Research on Green Vehicle Path Planning of AGVs with Simultaneous Pickup and Delivery in Intelligent Workshop

Abstract

In this study, we present and discuss a variant of the classic vehicle routing problem (VRP), the green automated guided vehicle (AGV) routing problem, which involves simultaneous pickup and delivery with time windows (GVRPSPDTW) in an intelligent workshop. The research object is AGV energy consumption. First, we conduct a comprehensive analysis of the mechanical forces present during AGV transportation and evaluate the overall operational efficiency of the workshop. Then, we construct a green vehicle path planning model to minimize the energy consumption during AGV transportation and the standby period. Hence, the problems considered in this study are modeled based on asymmetry, making the problem solving more complex. We also design a hybrid differential evolution algorithm based on large neighborhood search (DE-LNS) to increase the local search ability of the algorithm. To enhance the optimal quality of solutions, we design an adaptive scaling factor and use the squirrel migration operator to optimize the population. Last, extensive computational experiments, which are generated from the VRPSPDTW instances set and a real case of an intelligent workshop, are designed to evaluate and demonstrate the efficiency and effectiveness of the proposed model and algorithm. The experimental results show that DE-LNS yields competitive results, compared to advanced heuristic algorithms. The effectiveness and applicability of the proposed algorithm are verified. Additionally, the proposed model demonstrates significant energy-saving potential in workshop logistics. It can optimize energy consumption by 15.3% compared with the traditional VRPSPDTW model. Consequently, the model proposed in this research carries substantial implications for minimizing total energy consumption costs and exhibits promising prospects for real-world application in intelligent workshops.

1. Introduction

With the rapid development of Industry 4.0, the traditional forms of industrial work are facing unprecedented challenges. The construction of an intelligent workshop becomes an important starting point and a technical way for the manufacturing industry to promote digital transformation. Its core content is to improve the overall level of product production. As an important auxiliary link in the workshop, material distribution occupies a large proportion of the operation cost [1,2]. As one of the important carriers of workshop logistics, the material handling system composed of AGVs can greatly improve the production flexibility and competitiveness of enterprises.

According to a report by the International Energy Agency (IEA), the carbon dioxide emissions of the EU power industry increased by 28 million tons in 2022. However, fossil fuels, especially coal resources, support nearly half of the electricity demand. The production process of green manufacturing, energy saving, and emission reduction has become the main development trend of the modern manufacturing industry [3]. Its purpose is to achieve the goal of net zero emissions in 2050. When the AGV performs handling tasks, AGV is powered by its own battery, which needs to be supplemented after 8–10 h of operation [4]. Therefore, AGV execution tasks may be interrupted due to insufficient power, which will cause the production plan to be disrupted. Therefore, it is very important for the intelligent workshop to reasonably arrange and plan the path of reducing energy consumption for AGV. It can not only save costs for enterprises to achieve the development goal of energy conservation and emission reduction, but also improve the stability of material handling systems in intelligent workshops.

There are periodic logistics transportation tasks in the intelligent workshop. When the task information is released, AGVs start from the production scheduling center. They perform simultaneous pickup and delivery logistics transportation tasks on the workstation of the workshop along the planned route and return to the production scheduling center after completing the tasks. VRPSPD of AGV in intelligent workshops can be regarded as a variant of the typical vehicle routing problem (VRP). In many studies of the current green vehicle routing problem, travel distance is no longer the only influencing factor. Energy consumption or fuel consumption is closely related to the real-time payload weight of the vehicle body [5,6,7]. The AGV load is always changing during simultaneous pickup and delivery. The traction force of AGV will change with the change of load, in the process of driving. Therefore, we add the consideration of an AGV load change to the research of AGV green path planning in the intelligent workshop.

The production characteristics of an intelligent workshop are many batches and high output. If the AGV arrives at the workstation too early, there will be idle waiting causing energy waste. Moreover, the operator and the production machine work is suspended, resulting in production losses and delays in subsequent processing. We comprehensively consider the overall operation efficiency of the workshop and the development goal of energy conservation and further add time window constraints based on GVRPSPD. For this reason, G-VRPSPDTW (GVRPSPD With Time Windows) is the focus of this research.

Currently, the AGV path planning problem within intelligent workshops has garnered significant research attention as a subfield of the VRP problem. Numerous successful methods have been employed to address AGV path planning challenges [8,9,10]. However, in the majority of these studies, researchers tend to prioritize cost optimizations such as travel distance, travel time, number of AGVs, and penalty costs [11]. Regrettably, energy-saving path planning for AGVs during simultaneous pickup and delivery processes is seldom explored. Leveraging insights from contemporary scholarly research, we formulate a green vehicle path planning model with the optimization objective of minimizing energy consumption during AGV transportation and standby periods. At the same time, we devise a hybrid differential evolution algorithm and subsequently analyze the experimental outcomes.

This research focuses on the sustainable development of the intelligent manufacturing industry and the greater attention to the energy-saving and environmental protection application environment of the intelligent factory workshop. The energy-saving path planning of AGV simultaneous pickup and delivery is studied. We hope to minimize the energy consumption of the AGV fleet to complete the pickup and delivery tasks. In this research, our main contributions are summarized as follows:

• We established a mathematical model by analyzing the mechanical force of AGV and the overall efficiency of the workshop. The model takes into account the real-time load of AGV, the rated power of AGV and the time window of workstation, so it is more in line with the actual application environment.
• We propose a hybrid differential evolution algorithm. The algorithm combines the differential evolution algorithm with the large neighborhood search algorithm. The purpose is to increase the local search ability of the differential evolution algorithm, so that it can find the optimal solution faster, avoid falling into the local optimum, and improve the optimized solution.
• Our experiments show that the hybrid algorithm proposed in this paper is effective and applicable in solving the GVRPSPDTW problem of intelligent workshop. The model established in this paper can effectively reduce energy consumption compared with the traditional VRPSPDTW model.

The remainder of this article is structured as follows: Section 2 provides a review of green vehicle path planning and vehicle routing problem with simultaneous pickup and delivery with time windows. Section 3 introduces the problem, outlining its definition and mathematical formulation. In Section 4, we present the design of our differential evolution algorithm, incorporating novel enhancements such as squirrel search and a large neighborhood search. Section 5 outlines the experimental methodology utilized to validate the effectiveness and applicability of our algorithm, while also investigating the key factors impacting model optimization. Lastly, Section 6 offers a comprehensive conclusion and outlines potential avenues for future research endeavors.

2. Literature Review

The AGV path planning problem has been the subject of extensive research over the past few decades, and it remains a topic of great interest for scholars and manufacturing companies. However, it is worth noting that the GVRPSPDTW of AGVs in intelligent workshops has been little explored in the existing literature. Therefore, we conducted a thorough review of the literature related to the green vehicle routing problem and the simultaneous pickup and delivery vehicle routing problem with time window constraints.

2.1. Green Vehicle Route Problem

The transportation and transportation industry accounts for a large proportion of global carbon emissions. Therefore, it is very important to study energy conservation and emission reduction and environmental pollution reduction in the transportation and transportation industry. According to a report from the International Energy Agency (IEA), the transport sector has generated about 25% of global CO₂ emissions in recent years [12]. The concept of GVRP is proposed by considering the reduction of carbon dioxide emissions and the realization of green and energy-efficient logistics and transportation [13]. It is a branch of the classic VRP and belongs to the NP problem. The GVRP in the workshop is to find the best route set by scientifically planning the number of departures and transportation routes of the fleet to meet various constraints. the aim is to minimize energy consumption [14].

At present, the research on the green vehicle routing problem is mainly divided into two aspects: traditional fuel and new energy (electric energy). The research on GVRP of traditional fuel (fossil fuel)-driven vehicles are mostly concentrated in intercity logistics. The construction of the GVRP model is based on vehicle fuel consumption and carbon emissions as a measure, its goal is to reduce carbon dioxide emissions and improve environmental pollution and reduce energy consumption [15]. For example, Bektas [16] further extended the goal of traditional VRP, and he considered the problem of vehicle pollution emissions. The optimization objectives in this paper include vehicle cost, energy consumption, pollution gas emission and labor cost. Based on this, Demir [17] proposed an adaptive large neighborhood search algorithm to better solve the GVRP problem. In this paper, the actual geographical data is used to analyze the algorithm, in order to have greater applicability and authenticity for GVRP research. Antonio Giallanza [18] constructed a multi-objective fuzzy chance-constrained programming model to minimize total cost and carbon emissions and confirmed the robustness of the model by applying it to the case study of Hoh Xil. In recent years, power resources instead of traditional fuels have gradually become the focus of GVRP research and applications. Amiri Afsane [19] studied the GVRP problem of heterogeneous vehicles (traditional trucks and electric trucks). Transportation costs and greenhouse gases are considered combinatorial optimization objectives. Surendra Reddy Kancharla [20] considered the influence of load on battery consumption to calculate energy demand. He proposed an adaptive large neighborhood search algorithm with special operators to solve energy consumption and verified the effectiveness of the algorithm by testing on benchmark instances.

