Research on the Location Selection Problem of Electric Bicycle Battery Exchange Cabinets Based on an Improved Immune Algorithm

Abstract

The rise of new energy technologies has accelerated progress towards sustainable development, and many companies are beginning to invest in renewable resource-related facilities. Electric bicycles have always been an important mode of green transportation; however, they also have problems such as slow charging, difficult charging, and that burning and short circuiting may occur during charging. Electric bicycle battery exchange cabinets effectively solve these problems by exchanging low batteries with full batteries instead of charging. However, current battery exchange cabinets face the problems of insufficient construction and unreasonable site selection. Therefore, this paper proposes a location selection model for electric bicycle battery exchange cabinets based on point demand theory, aiming to maximize rider satisfaction and the service capacity of exchange cabinets. The immune algorithm is introduced to solve the location model; however, the traditional immune algorithm has some problems such as poor stability and slow convergence. In this paper, the mutation process of the traditional immune algorithm is improved by introducing multi-point mutation, guided mutation, and local search. Finally, based on the data of electric bicycle riders in Shanghai, we verify that the location model based on point demand theory performs well on two objective functions of rider satisfaction and battery exchange cabinet service capability. We also expand the application of point demand theory to location models. Then, by conducting experiments with different parameter groups, through sensitivity analysis and convergence analysis, we verified that the improved immune algorithm performs better than the traditional immune algorithm in its accuracy, search accuracy, stability, and convergence.

1. Introduction

In the wake of the COVID-19 outbreak, economic development in many countries around the world has been hit hard. For example, China implemented lockdowns, bans, and quarantines to control the spread of the pandemic. Although these measures limited the spread of the virus, they also restricted the movement of people and goods, significantly hindering the country’s economic and social development [1]. Similarly, COVID-19 has had a significant impact on the energy sector. It has not only disrupted the supply and demand balance of traditional energy commodities, but it has also affected the supply chain of new energy commodities, such as batteries and solar photovoltaic panels, especially on the intercontinental trade routes between China and other countries. Restrictions on the movement of goods and people also led to the suspension of many projects and reduced demand for energy products, which has reduced the ability of companies and governments to invest in energy projects [2]. Therefore, businesses need to weigh the pros and cons of investing in energy projects more carefully. Now, in the post-pandemic era, the global economy is gradually recovering. This also brings some opportunities for businesses. On the one hand, the demand for energy commodities has increased sharply due to the movement of people and commodities after the pandemic. On the other hand, the reshuffling of the energy market will create a higher-quality market. Thus, the energy industry will face the most significant return shocks [3]. In addition, while the development of traditional energy sources will be under greater pressure after the pandemic, in contrast, investing in renewable energy infrastructure as a forward-looking strategy has proven to have great potential for development [4].

The post-pandemic era has accelerated the pace of sustainable development, and the economic recovery of countries around the world will inevitably have an impact on sustainable development in the future. Therefore, we should consider green economic recovery, promote economic recovery, and improve the environment together [5]. Green transportation has always been regarded as an effective way to improve the environment, as it can reduce the large amount of oil consumption, relieve pressure on resources, and promote energy conservation and emission reductions. In recent years, electric vehicles have been a research hotspot in achieving environmental sustainability. Similarly, the choice of transportation for green travel has become an emerging topic. Electric bicycles have always been one of the options for green transportation [6]. Compared to other traditional four-wheeled vehicles, electric bicycles alleviate traffic congestion and reduce exhaust emissions. Additionally, for shorter distances within 5 km, electric bicycles are a more convenient means of transportation. Compared to non-electric bicycles, electric bicycles have advantages in terms of environmental protection and travel speed. Even shared bicycles can potentially harm the environment, as their delivery and distribution are carried out by trucks, which produce harmful gases that pollute the environment. Clearly, researching issues related to electric bicycles will be a key focus for countries looking to enhance green transportation and energy consumption [7].

Electric bicycles serve as a low-carbon, environmentally friendly, convenient, and affordable means of transportation, making them an excellent advocate for energy conservation, emission reduction, and green travel [8]. In China, the development of electric bicycles has clearly become mature. The government actively promotes electric bicycles as a mode of travel while considering the economic, social, and environmental benefits brought by electric bicycles, and promotes relevant policy research to encourage the exploration and development of shared electric bicycles and Internet-based electric bicycles while ensuring urban traffic safety. As of 2023, the number of electric bicycles in China has exceeded 450 million, with bicycle delivery personnel being one of the main user groups [9]. Although electric bicycles are favored for their ease of operation, economic and environmental benefits, and maneuverability, they also face issues such as slow charging, difficulties in charging, and risks of short circuits and burning during charging. These problems significantly hinder the development of electric bicycles [10].

Electric bicycle battery exchange cabinets combine Internet of Things technology with the sharing economy mindset. They are placed on street corners and in alleys, resembling rectangular cabinets over two meters tall with typically 12 compartments. These compartments can store batteries and charge them. Riders of electric bicycles can connect to the cabinet via a mobile app, allowing them to open a compartment and exchange their low battery with a full battery within three minutes. For delivery and logistics workers who ride long distances daily, ranging from 100 to 150 km, the typical lithium battery electric bike has a range of about 50 km per charge. Therefore, riders usually need to charge their batteries two to three times a day on average [11]. Slow charging batteries takes 4 to 6 h, while fast charging times range from 30 min to 2 h. This significantly impacts the riders’ work. Therefore, riders increasingly need public battery exchange cabinets that allow them to quickly replace low batteries with full batteries throughout their daily travels, thereby replacing the need for traditional charging [12]. At the same time, this battery exchange approach effectively avoids various safety hazards that may occur during the charging process of electric bicycles. While the cabinet is charging the batteries, it can also instantly report any abnormal battery statuses to backend supervisors for immediate handling. This capability helps effectively eliminate safety risks and prevents fire incidents [13].

