A Study on the Vehicle Routing Planning Method for Fresh Food Distribution

Abstract

Aimed at the high cost of cold chain distribution of fresh agricultural products within a specified time window, a joint optimization method based on a bi-level programming model for cold chain logistics is proposed for the location of front warehouses and distribution path planning. At the upper level of the bi-level programming model, k-means clustering analysis is used to obtain all accurate information about alternative locations for the front warehouse for site selection, thereby providing the corresponding foundation for the lower level algorithm. At the lower level of the model, a fusion algorithm of particle swarm optimization (PSO) and a genetic algorithm (GA) is used for solving. To accelerate the convergence speed of the population and lower the running time of the algorithm, the parameter values in the algorithm are determined adaptively. An adaptive hybrid algorithm combining the particle swarm optimization algorithm and the genetic algorithm (APSOGA) is used to reallocate the location information on backup points for the front-end warehouse, ultimately determining the facility location of the front-end warehouse and planning the end path from the front-end warehouse to the customer point, achieving joint optimization of the front-end warehouse’s location and path. A comparative analysis of algorithm optimization shows that using the APSOGA hybrid algorithm can reduce the total cost of the logistics network by 14.57% compared to a traditional single-algorithm PSO solution and reduce it by 5.21% compared to using a single GA. This proves the effectiveness of the APSOGA hybrid algorithm in solving location and path planning problems for cold chain logistics distribution companies.

1. Introduction

With the continuous improvement of the national economy and living standards, the demand for fresh products among residents is also rising, accelerating the cultivation of the public’s habits of online consumption and promoting the continuous development of the e-commerce industry. In 2021, the scale of fresh e-commerce transactions reached CNY 465.81 billion, an increase of 27.92% compared to the same period in 2020 [1]. As the fresh food e-commerce industry gradually matures, people are more interested in a convenient logistics model for online ordering and offline distribution. In response to the expansion of the market scale, a logistics model for fresh e-commerce has continuously been upgraded, and the front warehouse model has become a new choice for many e-commerce enterprises. Through online operations, offline customers’ orders are placed through service stations in the community, enabling goods to quickly reach consumers, solving the “last mile” problem in logistics distribution and improving logistics service quality [2,3].

Fresh products are prone to decay under normal temperatures due to certain characteristics. Therefore, the location of front warehouses and how to improve the efficiency of delivery and avoid the risk of deterioration have become key issues in the research on fresh product logistics. Reasonable location and distribution path planning for front warehouses can optimize the entire supply chain network, thereby minimizing the costs and maximizing the benefits. In order to provide convenient route planning services for the multi-point distribution of fresh agricultural products, a distribution route planning system for delivering fresh agricultural products in the shortest time is proposed [4]. Xu H X et al. [5] focused on summarizing an algorithm based on reinforcement learning and deep reinforcement learning methods for solving the path planning problem for autonomous vehicles; classified the algorithm according to value-based and policy-based methods; and analyzed the characteristics, benefits and drawbacks, and improvement measures for each type of algorithm. Zheng E Z and Lang F et al. [6,7] proposed planning methods for the distribution paths for emergency supplies. For vehicle path optimization, Ren T et al. [8] designed an ant colony algorithm that incorporates a knowledge model of tabu search operators and dynamic probability selection under a knowledge-based elite strategy to solve cold chain path optimization problems. F. Errico et al. [9] developed a method for accurately calculating the probability of route success, which uses a branch-and-cut algorithm to solve vehicle routing problems involving hard time windows and random service times. Ma et al. [10] used the ideal point method, a penalty function, and a Pareto optimal solution set to optimize vehicle delivery routes involving multiple objectives and multiple time windows, reducing the vehicle transportation costs, shortening the transportation distances, and reducing the number of transportation carriers. Avci et al. [11], Ma Y F et al. [12], and Guo Q et al. [13] combined tabu search and a differential algorithm based on 2-OPT ε—an approximate algorithm for performing constrained scheduling. HsiaoYH et al. [14] designed an adaptive biogeographical optimization algorithm for optimizing storage settings, involving multiple categories and temperatures in the logistics and distribution. Waisnawa I et al. [15] modeled a cold chain system for fruit and vegetable rich regions, analyzed the obstacles and problems in traditional fruit and vegetable distribution channels, and created new distribution networks. In response to the problem of location in the product supply chain, Xu et al. [16] adopted a pseudo reverse spider monkey optimization algorithm based on the Laplace distribution (LOBSMO) to solve multi-dimensional complex logistics distribution center location problems. Yan et al. [17] used triangular fuzzy numbers to represent the uncertainty of demand, weighted time, and cost and established a multi-level emergency logistics network. Han S et al. [18] proposed a mixed integer programming (MIP) algorithm for addressing the location of facilities in emergency logistics and meeting various material distribution needs in disasters.

