With energy and environmental issues becoming increasingly prominent, electric vehicles (EVs) have become the important transportation means in the logistics distribution. In the real-world urban road network, there often exist multiple paths between any two locations (depot, customer, and charging station) since the time-dependent travel times. That is, the travel speed of an EV on each path may be different during different time periods, and thus, this paper explicitly considers path selection between two locations in the time-dependent electric vehicle routing problem with time windows, denoted as path flexibility. Therefore, the integrated decision-making should include not only the routing plan but also the path selection, and the interested problem of this paper is a time-dependent electric vehicle routing problem with time windows and path flexibility (TDEVRP-PF). In order to determine the optimal path between any two locations, an optimization model is established with the goal of minimizing the distance and the battery energy consumption associated with travel speed and cargo load. On the basis of the optimal path model, a 0-1 mixed-integer programming model is then formulated to minimize the total travel distance. Hereinafter, an improved version of the variable neighborhood search (VNS) algorithm is utilized to solve the proposed models, in which multithreading technique is adopted to improve the solution efficiency significantly. Ultimately, several numerical experiments are carried out to test the performance of VNS with a view to the conclusion that the improved VNS is effective in finding high-quality distribution schemes consisted of the distribution routes, traveling paths, and charging plans, which are of practical significance to select and arrange EVs for logistics enterprises.
In the past two decades, with the rapid development of economy and e-commerce in China, the annual parcel deliveries have been growing continuously, bringing opportunities and challenges for the development of the logistics industry. For example, the annual business volume of express delivery service enterprises reached 50.71 billion pieces in 2018 with a year-on-year growth of 26.6% in China [
Logistics vehicles are characterized by high energy consumption, emissions, and pollution. The Organization for Economic Cooperation and Development points out that the freight traffic in major cities of developed countries accounts for 10%–15% of the total urban traffic volume, while the environmental pollution caused by freight vehicles accounts for 40%–60% of the total urban traffic volume [
With increasingly prominent energy and environmental problems, electric vehicles (EVs) stand out by virtue of their characteristics of environmental protection, energy-saving, and low cost, and they gradually become an important transportation means in the logistics industry. Hence, electric vehicle routing problem (EVRP) has become a popular research field, which attracts the attention of many scholars. Nevertheless, EVs always have some limitations such as limited battery capacity and long recharging times. With the rapid development of advanced technology, the shorter charging time and the improvement of battery endurance bring opportunities for the wide application of electric vehicles. Furthermore, this paper is in line with the goal of the Ministry of Transport of China to achieve the successful completion of the task of tackling the key problems of transportation pollution prevention and control in 2020, and it also has responded to the call of promoting scientific and technological innovation in transportation and the development of intelligent transportation, as well as intensive logistics.
To the best of our knowledge, Dantzig and Ramser [
Vehicle routing problem with time window (VRPTW) is a classical VRP problem with the time window constraint. Solomon [
Since the 21st century, the problem of environmental pollution and energy shortage has attracted people’s attention. When people began to look for more energy-saving and clean vehicles, green vehicles emerged at the historic moment. For the past few years, with people’s further attention, the continuous innovation of electric vehicle technology has made it become popular in recent years and become a mainstream of green vehicles. Consequently, a majority of large logistics enterprises have also begun to develop electric vehicle distribution. Therefore, EVRP, which is the combination of EVs and VRP, become a new research direction.
Lin et al. [
In the study of the EVRP, due to the great influence of congestion and service demand problems, there are a stream of studies on the electric vehicle problem with time widow, which we refer to as E-VRPTW. Penna et al. [
There is also a great deal of research studies on the charging of EVs. Schneider et al. [
In this paper, we are committed to planning reasonable distribution routes to save logistics costs and improve economic efficiency for enterprises. Concretely, we consider the EVRP with time windows and flexible paths. Besides, vehicles in this research are assumed to be fully charged in charging stations.
