An Analysis of Optimization for Car PBS Scheduling Based on Greedy Strategy State Transition Algorithm

Abstract

The differences in constraints between an automotive painting workshop and a final assembly workshop can lead to production scheduling disruptions, and using the Painted Body Store (PBS) in automotive manufacturing can optimize production scheduling to meet the production requirements of the final assembly workshop. This is achieved by establishing a multi-objective mixed-integer optimization scheduling model for PBS and solving the model using a state transition algorithm based on the greedy strategy; finally, a comparison was made with the classical genetic algorithm. The results show that the optimal objective function is achieved when the body sequence movement iterations for data sets N and R are 34 and 54, respectively. At this point, the highest objective function scores are 23.46 and 49.79, and vehicles entering the PBS scheduling system from the painting-to-assembly line require 31 and 38 sequence adjustments, respectively. Compared with the classical genetic algorithm, the simulation results show that the state transition algorithm based on the greedy strategy is significantly better; it can also perform well in discrete sequence movement operations, demonstrating strong global search capabilities and fast convergence. It provides a new approach for optimizing vehicle scheduling in the automotive painting-to-assembly line.

1. Introduction

Due to the different production preferences between the automotive painting workshops and the final assembly workshop, and the inability to carry out continuous production according to the same sequence in the workshop production schedule, it is necessary to establish an automotive manufacturing painting–final assembly sequencing buffer with a sequencing function in order to adjust the sequence of cars leaving the painting workshop and meet the constraints of the final assembly workshop’s incoming car sequence. Therefore, the study of the optimized scheduling of automotive PBS is of great significance in relation to reducing the costs of automotive manufacturing and improving productivity [1,2].

In recent years, due to the continuous development of intelligent optimization algorithms, many scholars have proposed different solutions to scheduling optimization problems in automotive manufacturing workshops [3,4,5,6,7,8,9,10,11,12,13]. They have studied and analyzed this problem using algorithms such as genetic algorithm (GA) [3,4], particle swarm optimization (PSO) [5,6,7], artificial bee colony algorithm (ABC) [8,9,10], and ant colony optimization (ACO) [11,12,13]. However, traditional intelligent optimization algorithms still have certain limitations, such as facing problems of large search spaces and long solution times when solving large-scale problems, resulting in difficulty in finding algorithmic solutions. Most traditional intelligent optimization algorithms perform well in low-dimensional problems, but their performance significantly deteriorates when the dimensionality increases. This can lead to phenomena such as stagnation or premature convergence, where the algorithm may stagnate at any random point and be unable to approach the strictly optimal solution. Traditional intelligent optimization algorithms may also become trapped in local optimal solutions when solving optimization problems, resulting in slow convergence speed and difficulty in achieving global optimal solutions.

Therefore, scholars have proposed some improved intelligent optimization algorithms or combined intelligent optimization algorithms to analyze and study PBS optimization scheduling in automobile manufacturing. For example, Boysen et al. [14] used the ant colony algorithm and beam search method to optimize the objective of the car reordering problem based on PBS. Moon et al. [15] proposed a storage/retrieval algorithm to simulate the dynamic operation of automobile PBS and verify the proposed system. Tian et al. [16] analyzed the scheduling of the PBS sequencing process, which contains upstream sequence inbound and downstream sequence outbound, through a two-stage algorithm. In the first stage, the discrete small-world optimization algorithm (DSWOA) was used to schedule the inbound sequence, and in the second stage, a heuristic algorithm was used to schedule the outbound sequence. The results show that the two-stage algorithm is applicable to the PBS sorting problem. Cordeau et al. [17] used the iterative taboo search heuristic method to sort automobiles and meet the requirements of the painting workshop and assembly line. This method is flexible, fast, and easy to implement. Shen et al. [18] proposed a new dynamic production scheduling model considering factors such as final assembly workshop, painting workshop, and painted body storage, and used no-domination sorting genetic algorithms-II to solve multi-objective optimization problems. Li et al. [19] established an improved ant colony algorithm to solve the scheduling of automobile mixed-flow assembly in the final assembly workshop, which has certain significance in relation to production guidance. Hosseini et al. [20] established an MIP mathematical model and used a new hybrid algorithm to solve the problem of re-sorting automobiles in PBS due to disturbances in the initial planning sequence.