The above research separates distribution vehicle routing optimization problems from distribution center location problems and can only obtain the results for a single type of problem, which cannot meet the requirements of overall optimization. In the process of distribution in cold chain logistics networks, considering collaborative optimization between these two aspects can maximize the benefits of the overall distribution system.

In recent years, domestic and foreign scholars have continuously deepened their research on agricultural product supply chains, comprehensively considering location and routing problems (LRPs), which are divided into single-layer location routing problems, double-layer location routing problems, and multi-layer location routing problems according to the node types. Yan et al. [19] studied a location routing problem with its time windows constrained by the volume of distribution centers, using a two-stage algorithm to determine the location of the distribution centers by introducing space–time distance through a k-means clustering algorithm and then using particle swarm optimization to obtain the optimal distribution path for each distribution center. Yang et al. [20] considered the complex dynamic requirements when conducting location route optimization and proposed a two-stage model. In the first stage, the initial distribution path was designed based on alternative front warehouse locations. After running for a while, the second stage entered the dynamic optimization stage, where real-time new demand points were optimized and calculated to adjust the distribution path. Guo et al. [21] designed a hybrid heuristic algorithm based on a savings algorithm and adaptive large neighborhood search for differences in the status of returned goods while allowing replenishment operations within the front warehouse and taking into account the location of the front warehouse. They proposed improved operators to effectively improve the algorithm speed, reduce the number of vehicles used and the logistics delivery costs.

To sum up, current domestic and foreign scholars often focus on the optimization of transportation routes for fresh product logistics network planning and ignore the impact of selecting appropriate locations to establish front warehouses on the entire logistics network distribution. Both location and route optimization are closer to the actual distribution mode, and considering only one aspect makes it difficult to achieve an optimal solution for overall logistics network planning. In studies that consider both the location of front warehouses and the transportation path, the problem of location is optimized, and the path optimization is performed after obtaining the determined location, without considering the interaction between them. This paper proposes to establish a mathematical model for solving the problem by using bi-level programming, where the upper layer is the problem of location and the lower layer is the problem of path optimization. The output results of the upper and lower layers interact with each other to minimize the total cost of the logistics system. The research aim of this article is to provide convenient path planning services for multi-point delivery of fresh products, shorten delivery time, ensure freshness and reduce the loss of fresh products. We take a fresh food company as an actual case to verify the effectiveness of the model and algorithm.

The paper is organized as follows. A description of the problem is given and modeling is constructed in Section 2. Section 3 is the algorithm design. Section 4 is the case verification and analysis. Finally, in Section 5, the conclusions are reported.

2. Problem Description and Model Construction

By studying location and path problems, a bi-level programming model is constructed to solve the location and path planning problems of a multi-center single vehicle type with time window constraints and an objective function of minimizing the total cost.

2.1. Problem Description

On the O2O e-commerce platform, fresh products account for over 50% of sales. Because of the time constraints in the delivery of fresh products, as well as customers’ requirements for quality and rapid delivery, it is common to establish a front warehouse to deliver to customers in demand within the coverage range. A single front warehouse covers an area within a radius of five kilometers. Compared to traditional distribution models, it directly delivers from the front warehouse to consumers, eliminating preservation and sales processes of intermediate wholesalers, supermarkets, and markets, achieving rapid delivery of fresh products and ensuring a positive shopping experience for customers. Currently, leading warehouses are mostly distributed in cities. By continuously establishing leading warehouses to cover a wider range of consumers, cold chain logistics costs can be effectively shared. A supply chain model of leading warehouses is shown in Figure 1. This article mainly studies how to maximize the benefits of logistics distribution through a two-level planning front warehouse location scheme and terminal path planning from the front warehouse to the point of the customer. Given the distribution vehicle information, volume, distribution and demand of customer service points, time windows, and other information, in addition to not exceeding the volume constraints of the front warehouse and load of the cold chain distribution vehicle, location selection is based on the distribution and demand of customer points. Distribution vehicles can service multiple customer points from the front warehouse, with each customer point being served by only one vehicle and only once.

2.2. Model Construction

To better solve the objectives of this article, the following assumptions are made:

(1)

Fresh products include fruits and vegetables, meat, and aquatic products. The degree of freshness is an important indicator that determines the value of these fresh products. Different types of products have different properties. This article only considers the category of fresh fruits and melons in the fresh category.

(2)

The distribution of customer points and demand are known and remain stable for a while without change.

(3)

The delivery vehicle maintains a constant speed in the delivery process.

(4)

Within the service time window, there are no new orders added for delivery vehicles.