The main contributions of the paper can be summarized as follows: This paper first considers the path flexibility between any two locations in the EVRP. In the process of distribution, the path selection mainly depends on the departure time from the locations and the congestion levels in the current road network. In this case, the electric vehicle may travel along different paths since the time-dependent speed in the real-world road condition, which is defined as path flexibility (PF) in this paper. In addition, with the consideration of path flexibility, it is able to save battery energy and thus plays an important role in cost savings. This paper provides two mathematical models to formulate the time-dependent electric vehicle routing problem with time windows and path flexibility (TDEVRPTW-PF). The first model is used to select the optimal path between any two locations including consumers and charging stations to achieve the shortest path and the least energy consumption. The second model gives the general travel route of multiple EVs’ distribution, taking path flexibility and time windows into consideration. The total goal is to minimize the total distance during the travel, to save costs, and to increase profits for enterprises. This paper proposes an effective VNS algorithm to solve TDEVRPTW-PF. By inserting virtual distribution centers into the traveling sequence to represent the routes of multiple vehicles, the results can be seen more intuitively. In addition, considering the time and space complexity of the algorithm, the traveling sequence and charging record are separated, and multithreading is adopted, which improves the efficiency significantly. When testing the charging conditions, two locations in advance are used to determine the charging conditions in order to ensure that the vehicle has enough electricity to continue serving and to provide protection for the vehicle’s endurance.
The remainder of this paper is organized as follows: Section
This paper focuses on the routing optimization of EVs with time windows and time-dependent travel times, and it also involves charging and battery energy consumption. Before formulating the multiple EVs’ routing optimization model with time windows and path flexibility, we first make the following assumptions:
The electric vehicles are assumed to be a homogeneous fleet. All vehicles hence have the same load and volume. Furthermore, all EVs are associated with the same maximum battery capacity and recharging rate.
The EV is always fully charged, and it can be recharged multiple times. Besides, a charging station can hold many EVs simultaneously, and we do not need to consider the queue time for charging.
According to the actual distribution situations of logistics enterprises, we divide the working time into several time periods, and therefore, the speed on each path between any two locations is constant in a certain time period, so it is expressed in terms of average speed.
Based on the above assumptions, the problem studied in this paper can be formulated as a fleet of EVs fully charged, departing from a distribution center, can serve a series of customers with different demands. Each vehicle delivers its goods to the customers within predetermined time windows. Moreover, in order to prevent an EV that needs to charge from not reaching the charging stations, the energy consumption analysis should be carried out in advance.
Next, we will give an example of how to analyze the energy consumption so as to ensure the vehicle that needs to charge can reach the charging station. Figure
For the sake of convenience, relevant notations used in the models are listed in Table
An illustration for analyzing the energy consumption and recharging process.
Relevant notations.
Notation | Definition |
---|---|
Road network graph | |
Distribution center that EVs depart from | |
Distribution center that EVs return to | |
Set of customer nodes | |
Set of charging stations | |
Set of locations, i.e., | |
Set of arcs connecting any two locations, | |
Set of paths connecting arc | |
A path in the set | |
Number of vehicles | |
Battery capacity of a vehicle | |
Maximum load of a vehicle | |
Maximum volume of a vehicle | |
Time window of customer node | |
The time when the vehicle arrives at | |
Service time of customer node | |
A big enough travel time | |
The demand weight of customer node | |
The demand volume of customer node | |
The remaining battery electricity at node | |
The remaining weight of cargoes at node | |
The weight of cargoes when the vehicle departs from the distribution center | |
The remaining volume of cargoes at node | |
The volume of cargoes when the vehicle departs from the distribution center | |
Number of time periods with constant speed | |
The charging rate | |
Distance of path | |
Travel speed in time period | |
Travel time on path | |
The total travel time from node | |
Energy consumption from node | |
Energy consumption from node | |
Energy consumption on the arc |
Referring to Zheng [
Analysis for energy consumption of electric vehicles.
Based on the above discussion, the energy consumed by the driving style and vehicle accessories is difficult to calculate. Therefore, the energy consumption of working by overcoming resistance is firstly calculated, and then the proportion of this energy consumption to total energy consumption is given based on experience and prediction. Finally, the total energy consumption can be obtained according to this proportion.