In summary, the majority of current research on PBS in automotive manufacturing focuses on designing algorithms for ordering the exit sequence of vehicles in the paint shop and the entry sequence in the assembly shop under the conditions of uncertain sequences. In actual automotive manufacturing workshops, the exit sequence of vehicles in the paint shop is usually known, and vehicles are painted and dried in a specific order to ensure paint quality and production efficiency. PBS reorders vehicles according to this sequence. This study takes into account the constraints of PBS, time data descriptions, paint shop exit sequences, and PBS area scheduling capabilities and limitations. Under the premise of satisfying the mixed model sequences and drive model sequences, the objective is to minimize the number of scheduling times and durations, and propose a state transition algorithm based on the greedy strategy for solving known exit sequences of the paint shop. Compared to other researchers’ work on PBS in automotive manufacturing, the novelty of this study lies in two aspects: on the one hand, there is relatively less research addressing the fixed exit sequence in the paint shop, whereas this study establishes specific models and solution methods for this problem; on the other hand, the state transition algorithm based on the greedy strategy can effectively solve the issue of traditional genetic algorithms falling into local optima due to changes in the original exit sequence of the paint shop. The state transition algorithm does not violate the vehicle constraints in the sequence, while possessing strong global search capabilities and rapid convergence characteristics, enabling it to quickly find convergent optimal solutions within the entire search space. Therefore, our work is essential to the field.

2. Description of the Problem

The automobile PBS system consists of seven areas, including the painting–PBS area, the car-receiving lateral-moving machine, six entrance lanes, one return lane (each lane has ten parking spaces), the car-delivery lateral-moving machine, and the PBS–final assembly car-receiving port. The distances between each lane are equal, with a one-meter gap between two adjacent lanes, and each lane has a width of two meters. The lateral-moving machine maintains a consistent speed during operation. The PBS receiving transverse vehicle transfer system transports vehicle bodies from the paint shop–PBS exit to the appropriate entry lane, and from the return lane to the appropriate entry lane. The receiving transverse vehicle transfer system delivers the first vehicle body in the current paint shop–PBS exit queue or a vehicle body at the 10th parking spot in the return lane to any of the 10 parking spots in the entry lane. The PBS delivery transverse vehicle transfer system transports the selected vehicle body from the entry lane to the PBS–assembly reception and the vehicle body requiring a sequence adjustment from the entry lane to the return lane. The delivery transverse vehicle transfer system sends the vehicle to the first parking spot in any entry lane to the first parking spot in the return lane or the PBS–assembly reception. Both the receiving and delivery transverse vehicle transfer systems can be set to idle status for a period of time.

Moreover, when any vehicle body moves from one parking spot to the next in the entry lane or return lane, it takes 9 s. During the entry process, when any vehicle body arrives at the paint shop–PBS exit, it is positioned at the center, facing entry lane 4. The receiving transverse vehicle transfer system transports the vehicle body from this position to different entry lanes’ 10th parking spots and returns to the initial position. For entry lanes 1-6, the time consumption is 18, 12, 6, 0, 12, and 18 s, respectively. During the exit process, the delivery transverse vehicle transfer system starts from the central initial position, and transports vehicle bodies from the first parking spot in each entry lane to the assembly–PBS reception, which is at the center and directly facing entry lane 4. For entry lanes 1-6, the time consumption is 18, 12, 6, 0, 12 and 18 s, respectively.

Optimizing the scheduling for automobile PBS involves various issues, which often require one consider the time consumed during the entry and exit processes and the time consumed when the car enters and leaves the return lane, and ensuring that the car-receiving lateral-moving machine and car-delivery lateral-moving machine comply with the constraint conditions during their actions. Based on these considerations, an optimized scheduling plan can be developed. Figure 1 shows a schematic diagram of the automobile PBS scheduling structure.

In addressing the optimization and scheduling problem in the buffer adjustment area of the automobile painting and assembly workshop, we set out the following research hypotheses: (1) The time taken for the body to move from the painting–PBS exit to the car reception cross-shuttle, from the cross-shuttle to entrance 4, and from the shuttle to the assembly reception point can be ignored. (2) The time taken for the body to be unloaded from the shuttle to parking space 10 or return lane 1, and loaded from parking space 1 or return lane 10 onto the shuttle, can be ignored. (3) When multiple bodies are waiting at parking space 1 and the shuttle is available, the first body to arrive at parking space 1 is given priority for processing according to the FIFO rule. (This refers to the scheduling and allocation of car bodies entering the PBS system, where car bodies that enter the system first are processed and completed by transfer machines in the order of their arrival. This constraint guarantees that tasks are handled sequentially according to their arrival time, thus maximizing production efficiency and task processing efficiency.) (4) The body can only change the assembly sequence when it is on the return lane. (5) The body is not hungry in the painting–PBS, and does not block PBS assembly. (6) A global optimal solution to the optimization problem addressed by the state transition algorithm exists; that is, the objective function has a lower bound.