After 2012, China’s food delivery industry entered a period of rapid growth with frequent capital investments and intense competition among enterprises. By 2016, the market was predominantly dominated by two major players, Elema and Meituan [14]. In the same year, a group of Chinese enterprises began independently developing battery exchange cabinets specifically for charging electric bicycle batteries. Companies providing battery exchange services also emerged within five years. As of now, products like those from these manufacturers like Tower Energy, Hello Bike, Didi Chuxing, and Qingju Bike in China, which represent battery exchange cabinets, have been launched into the market. They have effectively alleviated the battery exchange demand for food delivery riders [15]. Especially in the post-pandemic era, the demand for food delivery and outdoor activities continues to rise, leading to a gradual increase in demand for battery exchange cabinets. However, within the battery exchange cabinet industry, there is intense competition, with many enterprises currently experiencing a surge in capital influx. Many cities still face issues such as insufficient construction of battery exchange cabinets and unreasonable site selection, which greatly hinder the development of the electric bicycle industry. Therefore, it is crucial to address these issues by constructing an adequate number of proper battery exchange cabinets to meet the battery exchange demand of electric bicycles. This will further mitigate the problems of slow and difficult charging, enhance the purchasing willingness of electric bicycle riders, and promote the development of the electric bicycle industry.

The significance of this study lies in providing a solution for the site selection of battery exchange cabinets for electric bicycles, manifested in two main aspects. Firstly, this paper uses kernel density analysis to identify the hot spot distribution of rider demand. Secondly, we build a site selection model based on point demand theory, which aims to maximize rider satisfaction and maximize battery exchange cabinet service capacity. Compared to models based on a single objective, the objective function in this study is more comprehensive and reasonable, offering a method for determining demand volume based on point demand theory. Finally, we innovatively propose an improved immune algorithm that combines two different types of mutation operators and local search. It is compared with the traditional immune algorithm in terms of convergence speed, stability, and search accuracy.

2. Materials and Methods

2.1. Related Theory of Location Selection Method

The site selection theory adopted in this paper is a point demand-based site selection theory. This theory posits that riders’ demand originates at specific nodes, typically dividing all nodes within the site selection range into demand points and candidate points. By analyzing the relationship between demand points and candidate points, the optimal site selection scheme is determined. Research based on point demand site selection theory mainly includes P-median site selection studies, coverage site selection studies, and P-center site selection studies.

The P-median site selection theory was initially proposed in 1970 and is formulated as a mixed-integer model [16]. In recent studies, scholars have used and improved the P-median theoretical model to different degrees to make the model more suitable for solving the location problem. Murad et al. studied the optimization of healthcare service locations. They propose an MC-P-Median model. The model first used the Maximum Coverage (MC) location problem model to prioritize demand locations, then calculated the optimal hospital location through the P-median function. Finally, the model was verified using the Jeddah hospital location problem as an example [17]. M H Ramadhan et al. [18] studied warehouse locations by considering factors such as distance, customers, and cost. They used the combination of set coverage and P-median to construct the location model. Then, by comparing the size of the target value obtained by the two methods, they reached the conclusion that the P-median method has a better solving effect. Janjić et al. [19] explored the location problem of electric wake-up charging piles. In the case of determining the number of charging piles, they set up a new location model by improving the P-median iteration method and combining it with a weighted analytic hierarchy process. The objective function considers the installation cost of the charging pile, the distance of the car to the charging pile, and some actual conditions of the selected location. Finally, the effectiveness of the proposed method is verified by taking Nish City, Serbia, as an example. Hanif [20] studied the application of the centroid method and the P-median method for the site selection of a single non-averse facility for a utility store company. They aimed to minimize transportation costs and determine the optimal location for a warehouse.

The principle of covering location is usually divided into two types in the study of location problems: one is set cover, and the other is maximum cover. When studying the location of electric vehicle charging facilities, Davidov et al. [21] considered the driving path of electric vehicles, the time required for charging, and the parking time after users arrive at the charging facilities, and then established a discrete location model by using the set to cover the original.

Similarly, Zhou et al. [22] also studied the location of EV charging stations. From the perspective of station construction cost, they divided the cost into comprehensive economic cost and environmental cost. The site selection model based on maximum coverage site selection theory is established. Then, taking Ireland as an example, through the genetic algorithm, the model is solved and analyzed. Fang et al. [23] studied the site selection problem for public electric vehicle charging stations, considering the limited range and charging demands of pure electric vehicles. They established a Maximal Covering Location Problem model and solved it using a dual-level mathematical programming approach and a genetic algorithm-based procedure. Pan et al. [24] considered the spatial distribution of demand points, service coverage, and construction costs in different areas to influence the layout of charging pile site selection. They aimed to minimize the number of candidate points and costs within the service coverage area. They established a Set Covering Model and used the entropy weight method and greedy algorithm to solve it. Huang [25] studied the layout optimization of public charging stations in Jin Shui District, Zhengzhou City. The study utilized charging station data, parking lot data, statistical yearbook data, population data, and urban road network data. They employed a Set Covering Model and a P-median model with capacity constraints to optimize the layout of public charging stations in Jin Shui District.

The P-center location problem was proposed by Hakimi [26], based on the relationship between feasible candidate points and grid vertices, as well as inherent constraints. The P-center problem is divided into a vertex center and an absolute center. To provide a basis for further research and improvement of the P-center model, Liu et al. [27] proposed an iterative solving framework for computing the lower bounds of the P-center model. With the development of the times and the diffusion of social networks, to better handle the impact of social factors on facility location, Ma et al. [28] proposed a partitioning algorithm based on Voronoi diagrams. This algorithm aims to establish social constraints and integrate them with spatial constraints, thereby developing an enhanced P-center location model. Yi et al. [29] considered the impact of agglomeration effects in mountainous urban areas and proposed an improved P-center model to determine the optimal locations for charging facilities. Zhou et al. [30] analyzed the relationship between agglomeration effects in mountainous urban areas and the predicted distribution of charging demand, fitting correlation parameter values. They integrated these values with an improved P-center location model, using annual construction costs, electric vehicle operating costs, annual maintenance costs, electricity purchase costs, and expanded distribution network costs as optimization indicators. This was used to establish a site layout model for charging stations in mountainous urban areas.