The calculated process of energy consumption from node
Hereinafter, the notions in equation (
Notations used in formula (
Notations | Definition |
---|---|
Proportion of resistance energy consumption to total energy consumption | |
Acceleration of gravity | |
Tire rolling resistance coefficient | |
Air resistance coefficient | |
Air density | |
Frontal area against wind |
For the charging process, we assume that the charging time is constant, and the EV is always fully charged. Besides, it can be recharged multiple times. The dynamic performance of electric vehicles is reflected on the influence of dynamic speed and different path choices on the distribution process.
In order to determine the optimal path between any two locations including customers, charging stations, and distribution center, we will establish a 0-1 mixed-integer programming model to minimize the energy consumption and the distance.
Firstly, the decision variables can be defined as
Since a vehicle traveling from a location to another one may cover multiple time periods, its total time is the sum of travel time in these time periods. For example, we assume that the distance between two locations is 10 km, and the duration of the time period is one hour, i.e., 60 min, and thus, the step speed function is shown in Figure
Step speed function.
Correspondingly, the total energy consumption of an EV is the sum of the consumption in all of the time periods. Thus, we can get the equation as
To select the optimal path on arc (
The objective function (
The speeds on different paths between any two locations are not always the same during different time periods, which leads to different energy consumption. We can get the optimal path with least energy consumption and shortest distance between any two locations at any departure time from the above model. Accordingly, we can use the variables on the optimal path between
On the basis of the optimal path selection model, the time-dependent electric vehicle routing problem with time windows and path flexibility (TDEVRP-PF) can be formulated as
The objective function (
The proposed model is an NP-hard problem, and it involves many factors, such as multivehicle simultaneous delivery, hard time window, path flexibility, and charging on the route. Through comprehensively considering both advantages and disadvantages of various heuristic algorithms (tabu search algorithm, genetic algorithm, simulated annealing algorithm, variable neighborhood search algorithm, etc.), we finally choose the variable neighborhood search (VNS) algorithm which is more inclusive to solve the proposed model. Based on the idea of the VNS algorithm, we improve and perfect it according to the actual needs, so as to solve the proposed model effectively and efficiently. The objective of the algorithm is to minimize the total traveling distance while satisfying the constraints.
The rest of this section can be divided into four parts. The first part shows a way to solve the PF problem. According to the idea of solving the multiobjective decision model, we collect the data of the dynamic road network and select the optimal path between any two locations to generate the database for TDEVRPTW-PF. In the second part, we present a form of the initial solution for simultaneous distribution of multiple vehicles, which is an important part of the algorithm and a prerequisite for data experiments. The third part is to analyze the characteristics of the target problem (that is, the constraints in the presented model) and to filter the solutions. Finally, the whole VNS framework and the termination condition of the algorithm are introduced in the fourth part.
As shown in Figure
Path selection between two locations in different periods.
In order to select the dynamic path, we construct an optimal path selection model. In the algorithm for solving it, we store the average speed of multiple optional paths between any two locations corresponding to different time periods in the database. In other words, the path flexibility (PF) problem influenced by the time factor has been commendably solved through the optimal path selection model.
In the conventional VRP, vehicle routes are usually represented in the form of customer point sequences. Nevertheless, the solution form of traditional route planning cannot tackle the multiple electric vehicles’ problem appropriately for the VNS algorithm. In TDEVRPTW-PF, we aim at proposing a route planning scheme for multiple electric vehicles to distribute simultaneously, which can both meet the constraints and give a charging scheme. Therefore, we propose a concept of virtual distribution center and fit the drop-shaped distribution route of multiple vehicles into a circular distribution path, which is explained concretely in Figure
Generation of virtual distribution centers.
Representation of traditional VRP solutions is given in Figure
Representation of traditional solutions.
Representation of the solution after generating virtual distribution centers is given in Figure
Representation of the solution after the virtual distribution center is generated.
The sites in this study that constitute the distribution route include customer sites and charging station sites. The traditional representation of the solution to solve the VRP (as shown in Figure
When generating the initial traveling sequence, the total number of virtual distribution centers in the sequence is equal to the number of vehicles minus 1 (see Figure
In order to obtain an initial feasible solution, a feasibility test is carried out on the generated initial driving route (see Section
The process of generating the initial solution can be introduced as follows. Firstly, the customers are randomly arranged to 5-2-4-6-9-7-8-10-3-1. Additionally, at both ends, two 0’s are inserted to represent the actual distribution center. Then, according to the principle of nonadjacency, two 0’s which represent virtual distribution centers are inserted into the sequence separately. Finally, the initial solution includes charging record which is obtained through feasibility examination.
