance improvement of HDEGA does not only depend on the proposed population initialization method.

In order to analyze the details of the solutions obtained by each algorithm, Figure 5 illustrates three-dimensional plots depicting the results achieved by each algorithm on several test cases. As can be seen from Figure 5, the solutions obtained by HDEGA are closer to the origin than those obtained by the other five algorithms, and the solutions obtained by HDEGA are more evenly distributed. From Figure 5, we can see the distribution of results for each algorithm over 25 runs. The solutions obtained by HDEGA in 25 runs basically constitute a non-dominated solution set, but the solutions obtained by the other five algorithms are chaotic and not a non-dominated solution set. To a certain extent, it shows that the quality of solutions obtained by HDEGA is relatively stable, while the stability of solutions obtained by other algorithms is poor.

5.4. Ablation Experiment

To verify the effectiveness of the proposed initial population method, we compared HDEGA using the proposed initial population method with HDEGA without using the proposed initial population method (HDEGA-I). Figure 6 shows the HV values obtained by HDEGA and HDEGA-I in 21 cases. As can be seen from Figure 6, the HV values obtained by HDEGA in 21 cases are larger than those obtained by HDEGA-I, indicating the effectiveness of the proposed population initialization method. Figure 7 shows the boxplot of the standard deviations obtained by HDEGA and HDEGA-I. It can be seen that the mean value of the standard deviation obtained by HDEGA is smaller, indicating that the stability of HDEGA is better than that of HDEGA-I. From the above results, we can see that the proposed population initialization method not only improves the performance of the algorithm, but also makes the algorithm more stable.

5.5. Discussion

The proposed HDEGA utilizes different evolutionary algorithms to optimize production sequences in different workshops, and experimental results in 21 cases have proven the effectiveness of the proposed algorithm. The characteristics of production sequencing problems in different workshops are different, and different evolutionary algorithms have their own advantages in problems with different characteristics. Therefore, this paper adopts different evolutionary algorithms to solve production sequencing problems in different workshops. In the welding workshop, a genetic algorithm is used to optimize the production sequence of cars. At the same time, a differential evolution algorithm is used to optimize the car entry sequence in the buffer zone between the welding workshop and the painting workshop, as well as the car entry sequence in the buffer zone between the painting workshop and the assembly workshop. The experimental results indicate that the strategy of combining differential evolution algorithm with genetic algorithm for optimizing production sequences in multiple workshops is effective. In addition, the experimental results indicate that using heuristic rules to construct an initialization population can help obtain better quality solutions.

6. Conclusions

This paper studies the collaborative sequencing problem of the welding workshop, painting workshop, and assembly workshop, and takes the number of vehicle model changes in welding workshops, the number of color changes in painting workshops and the total number of overloads of assembly workshops as the optimization objectives. In order to solve the collaborative sequencing problem of three workshops, a hybrid evolutionary algorithm (HDEGA) is proposed. Considering that production sequence optimization problems in different workshops have their own characteristics, and different evolutionary algorithms have their own advantages. Adopting a single evolutionary algorithm is difficult to effectively solve the production sequence optimization problem of multiple workshops, while designing corresponding evolutionary algorithms based on the characteristics of the workshop production sequence optimization problem can more efficiently solve the problem. Therefore, a genetic algorithm is used to optimize the sequencing of welding workshops, while a differential evolutionary algorithm is used to optimize the inbound sequence of vehicles in the painting workshop and assembly workshop buffers, and heuristic rules are used to optimize the vehicle outbound in the buffers. In order to generate high-quality initial individuals, a heuristic-based initial solution generation method is proposed. A large number of experiments show that the performance of the proposed HDEGA is significantly better than that of the compared algorithms, which proves the effectiveness of the proposed algorithm. In addition, from the pseudo code of the algorithm, it can see that the code of HDEGA is simple to implement and easy to understand. The proposed method plays an important role in promoting the implementation of TPS and Heijunka, helping to improve automotive manufacturers production efficiency and reduce overall production costs.

Since the current experimental data are generated through simulation, in order to further verify the effectiveness of the proposed algorithm, future work will apply the proposed algorithm to real-world problems to verify the effectiveness of the proposed algorithm in practical application.