The GVRP problem in the workshop is to study the energy consumption of AGVs. Most AGVs used in intelligent workshops are also powered by batteries. At present, the research direction of scholars on AGV energy consumption is mainly divided into two aspects: One is to take the power demand of the AGV motor and its motion state as the research object [21,22]. With the increase in production demand in the intelligent workshop, AGVs need to complete deliveries of the workstations at the same time. The real-time change in AGV’s load weight caused by AGV loading and unloading of workstations has become another research focus. Some professional institutions and academic researchers have obtained a large number of statistical data on AGV energy consumption and its loading weight. These statistics show that these two variables are proportional [23]. Zhou [24] analyzed AGV energy consumption from AGV loading weight and travel distance and then constructed a multi-objective mathematical model to minimize total energy consumption and total weighted delay. He proposed a tabu-enhanced particle swarm optimization (TEPSO) algorithm to solve multi-objective problems and verified the effectiveness of the algorithm. Wang [25] took the body load and AGV driving speed during AGV transportation as the optimization objectives and established the AGV driving energy consumption model. He evaluated and weighed the two optimization objectives of distribution cost and energy consumption by a weighted index.

According to the above research, it can be concluded that the vehicle energy consumption index is closely related to the travel distance and the real-time load weight of the body. The reason is that they are the main parameters of the mechanical force of AGVs. In addition, the current research on GVRP mainly focuses on the path planning and vehicle scheduling of intercity logistics. Although some experts and scholars have studied the energy consumption of AGVs, the energy consumption characteristics of AGVs’ actual logistics transportation process and the overall operation efficiency of the workshop are rarely fully considered. For example, [21] considered the AGV motion state and time window constraints, but did not consider the impact of AGV load on energy consumption. Similarly, [24] considered the influence of AGV load and driving distance on energy consumption but did not consider the influence of workstation time window constraints on energy consumption. The environment of GVRP in intelligent workshops is different from that of intercity logistics. At present, the VRPSPDTW problem in the intelligent workshop has not been studied. Therefore, we propose a model, which considers two major factors of AGV energy consumption. namely, the main mechanical force and the overall operating efficiency of the workshop. In terms of mechanical force, we study its influencing factors including travel distance, real-time body load and AGV rated power. In terms of operating efficiency, we increase the workstation service time window constraint. The model can directly calculate the energy consumption of AGV performing the entire logistics transportation tasks in the intelligent workshop.

2.2. Vehicle Routing Problem with Simultaneous Pickup and Delivery with Time Windows

VRPSPD with a time window, namely VRPSPDTW, can be regarded as a combination of the VRPTW problem and VRPSPD problem. VRPSPDTW is widely used in modern logistics [26], therefore, it may be the most studied problem in the variants of VRPSPD. The production scheduling system usually limits the loading and unloading time of each workstation at a specific time interval in the intelligent workshop. Due to the actual correlation and inherent complexity of VRPSPDTW, it is a hot research direction of algorithm innovation and practical path planning in the field of logistics.

In the existing research, many path search algorithms are proposed based on mathematical models and environmental representations. They can be divided into two categories: exact algorithms and heuristic algorithms. Angelelli [27] was the first and so far the only scholar to solve the VRPSPDTW with an exact algorithm. The exact algorithm he designed was mainly the classical branch pricing method, but due to the limitations of the exact algorithm, the algorithm was only tested on 20 customers. Cao and Lai [28] first attempted to solve VRPSPDTW by implementing heuristics. They proposed an improved genetic algorithm, which contains an encoding for the customer order problem. The experimental results show that their proposed algorithm can effectively find the optimal or approximate optimal solution when the number of customers is eight. Later, in order to further solve the VRPSPDTW problem, Lai and Cao [29] designed an IDE. In the experiment, they used a case with eight customers and two vehicles. Wang and Chen [30] studied VRPSPDTW using co-evolutionary genetic algorithm. They designed a benchmark instance for solving VRPSPDTW based on the famous Solomon [31] to evaluate the performance of their proposed algorithm. Recently, Hof and Schneider [32] designed an adaptive large neighborhood search algorithm (ALNS-PR) to solve VRPSPD problems, including VRPSPDTW problems. In their published results, it can be seen that ALNS-PR can obtain a better solution than the previous benchmark results, and the effectiveness of the algorithm is verified. Wu Hongguang [33] proposed an ant colony optimization algorithm with a damage and repair strategy (ACO-DR) to solve the VRPSPDTW problem and verified the effectiveness of the algorithm.

In the past few decades, VRPSPDTW has been widely studied by scholars. However, as we know, the current research on VRPSPDTW in intelligent workshops is scant in the literature. Therefore, we reviewed the literature on AGV pickup and delivery in the workshop. Sungbum Jun [34] took minimizing the total delay of transportation requests as the optimization goal in the pickup and delivery problem of intelligent workshops. He proposed two constructive heuristic algorithms with high computational speed. The proposed algorithm can find a near-optimal solution in a reasonable time. Qiu [35] studied the routing problem of the AGV pickup and delivery process in the depot. He took minimizing energy consumption as the optimal goal and adopted the particle swarm optimization (PSO) algorithm based on the rapidity of the algorithm. The simulation results showed that the model has good performance and can effectively reduce the energy consumption of the system. Zou [11] established a multi-objective mixed integer linear programming model. Customer satisfaction and total driving cost were the optimization objectives of AGV pickup and delivery problems. He developed an effective multi-objective evolutionary algorithm to solve this problem.

From the brief review above, we can observe that many methods have been successfully applied to the pickup and delivery problems of AGVs in intelligent workshops. Most experts and scholars use heuristic algorithms to solve such problems. Due to the characteristics of the VRPSPDTW problem, we usually abstract the logistics transportation network in the intelligent workshop into a topological graph to plan the path and find the optimal path. With the increase of the difficulty of the path planning problem and the increase of the constraints in the corresponding model, it is a very difficult task to find the optimal path to get the optimal solution. For this reason, On the basis of the DE algorithm, we propose an effective hybrid differential evolution algorithm (DE-LNS) to solve the VRPSPDTW problem in intelligent workshops.

The comparison of the literature is shown in

Table 1. Tabular form of the related reference.

literature	Algorithm	Time Windows	Objective
			Energy Consumption	Cost	Other	Simultaneous Pickup and Delivery
Afsane et al. [19] (2023)	Hybrid			√	√
Chen et al. [15] (2023)	Hybrid	√	√	√
Gao et al. [22] (2022)	Heuristic	√	√		√
Zou et al. [23] (2022)	Heuristic		√	√	√	√
Zou et al. [11] (2021)	Heuristic			√	√	√
Jian et al. [10] (2021)	Heuristic			√	√
Zhang et al. [21] (2021)	Heuristic		√		√
Giallanza et al. [18] (2020)	Heuristic		√	√
Shi et al. [26] (2020)	Hybrid	√		√	√	√
Jun et al. [34] (2020)	Hybrid				√	√
Hof et al. [32] (2019)	Hybrid	√		√		√
Wu et al. [33] (2019)	Heuristic	√		√		√
Kancharla et al. [20] (2018)	Heuristic		√		√
Zhou et al. [24] (2018)	Heuristic		√		√
Qiu et al. [35] (2015)	Heuristic		√			√
Wang et al. [25] (2014)	Heuristic		√	√		√
Our study	Hybrid	√	√			√

.

3. Mathematical Model

A typical intelligent workshop is mainly composed of some workstations, material handling equipment (AGV), and a production scheduling center. In general, the actual layout of the workshop is asymmetric. This also increases the difficulty of problem solving. The workstation is mainly composed of two parts, one is the buffer area, and the other is the computer numerical control (CNC). The buffer is used to store raw materials and finished products. The continuous work of CNC machine tools will reduce the inventory of raw materials in the buffer area and increase the inventory of finished products, the production scheduling center formulates the AGV scheduling plan, then AGVs perform logistics transportation tasks. Each AGV starts from the production scheduling center along the route to the designated workstation for pickup and delivery as shown in Figure 1. To ensure the overall operation efficiency in the workshop, an AGV is required to not arrive at the workstation at the latest time beyond the specified time window and return to the production scheduling center after each transportation cycle. The problem is how to assign workstations to AGVs and how to determine the order of workstations in each AGV. The objective is to minimize the energy consumption of AGVs in performing logistics transportation tasks, including AGV driving energy consumption and standby energy consumption.

To effectively illustrate the problems studied, we made the following basic assumptions:

• The production inside the workshop is stable, and there will be no shutdown, failure, or other phenomena.
• The location of the workstation and the production scheduling center in the workshop is unchanged.
• The type of AGV in the workshop is the same, and its loading capacity is consistent.
• Before the dispatch command is issued, all AGVs are parked at the production scheduling center. When AGV performs logistics transportation tasks, there may be some traffic problems. In this study, the infrared obstacle avoidance module is installed on the AGV body, which can prevent the occurrence of traffic problems in the workshop.
• The speed of AGVs in turning and going straight is a uniform constant.

3.1. Notation

The summary of the notations used in this paper is shown in

Table 2. Notations.