In addition to the point demand theories, many scholars also study site location from the perspective of business profitability or serving users, which is also part of the research on point demand theory.

Scholars consider the profitability of enterprises in the location problem, which is usually achieved by reducing costs, such as the cost of site construction or the cost in the subsequent operation stage. Fareed et al. [31] considered the impact of electric vehicle load on power loss in the distribution network. They combined the land cost index and the electric vehicle charging demand index to construct the charging station investor index. Then, based on the decision index, an optimization model aiming to achieve the minimum power loss cost was constructed, and the optimal location of the charging station was determined. Gao et al. [32] considered the impact of existing charging stations on the location of new charging stations. By introducing intermediate variables, they constructed a segmented nonlinear function to describe the relationship between existing charging stations and new charging stations, considering the total investment cost of charging stations and the distance between users and charging stations. Finally, they developed a location model that could be based on adjustments to existing charging stations in each area. Min et al. [33] aimed to minimize daily food delivery costs using pure electric trucks. They built a first-stage site selection model for electric truck charging stations and defined relevant parameters to be input into the second-stage model. The second stage model is based on the fixed capacity model of the pure electric truck charging station, considering the annual comprehensive cost of the charging station. By integrating Phase 1 and Phase 2 models, they determined the site layout and capacity of all-electric truck charging stations.

From the perspective of serving users, considerations often include their time costs and convenience. Zhang et al. proposed a charging station location problem that considers user preferences and waiting times, formulated as a multi-objective two-level programming model. Zhang et al. [34] studied the location of charging piles and considered users’ preferences for charging piles and the waiting time for charging. They built a multi-objective, two-level programming model. The first-level model aims to minimize total costs to determine the locations and number of charging piles, while the second-level model aims to minimize total travel time to match the user’s preference. Xu et al. [35] thoroughly considered the impact of transportation convenience on users driving electric vehicles and the acceptance capacity of the distribution network. They established a fixed-capacity charging station location model that incorporates both convenience and acceptance capacity considerations. They solved this model using a combined approach of particle swarm optimization and Voronoi diagrams. Tran et al. [36] are studying the location of charging piles. They give priority to the driving distance between users and charging. Secondly, they combine the location model of charging pile construction design. Zhang et al. [37] also considered the user’s driving time to the charging pile when studying the location of charging stations, and proposed the concept of social cost, including the construction cost and operating cost of the charging pile. Finally, they built a site selection model and used an improved firefly algorithm to solve the model.

In summary, this paper considers the user experience of electric bicycle riders, with the goal of maximizing rider satisfaction and battery exchange cabinet service capacity. A location model based on point demand theory is established. Compared with the single objective location model, the objective function of this paper is more comprehensive and reasonable.

2.2. Location Selection Method-Related Algorithm

Some scholars use genetic algorithms to solve the point demand model. Although it has advantages in solving problems involving mixed discrete, continuous, and integer design variables, the immune algorithm has characteristics similar to the ability of the biological immune system to produce various specific antibodies and has advantages over genetic algorithms, such as preventing degeneration, accelerating convergence speed, and simplifying operators. Therefore, immune algorithms are more suitable for solving various types of location models, vehicle scheduling models, and many other different types of models [38]. Liu et al. [39] studied the location problem of medical waste treatment facilities and solved the multi-objective optimization model of the facility location using a traditional immune algorithm. Traditional immune algorithms suffer from slow convergence and low search accuracy. To address these issues, various improvements have been made to this algorithm. Yang et al. [40] combined the immune algorithm with an ant colony algorithm, innovatively proposing a new immune ant colony optimization (IACO) algorithm. They applied this algorithm to solve multi-objective task allocation problems for warehouse robot groups. The experimental results demonstrate that integrating these two algorithms can effectively reduce computational redundancy in immune algorithms and improve convergence speed. Xu [41] improved the immune algorithm by extracting methods from artificial immune theory, such as extracting and injecting vaccines. Based on the improved immune algorithm, he solved site selection problems and proved that the algorithm can reach the optimal solution faster through iteration. Wang et al. [42] introduced a new improvement to the classical immune algorithm by incorporating a drug-assisted process. This approach reduces the size of the search space, reaching the optimal solution of the model faster. They validated the feasibility of this approach by solving test functions. Xu et al. [43] introduced several new operators into the immune algorithm to modify the process of population generation, updating, and cross-variation. To improve algorithm performance and adapt it for solving their constructed site selection model for emergency medical facilities, Zhou et al. [44] made three specific design improvements to the immune algorithm. Firstly, they optimized the generation of the immune algorithm population based on the specific characteristics of the emergency medical facility site selection model. Secondly, they designed an antibody evaluation function. Lastly, they enhanced the crossover stage by designing two neighborhood search mechanisms to reduce antibody similarity. This ensured diversity among antibodies in each iteration, facilitating the algorithm to achieve global optimal solutions and preventing premature convergence.

In this paper, the improved immune algorithm is used to solve the location model. Scholars have improved the traditional immune algorithm in two main ways: one is to combine the immune algorithm with other intelligent algorithms and the other is to refine some internal calculation processes of the immune algorithm. This paper enhances the performance of the immune algorithm through the second approach, specifically by improving the mutation phase of the algorithm. This improvement involves introducing two types of mutation operators and operators for local search.

3. Problem Description

3.1. Basic Problems

The basic problem of this paper is how to determine the number and locations of electric bicycle battery exchange cabinets. As shown in Figure 1, the steps to solve the basic problem are mainly divided into two stages and four associated problems. The first stage is to collect battery data through battery sensors to determine the battery exchange demand of e-bike riders in the area, which is the content of the first associated problem. In the second stage, based on demand points, the hot spots of demand points are identified through nuclear density analysis, which is the content of the second associated problem. Then, based on point demand theory, a location model of electric bicycle battery exchange cabinets is constructed. This model is a bi-objective model that considers two factors that affect the selection of electric bicycle battery exchange cabinets, which is the content of the third associated problem. Then, to clearly indicate the location scheme, this paper also needs to delimit the location range of battery exchange points in the area, which is the fourth associated problem. Finally, the siting of these cabinets can also influence rider travel trajectories to some extent, leading to variations in the data collected by battery sensors correspondingly. It is essential to address the following associated problems:

• Identifying the battery exchange demand among electric bicycle riders in the area.