3. The Establishment of an Optimization and Scheduling Model for Automobile PBS

Given that the length of the vehicle queue is $C$ and the information of each vehicle in the queue is known, the position of each vehicle in the queue can be defined as $i(i=1,2,3,\cdots ,C)$. The vehicle’s sequence of exit from the painting workshop, denoted as $\pi =\left\{1,2,3,\cdots ,C\right\}$, is obtained based on the exit sequence sorting $\pi =\left\{1,2,3\right\}$. Let $\sigma$ be a bijective function that represents the vehicle after scheduling and adjustment in PBS, and ${\pi }^{\prime }=\left\{\sigma (1),\sigma (2),\cdots ,\sigma (C)\right\}$ be the assembly entry sequence. There exists a bijective function $\sigma$ such that $\sigma (1)=2$, $\sigma (2)=3$ and $\sigma (3)=1$, and the assembly entry sequence obtained after PBS scheduling and adjustment is denoted as ${\pi }^{\prime }=\left\{2,3,1\right\}$.

During computation, the binary variable $P_{ki}(i,k=1,2,3,\cdots ,C)$ is set as 0–1. For a given vehicle $i$ in the painting exit sequence, if it is located at position $k$ in the assembly entry sequence after being resorted by the PBS scheduling system, then $P_{ki}=1$; otherwise, $P_{ki}=0$. Additionally, for two vehicles $i$ and $j$ in the painting exit sequence where $i<j$, if they are located in the same lane and adjacent to each other, then $Q_{ij}=1$; otherwise, $Q_{ij}=0$.

3.1. Objective Function

According to the specific quantification logic of the optimization objectives in the problem, an initial score is set for each optimization objective. The scores are updated based on the following logic, multiplied by the corresponding coefficients, and then combined to obtain the final weighted total score (with a theoretical maximum score of 100 points). Optimization Objective 1 (weight coefficient 0.4, initial score 100 points): Examine the vehicle dispatch sequence to identify all hybrid car bodies. Calculate the number of non-hybrid car bodies between every consecutive pair of hybrid car bodies in the sequence, in the order of their appearance. Deduct 1 point for each pair that does not have exactly two non-hybrid car bodies. Optimization Objective 2 (weight coefficient 0.3, initial score 100 points): Divide the vehicle dispatch sequence into segments to determine if the ratio of four-wheel drive to two-wheel drive vehicles in each segment is 1:1. Deduct 1 point for each segment where the ratio is not met. The basis for dividing the sequence into segments is as follows: if the sequence starts with 4, divide it whenever the value changes from 2 to 4; if the sequence starts with 2, divide it whenever the value changes from 4 to 2. For example, if the dispatch sequence is 442242444224 (4 represents four-wheel drive vehicles, and 2 represents two-wheel drive vehicles), the segmentation result is 4422, 42, 44422 and 4. The last two segments, 44422 and 4, do not satisfy the 1:1 ratio, so we deduct 2 points. Optimization Objective 3 (weight coefficient 0.2, initial score 100 points): Count the number of times the return path is used and deduct 1 point for each use. Optimization Objective 4 (weight coefficient 0.1, initial score 100 points): Reduce the number of times the return path is used, with the final objective score being 100 minus the time penalty value.

• Hybrid vehicles with a spacing of two non-hybrid vehicles are preferred.

Let $L_0$ be the hybrid vehicle binary sequence of the original sequence (with dimensions $C\times 1$ and $C$ as the number of vehicles), $L_d$ be the hybrid vehicle binary sequence after adjustment (with dimensions $C\times 1$), and $L_d(i)$ be the value of the $i$th element in sequence $L_d$. Let $\mathit{l}\left(\mathit{i}\right)={\left[L_d(i)L_d(i+1)L_d(i+2)L_d(i+3)\right]}^T$ be a vector, and $\mathit{l}={\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T$ be the target sequence. The adjusted hybrid vehicle binary sequence $L_d$ (with dimensions $C\times 1$) is partitioned into block vectors starting with 1, resulting in $l_i(i=1,2,3,\cdots ,C)$ and $l_i$ with the format “1 0 *”, where “$*$” indicates that the preceding element 0 may or may not appear, or may appear one or more times. Therefore, the highest score for Optimization Objective 1 is as follows.

$$S_1={\displaystyle \sum _{i=1}^{C}equals(l_i,l)}-1$$

(1)

$$equals(l_i,l)=\left\{\begin{array}{c}\begin{array}{cc}1& l_i is identical to l\end{array}\\ \begin{array}{cc}0& l_i is not identical to l\end{array}\end{array}\right.$$

(2)

• The sequence of deploying four-wheel drive and two-wheel drive models tends to be 1:1.

Let $D$ be the adjusted two-wheel/four-wheel drive sequence (with dimensions $C\times 1$), partitioned into $h$ sub-sequences $D_i(i=1,2,3,\cdots ,h)$ based on the type of drive. Let $D_i$ be the vector (with dimensions $d_i\times 1$). Therefore, the highest score for Optimization Objective 2 is as follows:

$$S_2={\displaystyle \sum _{i=1}^{h\backslash 2}(2}\delta (d_{2i}-d_{2i-1})-2)-\mathrm{mod}(h,2)$$

(3)

where \ denotes integer division; $\delta (·)$ represents the pulse function; and $\mathrm{mod}(h,n)$ represents the remainder of m divided by n.