Notation	Description
Sets
$\mathrm{G}$	Undirected graph representing an intelligent workshop
$A$	Set of graph edges
$K$	Set of all AGVs
$S$	Ordered node set representing a path
$V_0$	Set of workstations
$V$	Node set of a graph
Parameters
$i,j$	$\mathrm{Node}, i,j\in V$
$x_i$	x-axis of location $i$
$y_i$	y-axis of location $i$
$n$	total number of workstations
$m$	Total number of AGVs
$k$	Any AGV in the AGV set $K$$, k\in \{1,2,3\dots ,m\}$
$Q$	load capacity of AGV
$Q_0$	Body weight of AGV
$P_t$	The motor power of AGV
$g$	Gravitational acceleration
$v$	velocity of AGV
$C_r$	coefficient of rolling friction
$[E{T}_i$$,L{T}_i$]	The time window of the workstation $i$
$s_i$	Service time of workstation $i$
$p_i$	pickup demand of workstation $i$
$d_i$	delivery demand of workstation $i$
$c_{ij}$	distance between node $i$ and $j$
$T_i$	arrival time of the workstation $i$
$U_{ij}$	amount of delivery demands carried on arc  $\forall i,j\in V,i\ne j$
$V_{ij}$	amount of delivery demands carried on arc $\forall i,j\in V,i\ne j$
Decision Variables
$x_{ijk}$	$x_{ijk}=\left\{\begin{array}{cc}1,& \mathrm{if} \mathrm{AGV} \mathrm{k} \mathrm{travels} \mathrm{from} \mathrm{node} \mathrm{i} \mathrm{to} \mathrm{node} \mathrm{j}\\ 0,& otherwise\end{array}\right.$

.

3.2. Energy Consumption Analysis

With the above description, we define the workshop GVRPSPDTW as an undirected graph $\mathrm{G}=(V,A)$, where $V_{}=\left\{0\right\}\cup \left\{V_0\right\}$. The node 0 in the set is represented as the production scheduling center, which is the starting point of AGV. $V_0=\{1,2,\cdots n\}$ is the workstation loading and unloading point. $A$ represents the set of all edges connecting each node, $A=\{(i,j)|i,j\in V\}$.

Suppose that an AGV travels from node $i$ to node $j$ the traveling distance and traveling time are respectively calculated by using the following formula:

$$c_{ij}=|x_i-x_j|+|y_i-y_j|$$

(1)

$$t_{ij}=d_{ij}/v$$

(2)

If there is an obstacle between $i$ and $j$, the distance between two points is calculated by using the Dijkstra algorithm.

The arrival time of node $j$ is calculated by using the following formula:

$$T_j=T_i+s_i+t_{ij}$$

(3)

When AGVs perform logistics transportation tasks in the intelligent workshop, they roll on a flat, concrete surface, such as that commonly encountered in factories. As discussed in Garcia [36], The energy consumption of such vehicles is mainly affected by three mechanical forces, namely traction ($F_{traction}$), friction ($F_{friction}$) and air resistance ($F_{airdrag}$), in Newton (N). Because the vehicle itself has a higher quality and the vehicle’s speed is lower, the air resistance is negligible, which means that the vehicle’s aerodynamics will not play a significant role in this AGV’s logistics transportation tasks. There is traction when there is movement between the object and the tangential surface. AGV traction for uniform motion depends on its rated power and travel speed, as shown in Formula (4). The $F_{friction}$ is expressed as the reaction force that hinders the movement of the object when it rolls on the surface. According to Formula (5), $F_{friction}$ varies with the weight load, rolling coefficient, and gravity acceleration of an AGV.

$$F_{traction}=P_t/v$$

(4)

$$F_{friction}=mg{C}_r$$

(5)

The research in [37] shows that any plan to minimize energy consumption should take into account the three parameters of load, number of stops, and travel distance. Because of the characteristics of the problem studied in this paper, an AGV performs pickup and delivery on all workstations in the workshop, and the number of stops is fixed. But the AGV load is constantly changing due to the nature of the VRPSPD problem. In order to solve the energy consumption change caused by load, they use a fuel consumption function (FCR) where the fuel consumption rate per unit volume is proportional to the driving distance and vehicle weight.

$$p(Q_1)=\alpha (Q_0+Q_1)+b$$

(6)

In this formula, $Q_0$ is the weight of the vehicle without load, $Q_1$ is the load carried by the vehicle, and $b$ is the automobile fuel consumption coefficient.

Let $Q_{}$ be the capacity of the vehicle. The FCR when fully loaded ($p^{*}$), and the FCR without load ($p_0$) are calculated by Equations (7) and (8), respectively.

$$p^{*}=\alpha (Q_0+Q)+b$$

(7)

$$p_0=\alpha Q_0+b$$

(8)

the mixed integer programming Formula (6) of GVRPSPD is constructed in [5].

$$Minimize{\displaystyle \sum _{i=0}^n{\displaystyle \sum _{j=0}^n{c}_0{d}_{ij}(p_0{x}_{ij}+\alpha (U_{ij}+V_{ij}))}}$$

(9)

In the formula, $c_0$ is the unit fuel cost. $\alpha$ is a parameter.

$$\alpha =(p^{*}-p_0)/Q$$

(10)

$U_{ij}$ and $V_{ij}$ are the total demand for delivery and pickup on an arc.

The model in [5] takes into account the real-time change in load when calculating energy consumption. However, the environment of intercity logistics is still different from that of an intelligent workshop. The author only models the path problem of one vehicle, and multiple AGVs perform tasks in the intelligent workshop.

For this reason, we combine the energy consumption calculation method proposed in the literature [5,37] with the GVRPSPDTW in the workshop and define the energy consumption $E_p$ of the AGV transportation process as:

$$E_p={\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{c}_{ij}{x}_{ijk}(((V_{ijk}-{\mathrm{U}}_{ijk})+{\mathrm{Q}}_0)}}}g{C}_r+P_t/v)$$

(11)

In the case of AGVs working normally, the sum of speed and power can be assumed to be known and constant because they are automatically managed by the production scheduling center. The rolling coefficient $C_r$ is also constant, as plants generally have a concrete floor. When AGV performs logistics transportation tasks, the parameters affecting AGV energy consumption are real-time changing load and driving distance.

Due to the limited battery capacity of the AGV body, if the AGV arrives at the workstation in advance, there will be a certain standby window period. Excessive standby time will not only cause the punctuality rate of AGV to performing the loading task to decrease, but also may lead to excessive consumption of AGV battery power and inability to support the completion of the remaining tasks. Conversely, the workstation cannot work because the cache is overloaded. Based on the above analysis, we define the service time window of the workstation as a semi-soft time window to ensure the comprehensiveness of the AGV energy consumption calculation. We allow an AGV to arrive at the workstation earlier but do not allow an AGV to reach the workstation beyond the established limit. The standby energy consumption $E_t$ is defined as follows:

$$E_t=P_t\max \{E{T}_i-T_{ik},0\}$$

(12)

3.3. Mathematical Model

The problem solved in this paper is how to solve the service order of each AGV arriving at the specified workstation to perform logistics transportation tasks. The output is also the order of AGV service to the workstation, and the variable is an integer. We hope to achieve the purpose of reducing energy consumption by changing the order of AGV access. For this reason, the objective of this study is to lower the energy consumption. The mathematical model was developed as follows:

$$\min E_{total}=E_p+E_t$$

(13)

The constraints are as follows:

$${\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=0}^n{x}_{ijk}=1}}, \forall j\in V_0$$

(14)

$${\displaystyle \sum _{i\in S}{\displaystyle \sum _{j\in S}{x}_{ijk}\le \left|S\right|-1}}, S\subseteq V_0, k\in K$$

(15)

$${\displaystyle \sum _{j=0}^n{x}_{0jk}}-{\displaystyle \sum _{j=0}^n{x}_{j0k}}=0, j\in V_0; \mathrm{k}\in K$$

(16)

$${\displaystyle \sum _{k=1}^m{\displaystyle \sum _{j=1}^n{x}_{0jk}\le m}}$$

(17)

$${\displaystyle \sum _{i=1}^n{T}_i\le L{T}_i}$$

(18)

$$0\le {\displaystyle \sum _{j=1}^n{x}_{ijk}(V_{ijk}-U_{ijk})}\le Q$$

(19)

$$U_{ijk}\le (Q-d_i)x_{ijk}, i\in V, j\in V_0$$

(20)

$$V_{ijk}\le (Q-p_j)x_{ijk}, i\in V_0, j\in V$$

(21)

$$U_{ijk}\ge d_j{x}_{ijk}$$

(22)

$$V_{ijk}\ge p_i{x}_{ijk}$$

(23)

$${\displaystyle \sum _{i\in V}{V}_{jik}}-{\displaystyle \sum _{i\in V}{V}_{ijk}}=p_j, j\in V_0$$

(24)

$${\displaystyle \sum _{i\in V}{U}_{ijk}}-{\displaystyle \sum _{i\in V}{U}_{jik}}=d_j, j\in V_0$$

(25)

$$x_{ijk}\in \{0,1\}, \forall i,j\in V, \forall k\in K,$$

(26)

In the above model, goal (12) shows the solution goal of the problem, which is to minimize the total energy consumption. It is separated into two sections: AGV transportation energy consumption and standby power consumption. Constraint (14) ensures that each workstation is only accessed by one AGV and only accessed once. Constraint (15) states that backward paths and possible loops are not allowed. Constraint (16) imposes that each AGV starts from the production scheduling center and returns back to it. Constraint (17) limits the number of AGVs used. Constraint (18) ensures that the service time of AGVs to workstation $i$ cannot exceed the $L{T}_i$ of $i$ point. i.e., the time window constraints. Constraint (19) ensures that the load weight of any workstation of the AGV transportation route does not exceed the maximum load of the body. Constraints (20)–(23) determines the upper and lower limits of pickup and delivery on arc $i\to j$. Constraints (24) and (25) deal with the balance of pickup and delivery demands on the route and the amount of load carried on arcs. Constraint (26) is a decision variable constraint.