This paper collects riding data of electric bicycles through internal battery sensors, including battery charge levels and current location information. Batteries may require exchange at any location, but the willingness to exchange varies with the battery’s current charge level. Therefore, we define the parameters $r$ and v. The parameter $r$ is the demand coefficient, the parameter $E$ is the battery charge at a particular moment, and the relationship between the two is the parameter $r=\left(100-E\right)$. Then, we perform a weighted average for each battery based on the demand coefficient, as shown in Equation (1). The point $\left(X,Y\right)$ represents the coordinates of the battery at a specific moment. The point $\left(\overline{X},\overline{Y}\right)$ denotes the weighted average coordinates of the battery across all moments. Each battery represents a rider and corresponds to a demand point within a specific area range.

$$\left(\overline{X},\overline{Y}\right)=\left({\displaystyle \frac{\sum rX}{N\sum r}},{\displaystyle \frac{\sum rY}{N\sum r}}\right)$$

(1)

2.

Analyzing the distribution differences and clustering of demand hotspots.

Kernel density analysis is used to measure and analyze the density of point or line features in their surrounding neighborhoods. It is one of the methods used to determine population density distribution in urban areas. Compared to traditional population density analysis, the advantage of kernel density analysis lies in its ability to intuitively reflect the distribution of discrete measurement values within continuous areas. It also overcomes the issue of uniform density values within statistical units. Therefore, this paper employs kernel density analysis to identify differences and clustering patterns in rider density hotspots within urban areas. According to the kernel density estimation method, we generate evenly spaced grid points within the coordinates of rider demand points to cover all rider locations. The density value for each grid point is computed using the kernel density estimation formula.

For the kernel function selection, the Gaussian function is commonly used as the kernel function, as shown in Equation (2):

$$K\left(\mu \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}{\mu }^2}$$

(2)

In this paper, the bandwidth parameter $h$ is set to control the width of kernel function, which determines the smoothness of kernel function. Smaller bandwidths produce more detailed density estimates (which can lead to overfitting), while larger bandwidths lead to smoother density estimates (which can lead to underfitting). Silverman’s law is commonly used to calculate bandwidth, as shown in Equation (3). The parameter $\sigma$ represents the variance in the sample data, and the parameter $n$ represents the number of samples in the data. Parameter a in Silverman’s rule adjusts the bandwidth to balance between bias and variance. The term $-1/5$ acts as a scaling factor to adjust the bandwidth $h$ with respect to the sample size $n$.

$$h=a\sigma n^{-{\displaystyle \frac{1}{5}}}$$

(3)

The density estimate at each grid point (x, y) is obtained by averaging the weighted kernel function values of all data points. The equation for the density estimates at each grid point $\left(x, y\right)$ is shown in Equation (4). The point $\left(x_i,y_i\right)$ represents the demand points for each rider. Then, based on the density estimates, a heatmap of the density distribution is plotted to identify hotspots where rider density is high.

$$f\left(x,y\right)={\displaystyle \frac{1}{n{h}^2}}K({\displaystyle \frac{\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2}}{h}})$$

(4)

3.

Factors influencing the siting of electric bicycle battery exchange cabinets

Nowadays, China is in the early stages of development for electric bicycle battery exchange cabinets. The service of electric bicycle battery exchange has not yet been fully popularized. From the point of view of promoting services and increasing riders’ willingness to replace batteries, when selecting sites for battery exchange cabinets, priority should be given to considering the riders’ perspectives and factors that encourage growth in the number of battery-exchanging riders. Therefore, this paper primarily considers the following two main factors.

The first factor is riders’ satisfaction with battery exchange cabinets, which refers to whether there is a nearby cabinet when riders need to exchange batteries, and the closer the cabinet is, the higher the riders’ satisfaction. The satisfaction function of electric bicycle riders describes the relationship between objective distance and riders’ subjective evaluation. The satisfaction function of electric bicycle riders is presented in Equation (5). According to point demand theory, the parameter $i$ is the demand point, which is the rider’s coordinate point. The parameter $j$ denote the candidate point, which is the cabinet’s candidate coordinate point. The parameter $L_i$ indicates the farthest distance from the power change cabinet when the rider is most satisfied. The parameter $U_i$ indicates the maximum change cabinet distance the rider can tolerate, and the parameter $d_{ij}$ is the distance from demand point to candidate point. The function $F(d_{ij})$ denotes the satisfaction evaluation value of electric bicycle riders at the demand point.

$$F\left(d_{ij}\right)=\left\{\begin{array}{c}1,d_{ij}<L_i\\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\cos \left({\displaystyle \frac{\pi }{U_i-L_i}}\left(d_{ij}-{\displaystyle \frac{U_i-L_i}{2}}\right)+{\displaystyle \frac{\pi }{2}}\right),L_i<d_{ij}<U_i\\ 0,d_{ij}>U_i\end{array}\right.$$

(5)

The second factor is the service capacity of a battery exchange cabinet, which indicates the number of demand points that fall within the scope of the service. The higher the service capacity of a battery exchange cabinet, the more riders it can serve. Unlike power infrastructure such as power plants and substations, electric bicycle battery exchange cabinets serve as public infrastructure for the public. In addition to meeting urban layout and grid planning requirements, they also consider the maximum distance riders can travel to exchange batteries when their electric bicycle battery levels are low. In this paper, this distance is defined as the service radius $R$. If a coordinate point is selected as the location for building a battery exchange cabinet and it is the closest cabinet to the demand point, and if the distance between the demand point and coordinate point is less than the service radius $R$, then the coordinate point is considered to serve the demand point. Therefore, a higher service capacity of the battery exchange cabinet means it can serve more riders, leading to greater profitability for the enterprise.

4.