The decision variables of the model are as follows.

$$X_{hi}=\left\{\begin{array}{l}1,\mathrm{front} \mathrm{warehouse}h\mathrm{servers} \mathrm{customer} \mathrm{point}i\\ 0,\mathrm{else}\end{array}\right\}$$

(1)

$$Y_h=\left\{\begin{array}{l}1,\mathrm{front} \mathrm{is} \mathrm{established} \mathrm{at} h\\ 0,\mathrm{else}\end{array}\right\}$$

(2)

$$x_{ij}^k=\left\{\begin{array}{l}1,\mathrm{vehicle} k \mathrm{tarvels} \mathrm{from} \mathrm{customer} \mathrm{point} i \mathrm{to} j\\ 0,\mathrm{else}\end{array}\right\}$$

(3)

$$x_{eh}^k=\left\{\begin{array}{l}1,\mathrm{vehicle} k \mathrm{tarvels} \mathrm{from} \mathrm{central} \mathrm{warehouse} e \mathrm{to} \mathrm{front} \mathrm{warehouse} h\\ 0,\mathrm{else}\end{array}\right\}$$

(4)

$$x_{k1}=\left\{\begin{array}{l}1,\mathrm{vehicle} k1 \mathrm{is} \mathrm{used}\\ 1,\mathrm{else}\end{array}\right\}$$

(5)

$$x_{k2}=\left\{\begin{array}{l}1,\mathrm{vehicle} k2 \mathrm{is} \mathrm{used}\\ 1,\mathrm{else}\end{array}\right\}$$

(6)

$$y_i^k=\left\{\begin{array}{l}1,\mathrm{customer} \mathrm{point} i \mathrm{is} \mathrm{serviced} \mathrm{by} \mathrm{vehicle} k\\ 0,\mathrm{else}\end{array}\right\}$$

(7)

Descriptions of the symbols are given in

Table 1. Descriptions of the symbols.

Symbols	Description of Symbol
$E$	The set of large warehouses in a city center, $\left\{e\right|e\in E\}$
$H$	The set of candidate points for the front warehouse, $\left\{h\right|h\in H\}$
$I$	The set of customer points for distribution, $\left\{i\right|i\in I\}$
$W=I\cup H$	The set of the customer point $i$ and the front warehouse $h$
$K$	The set of delivery vehicles, $\left\{k\right|k\in K\}$
$(i,j)$	The point from the distribution center to the front warehouse $i,j\in H,i\ne j$
$p_i$	The demand at the customer point (kg)
$O_h$	The capacity of the front warehouse (kg)
$C_h$	The unit fixed cost of the front warehouse $h$ (CNY)
$T_h$	The operating cost of the front warehouse $h$ (CNY)
$C_{eh}$	The unit distance transportation cost of trunk distribution vehicles from the central warehouse to the front warehouse (CNY/km)
$d_{eh}$	The distance from the central warehouse to the front warehouse of the main delivery vehicle
$d_{hi}$	The distance between the front warehouse and the customer point
$d_{hi\max }$	The farthest distance between customer points that regional vehicles can deliver to and the front warehouse
$t_{eh}$	The time from the central warehouse to the front warehouse for the main delivery vehicle
$t_i^k$	The service time of the vehicle $k$ at the customer point $i$
$t_i/t_j$	The time when the vehicle arrives at the customer point $i$ or $j$
$t_{di}$	The waiting time for the vehicle at the customer point $i$
$C_{k1}$	The fixed cost per unit distance of mainline vehicles (CNY/km)
$C_p$	The transportation cost per unit distance of branch line vehicles (CNY/km)
$C_{k2}$	The fixed cost per unit distance of branch line vehicles (CNY/km)
$C_{rk1}$	The unit time cooling cost of mainline vehicles (CNY/h)
${\theta }_1$	The unit loss cost from the large warehouse in the city center to the front warehouse
${\theta }_2$	The unit loss cost from the front warehouse to the customer point $i$
$F_k$	The vehicle capacity
${\omega }_t$	The cargo damage amount at time $t$ (kg)
$\delta$	The corruption rate of the goods
$g_{eh}$	The transportation volume from the large warehouse in the city center to the front warehouse h (kg)
$d_{ij}$	The distance from the customer point $i$ to the customer point $j$ (km)
$R_{ih}$	The distance between the front warehouse and the customer point
$\Delta t_i$	The service time of the vehicle at the customer point $i$
$E{T}_i$	The earliest expected arrival time at the customer point $i$
$L{T}_i$	The latest expected arrival time at the customer point $i$
${\beta }_1$	The penalty coefficient per unit time that the vehicle performs the service earlier than the earliest expected service start time of the customer point i
${\beta }_2$	The penalty coefficient per unit time that the vehicle performs the service later than the latest expected service start time of the customer point i

.