4. Algorithm Design

We design a differential evolution algorithm with a hybrid large neighborhood search (DE-LNS) for the complexity of GVRPSPDTW. The overall framework of DE-LNS is shown in Algorithm 1.

Algorithm 1 General Framework of DE-LNS
Input:	P, MaxGen, D, MaxN, f
Output:	Final external archive
1.	Initialize the population P, by fetch delivery problem feature
2.	gen=1
3.	while gen<= MaxGen do
4.	Calculate the fitness function f;
5.	$\mathbf{for} i=1\to P$ do
6.	Adaptive calculation of the value of $F_{{}_M}$ according to gen;
7.	$x_1$ = Mutation ($x$);
8.	$\mathrm{Legalization} \mathrm{of} x_1$;
9.	$x_2$ = Cross ($x$$, x_1$);
10	Greedy Selection;
11.	Sort $x$$, x_1$$, \mathrm{and} x_2$$, \mathrm{choose} x_{best}$ with the largest value;
12.	$x$$=x_{best}$
13.	end for
14.	The population is optimized according to squirrel search operator;
15.	Calculate f, Select excellent chromosome population;
16.	for$j=1\to D$ do
17.	$\mathbf{for} iter=1\to MaxN$ do
18.	The neighborhood solution perturbs individual $x$ and generates $y$
19.	$\mathbf{if} f(y)<f(x)$ then
20.	$x\leftarrow y$
21.	break
22.	else
23.	$iter=iter+1$;
24.	end if
25.	until(iter = MaxN)
26.	end for
27.	end for
28.	gen = gen + 1;
29.	end while
30.	return final external archive

The algorithm uses a differential evolution algorithm as a global search algorithm. Firstly, we design a mutation strategy with an adaptive scaling factor and an adaptive crossover operator. Then we refer to the foraging process of the squirrel search algorithm [38] to optimize and screen the population. Finally, we introduce a large neighborhood search as a search mechanism to perform local searches on excellent chromosomes. The large neighborhood search algorithm has a stronger ability in the local search stage and can overcome the shortcomings of the differential evolution algorithm. Among them, P is the population size, MaxGen is the maximum number of iterations, D is the optimal number of individuals in the population, MaxN is the number of iterations of large neighborhood search, and f is the fitness function.

4.1. Encoding and Decoding Design

According to the characteristics of GVRPSDPTW, the chromosome coding in this algorithm adopts the natural number coding mechanism. The total number of genes on one chromosome in the population is $n+m$. The workstation number is $\{1,2,\cdots ,n-1,n\}$, and the AGV number is $\left\{n+1,n+2,\cdots ,n+m\right\}$. We make the chromosome gene composition multiple gene fragments by inserting the AGV number in the workstation number. Each gene fragment represents a path of the AGV fleet in the logistics transportation task.

We give a simple example to solve the problem under consideration to express our method clearly. When $n=10,m=3$, the set of workstation numbers waiting for service is $\left\{1,2,3,4,5,6,7,8,9,10\right\}$, and the AGV number is $\{11,12,13\}$. A possible chromosome is $\left\{2,5,4,12,1,6,7,11,9,6,10,8,3,13\right\}$. The specific codec operation is shown in Figure 2.

4.2. Initialization Population

We combine the initial population with random traversal to generate the initial path according to the nature of the simultaneous pickup and delivery problem. We constrain each route according to the service time window and AGV loading status. The specific steps of Algorithm 2 are shown. Among them, N is the number of workstations, $V_0$ is the workstation set, Q is the maximum load of an AGV, and the left time window of ET workstation.

Algorithm 2 Population initialization based on VRPSPDTW
Input:	N, $V_0$, Q, ET
Output:	Constructing the set of storage paths K
1.	$\mathrm{Number} \mathrm{of} \mathrm{AGVs} \mathrm{required} AG{V}_{number}=1$;
2.	while$V_0\ne null$ do
3.	The current path is S;
4.	$\mathrm{Current} \mathrm{route} \mathrm{total} \mathrm{pickup} P_t=0$$, \mathrm{Current} \mathrm{route} \mathrm{total} \mathrm{delivery} D_t=0$;
5.	$\hspace{1em}\mathbf{while}(P_t\ne Q)\wedge (D_t\ne Q)\wedge (V_0\ne null)$ do
6.	randomly select a workstation $w$$\mathrm{from} V_0$;
7.	$\mathrm{Remove} \mathrm{workstation} w_i$$\mathrm{from} V_0$;
8.	$D_t=D_t+D_w$$, P_t=P_t+P_w$;
9.	$\mathbf{if} (D_t<=Q)\wedge (P_t<=Q)$ then
10	Add workstation s to the current route S;
11.	end if
12.	end while
13.	$\hspace{1em}K[end]\leftarrow S$,$S\leftarrow null$,$AG{V}_{number}=AG{V}_{number}+1$;
14.	end while
15.	for $i=0\to AG{V}_{number}$ do
16.	Sort path K by ET;
17.	end for
18.	return K

4.3. Mutation Operator

The mutation is to obtain a new solution different from the current population. The value of the scaling factor $F$ is crucial to the algorithm in the mutation strategy. If the value of $F$ is large, it is easy to obtain the optimal solution, but the convergence speed of the algorithm will decrease. Conversely, the convergence speed of the algorithm becomes faster, but it is prone to stagnation. Therefore, the scaling factor $F$ should maintain a large value at the beginning and then change to a smaller value. We refer to the scaling factor designed by Zhou [39] and design an adaptive scaling factor based on the logistic function. The specific formula of scaling factor $F$ is:

$$F=\frac{F_{start}\ast \lambda }{1+F_{start}\ast (\lambda -1)}$$

(27)

$F_{start}$ in the formula is the initial value, $\lambda =e^{1-\frac{M\mathrm{ax}Gen}{M\mathrm{ax}Gen+1-gen}}$, and $gen$ is the number of current iterations. In the early stage of the algorithm, the scaling factor number is large, and the purpose is to maintain the diversity of the population. The scaling factor is gradually reduced, and the convergence speed and local optimization ability are improved in the iterative process, as shown in Figure 3. The value of $F$ is in a reasonable range during the iteration process. When the value of $F_{start}$ is different, the value range of $F$ is different. The larger the value range is, the more conducive it is to enhancing the diversity of individuals. In order to ensure the comprehensiveness of $F$ in the iterative process, the initial value of $F_{start}$ is 0.9.

We use Formula (28) to mutate the current individual according to the scaling factor obtained from each generation.

$$x_i^{}=x_i^{}+F\ast (x_{r1}^{}-x_{r2}^{})$$

(28)

In the formula, $x_i^{}$ is the current mutation individual, and $x_{r1},x_{r2}$ is any two different individuals in the population. At this time, $x_i^{}$ may be illegal coding because there may be negative digits and value repetition problems. For this reason, we legalize $x_i^{}$. First, if the gene $g$ in the mutated individual is greater than $P$, then $g=g+P$; if $g$ is less than 0, then $g=g+P$. Then, the same gene is deleted. Finally, the missing gene fragment is completed. A possible mutation process is shown in the Figure 4.

4.4. Crossover Operator

The crossover operator is similar to the mutation operator. We also use an adaptive crossover operator in the algorithm. However, the crossover operator is opposite to the scaling factor, and the crossover operator needs to increase with iteration. The smaller crossover operator can avoid destroying the mutant individual. A larger crossover operator can avoid the algorithm from falling into the local optimum in the later stage of evolution and can increase the population diversity. as shown in the Figure 5. The value of $CR$ is in a reasonable range during the iteration process. When the value of $C{R}_{start}$ is different, the value range of $CR$ is different. The larger the value range is, the more conducive it is to enhancing the diversity of individuals. In order to ensure the comprehensiveness of $CR$ in the iterative process, the initial value of $C{R}_{start}$ is 0.9.

Therefore, the self-adaptive crossover operator is shown as follows:

$$CR=\frac{C{R}_{start}*\lambda }{1+C{R}_{start}*(\lambda -1)}$$

(29)

The $C{R}_{start}$ in the formula is the initial value, $\lambda =e^{1-\frac{M\mathrm{ax}Gen}{gen+1}}$.

We use the OX crossover operator to expand the search range of the solution space. The purpose is to generate different AGV task allocation schemes. As shown in Figure 6, taking the generation of offspring 2 as an example, we randomly selected the crossover position of two chromosomes to perform the crossover operation. The sequence 4-5-6-7-8 in parent 1 is the first segment of offspring 2. The sequence of parent 2 after screening points 4, 5, 6, 7 and 8 was used as the second segment of offspring 2. Similarly, offspring 1 is generated.

4.5. Squirrel Migration Operator

The squirrel migration operator designed in this paper is based on the original squirrel search algorithm (SSA). We divide all individuals in the population into three different categories of squirrels according to their fitness function. The best squirrel remains unchanged; it is the best individual in the current population. The number of sub-optimal squirrels accounted for 10% of the total. Sub-optimal squirrels use one of the best squirrels as a model, and the OX crossover operation is performed between them. The remaining number of individuals are common squirrels. In the second case, common squirrels used one of the sub-optimal squirrels as a model [40], and the OX crossover operation is performed between them. We perform crossovers on different species of squirrels. Algorithm 3 shows the migration process. The squirrel migration operator of the SSA algorithm is used to optimize the population solved by the DE algorithm. At the same time, the migration operation can filter the population and retain some of the optimal solution sets for large neighborhood search operations.