The initial solution consists of two parts: one is the traveling route and the other is charging record. Now that 0 represents the virtual distribution center, the sequence between each two 0’s denotes a traveling route of a single vehicle. In this case, the sequence (0-5-2-4-0-6-9-7-0-8-10-3-1-0) means the first vehicle A serves customers 5, 2, and 4 in sequence. Likewise, vehicle B serves customers 6, 9, and 7, and vehicle C serves customers 8, 10, 3, and 1 in order. Furthermore, according to the charging history of vehicles, after serving customer 9, vehicle B has to go to charging station 12 to recharge, and vehicle C ought to get energy in charging station 15 after serving customer 3.
In order to examine the feasibility of the generated solution, a checking procedure is applied to distinguish whether the initial sequence is feasible or not, which can be divided into the following four parts (see the left part of Figure
A flowchart of VNS to solve TDEVRPTW-PF.
For an initial sequence, if two 0’s are adjacent to each other in a traveling route sequence, it indicates that the actual number of vehicles is reduced, and the number of vehicles in such an initial sequence does not satisfy the constraints, namely, it is an unfeasible sequence.
The total weight of freight to be delivered at all customer sites between the two adjacent 0’s of a traveling route sequences is compared with the maximum load of the vehicle. If the load is exceeding the maximum load, the sequence is not feasible. The measurement of volume is similar to that of load.
We refer to Zheng [
As shown in Figure According to the energy consumption during traveling to determine whether the vehicle needs to charge. As is shown in Figure Determine whether the route sequence is feasible according to the current electric quantity during traveling. When the residual battery capacity of the current vehicle is unable to reach the charge station nearest to the current point, that is, when SOC < E1, it is judged as an unfeasible solution.
Charging criterion.
We update the current time every time when vehicles reach a customer site through the time advance mechanism. Comparing the current time with the time window of the customer, if it exceeds the time window range, it is determined as an unfeasible sequence.
Mladenovic and Hansen [
The traditional VNS algorithm has a considerable adaptability to the TSP and other single-vehicle problems, but it is unable to solve multivehicle problems. Therefore, by generating virtual distribution centers, we propose a solution suitable for multivehicle distribution, which has been described in detail in Section
The neighborhood structure is crucial in the VNS algorithm. The purpose of the neighborhood structure is to constantly transform the current feasible solution so that a large number of other solutions can be derived from the feasible solution in the neighborhood. In the neighborhood structure, the core of the algorithm to transform the feasible solution is the neighborhood operations. Thus, in our algorithm structure, two complementary and effective neighborhood operations are selected after screening so that the operations can be fully traversed in the neighborhood.
Neighborhood operation 1.
Neighborhood operation 2.
The following describes the procedure of the neighborhood structure algorithm by taking neighborhood structure 1 as an example: Step 1: enter a feasible solution Step 2: the position in the sequence in which the initial operation is performed is changed to Step 3: set the maximum number Step 4: Step 5: (1) if the position of (2) If the position of (3) Perform neighborhood operation 1 on the two positions Step 6: determine whether Step 7: determine whether Step 8:
VND is a major component of the VNS algorithm. Its main function is to perform neighborhood operation on the feasible solutions of the input, generate the approximate optimal solutions within the range of multiple neighborhoods of the current solution by turn, and select the new approximate optimal solution through comparison. Two types of neighborhood structures and the corresponding operations have been introduced in Section
The specific operation steps of VND are as follows: Step 1: enter a feasible solution and assign it to the current solution. Step 2: generate neighborhood 1 by performing neighborhood structure 1 on the current solution, and screen out the local approximate optimal solution in neighborhood 1. Step 3: compare the local approximate optimal solution in neighborhood 1 with the objective function value of the current solution. If the objective function of the local approximate optimal solution is better, replace the current solution with the local approximate optimal solution of neighborhood 1 and return to Step 2; otherwise, proceed to Step 4. Step 4: generate neighborhood 2 by performing neighborhood structure 2 on the current solution, and screen out the local approximate optimal solution in neighborhood 2. Step 5: compare the local approximate optimal solution in neighborhood 2 with the objective function value of the current solution. If the objective function of the local approximate optimal solution value is better, replace the current solution with the local approximate optimal solution of neighborhood 2, and return to Step 2; otherwise, perform Step 6. Step 6: current approximate optimal solution = current solution. Output the current solution.