The bi-level programming mathematical model is constructed as follows.

Upper model. In the cold chain logistics network for distributing fresh agricultural products, the location and quantity of the front warehouses are determined based on the degree of customer dispersion and the distribution of fresh agricultural products from the primary network node to the secondary network node. A reasonable location can achieve full coverage of nearby demand points and greatly improve vehicle delivery efficiency. The upper objective function includes the cost of building a front warehouse, vehicle transportation costs, refrigeration costs, cargo damage costs and time window penalty costs. The objective is to minimize the total cost of the system. The mathematical model is shown in Equation (8).

$$\begin{array}{c}\min Z={\displaystyle \sum _{h\in H}{\displaystyle \sum _{i\in I}{Y}_h(C_h+T_h)+{\displaystyle \sum _{e\in E}{\displaystyle \sum _{h\in H}{C}_{eh}{d}_{eh}{Y}_h}}}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{e\in E}{C}_{rk1}}}{t}_{eh}{x}_k+\\ {\displaystyle \sum _{k\in K}{C}_{k1}}{x}_{eh}^k+{\theta }_1{\omega }_t{\displaystyle \sum _{k\in K}{\displaystyle \sum _{h\in H}{x}_{k1}{g}_{eh}(1-e^{-\delta t_{eh}})}}+{\theta }_2{\omega }_t{\displaystyle \sum _{h\in H}{\displaystyle \sum _{i\in I}{x}_{k2}{g}_{h\mathrm{i}}(1-e^{-\delta t_{hi}})}}\\ +{\displaystyle \sum _{k\in K}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in I}{C}_{k2}}}}{x}_{ij}^k+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{i,j\in H\cup I}{C}_p{d}_{ij}}}{x}_{ij}^k+{\displaystyle \sum _{i\in I}{\beta }_1\max \left\{E{T}_i-t_t,0\right\}}+{\displaystyle \sum _{i\in I}{\beta }_1\max \left\{t_t-L{T}_i,0\right\}}\end{array}$$

(8)

Constraints s.t.:

$$1\le {\displaystyle \sum _{h\in H}{Y}_h}$$

(9)

$${\displaystyle \sum _{h\in H}{R}_{ih}{Y}_h}\ge 1{\forall }_i\in I$$

(10)

$$Y_h=\{0,1\}\forall h\in H$$

(11)

Equation (9) indicates that the demand point is serviced by the front warehouse at least once. Equation (10) represents the construction of at least one front warehouse to cover demand points. Equation (11) represents the value constraint of the decision variable $Y_h$.

Lower layer planning. Based on the location scheme of the initial front warehouse and the customer point division of the upper layer model, decisions are made on the terminal distribution path for each customer point in the region. The distribution cost includes the terminal vehicle transportation cost, the fixed cost when the electric vehicle starts, and the time window penalty cost. The mathematical model is given in Equation (12) to minimize the total cost of vehicles delivered along the specified route.

$$\begin{array}{l}\min z={\displaystyle \sum _{k\in K}{\displaystyle \sum _{i,j\in H\cup I}{C}_p{d}_{ij}}}{x}_{ij}^k+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in I}{C}_{k2}}}}{x}_{ij}^k+{\theta }_2{\omega }_t{\displaystyle \sum _{h\in H}{\displaystyle \sum _{i\in I}{x}_{k2}{g}_{h\mathrm{i}}(1-e^{-\delta t_{hi}})}}\\ +{\displaystyle \sum _{i\in I}{\beta }_1\max \left\{E{T}_i-t_t,0\right\}}+{\displaystyle \sum _{i\in I}{\beta }_1\max \left\{t_t-L{T}_i,0\right\}}\end{array}$$

(12)

Constraints s.t.:

$$X_{hi}\le Y_h{\forall }_h\in H$$

(13)

$${\displaystyle \sum _{j\in V}{\displaystyle \sum _{k\in K}{x}_{ij}^k}}=1{\forall }_i\in I$$

(14)

$${\displaystyle \sum _{u\in I}{\displaystyle \sum _{i\in V}{p}_u}}{x}_{ij}^k\le Q^k{\forall }_k\in K$$

(15)

$${\displaystyle \sum _{i\in I}{X}_{hi}^k}={\displaystyle \sum _{j\in I}{X}_{jh}^k}{\forall }_k\in K, {\forall }_h\in H$$

(16)

$${\displaystyle \sum _{h\in H}{Y}_{hi}}=1{\forall }_i\in I$$

(17)

$$x_{h_1{h}_2}^k=0$$

(18)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in I}{x}_{ij}^k}}=|S_k|-1{\forall }_k\in K$$