Algorithm 3 pseudocode for squirrel migration operator
Input:	population P
Output:	The top 25% individuals
1.	Calculate the fitness function $f$ of all individuals;
2.	$\mathrm{Individuals} \mathrm{are} \mathrm{divided} \mathrm{into} x_{best}$, $x_{good}$$, x_{ordiary}$ three types of squirrels;
3.	$x_{best}$$\mathrm{and} x_{good}$ perform OX crossover
4.	$x_{ordiary}$$\mathrm{and} x_{good}$ perform OX crossover
5.	Calculate f of all new individuals;
6.	Sort new and old individuals according to $f$;
7.	return The top 25% individuals

4.6. Large Neighborhood Search

The differential evolution algorithm easily falls into local optimal solutions in the process of solving. We use the large neighborhood search algorithm to perform local search operations on the filtered excellent chromosomes. The purpose is to improve the quality of the solution. For the consideration of the characteristics of the problem, we use the method of removing and inserting to meet all constraints. The point is to destroy and recreate some parts of the solution set to search for new feasible solutions. This process is to provide a better mechanism to escape the local optimal solution created by DE.

4.6.1. Removal Operator

For the AGV path solution set of the GVRPSPDTW problem in the intelligent workshop, we use the Shaw removal (SR) method. The general idea of the SR operator is to remove multiple sets of points. The similarity between two workstations $i$ and $j$ is defined by the correlation weight $w(i,j)$. This includes two items: distance item $C_{ij}^{\prime }$ and relation item $R_{ij}^{\prime }$. We use the distance $c_{ij}$ between workstations $i$ and $j$ to define their correlation $C_{ij}^{\prime }$. The lower the $C_{ij}^{\prime }$, the greater the relationship between the two workstations. The determination of $R_{ij}^{\prime }$ is the same as that of $C_{ij}^{\prime }$, because the relationship between the two workstations $i$ and $j$ on the same transport path is greater than the relationship on different paths.

The definition of $w(i,j)$ is as follows:

$$w(i,j)=\frac{1}{C_{ij}^{\prime }+R_{ij}^{\prime }}$$

(30)

where $C_{ij}^{\prime }$ is obtained by $C_{ij}^{\prime }=\frac{d_{ij}}{\max d_{ij}}$, and $C_{ij}^{\prime }$ is a range from the interval $[0,1]$, If $i$ and $j$ are on the same path, the value is 1, otherwise it is 0.

4.6.2. Insertion Operator

The greedy repair operator is the most commonly used insertion operator method. However, the greedy insert operators only consider improvements resulting from inserting a contract into its best position. This is its obvious disadvantage, and it does not consider the impact of this insertion on subsequent decisions. The regret insertion operator tries to circumvent this problem [41,42]. Let $\mathrm{\Delta }{f}_i^n$ denote the change in the objective function value caused by the insertion of the best position of $n$ into workstation $i$. It means $\mathrm{\Delta }{f}_i^n\ge \mathrm{\Delta }{f}_i^{n^{\prime }}$ for $n\le n^{\prime }$. We consider a regret Regret-2 operator [43] in our study. The first inserted workstation $i$ satisfies:

$$i^{*}=\arg \max (\mathrm{\Delta }{f}_i^1-\mathrm{\Delta }{f}_i^2)$$

(31)

We insert workstation $i$ at the best position, and the process iterates until the required number of workstations is inserted.

The flow-chart of the DE-LNS algorithm structure can be seen in Figure 7.

5. Experimental Studies

The algorithm proposed in this paper is coded by MATLAB language and was tested on a PC with Intel Core (TM) i7-8700 3.20 GHz CPU, 16 GB RAM, and Windows 10 OS. GVRPSPDTW in the intelligent workshop is an important extension of VRPSPDTW. Therefore, the computational experiment part of this paper is divided into two stages, respectively in terms of algorithm performance effect and model energy saving effect. In the analysis of algorithm effectiveness, we use the international benchmark sets of VRPSPDTW proposed by Wang and Chen [30] to test the efficiency and performance of the DE-LNS algorithm. At the same time, we use the data of the real workshop to conduct experiments to measure the applicability of the algorithm in solving the GVRPSPDTW of the intelligent workshop. In the energy saving effect, we conduct simulation experiments with the traditional VRPSPDTW model, and compare the calculated energy consumption to verify the energy-saving performance of the model.

The experimental parameters of the algorithm are set as follows: population size $P=100,\mathrm{D}=25$. The maximum number of iterations $GenMax=500$. The maximum number of iterations of the large neighborhood search algorithm is $MaxN=50$, $F_{start}=0.9$, and $C{R}_{start}=0.9$. Among them, $F_{start}$ and $C{R}_{start}$ are obtained from Section 4.3 and Section 4.4. In the model, $g=9.8$ N/kg, $C_r=0.71$. The $C_r$ value is set according to the workshop floor material.

5.1. Instances and Results Comparison

The international benchmark sets of VRPSPDTW contain 9 small-scale instances and 56 medium-scale instances. It can be divided into categories 1 and 2 based on the width of the time window. Class 1 indicates that the time window required by the customer is narrow, while class 2 indicates that the time window is wide. It can also be divided into R, C, and RC categories based on the distribution of customer points. R class indicates that the geographical distribution of customer points is random and scattered, the C class indicates that the geographical distribution of customer points is aggregated, and the RC class is a mixture of the above two types. All of the small-scale instances are RC category. The number of customer points is 10, 25, and 50, respectively. The medium-sized instances include 23 R categories, 17 C categories and 16 RC categories. The set classification is shown in

Table 3. VRPSPDTW instance set classification.

	Cdp	Rdp	Rcdp
1	The customer distribution is concentrated, the time window is narrow and the vehicle load is small.	The customer distribution is random, the time window is narrow and the vehicle load is small.	The customer distribution is mixed, the time window is narrow and the vehicle load is small.
2	The customer distribution is distributed, the time window is wide and the vehicle load is heavy.	The customer distribution is random, the time window is wide and the vehicle load is heavy.	The customer distribution is mixed, the time window is wide and the vehicle load is heavy.

:

In this experiment, all these instances are defined in a two-dimensional Euclidean space. This means that all nodes (customer or depot) has their own unique coordinate $(x_i,y_i)$, and the distance between two nodes $i$ and $j$ is Euclidean distance $\sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$. The states in the VRPSPDTW example set can be further divided into six types (subsets) according to the distribution of customer locations and the strength of time window constraints and capacity constraints.

In this study, we solve the above six sets of instances under three different customer scales. The purpose is to illustrate the effectiveness of DE-LNS in solving VRPSPDTW. To ensure the comprehensiveness and completeness of numerical experiments, we consider three state-of-the-art algorithms, including Co-GA [30], p-SA [44], and ALNS-PR [32]. Each of these algorithms is tested on some or all instances in the Wang and Chen set. We analyze the experimental mathematical results obtained by DE-LNS, as shown in

Table 4. The results of small-scale instances are summarized.

Instance	CPLEX		CO-GA		P-SA		DE-LNS
	NV	TD	NV	TD	NV	TD	Avg. NV	Avg. TD	Best. NV	Best. TD
rcdp1001	3	348.98	3	348.98	3	348.98	3	348.98	0	348.98
rcdp1004	2	216.69	2	216.69	2	216.69	2	216.69	0	216.69
rcdp1007	2	310.81	2	310.81	2	310.81	2	310.81	0	310.81
rcdp2501	5	551.05	5	551.05	5	552.21	5	551.05	0	551.05
rcdp2504	7	738.32	4	473.46	4	473.46	4	473.46	0	473.46
rcdp2507	7	634.20	5	540.87	5	540.87	5	540.87	0	540.87
rcdp5001	9	994.18	9	994.18	9	994.7	9	994.18	0	994.18
rcdp5004	14	1961.53	6	725.59	6	725.59	6	725.59	0	725.59
rcdp5007	13	1814.33	7	809.72	7	810.04	7	809.72	0	809.72

and

Table 5. Summary of the results of the medium-scale instance.