After VND operation is performed on the current approximate optimal solution, the VND output needs to be compared with the global approximate optimal solution. If the value of the objective function of the current approximate optimal solution in many iterations is greater than that of the global approximate optimal solution, we may get in the situation of the local approximate optimal solution. To skip from local optimization, large transformations are required for the current approximate optimal solution, which is the main significance of the shaking procedure.
For a simple example, as shown in Figure
An instance for the shaking procedure.
The components of the VNS algorithm have been briefly introduced above. Figure Step 1: generate the initial route sequence according to the principle. Step 2: check the sum of vehicles; if the conditions are mismatched, return to Step 1. Step 3: check whether the maximum load and volume of the vehicle meet the requirements. If not, return to Step 1. Step 4: generate charging history and check the electric quantity. If the condition is not met, return to Step 1. Step 5: check the time window. If the condition is not met, return to Step 1; if met, take it as a feasible solution. Step 6: set an iteration counter Step 7: Step 8: determine whether the new feasible solution is better than the current feasible solution. If better, current feasible solution = new feasible solution. Step 9: perform shaking to avoid getting in the local approximate optimal solution. Step 10: if
In order to improve the optimization degree and stability of the global approximate optimal solution obtained by the VNS algorithm, we generate multiple initial sequences according to the above principles and operate VNS at the same time. By comparing the global approximate optimal solution generated by each group, the best one is taken as the final optimal solution. For the purpose of improving the time efficiency of the whole process, we use multithreading as the main framework of double-optimization when programming and achieved a good result in subsequent experiments and data analysis.
In this section, a large number of numerical experiments are conducted to evaluate the performance of our modified VNS algorithm. In the first part, we present the development tools, the experimental environment, and the parameter setting of the numerical experiments. In the second part, a set of solutions are generated under the initial conditions. Then, we evaluate the performance of the solutions and explain the traveling routes in detail. In addition, we also analyze the efficiency of the improved algorithm and the degree of solution optimization. Finally, the sensitivity analyses of battery capacity, load, volume, and the number of EVs are demonstrated in the third part.
Our modified VNS is implemented as a code in C++ in Microsoft Visual Studio 2017 software on the Windows 10.0 platform of a personal computer with Intel(R) Core (TM)i7-6500U CPU @ 2.50 GHz and 4 gigabyte memory.
Our data are randomly generated to simulate real distribution scenarios. We randomly select 30 customers, 12 charging stations, and 5 EVs as our experimental instances. Table
Basic parameters on TDEVRPTW-PF instances.
Customers | Charging stations | Electric vehicles | Work time |
---|---|---|---|
30 | 12 | 5 | 6:00–22:00 |
Parameters of EV energy consumption analysis.
Tire resistance coefficient | Energy conversion efficiency | Air drag coefficient | Air density | Gravity coefficient |
---|---|---|---|---|
0.02 | 12 | 0.6 | 1.29 | 9.8 |
Parameters of EVs.
Maximum load | Maximum volume | Battery capacity | Maximum speed of EVs | Frontal area | Weight |
---|---|---|---|---|---|
1500 kg | 22 | 200 kwh | 100 km/h | 5 | 10,000 kg |
Parameters of customers.
Customer | Time window | Weight (kg) | Volume (m3) |
---|---|---|---|
1 | 6:00–22:00 | 80 | 1.7 |
The geographical distribution of customers, charge stations, and the distribution center is shown in Figure
The distribution of the customers and the charging station in Zhengzhou.
The basic conditions for conducting numerical experiments are given in the previous part. In this part, the speed is made to fluctuate with time and path, which is limited to the range from 50 km/h to 60 km/h. After that, the VNS algorithm program is run with the maximum number of iterations set, 300 times, and the approximate optimal solution is obtained in Table
Results of the algorithm (total distance = 1512.4 km).