(19)

$${\displaystyle \sum _{g\in V}{\mathrm{x}}_{i\mathrm{g}}^k}={\displaystyle \sum _{g\in V}{\mathrm{x}}_{gj}^k}{\forall }_k\in K, {\forall }_{i,j}\in V$$

(20)

$$0\le {\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{X}_{hi}}}{P}_i\le O_h{Y}_h$$

(21)

$$t_j^k={\displaystyle \sum _{k\in K}{x}_{ij}^k}(t_i^k+{\displaystyle \frac{d_{ij}}{v}}+\Delta t_i){\forall }_{i,j}\in V$$

(22)

$$t_{di}=\max (E{T}_i-t_i,0)\hspace{1em}\hspace{1em}{\forall }_i\in I$$

(23)

$$x_{ij}^k=0,1{\forall }_{i,\mathrm{j}}\in V,\hspace{1em}{\forall }_k\in K$$

(24)

$${\mathrm{t}}_i\ge E_{\mathrm{i}}\hspace{1em}{\forall }_i\in I$$

(25)

$${\mathrm{t}}_i\le L_{\mathrm{i}}\hspace{1em}{\forall }_i\in I$$

(26)

Equation (13) indicates that the front warehouse can only be selected to provide services to the customer points. Equation (14) indicates that each customer point has been served by only one vehicle once and will not repeat the service. Equation (15) indicates that the total amount of goods required by customers on each line where vehicle k is installed does not exceed the maximum carrying capacity of the vehicle. Equation (16) indicates that each vehicle departs from the front warehouse and returns to the front warehouse. Equation (17) indicates that the same customer point has only one front warehouse to provide services. Equation (18) indicates that there is no collaborative relationship between any two front warehouses. Equation (19) represents the elimination sub-loop. Equation (20) indicates that vehicles in each path are continuously transported to adjacent customer points. Equation (21) represents the volume constraint of the front warehouse. Equation (22) represents the time change of the vehicle between customer points. Equation (23) indicates the waiting time required by the vehicle to arrive before the customer’s time window. Equation (24) represents a value constraint on whether the decision variable uses the vehicle $k$ for distribution between two customer points. Equations (25) and (26) jointly ensure that the vehicle arrives within the customer’s time window.

In summary, the upper level model is the dominant layer in the two-layer model, which completes facility location decisions and determines the starting point of vehicles in the lower level path planning. The lower level model is a subordinate layer, which is the follower of the dominant layer. It allocates customer points according to time and distance; solves the single vehicle path problem corresponding to the front warehouse; determines vehicle path planning; affects distribution costs, fresh arrival time and the resulting cargo damage costs; and reacts to upper level decisions. Finally, the global model of master–slave optimization is constructed with the iterative goal of minimizing the total cost of the fresh logistics network system in the dominant layer.

3. Algorithm Design

The location routing problem (LRP) is a complex system problem composed of Facility Location Problem (FLP) and vehicle routing problem (VRP) and there is a dependency relationship between FLP and VRP [22]. The network level of the front warehouse location and path planning studied in this paper belongs to two-level LRP (2E-LRP), where the upper level location is used to solve a single objective multiple constraint problem and the lower level path planning is used to solve a multiple time window path problem. This paper uses k-means-based considerations to introduce temporal and spatial distance factors to cluster customer points and determine the initial set of candidate points for the front warehouse.

Cluster analysis is an unsupervised learning method that has the advantages of fast computation speed and low computational complexity, making it suitable for processing large-scale data. We divide the samples according to certain rules, gather highly correlated samples in the same cluster and divide irrelevant samples into different clusters to analyze the differences in the properties and correlations between samples.

This chapter conducts cluster analysis on customer points that require front services, providing a good set of candidate point locations for future front warehouse site selection.

3.1. Generation of Candidate Points for Front Warehouse

In the front warehouse mode, demand points are scattered everywhere, and the level of demand of the points is determined by forecasting based on the level of demand of the customer points over the years. Using the improved k-means algorithm, the traditional method of calculating the distance between the centroid and the sample is replaced by calculating the distance between the longitude and latitude of two points in the actual front warehouse problem, replacing the Euclidean distance in the traditional algorithm. Traditional site selection uses the maximum coverage model to establish a mathematical model. It is necessary to determine each location of alternative points in advance and then select a location from these alternative points. In this paper, the set coverage model is used to obtain the front warehouse reserve points through improved k-means clustering analysis. For this, a flowchart is shown in Figure 2. The selection of $K$ values in cluster analysis is evaluated and analyzed based on the clustering effect. The process of k-means clustering analysis is as follows.

Step 1: Import customer point information.

Step 2: Calculate the distance between the customer point and the cluster center.

Step 3: Assign customer points to the nearest cluster point.

Step 4: Update the list of customer points.