Instance	CO-GA		P-SA		ALNS-PR		DE-LNS				Gap 1		Gap 2		Gap 3
	NV	TD	NV	TD	NV	TD	Avg. NV	Avg. TD	Best. NV	Best. TD	NV	TD%	NV	TD%	NV	TD%
Rdp101	19	1653.53	19	1660.98	19	1650.80	19.0	1651.10	19	1651.10	0	−0.15	0	−0.15	0	+0.02
Rdp102	17	1488.04	17	1491.75	17	1486.12	17.0	1473.66	17	1473.66	0	−0.97	0	−1.21	0	−0.84
Rdp103	14	1216.16	14	1226.77	13	1297.01	14.0	1223.14	14	1223.14	0	+0.57	0	−0.31	+1	−5.69
Rdp104	10	1015.41	10	1000.65	10	984.81	10.0	1010.60	10	1010.60	0	−0.47	0	+0.99	0	+2.61
Rdp105	15	1375.31	14	1399.81	14	1377.11	14.0	1380.06	14	1380.06	−1	+0.35	0	−1.41	0	+0.21
Rdp106	13	1255.48	12	1275.69	12	1252.03	12.4	1257.32	12	1262.28	−1	+0.54	0	−1.05	0	+0.82
Rdp107	11	1087.95	11	1082.92	10	1121.86	10.0	1124.90	10	1124.90	−1	+3.39	−1	+3.87	0	+0.27
Rdp108	10	967.49	10	962.48	9	965.54	9.4	964.31	9	965.54	−1	−0.20	−1	0.32	0	0.00
Rdp109	12	1160.00	12	1181.92	11	1194.73	12.0	1180.04	12	1177.21	0	+1.48	0	−0.40	+1	−1.47
Rdp110	12	1116.99	11	1106.52	10	1148.20	11.3	1109.66	11	1106.52	−1	−0.94	0	0.00	+1	−3.63
Rdp111	11	1065.27	11	1073.62	10	1098.84	11.0	1065.27	11	1065.27	0	0.00	0	−0.78	+1	−3.06
Rdp112	10	974.03	10	966.06	9	1010.42	10.0	959.85	10	953.63	0	−2.09	0	−1.29	+1	−5.62
Cdp101	11	1001.97	11	992.88	11	976.04	11.0	953.69	11	943.32	0	−5.85	0	−4.99	0	−3.35
Cdp102	10	961.38	10	955.31	10	941.49	10.0	959.77	10	959.37	0	−0.21	0	+0.42	0	+1.89
Cdp103	10	897.65	10	958.66	10	892.98	10.0	906.20	10	906.20	0	+0.95	0	−5.47	0	+1.48
Cdp104	10	878.93	10	944.73	10	871.40	10.0	877.77	10	876.62	0	−0.26	0	−7.21	0	+0.59
Cdp105	11	983.10	11	989.86	10	1053.12	11.0	950.01	11	941.74	0	−4.21	0	−4.86	+1	−10.58
Cdp106	11	878.29	11	878.29	10	967.71	10.0	967.71	10	967.71	−1	+10.18	−1	+10.18	0	0.00
Cdp107	11	913.81	11	911.90	10	987.64	11.0	913.81	11	913.81	0	0.00	0	+0.21	+1	−7.48
Cdp108	10	951.24	10	1063.73	10	932.88	10.0	970.63	10	932.88	0	−1.93	0	−12.3	0	0.00
Cdp109	10	940.49	10	947.90	10	910.95	10.0	903.38	10	895.81	0	−4.75	0	−5.49	0	−1.66
RCdp101	15	1652.90	15	1659.59	14	1776.58	14.3	1710.82	14	1708.21	−1	+3.35	−1	+2.93	0	−3.85
RCdp102	14	1497.05	13	1522.76	12	1583.62	13.0	1522.76	13	1522.76	−1	+1.72	0	0.00	+1	−3.84
RCdp103	12	1338.76	11	1344.62	11	1283.52	11.0	1136.05	11	1336.05	−1	−0.20	0	−0.64	0	+4.09
RCdp104	11	1188.49	10	1268.43	10	1171.65	11.0	1182.91	11	1182.91	0	−0.47	+1	−6.47	+1	+0.96
RCdp105	14	1581.26	14	1581.54	14	1548.96	14.0	1565.77	14	1565.77	0	−0.98	0	−1.00	0	+1.08
RCdp106	13	1422.87	13	1418.16	12	1392.47	12.1	1409.17	12	1408.19	−1	−1.03	−1	−0.70	0	+1.13
RCdp107	12	1282.10	11	1360.17	11	1255.06	11.0	1255.06	11	1255.06	−1	−2.11	0	−7.73	0	0.00
RCdp108	11	1175.04	11	1169.57	10	1198.36	10.0	1207.06	10	1207.06	−1	+2.73	−1	+3.21	0	+0.73
Rdp201	4	1280.44	4	1286.55	4	1253.23	4.0	1253.23	4	1253.23	0	−2.12	0	−2.59	0	0.00
Rdp202	4	1100.92	4	1150.31	3	1191.70	4.0	1093.81	4	1093.81	0	−0.65	0	−4.91	+1	−8.21
Rdp203	3	950.79	3	997.84	3	946.28	3.0	956.56	3	956.56	0	+0.61	0	−4.14	0	+1.09
Rdp204	3	775.23	2	848.01	2	833.09	2.0	837.57	2	833.09	−1	+7.46	0	−1.76	0	0.00
Rdp205	3	1064.43	3	1046.06	3	994.43	3.0	1030.81	3	1029.12	0	−3.32	0	−1.62	0	+3.49
Rdp206	3	961.32	3	959.94	3	913.68	3.0	955.98	3	955.98	0	−0.56	0	−0.41	0	+4.62
Rdp207	3	835.01	2	899.82	2	890.61	2.0	890.61	2	890.61	−1	+6.66	0	−1.02	0	0.00
Rdp208	3	718.51	2	739.06	2	726.82	2.1	725.99	2	726.82	−1	+1.16	0	−1.66	0	0.00
Rdp209	3	930.26	3	947.80	3	909.16	3.0	936.41	3	936.41	0	+0.66	0	−1.20	0	+2.30
Rdp210	3	983.75	3	1005.11	3	939.37	3.0	976.18	3	974.29	0	−0.96	0	−3.10	0	+3.72
Rdp211	3	839.61	3	812.44	2	904.44	3.0	812.44	3	812.44	0	−3.23	0	0.00	+1	−10.17
Cdp201	3	591.56	3	591.56	3	591.56	3.0	591.56	3	591.56	0	0.00	0	0.00	0	0.00
Cdp202	3	591.56	3	591.56	3	591.56	3.0	591.56	3	591.56	0	0.00	0	0.00	0	0.00
Cdp203	3	591.17	3	591.17	3	591.17	3.0	591.17	3	591.17	0	0.00	0	0.00	0	0.00
Cdp204	3	590.60	3	594.07	3	590.60	3.0	590.60	3	590.60	0	0.00	0	0.00	0	0.00
Cdp205	3	588.88	3	588.88	3	588.88	3.0	588.88	3	588.88	0	0.00	0	0.00	0	0.00
Cdp206	3	588.49	3	588.49	3	588.49	3.0	588.49	3	588.49	0	0.00	0	0.00	0	0.00
Cdp207	3	588.29	3	588.29	3	588.29	3.0	588.29	3	588.29	0	0.00	0	0.00	0	0.00
Cdp208	3	588.32	3	599.32	3	588.32	3.0	588.32	3	588.32	0	0.00	0	0.00	0	0.00
RCdp201	4	1587.92	4	1513.72	4	1406.94	4.0	1432.56	4	1412.45	0	−11.05	0	−6.69	0	+0.39
RCdp202	4	1211.12	4	1273.26	3	1414.55	4.0	1162.80	4	1162.80	0	−3.99	0	−8.68	+1	−17.78
RCdp203	4	964.65	3	1123.58	3	1050.64	3.0	1075.37	3	1075.37	−1	+11.48	0	−4.29	0	+2.35
RCdp204	3	822.02	3	897.14	3	798.46	3.0	804.71	3	804.71	0	−2.11	0	−10.30	0	+0.78
RCdp205	4	1410.18	4	1371.08	4	1297.65	4.0	1326.04	4	1326.04	0	−5.97	0	−3.29	0	+2.19
RCdp206	3	1176.85	3	1166.88	3	1146.32	3.0	1139.60	3	1137.92	0	−3.31	0	−2.48	0	−0.73
RCdp207	4	1036.59	3	1089.85	3	1061.84	4.0	1035.15	4	1035.15	0	−0.14	+1	−5.02	+1	−2.51
RCdp208	3	878.57	3	862.89	3	828.14	3.0	850.86	3	850.86	0	−3.15	0	−1.39	0	2.74

. We take the number of vehicles (NV) and total travel distance (TD) as the main optimization objectives. The gap in NVs is expressed as an integer, and the gap in TD is expressed as a percentage. Minimizing NV has a higher priority than minimizing TD in Wang and Chen’s instances. For this reason, we refer to the objective function of Shengcai Liu [45] to solve the example, as shown in Formula (32).

$$TC(S)\triangleq u_1\ast NV+u_2\ast {\displaystyle \sum _{i=1}^{NV}TD}(R_i)$$

(32)

The total cost of $S$ is recorded as $TC(S)$, divided into two parts: One is the cost of the vehicle is $u_{{}_1}\ast NV$, and the other is the cost of driving distance. The driving cost per unit distance is $u_2$, The cost of each vehicle is $u_{{}_1}$. and $R_i$ is the customer point in the route. The optimization objective is to minimize $\mathrm{TC}(S)$.

For all instances in the wang and chen set, minimizing NV is the main goal and minimizing TD is the second one. In order to meet this point, in the objective function (32), the ratio of the cost of each vehicle ($u_{{}_1}$) to driving cost per unit distance ($u_2$), i.e., $u_1/u_2$, should be set to a large enough number. Following [45], in the experiments we set $u_{{}_1}$ and $u_2$ to 2000 and 1, respectively.

In this experiment, we use the value of $TC$ as the evaluation index of the effectiveness of the algorithm. If $TC$ of one solution is less than that of another, then the solution is better than the other, and the algorithm is better.

Table 4 lists the results of small-scale instances. The table includes the commercial mathematical software CPLEX (which was used in [30] to find the optimal solution for small-scale cases), the optimal experimental results of CO-GA, P-SA, and ALNS-PR, and the average and optimal values of the DE-LNS algorithm running 20 times. Since ALNS-PR does not report results from small-scale customers, it is not shown in the table. It can be seen from the data in the table that CPLEX successfully solved five examples, and DE-LNS and CO-GA also found the optimal solution, while P-SA performed slightly worse on rcdp2501. For the other four examples, CPLEX terminates early due to the ‘memory overflow’ condition, while the other three heuristic algorithms achieve better solutions. The solution of the DE-LNS algorithm is better than that of P-SA on rcdp5001 and rcdp5007. It is worth mentioning that DE-LNS performed very stably in these instances, achieving a standard deviation of 0 across 20 independent runs.