• The frequency of returning paths tends towards 0.

By combining the use of the vehicle transfer machine and the car transfer machine according to the first-in-first-out (FIFO) principle, we can infer that using the returning path is the only way to change the vehicle sequence. Each use of the returning path deducts 1 point, which not only penalizes the returning path objective value but also affects the scheduling time. Here, we consider only returning operations that are performed on each vehicle body no more than once. The number of times the returning path is used is equivalent to the number of local extreme values between the original sequence $V_0$ and the transformed sequence $V_d$. Therefore, the highest score for optimization objective 3 is as follows:

$$S_3=-{{\displaystyle \sum ‖Peak(V_0-V_d)‖}}_0$$

(4)

$$Peak(x(i))=\left\{\begin{array}{c}\begin{array}{cc}1& x(i) is the local maximum value of vector x\end{array}\\ \begin{array}{cc}0& x(i) is not the local maximum value of vector x\end{array}\end{array}\right.$$

(5)

where $Peak(x(i))$ represents whether the $i$th element of vector $x$ is a local maximum (crest).

• The preference is that the total scheduling time be as short as possible.

Let $f$ denote the number of times the vehicle body enters the returning path; the time spent using the return path once includes 9 s per interval in moving from one parking spot to the next, with a total of nine intervals. Additionally, the time required for the delivery transfer machine to transport the car body to the 1st parking spot on the return path and for the receiving transfer machine to transport the car body from the return path to the 10th parking spot in entry lane 4 is 6 s each, thus the additional time cost is $t=\left(6\times 2+9\times 9\right)\times f$.

$$S_4=100-t$$

(6)

3.2. Constraint Conditions

• Each vehicle is parked either at the front end of any entering lane or at a position adjacent to the entering lane where there is a preceding vehicle closer to the upstream end.

$${\displaystyle \sum _{i=0}^{j-1}{Q}_{ij}}=1\begin{array}{cc}& j=1,2,3,\cdots ,C\end{array}$$

(7)

where $Q_{ij}$ is a binary variable taking values of 0 or 1, while $Q_{ij}=1$ represents the adjacent arrangement of vehicles $i$ and $j$ (where $i<j$) in the same lane in the PBS scheduling system, and $Q_{ij}=0$ means the converse.

• In the PBS scheduling system, a maximum of one vehicle can be arranged for transportation in the adjacent space behind a vehicle.

$${\displaystyle \sum _{j=i+1}^C{Q}_{ij}\le 1\begin{array}{cc}& i=1,2,3,\cdots ,C-1\end{array}}$$

(8)

• The number of vehicles entering the PBS scheduling system for the first time does not exceed the total number of lanes in the system, and the lower bound of the number of lanes occupied by the vehicles can be expressed as follows.

$$⌈\frac{C}{p}⌉\le {\displaystyle \sum _{j=1}^C{Q}_{0j}}\le m$$

(9)

where $p$ represents the parking spaces on each lane, and $m$ represents the number of lanes in the PBS system.

• In the PBS scheduling system, the value of i, which represents the number of vehicles in the same lane, is equal to the actual number of vehicles in the same lane.

$$\left\{\begin{array}{c}{\displaystyle \sum _{i=0}^{j-1}{N}_{ij}-{\displaystyle \sum _{i=j+1}^C{N}_{ji}=1\begin{array}{cc}& j=1,2,3,\cdots ,C-1\end{array}}}\\ {\displaystyle \sum _{i=0}^{C-1}{N}_{iC}=1}\end{array}\right.$$

(10)

where $N_{ij}$ represents the number of vehicles in the same lane in the PBS scheduling system when vehicles $i$ and $j$ are arranged closely together in front and behind, and vehicle $i$ is in the same lane.

• The number of vehicles in each lane (including the entry lane and return lane) in the PBS scheduling system does not exceed its total capacity.

$$Q_{ij}\le N_{ij}\le p\cdot Q_{ij}\begin{array}{cc}& i=0,1,2,\cdots ,C-1\begin{array}{cc};& j=i+1,\cdots ,C\end{array}\end{array}$$

(11)

• If the vehicles entering the PBS scheduling system do not pass through the return lane, they will follow the first-in-first-out (FIFO) rule when they arrive at the final assembly entry sequence.

$${\displaystyle \sum _{k=1}^{C}k\cdot P_{ki}}\le {\displaystyle \sum _{k=1}^{C}k\cdot P_{kj}}+C\cdot (1-Q_{ij})\begin{array}{cc}& i=0,1,2,\cdots ,C-1\begin{array}{cc};& j=i+1,\cdots ,C\end{array}\end{array}$$

(12)