Step 5: If the list of customer points in the region has changed, return to step 3.

Step 6: Output cluster point center position.

Step 7: Update the center position of the clustering point until the customer point list remains unchanged.

Step 8: Output optimal cluster point center position.

The selection of $K$ values in k-means clustering analysis is evaluated and analyzed based on the clustering effect. We try to make the distance within each cluster as close as possible and the distance between each cluster as far as possible. When the value of k is different, we set K to 5, 6, 7… 10 and calculate the inter-class distance, intra-class distance, number of samples in the maximum cluster and number of samples in the minimum cluster for different clustering results under each K value. The k-means clustering parameter analysis is shown in

Table 2. K-means cluster parameter analysis.

$\mathit{K}$	5	6	7	8	9	10	11	12	13	14	15	16
Intra class distance	2.37	1.93	1.73	1.62	1.42	1.36	1.31	1.26	1.21	1.19	1.08	1.12
Inter class distance	0.7	0.72	0.75	0.73	0.78		0.7
The number of samples in the minimum cluster	11	11	11	4	7	5	5	11	1	1	3	1
The number of samples in the maximum cluster	24	25	18	25	17	17	17	25	13	12	11	13

.

From Table 2, it can be seen that the K value with the smallest intra-class distance, the largest inter-class distance and the smallest difference in sample size between the maximum and minimum clusters is 15.

3.2. Combining GA with PSO to Solve Double-Layer Model

The fusion of multiple heuristic algorithms can compensate for the shortcomings of a single algorithm. This paper combines the crossover and mutation operations in a genetic algorithm (GA) with particle swarm optimization (PSO) to improve the convergence speed of the GA and avoid defects of the PSO algorithm falling into local optima. By utilizing crossover and mutation operations in the GA, randomness and diversity are provided for particles in the PSO, while replacing the worst particles with the best chromosomes in offspring, resulting in continuous population evolution. The adaptive hybrid algorithm combining the PSO algorithm and GA (APSOGA) is used to reallocate the location information of backup points in the front warehouse, ultimately determining the facility location of the front warehouse and planning the end path from the front warehouse to the customer point, achieving joint optimization of the front warehouse location and path.

(1)

Generation of initial solution.

The PSO algorithm uses the floating point encoding method to generate two groups of data according to the number of demand points. One group of data is used to determine the corresponding distribution point of the demand point, and the other group of data is used to determine the distribution order of each demand point, to obtain the initial particle position and speed and to calculate the fitness of all particles. This is denoted as $X_i=\left\{{\mathrm{x}}_1,x_2,x_3,\dots x_{n-1},x_n\right\}(i\in \left[1,n\right])$, where $X_n$ is the particle position, $V_i$ is the particle velocity and $n$ is the number of particles.

(2)

Update particle position.

In the initial solution, the local optimal particle position (Pbest) and the global optimal particle position (Gbest) of all the current particles are calculated, corresponding to different arrangement orders of customer points. Particle $P_i$ is the local optimal particle with the smallest fitness among the current population particles, that is

$$F(P_i)={\min }_{i=1}^n\left\{F(Pbes{t}_i)\right\},i\in n$$

(27)

$F(P_i)$ iteratively updates the position of the particle, and its position and velocity are calculated according to Formulas (28) and (29)

$$X_i^{k+1}=X_i^k+V_i^k$$

(28)

$$V_i^{k+1}=\mathrm{\omega }{V}_i^k+C_1\times {\mathrm{rand}}_{(1)}\times ({\mathrm{X}}_{\mathit{Pbest}}^k-X_i^k)+C_2\times {\mathrm{rand}}_{(2)}\times ({\mathrm{X}}_{\mathit{Gbest}}^k-X_i^k)$$

(29)

where $V_i^k$ represents the rate of particle $k$, ${\mathrm{rand}}_{(1)}$ and ${\mathrm{rand}}_{(2)}$ represent random numbers within the range of 0 to 1, and $C_1$ and $C_2$ represent learning factor constants. $\omega$ represents the inertia weight, reflecting the ability of particles to inherit previous velocities. After updating the fitness value of the particle, we can recalculate the fitness function value. If the current particle fitness function value is better than $Pbes{t}_i$, we can replace it and use the replacement Formula (30) to update the position of particle.

$$F(Pbes{t}_i)=\left\{\begin{array}{l}F(P_i),\begin{array}{cc}& \end{array}F(P_i)<F(Pbes{t}_i)\\ F(Pbes{t}_i),\begin{array}{ll}& others\end{array}\end{array}\right\}$$

(30)

(3)

Selection and crossover of GA

We select a particle (Gbest) with the best fitness on a chromosome and a random particle in the existing population to perform a cross operation, generate a random intersection, generate two new offspring particles, calculate the fitness values of the two offspring particles and obtain two new paths.