The delineation of battery exchange point siting ranges in space

According to point demand theory, we simplify the site selection area into a rectangular region of specified length and width, establishing a coordinate system with the bottom-left corner of the rectangle as the origin. Then, we map the latitude and longitude of the demand points into the system. Every coordinate point in space is considered a potential candidate point. The problem in this paper transforms into a site selection issue where $p$ battery exchange cabinets need to be selected from $J$ candidate points within a specific rectangular area. Site selection solutions must align with rider satisfaction and the service capacity of the battery exchange cabinets.

3.2. Basic Assumptions

To prioritize maximizing rider satisfaction and maximizing the service capacity of battery exchange cabinets for electric bicycle riders, we establish a siting model. To facilitate the solution of the model, we make the following assumptions:

• According to the principle of proximity, riders prefer to go to the nearest battery exchange cabinet to their current location.
• In the study area, rider demand points remain the same every day.
• Due to the small footprint of battery exchange cabinets, they can be set up in most streets and alleys. Therefore, this paper does not consider whether the selected locations for setting up battery exchange cabinets meet the conditions for construction.
• The number of battery exchange cabinets set up in the network is not considered, i.e., ensuring they can meet the battery exchange demand of all riders within their service range.
• Battery data are collected from internal sensors, which accurately reflect riders’ cycling conditions.

Some parameters in the model have been simplified and replaced accordingly, as follows:

• The distance between riders and battery exchange cabinets is their Euclidean distance, shown in Equation (6). The point $\left(x_i,y_i\right)$ and the point $\left(x_j,y_j\right)$ are the coordinates of the demand point and candidate point, respectively.

  $$d_{ij}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}$$

  (6)
• During data collection, battery levels are uniformly distributed within a specified range. We know that riders have no intention to exchange batteries when their battery level is above a certain threshold. Therefore, we assume that riders have no battery exchange demand when their battery level is greater than 60%. Data points where the battery level exceeds 60% are excluded from the original dataset. Although this exclusion may cause slight deviations in the latitude and longitude of demand points, we believe it will not significantly affect the final results of this study.
• This paper assumes that $R$ is the maximum tolerable service radius for all e-bike riders.

4. Model Construction and Algorithm Design

4.1. Model Construction

The decision variables in this paper are $Z_j$ and $Y_{ij}$, both of which are binary variables (0 or 1). If the variable $Z_j$ is 1, it indicates that the candidate point is selected as the site for establishing a battery exchange cabinet. If the variable $Z_j$ is 0, it means the candidate point is not selected as a cabinet location. If the candidate point serves the demand point, then the variable $Y_{ij}$ is 1; otherwise, the variable $Y_{ij}$ is 0. The demand point must be within the service range of the candidate point. Therefore, the variables $Y_{ij},Z_jϵ\left\{0,1\right\}\left(\forall iϵn_i,\forall jϵn_j\right)$. The model’s symbols and indicators are shown in the following

Table 1. Model variables and parameters.

Symbol	Description
$i$	Represents the identification index of demand points, $iϵn_i$.
$n_i$	Set representing the collection of demand points.
$j$	Represents the identification index of candidate points, $\mathsf{j}ϵn_j$.
$n_j$	Set representing the collection of candidate points.
$I$	Represents the number of demand points.
$J$	Represents the number of candidate points.
$P$	Represents the number of exchange cabinets (antibody characteristics in immune algorithm).
$d_{ij}$	Distance between demand points.
$R$	The service radius of battery exchange cabinet.
$L_i$	The farthest distance from the power change cabinet when the rider is most satisfied.
$U_i$	Indicates the maximum change cabinet distance the rider can tolerate.
$P_s$	Population size in the immune algorithm.
$g$	Generations of population evolution in the immune algorithm.
$r_c$	Clone rate in the immune algorithm.
$w_1$	The weight of the objective function $f_1$.
$w_2$	The weight of the objective function $f_2$.
$r_m$	Mutation ratio in the immune algorithm.
$Z_j$	Decision variables.
$Y_{ij}$	Decision variables.

.

Based on the above descriptions, the expression for establishing a battery exchange cabinet siting model based on satisfaction and service capacity is as follows.

Objective function:

$$max{f}_1={\displaystyle \frac{{{\displaystyle \sum }}_{j=1}^J{{\displaystyle \sum }}_{i=1}^{I}F\left(d_{ij}\right)Y_{ij}{Z}_j}{I}}$$

(7)

$$max f_2={\displaystyle \frac{{{\displaystyle \sum }}_{j=1}^J{{\displaystyle \sum }}_{i=1}^I{Y}_{ij}{Z}_j}{I}}$$

(8)

Constraint conditions:

$${\displaystyle \sum }_{j=1}^J{Y}_{ij}=1 \left(\forall iϵn_i,\forall j\in n_j\right)$$

(9)

$${\displaystyle \sum }_{j=1}^J{Z}_j=P \forall jϵn_j$$

(10)

$$d_{ij}\le R\left(Z_j=1,Y_{ij}=1\right)$$

(11)

$$Z_j,Y_{ij}ϵ\left\{0,1\right\}\left(\forall iϵn_i,\forall jϵn_j\right)$$

(12)

In the siting model, Equation (7) represents maximizing average rider satisfaction. Equation (8) represents maximizing average battery exchange cabinet service capacity. Due to redundancy in the decision variables, certain constraints are imposed as follows:

(a)

It is ensured that each battery exchange cabinet’s candidate point serves multiple demand points and that each demand point is served by only one candidate point of a battery exchange cabinet. The constraints are formulated as in Equation (9).

(b)

In Equation (10), this paper selects p candidate points as the coordinate points set up by the battery exchange cabinet.

(c)

If the candidate point serves the demand point, then the distance between the demand point and the candidate point is less than radius $R$, as shown in Equation (11).

4.2. Improved Immune Algorithm

The improved immune algorithm addresses the challenging issue of premature convergence during iteration processes while maintaining population diversity. However, traditional immune algorithms suffer from slow convergence speed, low search accuracy, and poor stability, especially evident in large-scale computational problems like battery exchange cabinet siting. To tackle these issues, the following improvements were made to the traditional immune algorithm:

(a)

A multi-point mutation operator is introduced in the mutation process. Compared with the single point variation and small number of mutations in the traditional immune algorithm, more changes are introduced in the generation of new candidate solutions in the early stage of population evolution, which increases the chance of generating the optimal solution in the iterative process and speeds up algorithm convergence. In the later stage of population evolution, the algorithm is prevented from falling into the local optimal solution and prematurely converging.