EVs | Traveling sequence | Remaining electricity | Charging record |
---|---|---|---|
1 | 0 27 7 10 25 23 24 22 0 | 200 150 140 125 102 94 81 50 200 | 14 ⟶ 34 |
2 | 0 8 6 12 19 11 2 14 16 26 0 | 200 146 127 98 84 74 54 32 174 128 200 | |
3 | 0 18 20 15 17 1 3 21 5 0 | 200 156 140 125 113 103 81 73 57 200 | |
4 | 0 30 0 | 200 195 200 | |
5 | 0 13 9 28 4 29 0 | 200 166 140 116 110 96 200 |
We use the generated TDEVRPTW-PF test instances to analyze the performance of our VNS algorithm. Figure
Distribution route.
However, we also find some seemingly unreasonable routes at the same time, such as EV 5 chooses to deliver from customer 13 to customer 9 rather than to customer 14. Instead, EV 2 is needed to return from customer 2 and then head to customer 14. By exploring the reasons, we arrange customer 14 to be behind customer 13. Then, after checking the new sequences, it is found that EV 2 reaches customer 16 at 11:54 which is not in its service time window (12:00–22:00). Similarly, we try to arrange customer 16 to be behind customer 18 and customer 26 to be behind customer 29, however, all of which do not accord with its service time window when arriving at customer 16, and the underlying reason is that all vehicles are set to start to deliver at the same time as stipulated in our preconditions.
On the basis of a series of experiments, we draw a conclusion that the factors affecting the performance of the solutions are mainly divided into two categories. One is the constraint conditions, which include the hard time windows of the customers, the vehicle load constraint, and the vehicle volume constraint. The other is the VNS algorithm itself, which is based on the maximum number of iterations of the algorithm and the maximum number of iterations in the neighborhood. The partial irrationality of the solution affected by the factor of the first type is immutable, but that influenced by the second type can be changed, which will be explained in detail in the next part.
It can be seen from experiments on TDEVRPTW-PF instances that our improved VNS algorithm has been able to generate the solution we want to solve TDEVRPTW-PF. However, the problem we need to solve is to make the process more efficient and to optimize the distribution scheme. By adjusting the important parameters of the improved algorithm, the further experiments are shown in Table
Experiment results.
TN | MIA | MIN | OFV (km) | RT (min) | AAOS (km) | RAOS (km) |
---|---|---|---|---|---|---|
1 | 300 | 30 | 1672 | 2.35 | 1644.6 | 41.8 |
60 | 1630.2 | 3.86 | ||||
90 | 1631.6 | 4.85 | ||||
200 | 30 | 1608.7 | 2.1 | 1635.00 | 41.5 | |
60 | 1646.1 | 2.21 | ||||
90 | 1650.2 | 3.68 | ||||
100 | 30 | 1740.8 | 0.76 | 1705.47 | 295.6 | |
60 | 1816.2 | 1.58 | ||||
90 | 1559.4 | 1.85 | ||||
10 | 300 | 30 | 1520.6 | 7.64 | 1530.8 | 22.8 |
60 | 1528.5 | 10.2 | ||||
90 | 1543.4 | 12.21 | ||||
30 | 1537 | 4.12 | 1546.23 | 17.8 | ||
60 | 1554.8 | 5.32 | ||||
90 | 1546.9 | 5.93 | ||||
200 | 30 | 1548.4 | 1.56 | 1584.07 | 55.5 | |
60 | 1603.9 | 2.08 | ||||
90 | 1599.9 | 3.11 |
Note: TN: thread number; MIA: maximum iterations of the algorithm; MIN: maximum iterations in the neighborhood; OFV: objective function value; RT: running time; AAOS: average of the approximate optimal solutions; and RAOS: range of the approximate optimal solutions.
From the analysis of the experimental results in Table
In the case of experimental environment and parameter setting of the above examples, the sensitivity analysis is carried out by changing the battery capacity (electricity when an EV is fully charged) of EVs. Table
Sensitivity analysis of electricity.