(4)

Mutation of GA

For particles in the offspring chromosome, to avoid high similarity with the parent particle, two solutions of an offspring particle are randomly selected for exchange to obtain a new mutated offspring.

(5)

Calculation of fitness value

The optimal solution is found by calculating the fitness value of the particle. For the objective function, minimizing the cost of the cold chain logistics network is more in line with the expectations of the enterprise, so obtaining solutions with smaller objective function values will be prioritized. The function fitness $F=\min Z$.

The overall algorithm steps are as follows (as shown in Figure 3).

Step 1: Read the data and set the number of iterations to 1.

Step 2: Encode to generate an initial population and transfer the initial location plan to the lower level model.

Step 3: Solve the lower level model, allocate customer points based on the spatio-temporal distance and solve the vehicle routing problem for a single parking lot corresponding to the front warehouse.

Step 4: Calculate the fitness value of the upper level model algorithm based on customer point allocation and corresponding vehicle path results.

Step 5: Conduct selection, crossover and mutation of the upper level algorithm (location selection based on 0–1 coding), using the tournament selection method to perform single-point crossover and two-point mutation to obtain new offspring.

Step 6: Repeat steps 3 to 5, solve the lower level model in sequence and calculate the fitness value of the function.

Step 7: Increase the number of iterations by 1 and repeat the above steps until the algorithm reaches the termination condition. Output the optimal solution with the lowest total cost.

4. Example Analysis

4.1. Data Collection

This paper verifies the effectiveness of the mathematical model through example analysis. Taking Dingdong Maicai, Shahe district, as an example, ShaheDistrict, a city was selected as a research area. The longitude and latitude coordinates were picked up through Baidu map coordinate points, and 68 customer points in the residential area were selected. The refrigerated delivery truck departs from the city center warehouse to deliver goods to various front warehouses. According to the needs and locations of customer points, electric vehicles depart from corresponding front warehouses for delivery services. The basic information of each customer point includes community name, latitude and longitude coordinates, demand for customer points, time window restrictions and service time. The starting and ending service time of the front warehouse is [540,1260], starting from zero and the delivery time should be within the service time window as much as possible. When the service time window is exceeded, penalty costs will be incurred. The supply of fresh products rather than a single category of fruits, meat and seafood is believed to have little change in consumer demand over a while. The basic information of some customer points is shown in

Table 3. The basic information of customer point.

No.	Name	Latitude	Longitude	Demand	Start of Time Window/min	End of Time Window/min	Service Time/min
1	N01	22.559823	114.227134	1520.04	533	593	18
2	N02	22.560746	114.228704	501.9	729	789	10
3	N03	22.563047	114.228713	669.2	894	954	19
4	N04	22.563074	114.229686	1720.8	991	1051	20
5	N05	22.559611	114.229892	573.6	789	858	20
…	…	…	…	…	…	…	…
64	N64	22.601627	114.307327	348.94	1077	1137	10
65	N64	22.60637	114.312096	2652.9	627	687	14
66	N66	22.606548	114.31533	1467.46	704	764	10
67	N67	22.606693	114.316908	2055.4	551	611	19
68	N68	22.604831	114.321011	2260.94	930	990	12

.

4.2. Site Selection Planning for Front Warehouses

Using the K-means algorithm to perform cluster analysis on 68 customer points, the cluster center serves as an alternative point for establishing the front warehouse and the initial plan for selecting the location of the front warehouse is obtained. The cluster analysis results are shown in Figure 4. In the figure, the horizontal axis represents longitude and the vertical axis represents latitude.

Eight alternative points are passed into the lower level model as the initial solution, and after obtaining the allocation of customer points and vehicle paths, they are returned to the upper level model. According to the capacity and time window constraints of the front warehouse, we adjust the location plan of the front warehouse and determine the location of the front warehouse from eight alternative front warehouses, as shown in

Table 4. Position of front warehouses.

No.	Latitude	Longitude	Choose or Not
1	22.5938	114.2725	Yes
2	22.5556	114.2401	Yes
3	22.6057	114.3118	Yes
4	22.5640	114.2508	No
5	22.5658	114.2464	Yes
6	22.5618	114.2339	No
7	22.5949	114.2536	Yes
8	22.5605	114.2320	Yes

.

4.3. Path Planning of Front Warehouse

The path planning of the front warehouse includes mainline distribution and branch distribution. Main line delivery is the allocation of refrigerated trucks from the large warehouse in the city center to supply fresh products to the front warehouse, and each front warehouse is served by a refrigerated truck, as shown in Figure 5.