(b)

Guided mutation is added to the mutation operator of the traditional immune algorithm, mainly to improve the orientation of the mutation at the initial stage of population evolution and guide the antibody towards the optimal solution region, which can greatly increase the initial convergence of the algorithm compared with the traditional immune algorithm.

(c)

In the improved immune algorithm, the solution is fine-tuned and optimized by local search and setting termination conditions. In the late stage of population evolution, traditional immune algorithms often fall into the dilemma of local optimization. Local search can greatly improve this situation and speed up the convergence of the algorithm.

Based on the characteristics of the constructed model, an immune algorithm for battery exchange cabinet siting was designed, incorporating population generation, affinity functions, two types of mutation operators, and a local search operator. The specific calculation process of the improved immune algorithm is shown in Figure 2, described in Algorithm 1.

Algorithm 1: Improved immune algorithm

Input: Population size $P_s$, antibody characteristics (number of battery exchange cabinets) $P$, generations of population evolution $g$, clone rate $r_c$, mutation ratio $r_m$. Output: Site selection scheme for electric bicycle battery exchange cabinets

Step 1. Initialize Population     According to the point demand theory, among the candidate points with a total amount of $j$, we select the candidate point with a number of $P$ as the location of the battery exchange cabinet. These candidate sites form an antibody. Therefore, antibodies are a feasible solution to the location problem, and the number of candidate points composed of antibodies is $P$. These candidates have zero similarity to each other. A population is formed by the clumping together of q antibodies. The immune algorithm iterates the optimal antibody through continuous population multiplication. During antibody generation, we define the distance between each candidate point and at least one demand point to be within $R$, which can greatly improve the convergence speed of the algorithm. Step 2. Population evolution     Evolve the population over multiple generations, with the specific number of iterations determined by parameter $g$. In each generation, perform the following operations:     Step 2.1. Affinity calculation         The electric bicycle battery exchange cabinet siting model is a multi-objective nonlinear model, which can be handled using linear weighting methods. The affinity value is defined as $w_1{f}_1^{\prime }+w_2{f}_2^{\prime }$, where $f_1^{\prime }$ and $f_2^{\prime }$ are the normalized objective values of rider satisfaction and the service capacity of battery exchange cabinets, respectively. $w_1$ and $w_2$ are the weights for each objective, satisfying $w_1,w_2ϵ\left(0,1\right],w_1+w_2=1$. The normalization of $f_1^{\prime }$ and $f_2^{\prime }$ is calculated as shown in the equation below. $maxf$ and $minf$ represent the maximum and minimum values that $f$ can attain.

$f^{\prime }={\displaystyle \frac{f-minf}{maxf-minf}}$

Step 2.2. Update optimal antibody         Identify the antibody with the highest affinity from the population. Compare the affinity of this antibody with the current optimal antibody and update the optimal site selection scheme if the affinity exceeds the current best solution.     Step 2.3. Clonal elite antibody         Sort antibodies based on their affinity values. Select $P_s\ast \times r_c$ proportion of antibodies as elite antibodies. Clone each elite antibody, generating a few clones equal to $(1-r_c)/r_c$.     Step 2.4. Multi-point Mutation and Guided Mutation         Perform multi-point mutation on half of the cloned antibodies, selecting new candidate points to replace the old ones. Conduct guided mutation on the other half of the cloned antibodies, mutating based on the current optimal solution to increase directionality.     Step 2.5. Local Search         Perform a search on elite antibodies by aggregating searches within the elite antibody domain to enhance the directionality of mutation.     Step 2.6. Termination Condition Evaluation         At the end of each generation, check if the termination condition is met. If satisfied, terminate the algorithm early to reduce unnecessary computation.     Step 2.7. Update Population         Combine the newly generated clone antibodies of selected antibodies into a new population. Add elite antibodies to the new population to ensure that the optimal solution is retained within the population. Step 3. Population iteration     Repeat Step 2 until the number of iterations reaches the generations of population evolution $g$.

4.3. Multi-Point Mutation, Guided Mutation, and Local Search

Multi-point mutation is an extension of single-point mutation in traditional immune algorithms, where mutations occur at multiple gene positions to enhance individual diversity and algorithm exploration capabilities. By mutating at multiple positions, it becomes possible to alter the genotype of individuals over a larger range, thereby improving the algorithm’s ability to escape local optima. In this study, half of the cloned antibodies are subjected to multi-point mutation. Set the multi-point mutation rate as $r_m$, and calculate the number of mutated antibody genes as $\mathrm{P}\ast r_m$. Randomly select $\mathrm{P}\ast r_m$ antibody genes, ensuring that the randomly generated points fall within the service range of the battery exchange cabinet, and replace the old points with the new ones. The process of multi-point mutation is illustrated in Figure 3, where we assume a randomly selected set $\left[\left(x_2,y_2\right),\dots \dots \left(x_t,y_t\right)\right]$ comprising $\mathrm{P}\ast r_m$ genes. Then, $\left[\left(x_2^{\prime },y_2^{\prime }\right),\dots \dots \left(x_t^{\prime },y_t^{\prime }\right)\right]$ is randomly generated to replace the original gene set, thereby creating new antibodies.

Guided mutation builds upon traditional single-point mutation by targeting the remaining half of clones that have not undergone multi-point mutation. It sets a mutation magnitude $c$ with a tendency towards the center, making the mutation process more targeted and capable of converging faster to the global optimum. The mutation process is illustrated in Figure 4, where, for example, if a gene point $\left(x_t,y_t\right)$ is randomly selected, and if $x_t$ is greater than the average value $\overline{x}$, it reduces by $c$ to move closer to the center; conversely, if $x_t$ is less than $\overline{x}$, it increases by $c$. Similarly, this applies to $y_t$.