3.3. Optimization Scheduling Model for PBS in the Automotive Industry

Based on the weight coefficients of each optimization objective described in Section 3.1, it can be inferred that the maximum score of Optimization Objective 1 (sorting relationship between hybrid and non-hybrid models) is taken as objective function A and multiplied by the corresponding weight coefficient of 0.4. The maximum score of Optimization Objective 2 (the proportion between four-wheel drive and two-wheel drive models) is taken as objective function B, and multiplied by the corresponding weight coefficient of 0.3. The maximum score of Optimization Objective 3 (minimum usage of return lane) is taken as objective function C and multiplied by the corresponding weight coefficient of 0.2. The maximum score of Optimization Objective 4 (minimum total scheduling time) is taken as objective function D and multiplied by the corresponding weight coefficient of 0.1. The weighted maximum value of the four objective functions is taken as the total objective function to be solved.

$$\max S=0.4S_1+0.3S_2+0.2S_3+0.1S_4$$

(13)

$$\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{\displaystyle \sum _{i=0}^{j-1}}{Q}_{ij}=1 j=1,2,3,\cdots ,C\\ {\displaystyle \sum _{j=i+1}^c}{Q}_{ij}\le 1 i=1,2,3,\cdots ,C-1\\ ⌈\frac{C}{p}⌉\le {\displaystyle \sum _{j=1}^c}{Q}_{0j}\le m\\ {\displaystyle \sum _{i=0}^{j-1}}{N}_{ij}-{\displaystyle \sum _{i=j+1}^c}{N}_{ji}=1 j=1,2,3,\cdots ,C-1\\ {\displaystyle \sum _{i=0}^{C-1}}{N}_{iC}=1\\ Q_{ij}\le N_{ij}\le p·Q_{ij} i=0,1,2,\cdots ,C-1; j=i+1,\cdots ,C\\ {\displaystyle \sum _{k=1}^c}k·P_{ki}\le {\displaystyle \sum _{k=1}^c}k·P_{kj}+C·\left(1-Q_{ij}\right) i=0,1,2,\cdots ,C-1; j=i+1,\cdots ,C\end{array}\right.$$

(14)

4. Solution Based on Greedy Strategy State Transition Algorithm for Optimization

The reordering problem of paint bodyshop (PBS) buffer scheduling in automobile manufacturing is an NP-hard (non-deterministic polynomial-time hard) problem [21,22], and the solution space of such problems rapidly expands with increases in problem size. Therefore, it is often difficult or even impossible to obtain the optimal solution to this problem within a limited time. This paper proposes a state transition algorithm using the greedy approach to obtain the optimal solution for the PBS optimization scheduling problem in the context of maximizing the overall objective function.

4.1. Construction of Greedy Strategy

As shown by the schematic diagram of the car PBS dispatch structure, in the case of having only one receiving transfer machine and one delivery transfer machine, when there is a car body at the 10th parking spot on the return path and the receiving transfer machine is idle, the car body at the 10th parking spot on the return path should be prioritized. Additionally, when there is a car body waiting at the first parking spot in the entry lane and the delivery transfer machine is idle, the car body that arrived first at the first parking spot should be prioritized. Therefore, the six entry lanes mentioned can be considered as one final entry sequence and one final return path exit sequence. The input is the final entry sequence of one entry lane, and the output is the exit sequence of one return path. Consequently, when all car bodies occupy the entire entry lane and all return paths, a maximum of 20 car bodies can be accommodated (10 car bodies in one entry lane and 10 car bodies in one return path).

Following the greedy approach, only one round of swapping is considered. Starting with the first of the 20 sequences, the optimal value that satisfies the objective function is recorded, and the process is repeated sequentially until a locally optimal value that satisfies the objective function is found, the illustration of traversal based on the greedy strategy is shown in Figure 2. To quickly find a feasible initial solution, a heuristic rule based on the greedy approach is proposed, where the filling stage of the vehicles entering the PBS buffer and the releasing stage of the vehicles leaving the PBS buffer both search for the locally optimal solution based on the current state. Ultimately, the downstream vehicle sequence can be obtained.

Given that the weights for multi-objective optimization are fixed, it is unnecessary to search for the optimal weight allocation on the Pareto optimal front. Additionally, there is a FIFO constraint, and returning to the previous stage is the only way to adjust the vehicle sequence. The operation that adjusts the sequence by returning to the previous stage is referred to as a “shift” operation. Furthermore, there are limitations to the shift operation of the vehicles, which include the following:

(1)

Limitation on the number of moves—a maximum of one move per vehicle is allowed, i.e., each vehicle can run on the return stage at most once;

(2)

Limitation on the direction of moves—when the vehicles move forward according to the predetermined sequence L, the current vehicle that requires a shift operation cannot insert itself into the sequence of vehicles in front of it;

(3)

Limitation on the range of shifts—during the process of entering and exiting the PBS buffer, no more than 20 vehicles can reach the final assembly workshop. Therefore, the range of moves is limited to [1, 20).