Two electric vehicles can be used in each front warehouse area to complete delivery services. The electric vehicle starts from the front warehouse, selects the appropriate number of vehicles to load goods based on the corresponding customer demand and electric vehicle capacity in the area and delivers goods to the customer points along the optimal path in the sequence. After completing the delivery, it returns to the front warehouse, forming a closed-loop route. We design a delivery path to minimize the delivery cost at a lower level and solve it using the APSOGA fusion algorithm. The algorithm parameters are $c_1=0.5$ and $c_2=0.5$, the maximum number of iterations is $\lambda =100$, and other parameters are fresh product unit mass loss ${\omega }_t=0.02$, fresh product corruption rate $\delta =5\%$ and mainline unit time cooling cost $C_{rk1}=1.2$. After calculation, the mileage traveled by all electric vehicles for distribution is 4895.211 km, and the distribution cost is CNY 158,536.6. The distribution mileage and cost of electric vehicles in each region are shown in

Table 5. Distribution routes and electric vehicle mileage within each region.

No.	Distribution Routes	Transportation Mileage/km	Delivery Cost/CNY
1	7-28-29-35-18-32-25-7 7-27-24-26-23-37-7	288.518 322.272	4327.764 4834.075
2	2-21-20-19-9-16-15-13-12-10-8-2 2-22-14-11-17-2	459.597 215.979	6893.953 3239.684
3	6-36-44-49-31-40-48-6 6-33-45-47-30-34-38-6	344.550 346.577	51,682.456 51,986.601
4	4-55-54-51-52-53-52-43-42-41-4 4-46-56-39-54-4	590.014 429.683	8850.211 6445.242
5	5-67-65-62-61-66-58-5 5-63-60-59-57-68-64-5	624.889 666.192	9373.334 9992.882
6	3-69-70-72-74-3 3-71-73-75-3	287.399 319.541	431.098 479.311

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4.4. Algorithm Optimization Comparison

To verify the effectiveness of the APSOGA fusion intelligent optimization algorithm, a two-stage optimization was performed on the above example. The number of front warehouses and alternative points are determined through K-means clustering analysis, and based on this, the location and end path optimization of front warehouses are carried out. The final iteration goal is the total upper cost. A comparison of the solution results between the APSOGA fusion algorithm and the traditional single intelligent algorithm is shown in Figure 6. In Figure 6, the horizontal axis represents the number of iterations, and the vertical axis represents the objective function value.

The optimal solution curve in Figure 6 represents the convergence of different intelligent optimization algorithms for the same example. The total objective function values calculated by a single intelligent optimization algorithm, PSO and GA, are CNY 1.0614 million and CNY 974,700, respectively, while the total objective function obtained by the APSOGA fusion algorithm is CNY 925,600. Compared to a single algorithm, the total cost decreased by CNY 134,900 and CNY 48,200, respectively. Therefore, compared to solving with a single heuristic algorithm, the adaptive algorithm has a better solving effect. Compared with the other five heuristic algorithms, APSOGA has the lowest total cost, reducing the total costs by 25.75% and 25.74%, respectively, compared to using the adaptive particle swarm optimization algorithm (APSO) and the adaptive genetic algorithm (AGA), and reducing the total cost by 25.84% compared to using the GA. This confirms the effectiveness of the APSOGA in solving this type of LRP. The APSOGA has a faster solving speed than the traditional algorithms PSO and GA, as shown in

Table 6. Performance comparison of algorithms.

Algorithms	PSO	GA	APSOGA
Run time/S	8.298	8.434	8.154

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5. Conclusions

This article proposes an APSOGA fusion algorithm based on a bi-level programming model to solve the problem of front warehouse location and path planning in fresh agricultural product logistics networks. The study investigates the impact of time window constraints and upper and lower level mutual decision-making on front warehouse location path optimization. A K-means clustering algorithm was designed for the coordinates of customer points within the region, using the distance between longitude and latitude coordinates as a similarity measure to find various centroids in the coordinates, namely the front warehouse backup points mentioned earlier. An APSOGA fusion algorithm is used to solve the double-layer model. The upper layer considers whether to establish a front warehouse at that point, which affects the allocation results of the lower layer’s customer points. The lower layer model considers the vehicle routing problem for multiple parking lots with time windows and returns the results to the upper layer model, iterating the objective function of the upper model as the overall objective, while meeting the needs of various customer points and the maximum capacity of the front warehouse to minimize the total logistics cost of the objective function, which is CNY 925,600. The APSOGA fusion algorithm reduces the total cost of the logistics network by 14.57% compared to the traditional single algorithm PSO solution and by 5.21% compared to the single GA. This proves the effectiveness of the APSOGA fusion algorithm in solving the location and path planning problems of fresh food companies. This provides a reference for solving the problem of fresh logistics network distribution in the front warehouse mode.