The article adopts local search for elite individuals, playing a role in fine-tuning and optimizing solutions within the improved immune algorithm, accelerating the convergence speed of the algorithm and escaping local optima. The local search algorithm process is shown in Figure 5. A gene point is randomly selected, assuming $\left(x_t,y_t\right)$ with an affinity value $f_t$. Within its neighborhood range, with a change magnitude of a, $\left(x_t,y_t\right)$ transforms into eight neighboring solutions: $\left(x_t+a,y_t\right)$, $\left(x_t-a,y_t\right)$, $\left(x_t,y_t+a\right)$, $\left(x_t,y_t-a\right)$, $\left(x_t+a/2,y_t-a/2\right)$, $\left(x_t-a/2,y_t-a/2\right)$, $\left(x_t+a/2,y_t+a/2\right)$, $\left(x_t-a/2,y_t+a/2\right)$. If the affinity of a neighboring solution exceeds that of the current solution, it accepts replacing the current solution. Repeat this process until reaching the termination condition of the search. There are two kinds of termination conditions: one is that the optimal solution is found through local search, the other is that the local search has been carried out for each position. Typically, the algorithm terminates when one of these conditions is met.

5. Computing Example

5.1. Data Pre-Processing

The example analysis in this paper takes electric bicycles in Shanghai as the research object. The data were provided by a leading Chinese company related to shared batteries. The data collected by the sensors in the electric bicycle batteries under the company’s name mainly include information such as battery power, status, and location. Second, we simplified the land area of Shanghai into a rectangle 120 km long and 100 km wide. Then, each battery is regarded as a battery exchange demand point, and after calculating the weighted average, it is found that there are 19,000 demand points. The distribution of demand points in Shanghai is shown in Figure 6. The area is divided into 16 sections corresponding to the 16 administrative districts of Shanghai. The darker the color, the more demand points there are in that area, with the deepest brown representing the area with the highest number of demand points. Conversely, lighter colors indicate fewer demand points. The demand points are concentrated in the central and eastern districts.

5.2. Nuclear Density Analysis

A kernel density analysis was conducted for Shanghai, using the natural breaks classification method to categorize different levels of rider density. The rider density distribution is illustrated in Figure 7. Darker areas represent higher rider density. In comparison with Figure 6, it can be observed that the densest areas are also concentrated in the central and eastern parts of the city. These areas coincide with the most economically developed regions of Shanghai.

5.3. Theoretical Model Analysis

Nowadays, the electric bicycle battery exchange service industry is in its early stages of development, and battery exchange services are not yet widespread. At this stage, the objective value established by the site selection model serves as a crucial criterion for evaluating the quality of electric bicycle battery exchange cabinet location schemes. We set the variable parameters. The parameter $L_i$ is $2 \mathrm{km}$, and the parameter $U_i$ is $1 \mathrm{km}$. The service radius $R$ of the battery exchange cabinet is $2 \mathrm{km}$. Parameters ${\omega }_1$ and ${\omega }_2$ are both set to 0.5. The parameters for the immune algorithm are set as follows: Parameters $n$ and $g$ both are 1600. The parameter $p$ is 800. The parameter $r_c$ is 0.2 and the parameter $r_m$ is 0.1. Then, to determine the number of battery exchange cabinets to construct, sensitivity analysis of the site selection schemes is conducted, as shown in

Table 2. Quantitative sensitivity analysis of electric bicycle battery exchange cabinets.

Number of Cabinets	$\mathbf{Comprehensive} \mathbf{Objective} \mathit{f}$	$\mathbf{Rider} \mathbf{Satisfaction} {\mathit{f}}_1^{\prime }$ Value	$\mathbf{Battery} \mathbf{Exchange} \mathbf{Station} \mathbf{Service} \mathbf{Capability} {\mathit{f}}_2^{\prime }$ Value	$\mathbf{Difference} \mathbf{Value} \mathbf{of} \mathit{f}$
790	0.948688589	0.920485882	0.976891297
800	0.957488589	0.929024274	0.985952904	0.0088
810	0.960188589	0.931644009	0.98873317	0.0027
820	0.961288589	0.932711308	0.989865871	0.0011
830	0.961988589	0.933390498	0.99058668	0.0007
840	0.962588589	0.933972661	0.991204517	0.0006

. As the number of battery exchange cabinets increases, the objective value of the site selection schemes improves continuously from the users’ perspective. In addition, an average of two battery exchange cabinets are installed at each exchange cabinet site, and each exchange cabinet can store 12 batteries. Given the demand for 19,000 batteries, an average of 791 exchange cabinets are required. To balance these factors, we chose a scheme that maximizes the difference value of comprehensive objective $f$. Finally, we selected 800 candidate sites for setting up electric bicycle battery exchange cabinets.

Using asterisks to represent battery exchange cabinets and shading at the bottom to indicate the density range of demand points, Figure 8 illustrates the site selection plan for electric bicycle battery exchange cabinets. Combining Table 2 and Figure 8, when the parameter $P$ is 800, the user satisfaction reaches 92.90%, and the battery exchange cabinet service capacity reaches 98.6%. The comprehensive objective function value also exceeds 95%. It shows that most riders are very satisfied with the location scheme, and most riders are within the scope of battery exchange cabinet service. Meanwhile, we can also observe from Figure 8 that the exchange cabinets effectively cover all areas of battery exchange demand, enhancing service capability. The plan meets the battery exchange demand of all riders, and the greater the rider density of the area, the more battery exchange cabinet location points. Therefore, riders’ demand for battery exchange is met and riders’ satisfaction is improved. The results show that the point demand theory method of combining the two objectives is very effective in the application of a site selection scheme. It is also an extension of the application of the point demand theory to the location problem.