In general, the algorithms’ performance on instances of larger scales is of more interest. The medium-scale instances were first solved by Co-GA, but later p-SA and ALNS-PR algorithms refreshed most of the results. Table 5 gives the results of the DE-LNS algorithm and the best results of the three comparison algorithms obtained from the literature on medium-scale instances. We evaluate the performance of DE-LNS mainly from two perspectives, namely, the best and average performance achieved in 20 independent runs. Gap1, Gap2, and Gap3 in the table represent the deviation of the optimal solution between DE-LNS and CO-GA, P-SA, and ALNS-PR, respectively.

DE-LNS obtains better results on 41 instances compared with Co-GA and obtains approximately equal solutions to Co-GA on 15 instances, and there is no poor solution. More precisely, there are currently 16 instances that have been optimized in terms of vehicle number (the first optimization objective) in 56 instances, and the number of vehicles in the remaining 40 instances is the same as that of the Co-GA algorithm. For instances with the same number of vehicles, the DE-LNS algorithm improves the travel distance of 25 instances, and the other 15 instances obtain the same or approximately equal travel distance as CO-GA. From the data in the table, it can be found that the travel distance in some cases increases to a certain extent when the number of vehicles is optimized. This is a normal phenomenon because the number of vehicles is the first optimization goal, and its complete reduction may sacrifice optimization in terms of travel distance.

The DE-LNS algorithm achieves better solutions in 40 instances compared with the p-SA algorithm in 56 instances, and the same or approximate results are obtained in 16 instances. Specifically, the DE-LNS algorithm optimizes the number of vehicles in 6 instances and the travel distance in 35 instances. The number of vehicles in the remaining 14 instances is the same as that of the p-SA algorithm, and only 2 instances have one more vehicle. A total of 10 solutions are the same as the results of the p-SA algorithm in 16 identical or approximate solutions, and the deviation of the remaining 6 solutions from the p-SA algorithm is less than 1%. Compared with the results obtained by the ALNS-PR algorithm, DE-LNS performs well, and it optimizes five instances. On the remaining 51 instances, the DE-LNS algorithm achieves the same or a close solution as the ALNS-PR algorithm, and there is no worse solution.

From the data in the table, it can be found that DE-LNS obtained the best solution in 21 of the 56 instances. Specifically, these instances are evenly distributed in all instances, including 6 in Rdp, 12 in Cdp, and 3 in Rcdp. These results show that DE-LNS has strong robustness and stability. It is worth mentioning that the average performance of 20 independent runs of DE-LNS is the same as its optimal performance in 37 instances out of 56 instances. The average performance of DE-LNS is still significantly better than that of Co-GA and p-SA. The results obtained by DE-LNS still have great competitiveness compared with the ALNS-PR algorithm. Such results demonstrate that DE-LNS performs very stably, although it is a randomized algorithm in nature. In conclusion, all of the above observations prove that the DE-LNS algorithm has strong robustness and optimization ability and provides a high-quality solution for the VRPSPDTW problem.

5.2. Real Case Application

We take the production data in the actual production process of an intelligent workshop as an experimental case and then simulate the process of AGVs performing logistics transportation tasks. The purpose is to verify the benefits of the proposed green vehicle path planning model and to evaluate the practical applicability of DE-LNS. The workshop is mainly composed of automation workstations (material buffer and CNC), material handling equipment (AGV), a production scheduling center, and a cargo storage center (depot). The workshop topology map we established is shown in Figure 8. The map is asymmetric, and it consists of 27 nodes and 32 edges. The number 0 is the production scheduling center, and numbers 1–26 are the workstation.

There are 10 AGVs in the workshop. The AGV specific parameters required for this experiment include the AGV’s own weight and the maximum load of the body, as well as the rated power and speed. These parameters are derived to calculate the energy consumption of AGV driving. The parameter data comes from the AGV working in the actual workshop. The parameters are shown in

Table 6. Parameters of AGV.

Body Weight/kg	Optimum Capacity/kg	Nominal Power/W	Speed/m/s
50	200	200	1

.

Each edge of the workshop presents a transport path. The topology map of the workshop is displayed as 26 points. We use the production data of the workshop to verify the energy saving effect of the model and the applicability of the algorithm. Points are represented in the following

Table 7. Logistics Transportation Information.

ID	Coordinate	Pickups (kg)	Deliveries (kg)	$\mathit{E}{\mathit{T}}_{\mathit{n}}$	$\mathit{L}{\mathit{T}}_{\mathit{n}}$	Service Time (min)
0	(0, 27)	-	-	8:00	-	-
1	(0, 46)	10	20	8:00	8:05	1
2	(10, 46)	20	15	8:01	8:05	1
3	(27, 46)	20	10	8:01	8:06	1
4	(43, 46)	20	19	8:01	8:06	1
5	(43, 53)	31	20	8:01	8:06	1
6	(53, 53)	35	22	8:02	8:06	1
7	(15, 37)	23	10	8:02	8:09	1
8	(27, 37)	22	20	8:01	8:08	1
9	(37, 37)	20	14	8:01	8:08	1
10	(10, 27)	10	23	8:01	8:10	1
11	(15, 27)	10	30	8:02	8:10	1
12	(27, 27)	25	20	8:02	8:10	1
13	(37, 27)	20	14	8:02	8:10	1
14	(53, 27)	36	23	8:02	8:10	1
15	(10, 17)	21	10	8:01	8:08	1
16	(27, 17)	18	20	8:01	8:08	1
17	(43, 17)	26	26	8:02	8:10	1
18	(0, 12)	17	24	8:03	8:20	1
19	(10, 7)	33	28	8:02	8:20	1
20	(20, 7)	15	15	8:02	8:20	1
21	(32, 5)	10	20	8:05	8:20	1
22	(43, 5)	15	30	8:02	8:20	1
23	(53, 5)	22	27	8:02	8:20	1
24	(0, 0)	26	20	8:03	8:25	1
25	(20, 0)	26	40	8:03	8:25	1
26	(32, 0)	10	20	8:04	8:25	1

. ID is the number of the workstation and the production scheduling center, coordinate is their coordinates in the topology network, and pickups and deliveries are the weight of the goods taken and delivered respectively. $E{T}_n$ and $L{T}_n$ are the time window of the point. Service time is the service time of AGVs for each point.

We extract the specific data of a logistics transportation task and abstract the task execution time as a time window. Because of the asymmetry of the graph, the shortest distance between any two nodes $i$ and $j$ in the map was obtained by using the Dijkstra algorithm in advance. There is no production priority relationship between workstations, and each AGV corresponds to a specified transportation path.

5.2.1. Performance of DE-LNS on GVRPSPDTW in Intelligent Workshop

We analyze the calculation results of DE-LNS and current advanced heuristic algorithms under the case conditions designed in this paper. The purpose is to verify the effectiveness of DE-LNS in the context of real cases and the performance of De-LNS on GVRPSPDTW in the workshop.

The experimental conditions and parameters of the algorithm are the same. Each algorithm repeats the calculation 20 times, and then statistical analysis is performed. We take the optimal results from 20 calculations as shown in

Table 8. The optimal solution of each algorithm running results.

	Route	Total Energy Consumed /kJ	Transport Energy Consumption /kJ	Standby Energy Consumption /kJ
GA	1:0-1-2-7-0 2:0-8-9-12-11-10-0 3:0-25-20-21-23-22-26-0 4:0-16-17-14-13-0 5:0-5-6-4-3-0	679.31	647.58	31.73
DE	1:0-1-2-7-11-10-0 2:0-15-19-24-18-0 3:0-20-25-26-21-16-0 4:0-3-5-6-4-9-8-0 5:0-12-13-17-14-23-22-0	685.04	636.60	48.44
VNS	1:0-15-19-24-18-0 2:0-17-14-23-22-0 3:0-20-25-26-21-16-0 4:0-1-2-7-0 5:0-5-6-4-3-0 6:0-8-9-13-12-11-10-0	654.34	623.81	30.53
LNS	1:0-1-2-7-11-10-0 2:0-15-19-24-18-0 3:0-20-25-26-21-16-0 4:0-3-5-6-4-9-8-0 5:0-12-13-17-14-23-22-0	654.45	630.96	23.49
DE-VNS	1:0-1-2-7-10-0 2:0-3-5-6-4-9-8-0 3:0-20-25-26-21-16-12-11-0 4:0-17-22-23-14-13-0 5:0-15-19-24-18-0	660.10	631.25	28.85
DE-LNS	1:0-1-2-7-0 2:0-16-17-13-12-11-10-0 3:0-15-19-24-18-0 4:0-20-25-26-21-22-23-14-0 5:0-3-5-6-4-9-8-0	653.93	623.02	30.91

. The optimized curve is shown in Figure 9.