4.1.1. Fill Rule

The filling process of the vehicles mainly follows the upstream vehicle sequence. According to the order in which the vehicles enter the PBS buffer, each vehicle is selected to move into the PBS buffer on the corresponding lane. When each lane is available, the upstream vehicle sequence is given priority to use lane 4. Vehicle $i\left(i=1,2,3,\cdots ,318\right)$ selects its lane according to the following rules:

(1)

If the position of vehicle $i$ in the target vehicle sequence matches that of the last vehicle $j$ on a particular lane, i.e., $p_i=p_j+1$, and there is an available parking space on that lane, then vehicle $i$ will be selected to enter that lane;

(2)

If there is an unfilled lane in the PBS buffer, and the position of the last vehicle in that lane in the target vehicle sequence satisfies $p_i-p_j>1$, then the lane with a minimum value of $p_i-p_j$ will be selected for the vehicle to enter;

(3)

If there is an unfilled lane in the PBS buffer, and the last vehicle in that lane has a target sequence position $p_i$ that satisfies $p_i-p_j<0$, then the lane $p_j-p_i$ will be selected for the vehicle to enter.

4.1.2. Release Rule

During the vehicle release phase, a downstream vehicle sequence is constructed based on a greedy strategy. To release a vehicle from the PBS buffer, a candidate set is formed from the first vehicle in each non-empty lane. If the $n$th vehicle in the PBS buffer is to be released, the following rules are applied: if a vehicle $i$ in the candidate set has a target vehicle sequence position $p_i$ equal to $n$, then the vehicle $i$ is released to ensure the initial planned position of the vehicles. Otherwise, the vehicle with the smallest value of the target vehicle sequence position $p_i$ is selected to move to the return lane.

4.2. Construction of a State Transition Algorithm Based on Greedy Strategy

The state transition algorithm is a novel intelligent optimization algorithm based on structural learning, and was proposed by Zhou et al. [23] in 2012. This algorithm has an excellent global search ability and fast convergence characteristics [24]. The state transition algorithm constructed based on the greedy strategy can effectively solve the optimization scheduling problem of the PBS for vehicles.

4.2.1. Rules of the State Transition Algorithm Using the Mobile Transformation Operator

The Metropolis criterion $\mathsf{\Delta }=0.1$ was set for the algorithm, with a population size of 100 and a maximum iteration of 300. The population dimension is the number of vehicles to be solved. Two forms of mobile transformation operators for modifying the vehicle sequence are shown in Figure 3.

According to Figure 3, PreA and PreB are the mobile operations of the previous vehicle, indicating that the vehicle of sequence PreA is inserted into the sequence of vehicle PreB; A and B are the mobile operations of the current vehicle, indicating that the vehicle of sequence A is inserted into the sequence of vehicle B.

For the first form (Type 1), the PreA vehicle sequence and the A vehicle sequence are on the return lane, and the two movement operations are relatively close in time. The following relationship pertains between the positions of PreA, PreB, A, and B: PreA < A < PreB < B. To ensure only one use of the return lane, the following conditions must be satisfied: B-PreB ≤ A-PreA and B-A ≤ 20.

For the second form (Type 2), only the PreA vehicle sequence is on the return lane, and in this case, the two movement operations are more dispersed in time. The following relationship holds between the positions: PreA < PreB < A < B. Only the following condition needs to be satisfied: B-A ≤ 20.

The discrete state transition algorithm has four typical discrete state transformation operators [25]. For the optimization scheduling problem of PBS, the mobile transformation operator is used, which also has the capacity for global search. The mobile transformation operator is represented as shown in Equation (15), and the schematic diagram of the mobile transformation operator is shown in Figure 4.

$$x_{k+1}=A_k^{shift}\left(m_b\right)x_k$$

(15)

where $A_k^{shift}\in Z^{n\times n}$ is a 0–1 matrix with a mobile transformation function, which can move $m_b$ consecutive elements in $x_k$ to the back of another random position.

4.2.2. Design Steps for the Shift Transformation Operator State Transition Algorithm

• Step 1—Neighborhood and sampling.

The set of candidate solutions generated by the state transition operator acting on the current $x_k$ forms a neighborhood $N_{x_k}^{operator}$, where the operator is a specific operator. The pseudocode for the random sampling strategy in the successive state transition Algorithm 1 is shown below:

Algorithm 1: The successive state transition algorithm
(1)	$\mathbf{for} i\leftarrow 1,SE \mathit{do}$
(2)	$State\left(:,i\right)\leftarrow {\mathit{Best}}_k+\alpha \frac{1}{n{‖{\mathit{Best}}_k‖}^2}{R}_r{\mathit{Best}}_k$
(3)	$\mathbf{end} \mathbf{for}$

where ${\mathit{Best}}_k$ represents the current optimal solution; $SE$ represents the search intensity and sequence size; $State$ represents the size of the sequence generated by the mobile transformation under ${\mathit{Best}}_k$, which forms the set of states $SE$.