5.4. Comparison of the Two Algorithms

In this paper, sensitivity analysis and convergence analysis are used to compare the accuracy, search accuracy, stability, and convergence of the two models. To simplify the analysis in Section 5.4, we use the value of the comprehensive objective $f$ to replace the value of the two objective functions, and the comparative analysis of the algorithms in this section only refers to the value of the comprehensive objective $f$. Similarly, to make the verification more accurate, this paper sets $2\times 3$ different parameter combinations, the parameter $r_c$ is set to 0.1 or 0.2, and the parameter $r_m$ is set to 0.1, 0.2, or 0.3. The two algorithms are each run 10 times under each parameter combination, totaling $2\times 3\times 10\times 2=120$.

(1)

Sensitivity analysis

We compared the accuracy, search accuracy, and stability of the algorithms through sensitivity analysis, and calculated the optimal value, average value, and standard deviation of the comprehensive value of objective $f$ combined with the results of each run, as shown in

Table 3. Sensitivity analysis for different parameters.

		${\mathit{r}}_{\mathit{m}}=$0.1			${\mathit{r}}_{\mathit{m}}=$0.2			${\mathit{r}}_{\mathit{m}}=$0.3
		Optimal Value	Average Value	Standard Deviation	Optimal Value	Average Value	Standard Deviation	Optimal Value	Average Value	Standard Deviation
$r_c=$0.1	Traditional	0.9537	0.9500	0.0031	0.9309	0.9224	0.0048	0.9110	0.9067	0.0039
	Improved	0.9602	0.9578	0.0018	0.9382	0.9339	0.0025	0.9244	0.9205	0.0050
$r_c=$0.2	Traditional	0.9540	0.9501	0.0046	0.9309	0.9205	0.0061	0.9076	0.9041	0.0031
	Improved	0.9635	0.9614	0.0030	0.9404	0.9349	0.0033	0.9156	0.9124	0.0022

. Accuracy is determined by comparing the optimal value of the two algorithms, where the larger the optimal value, the more accurate the algorithm. Search accuracy is measured by the difference between the average value and the optimal value. The smaller the difference, the closer the average solution is to the optimal solution, the greater the probability of the algorithm reaching the optimal solution, and the higher the search progress of the algorithm. The stability of the algorithm is measured by standard deviation, where the smaller the standard deviation, the more stable the algorithm.

First, we find that under the conditions of $r_c$ = 0.2 and $r_m$ = 0.1, the optimal value and average value of the two algorithms are the best. Under these conditions, the accuracy of the improved immune algorithm is increased by 1%, the search progress is increased by 46%, and the stability of the algorithm is increased by 34%. Secondly, we comprehensively calculated the average values under the six combinations, and the accuracy of the improved immune algorithm increased by 0.97%, the search progress increased by 32%, and the stability of the algorithm increased by 28%. We also found that under the conditions of $r_c$ = 0.1 and $r_m$ = 0.3, the stability of the improved immune algorithm is 28% lower than that of the traditional immune algorithm. Finally, the above analysis shows that the improved immune algorithm is superior to the traditional immune algorithm in the above three aspects.

(2)

Convergence analysis

In this paper, the convergence of the two algorithms is compared through convergence analysis. In Equation (13), the parameter $g$ represents the number of generations the algorithm needs, the parameter $t$ represents the index of the number of generations, values $f_{best}^{t+1}$ and $f_{best}^t$ are the optimal values under t + 1 and t generations, and then the rate of change of each generation needs to be summed. If the algorithm converges earlier, $f_{best}^t$ will be lower, and thus the convergence rate will be higher. Similarly, if the algorithm’s final optimal target value is larger, the difference between values $f_{best}^{t+1}$ and $f_{best}^t$ will be larger, and the convergence rate will be improved. Therefore, the convergence rate can effectively measure the convergence speed of the algorithm and the final optimal value span. In this paper, the average convergence rate of the 10 algorithm iterations run under each combination is shown in Figure 9. We find that both algorithms perform best when $r_c$= 0.2 and $r_m$ = 0.1, with convergence being improved by 7.29%, and perform the worst when $r_c$ = 0.1 and $r_m$ = 0.3. Moreover, the average convergence rate of the improved immune algorithm under the six parameter combinations is higher than that of the traditional immune algorithm, with an average increase of 12.4%, which proves that the convergence of the improved immune algorithm is better than that of the traditional immune algorithm under the influence of multi-point variation and guided variation.

$$Convergence Rate={\displaystyle \sum }_{t=1}^{g-1}{\displaystyle \frac{\left|f_{best}^{t+1}-f_{best}^t\right|}{\left|f_{best}^t\right|}}$$

(13)

6. Conclusions and Future Research

This paper proposes an electric bicycle battery exchange cabinet site selection model that considers rider satisfaction and the service capability of exchange cabinets and expands on existing literature. This paper is divided into three steps. Firstly, the study uses demand point theory to establish the site selection model, considering rider satisfaction and the service capability of the exchange cabinets. Secondly, it introduces an improved immune algorithm for model solving and enhances the algorithm by adding two mutation operators and one local search operator, thereby improving convergence speed, stability, and search accuracy. Thirdly, it provides a computational instance for electric bicycle riders in Shanghai, validating the advantages of the improved immune algorithm through comparison with the traditional immune algorithm. This offers enterprises optimal solutions for exchange cabinet site selection.

Through example calculations, we verify that the location model based on point demand theory performs well on two objective functions of rider satisfaction and battery exchange cabinet service capability. The battery exchange cabinets effectively cover all areas where battery exchange demand points are distributed, achieving the maximum service capability objective for the cabinets. Moreover, the cabinets are distributed more heavily in areas with higher demand points, maximizing rider satisfaction. We expand the application practice of point demand theory in the location selection model. Then, experiments with different parameter groups, through sensitivity analysis and convergence analysis, verify that the improved immune algorithm performs better than the traditional immune algorithm in accuracy, search accuracy, stability and convergence.

In future research, various simplifying assumptions (as discussed in Section 3) could be relaxed to better represent the problem. For instance, actual route distances based on POI data could replace Euclidean distances between riders and exchange cabinets as modeled in this study. Future work will aim to more scientifically and practically design applicable site selection models or more efficient algorithms. Furthermore, considering more practical factors in exchange cabinet site selection processes, such as enterprise perspectives on exchange cabinet construction costs, operational costs, and profitability, will be crucial.