Battery capacity (kwh) | Total distance (km) | Charging times |
---|---|---|
140 | 1768.4 | 3 |
160 | 1676.4 | 3 |
180 | 1669.8 | 2 |
200 | 1554.3 | 2 |
240 | 1521.1 | 0 |
260 | 1407.1 | 0 |
280 | 1413.8 | 0 |
300 | 1435.9 | 0 |
Experimental results drawn in a two-dimensional coordinate system indicate that the objective function value of the solution decreases as the battery capacity of EVs increases in the range from 120 kwh to 260 kwh. There is a strong correlation between the objective function value of the solution and the battery capacity of EVs, and the correlation coefficient is 0.9683. The reason is that when the number of vehicles is less, the average number of customers that each vehicle needs to serve will increase, which will increase the charging times and the total traveling distance of the scheme. When the battery capacity of an EV is greater than or equal to 260 kwh, the objective function value of the solution fluctuates steadily within a certain range, and an EV can even complete its distribution task without charging. The decreasing trend of the objective function value of the approximate optimal solution also disappears. In terms of the robustness of the solution, the volatility of each approximate optimal solution is about 5%, which is within the acceptable range when the battery capacity is constant.
Through the sensitivity analysis of the battery capacity of electric vehicles, we get a suggestion that according to the actual distribution requirements, logistics enterprises should select EVs with the battery capacity at the inflection point of the C-E curve in Figure
Sensitivity analysis of electricity (C-E curve).
In addition to electricity, the maximum load and maximum volume of EVs also affect the performance of the solution. Because of the similar properties, we select the maximum volume to perform the sensitivity analysis. The experimental results in Table
Sensitivity analysis of the maximum volume.
Volume (m3) | Total distance (km) | Charging times |
---|---|---|
20 | 1572.5 | 1 |
22 | 1554.3 | 2 |
24 | 1538.4 | 1 |
26 | 1533.9 | 2 |
28 | 1495 | 2 |
This is of practical significance in vehicle procurement, and we can get the best vehicle volume level by analyzing the inflection point of the C-V curve in Figure
Sensitivity analysis of the maximum volume (C-V curve).
The instances given in Section
Sensitivity analysis of the number of EVs.
Number of EVs | Total distance (km) | Charging times |
---|---|---|
3 | 2027.6 | 3 |
4 | 1730.4 | 2 |
5 | 1554.3 | 2 |
6 | 1610.7 | 1 |
7 | 1654.7 | 2 |
Sensitivity analysis of the number of EVs (C-N curve).
In this paper, we present TDEVRPTW-PF with the consideration of time windows, charging, energy consumption, and path flexibility. In order to determine the optimal path between any two locations, a 0-1 mixed-integer programming model is established, minimizing the energy consumption and the distance. Based on the optimal path selection model, the other model is established to solve multiple electric vehicles’ routing problems with time windows.
Then, we choose the VNS algorithm with good effect on the NP-hard problem in the heuristic algorithm as our algorithm framework. We modify and optimize it on the basis of TDEVRPTW-PF. After that, programming and debugging of the program are completed based on the improved VNS.
In the end, a large number of numerical experiments are carried out by running the program. We explain the experimental instances and analyze the efficiency of the algorithm. In addition, the sensitivity analyses are also carried out. By adjusting some parameters such as battery capacity, maximum load, maximum volume, and the number of planned EVs, we draw some conclusions that have practical significance on purchasing operation and vehicle arrangement for logistics enterprises.
However, as a vehicle routing problem, the optimization of TDEVRPTW-PF is never-ending. In the future, further studies may concentrate on energy consumption analysis and the remaining electricity estimation of EVs. Furthermore, the diverse types of the vehicles need to be considered. Further research on this topic may also address that detailed work needs to be executed such as the different departure times of EVs and the insertion of the random distribution task. All of the studies are aimed at making the model established closer to the reality, and the results of scientific research are more capable of serving the society.
The data used to support the findings of this study are included within the paper. Further details may be available to readers from the corresponding author upon request.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This research work was supported by the National Natural Science Foundation of China (no. 71801018) and the Fundamental Research Funds for the Central Universities (no. 2018RC39).