• Step 2—Selection and update.

In the state transition algorithm, it is necessary to select the “optimal solution” from the set of states of $SE$ sequences, then compare it with the quality of the optimal solution ${\mathit{Best}}_k$ and update it.

$$\mathit{Best}=\left\{\begin{array}{l}\mathit{newBest}, \mathit{if} f\left(\mathit{newBest}\right)<f\left(\mathit{Best}\right)\\ \mathit{Best}, else\end{array}\right.$$

where $\mathit{n}\mathit{e}\mathit{w}\mathit{B}\mathit{e}\mathit{s}\mathit{t}$ is the “optimal solution” selected from the set of states.

• Step 3—Alternating swap.

To find the global optimal solution in the shortest possible time, the state transition algorithm includes global search operators and local search operators, which are alternately used. The pseudocode for the continuous state transition Algorithm 2 is as follows:

Algorithm 2: The continuous state transition algorithm

$\begin{array}{l}(1)\mathit{Best}\leftarrow {\mathit{Best}}_0\\ (2)\mathbf{repeat}\\ (3) \mathbf{if} \alpha <{\alpha }_{\min } \mathbf{then}\\ (4) \alpha \leftarrow {\alpha }_{\max }\\ (5) \mathbf{end} \mathbf{if}\\ (6) \mathit{Best}\leftarrow \mathrm{shift}\left(funfcn,\mathit{Best},SE,\beta ,\gamma \right)\\ (7) \alpha \leftarrow \frac{\alpha }{fc}\\ (8)\mathbf{until} some termination condition is met\end{array}$

The flowchart used for solving the car PBS optimization scheduling problem using the state transition algorithm based on greedy strategy is shown in Figure 5.

5. Experimental Simulation and Result Analysis

5.1. Experimental Simulation

The algorithm in this paper was implemented using Matlab programming. The data were obtained from the publicly available dataset “2022 huawei cup China postgraduate mathematical modeling competition”, which includes two sets of data—normal data (N) and adjusted data—for testing the model and algorithm adaptability (R) after desensitization of the production data. Based on the statistical analysis, the number of car bodies in the entry sequences of both dataset N and dataset R provided in the competition is 318 vehicles each, including 106 fuel vehicles and 212 hybrid vehicles in dataset N, and 159 fuel vehicles and 159 hybrid vehicles in dataset R. In addition, there were 289 two-wheel drive vehicles and 29 four-wheel drive vehicles in both datasets. The power type of the vehicles was encoded as “0” for hybrid and “1” for fuel for ease of subsequent data processing, while the drive type was encoded as “2” for two-wheel drive and “4” for four-wheel drive. Further details are shown in Figure 6.

5.2. Simulation Results and Analysis

By using a state transition algorithm based on the greedy strategy to solve the optimization and scheduling model of automobile PBS, the power parameter sequence of vehicles in the painting production line after optimization and scheduling, the final assembly sequence of vehicles, the optimal iteration number, the optimal score of the multi-objective optimization model, the moving sequence of different vehicle types, and the area codes of different vehicle types at different time points can be obtained for both datasets N and R.

5.2.1. Solution Results for Dataset N

By subtracting the power parameter sequence of the final assembly sequence obtained from the painting production line sequence, one can determine which vehicle bodies have been moved. The difference between the input and output sequences is shown in Figure 7. Based on the greedy strategy state transition algorithm, the calculation of the average number of iterations is illustrated in Figure 8. It can be observed that when the average number of iterations reaches 34, the score of the optimization plan will have already approached a stable state. At this point, the average computation time required is 19.07 s, and the optimal average score result for the multi-objective optimization model is 23.46.

From Figure 7, it can be seen that the difference between the input and output sequences appears in the form of 31 peaks, indicating the number of times the painting production line sequence was moved while meeting the requirements of the final assembly production workshop. The detailed vehicle movement process is shown in

Table 1. Table of the movement of the painting production line sequence for dataset N in the PBS scheduling system.

1→4	6→15	22→25	31→35
44→53	54→63	56→65	58→67
71→80	97→106	100→106	102→111
107→116	126→135	134→143	140→159
151→158	166→172	178→187	204→213
226→231	233→242	234→238	235→244
250→264	255→264	270→272	283→288
292→301	295→304	312→318

A→B represents the movement of a vehicle body from its original position in sequence A to a new position in sequence B.

. In the table, “1→4” indicates that the position of vehicle body number 1 in the original sequence has been moved to the position of vehicle body number 4 in the new sequence, “6→15” indicates that the position of vehicle body number 6 in the original sequence has been moved to the position of vehicle body number 15 in the new sequence, and so on.