According to the experimental data, the calculation result of the DE-LNS algorithm is the best. The energy consumption of the transportation process obtained by the DE-LNS algorithm in the actual AGV logistics transportation task is 3.8% lower than that of GA and 2.1% lower than that of the DE algorithm. The standby energy consumption is optimized by 2.6% compared with GA and 36.2% compared with the DE algorithm. Although the results of the DE-LNS algorithm are larger than those of the VNS algorithm, LNS algorithm, and DE-VNS algorithm in terms of standby energy consumption, there are more optimizations in total energy consumption and transportation energy consumption. The total energy consumption calculated by the DE-LNS algorithm is 0.06% lower than that of VNS, 0.08% lower than that of LNS, and 9.34% lower than that of DE-VNS. Note that the VNS algorithm calculates more AGVs.

The reason is that the algorithm designed in this paper uses more optimization methods. We have improved the defects of the DE algorithm, such as premature convergence and search stagnation, and made up for the defects of DE through the LNS algorithm. At the same time, we optimized and screened the population by the squirrel migration strategy. Based on the above reasons, the DE-LNS algorithm optimizes the service order of AGV to workstations in the total transportation route, and the route can control the service time of AGVs to more workstations within its corresponding time window. Therefore, the DE-LNS algorithm designed in this paper has higher solution quality in optimizing the total energy consumption of AGVs. The solution also shows good optimization ability for actual production cases. For this reason, the effectiveness and applicability of the algorithm are verified.

From the optimization curve in Figure 9, it can be seen that the total energy consumption of an AGV in performing logistics transportation tasks gradually decreases with the increase in the number of iterations. The optimal solutions of GA, DE, and DE-VNS are obtained in 434, 336 and 202 generations, respectively. The convergence speed of DE-LNS is 58.1%, 45.8%, and 9.9% higher than that of GA, DE, and DE-VNS, respectively. The LNS algorithm obtains the optimal solution in 181 generations, and the convergence speed is almost the same as that of DE-LNS. Although the VNS algorithm obtains the optimal solution in 174 generations, the calculation result is worse because the VNS algorithm falls into the local optimal solution. In the experimental environment of this actual case, the energy consumption value and convergence speed calculated by the algorithm are used as evaluation indexes to evaluate the effectiveness and applicability of the algorithm. The reason is that DE-LNS designed in this paper designs a variety of strategies to improve the convergence speed of the algorithm. We design an adaptive scaling factor based on the logistic function, which gradually decreases during the iteration process to improve the convergence speed and local optimization ability. The squirrel migration strategy was used to optimize the population on a large scale and screen out some elite individuals. The powerful local search ability of LNS is used to destroy and repair these individuals, and finally, a series of new sets are obtained. This process can help the algorithm quickly find the optimal solution set and accelerate the convergence of the algorithm.

5.2.2. Energy Consumption Cost Comparison of the GVRPSPDTW and the VRPSPDTW

In this section, we study the impact of the green vehicle path planning model designed in this paper on energy consumption costs. Therefore, we compare VRPSPDTW and GVRSPDTW in a real case. The energy consumption value and driving distance calculated by the same algorithm are used as the evaluation indexes of the model. When the objective function (8) is replaced by the minimum total travel distance, GVRPSPDTW is reduced to VRPSPDTW. As shown in Formula (33),

$$\begin{array}{ll}\min Z=& {\displaystyle \sum _{k=1}^m{\displaystyle \sum _{j=0}^n{\displaystyle \sum _{i=0}^n{x}_{ijk}{c}_{ij}+}}}\\ & {\displaystyle \sum _{i=1}^{n}C{T}_i\max \left\{\left({\mathrm{ET}}_i-T_{ik}\right),0\right\}}+\\ & {\displaystyle \sum _{i=1}^{n}C{L}_i\max \left\{\left(T_{ik}-{\mathrm{EL}}_i\right),0\right\}}\end{array}$$

(33)

In the formula, $C{T}_i$ is the unit loss cost when an AGV arrives earlier than the time window, and $C{L}_i$ is the unit time penalty cost when an AGV arrives later than the time window. The constraints of the two models are the same. We hope to reduce how much energy consumption can be reduced by observing the increase of AGV driving distance. In order to achieve this goal, we calculate the optimal total travel distance and the optimal the total energy consumption obtained by GVRPSPDTW and VRPSPDTW respectively. We design 5 experimental cases in the logistics transportation tasks of different time periods in the workshop and use the DE-LNS algorithm to perform 20 experiments on the case to obtain the best solution. The results are shown in

Table 9. GVRPSPDTW Optimal Solution.

	Route	Energy Consumption /kJ	Distance /m	Total Time /min
Case 1	1:0-1-2-7-0 2:0-16-17-13-12-11-10-0 3:0-15-19-24-18-0 4:0-20-25-26-21-22-23-14-0 5:0-3-5-6-4-9-8-0	653.93	487.02	9.94
Case 2	1:0-1-2-7-11-10-0 2:0-3-5-6-4-9-8-0 3:0-20-25-26-21-16-0 4:0-12-13-14-17-23-22-0 5:0-18-24-19-15-0	673.87	485.08	9.90
Case 3	1:0-1-2-7-11-10-0 2:0-20-25-26-21-16-0 3:0-18-24-19-15-0 4:0-3-5-6-4-9-8-0 5:0-12-13-14-17-23-22-0	662.84	485.08	9.90
Case 4	1:0-1-2-7-11-10-0 2:0-3-6-5-4-9-8-0 3:0-12-13-14-17-23-22-0 4:0-20-25-26-21-16-0 5:0-15-19-24-18-0	641.52	489.33	9.91
Case 5	1:0-1-2-7-11-10-0 2:0-3-5-6-4-9-8-0 3:0-18-24-19-15-0 4:0-12-13-14-17-23-22-0 5:0-20-25-26-21-16-0	654.72	485.08	9.90
Avg		657.38	486.32	9.91

and

Table 10. VRPSPDTW Optimal Solution.

	Route	Energy Consumption /kJ	Distance /m	Total Time /min
Case 1	1:0-1-2-7-11-10-0 2:0-8-3-5-6-4-9-12-0 3:0-15-20-25-19-24-18-0 4:0-16-13-14-17-23-22-26-21-0	701.72	435.03	11.10
Case 2	1:0- 1-2-5-6-4-9-13-12-0 2:0-7-3-8-11-10-0 3:0-16-17-14-23-22-26-21-20-0 4:0- 15-19-25-24-18-0	759.11	447.22	10.99
Case 3	1:0-1-2-7-8-12-11-10-0 2:0-3-9-4-5-6-14-13-0 3:0-16-17-23-22-21-26-25-20-0 4:0-18-24-19-15-0	726.18	438.63	10.75
Case 4	1:0-18-24-19-15-0 2:0-16-17-23-22-21-26-25-20-0 3:0-1-2-3-8-7-0 4:0-9-6-5-4-14-13-12-11-10-0	658.02	434.50	11.90
Case 5	1:0-3-9-4-5-6-14-13-12-0 2:0-18-24-19-15-0 3:0-1-2-7-8-11-10-0 4:0-16-17-23-22-21-26-25-20-0	727.97	432.25	13.74
Avg		714.60	437.53	11.70

.

According to the experimental data, although the total travel distance calculated by the green vehicle path planning model increased by 10.03% on average, the total energy consumption decreased by 8.01% on average. The reason is that the green vehicle path planning model not only considers the influence of the AGV driving path on energy consumption, but also deeply analyzes the mechanical force affecting energy consumption during AGV driving. We add the consideration of real-time changing load and AGV power. The results show that the green vehicle path planning model designed in this paper is very effective in terms of energy consumption. In terms of total transportation time, the green mathematical model is 15.3% lower than the VRPSPDTW model. This result shows that the green vehicle path planning model not only achieves the purpose of reducing energy consumption but also improves the overall efficiency of AGV logistics transportation tasks in the workshop. For this reason, the green vehicle path planning model can have great application value in an actual intelligent workshop.

6. Conclusions and Future Research Work

In this paper, the GVRPSPDTW in the intelligent workshop is studied, and the conclusions are as follows.

(1)

We build a green AGV path planning model. The combined optimization goal is to minimize the energy consumption of the AGV transportation process and standby energy consumption. The model can objectively and comprehensively reflect the total energy consumption of AGVs in an intelligent manufacturing workshop.

(2)

We design a differential evolution algorithm with large neighborhood search to solve the problem studied in this paper. The algorithm can improve the quality of the initial solution by improving the neighborhood searchability.

(3)

DE-LNS adopts adaptive crossover and mutation strategies in evolutionary operations. We design an adaptive scaling factor based on a logistic function and adaptive crossover operator to ensure effective convergence and robustness. We verify the effectiveness of the algorithm through the VRPSPDTW international standard dataset.

(4)

The experiments in this study show that the designed algorithm still has strong applicability to the AGV routing problem in an actual intelligent workshop. The solution quality of DE-LNS is better than that of advanced heuristic algorithms. The green mathematical model we established is very effective for the energy consumption of AGV’s logistics transportation tasks. There is no doubt that AGVs, as the main transportation equipment for intelligent workshops, have great energy-saving potential.

Therefore, this paper provides a potential basis for future research. At the same time, in order to better solve the problem, this paper does not consider the speed change caused by AGV turning. To further realize the development goal of energy saving and emission reduction in the workshop, we will consider the impact of AGV speed changes on energy consumption in future work. Of course, whether different types of AGV can save more energy is also one of the focuses of our research. We will conduct in-depth research on the integrated scheduling of processing resources and transportation resources from the perspective of energy saving. The GVRPSPDTW problem of multi-process processing in intelligent workshops is also one of our important research contents. In the future, the solution algorithm will be further optimized.