5.2.2. Solution Results for Dataset R

By subtracting the final assembly input sequence from the painting output sequence, the difference plot between the input and output sequences for dataset R (Figure 9) can be obtained. Based on the greedy strategy state transition algorithm, the calculation of the average number of iterations is illustrated in Figure 10. It can be observed that when the average number of iterations reaches 54, the score of the optimization plan will have already approached a stable state. At this point, the average computation time required is 28.13 s, and the optimal average score result for the multi-objective optimization model is 49.79.

Based on Figure 9, it can be observed that the difference between the input and output sequences appears in the form of 38 peaks, indicating the number of times the painting production line sequence was moved while meeting the requirements of the final assembly production. The detailed vehicle movement process is presented in

Table 2. Table of the movement of the painting production line sequence for dataset R in the PBS scheduling system.

2→5	5→14	13→20	17→26
31→34	44→51	46→55	53→58
62→70	66→85	71→80	73→92
76→88	100→106	102→111	118→126
121→129	134→143	154→163	155→164
159→168	172→181	177→186	193→202
196→205	219→235	223→232	230→239
244→252	268→270	272→283	276→284
278→287	281→290	287→296	297→309
304→312

. In the table, “2→5” represents the movement of vehicle body number 2 from its original position in the sequence to the new position of vehicle body number 5, “5→14” represents the movement of vehicle body number 5 from its original position in sequence to the new position of vehicle body number 14, and so on.

5.3. Comparative Analysis of Algorithms

Based on the aforementioned case study, a comparative analysis of the proposed greedy strategy state transition algorithm and the genetic algorithm is conducted. The superiority of the algorithm presented in this paper is verified through the obtained objective function scores, the average number of iterations, and the average iteration time.

Since the crossover operator in the classical genetic algorithm involves partial sequence exchanges between multiple individuals, and the mutation operator may cause changes to the individual vehicle’s characteristics (such as vehicle type and drive mode), it alters the original vehicle dispatch sequence in the paint shop. Therefore, when comparing the algorithms, this paper makes some improvements to the classical genetic algorithm by redesigning the mutation and crossover genetic operators as swap operators. The specific operation of the swap operator is to select two vehicles in the overall vehicle sequence and exchange their positions, thus avoiding violations of vehicle type quantity constraints and vehicle number constraints during the calculation process.

By using Matlab programming, the objective function scores, average number of iterations, and average iteration time for dataset N and dataset R can be calculated under both the genetic algorithm and the greedy strategy state transition algorithm. The results are shown in

Table 3. Comparative analysis table of different algorithms.

Types of Algorithms	Dataset	Objective Function Score	Average Number of Iterations (30 Times)	Average Iteration Time (30 Times)
Genetic algorithm (GA)	N	15.37	26	14.38
	R	34.51	37	21.09
Greedy strategy state transition algorithm (GSSTA)	N	23.46	34	19.07
	R	49.79	54	28.13

.

As shown in Table 3, regardless of whether it is the normal dataset N obtained after desensitizing production data or the adjusted dataset R designed for testing the adaptability of the model and algorithm, the objective function scores obtained by the algorithm proposed in this paper are higher than those of the genetic algorithm. The genetic algorithm has a smaller average number of iterations compared to the algorithm presented in this paper. Based on the objective function scores and the number of iterations, it can be inferred that the genetic algorithm eventually falls into a local optimal solution, while the greedy strategy state transition algorithm is closer to the global optimal solution. The iteration diagrams for dataset N and dataset R under different algorithms are shown in Figure 11, which provides an intuitive view of the average number of iterations and the objective function scores. Therefore, based on the comparison of the results, it can be concluded that the algorithm proposed in this paper performs better overall than the classical genetic algorithm.

6. Conclusions and Future Outlooks

In this paper, a PBS multi-objective mixed-integer optimization scheduling model is constructed for the scheduling optimization problem in the buffer sequencing area of the automobile manufacturing paint and assembly processes, with the goal of maximizing the optimization target score. A solution based on a greedy strategy state transition algorithm is proposed. Simulation experiments show that this algorithm can easily adjust the exit sequence of the paint shop to meet the entry sequence constraints of the assembly shop, while also easily solving for the optimal score of the overall objective function. In addition, compared to the classical genetic algorithm, the solution result of this algorithm tends to be closer to the global optimal solution, and avoids falling into local optimal solutions. The greedy strategy state transition algorithm has strong global search capabilities, fast convergence, and other advantages, making it well-suited for application in discrete sequence moving operations. This research method can also be extended to the placement of shared bikes in a specific urban area, which helps reduce energy consumption and operating costs.

However, as a type of intelligent algorithm, the solution process of the algorithm proposed in this paper still has a certain degree of randomness, which means it is difficult to ensure that the results obtained each time are close enough to the optimal solution. This is one of the limitations of this research. Given the relatively short development history of STA, future research work can further analyze the theoretical and practical aspects of GSSTA to achieve more satisfactory results. Additionally, in the optimization of the PBS scheduling system, the impact of different numbers of return paths on the objective function scores can be further investigated.