1. Introduction

The logistics industry is a composite service industry that integrates transportation, storage, and information industries. It covers a wide range of fields, absorbs a large number of jobs, and plays a major role in promoting production and stimulating consumption. It is an important part of the national economy. How to effectively reduce logistics costs, improve logistics delivery efficiency, provide customers with differentiated services, and improve customer satisfaction are important management issues facing logistics companies. The vehicle routing problem is the core problem in the optimization of logistics distribution, as well as the key problem of smart logistics. Scholars have carried out extensive research on this problem and have achieved fruitful results.

Vehicle routing problem (VRP) is a classic combinational optimization problem with NP. With the rise of modern logistics industry, it has become a key component of intelligent logistics. Since Dantzig and Ramser [1] put forward and studied this problem in depth, scholars have conducted extensive research on the VRP problem and several important variants of this problem and achieved a lot of results. Many of them are based on the actual logistics distribution of scholars who added more constraints so that the research results can be better applied to the actual production.

In VRP variants’ problems, as vehicle routing problem with time windows and multiobjective constraints (MOCVRPTW) considers the constraints of customer time windows, customer satisfaction, delivery cost, etc., it is more in line with the “customer first” and in line with the production reality of logistics distribution, which has attracted the attention of academia and industry.

In the actual logistics distribution process, it is necessary to consider not only the uncertainty of vehicle travel time in the distribution process but also how to save the distribution cost to the greatest extent on the basis of meeting the personalized needs of customers. In this paper, a multiobjective solution method under the constraints of grey time windows is proposed. In terms of the processing of grey time windows, the estimation method of grey time windows under 95% confidence level is given. In terms of satisfaction and distribution cost control, the fuzzy gradient function of customer satisfaction is introduced to set different satisfaction thresholds according to customer classification, and then, the Pareto solution set of different examples is obtained according to the threshold. In order to verify the solvability of the model, based on Solomon’s example, this paper extends the customer hard time windows into soft time windows and carries out two groups of numerical experiments. The results show that the fluctuation of distribution cost and customer satisfaction with grey time windows is small. Under different satisfaction threshold conditions, the distribution cost is increased significantly with the increase of satisfaction threshold.

This paper expands the scope of random travel time research and provides decision-making reference for logistics companies. However, this research also has some shortcomings. There is no clear method for customer classification, and the method for setting satisfaction thresholds for different customer categories has not been given. This is also the focus of the author′s next research.

The rest of this paper is arranged as follows. After Section 1, related literature will be reviewed in Section 2, and problems are described and modeled in detail in Section 3. Section 4 presents the grey distribution method for generating time windows. Section 5 describes albino customer satisfaction and grey distribution time. Section 6 presents the methods and results of numerical experiment analysis. Finally, the paper makes a summary in Section 7.

2. Literature Review

Scholars generally refer to the problem of uncertain driving time of distribution vehicles as vehicle routing problem with stochastic travel times (VRPSTT). It mainly studies the change of travel time due to traffic condition, road repair, weather, and other factors, and the change of the travel time satisfies some statistical rule of the vehicle routing problem. Laporte et al. [2] established the opportunity constraint and compensation model of the problem. They believed that the punishment for vehicles should be proportional to the length of delay and designed a branch cutting algorithm to solve the problem. Park et al. [3] constructed three different heuristic algorithms to solve this problem based on a modified deterministic VRP solution algorithm. Subsequently, Qiang et al. [4] and Yang et al. [5], respectively, extended and solved the problem on the basis of Laporte research. Zhang et al. [6] defined the uncertainty of travel time as fuzzy travel time, built a mathematical model with fuzzy travel time, and designed a new hybrid genetic algorithm by effectively combining fuzzy logic and fuzzy control method with the genetic algorithm of traditional vehicle routing problem. Li et al. [7] proposed two models of opportunity-constrained programming and stochastic programming with correction for this problem and designed a hybrid algorithm based on tabu search to solve the model. Tao et al. [8] established an opportunity-constrained programming model for random travel time with simultaneous delivery and pickup and designed a decentralized search algorithm solution strategy. Hou et al. [9] also considered the random vehicle routing problem with uncertain demand and uncertain travel time and proposed an improved genetic algorithm with an adaptive mechanism. Tas et al. [10] studied the vehicle routing problem with soft time windows for random travel time and solved the problem based on tabu search algorithm. Li et al. [11] comprehensively considered the total distance and time of vehicles, as well as the equalization of the distance and time of each vehicle, and designed a multiobjective hybrid genetic algorithm combining simulated annealing and genetic algorithm to solve this problem.

In recent years, the rise of e-commerce has led to the vigorous development of the logistics industry, and the research on this issue has maintained a trend of continuous growth. Binart et al. [12] studied this issue with the overall goal of serving the most customers in the shortest time. Miranda and Conceicao [13] focused on the problem that the waiting time after the vehicle arrives at the customer in advance may affect the follow-up distribution arrangement and designed a new method to estimate the time when the vehicle arrives at each customer and the possibility that the vehicle complies with the customer’s time windows. Errico et al. [14] proposed a two-stage strategy for this problem and designed an accurate branch cutting algorithm. Gomez et al. [15] relaxed the hypothesis of random travel time distribution of scholars in this question and designed a new solution strategy that could adapt to different travel time distributions. Wang et al. [16] first classified or divided customers into service areas and then designed two heuristic algorithms by introducing intimacy. Miranda et al. [17] studied the multiobjective random traveling time vehicle routing problem with both cost and service level and constructed the multiobjective memetic algorithm and multiobjective iterative local search algorithm. Jianli et al. [18] abstracted the freight distribution routing problem of railway logistics center into the optimization of the batch distribution vehicle routing problem with random travel time and service time, established a stochastic programming model with correction, and designed the corresponding algorithm to solve the model. Guimarans et al. [19] studied the random travel time vehicle routing problem with two-bit packing and designed the path optimization algorithm with Monte Carlo simulation and bias randomness. In order to study the collection of urban garbage, Markovi et al. [20] established a model of simultaneous delivery with random demand and travel time. Hashemi Doulabi et al. [21] solved the VRPSTT problem with synchronous access and service time based on the branch and bound method. Liu [22] studied the side-sharing services and proposed an arc-based vehicle-based integer linear programming model in space-time-state networks, which is solved by Dantzig–Wolfe decomposition.

3. Problem Description and Modeling

4. Grey Delivery Time Windows

Under normal circumstances, the time to reach the destination is often different due to road traffic conditions or nonuniform speed of the vehicle, resulting in uncertainty in the time for the vehicle to reach the customer, but the range of arrival time can be estimated from historical data. Therefore, this paper assumes that the delivery time is grey, and the travel budget time equation in [24] generates vehicle delivery grey time. The specific method is as follows.

The time from node i to node j is affected by many independent random factors. According to the central limit theorem, the sum of the superimposed effects of many small independent factors obeys or approximately obeys a normal distribution [18, 25]. It is assumed that the actual arrival time t_ij obeys a normal distribution, that is, $$, where $$ represents the variance of the travel time of the vehicle on the road section ij.

5. Customer Satisfaction

5.1. Fuzzify Customer Time Windows

In the delivery process, customer satisfaction is mainly reflected in whether the vehicle can arrive on time. If the vehicle arrives within the customer’s expected time, the customer’s satisfaction is the highest, and the delayed arrival will cause a decrease in customer satisfaction. In order to reflect the customer’s expected time more accurately, combined with time elements and the actual needs of the customer, the customer’s expected time windows is first blurred, and then, the fuzzy gradient function is used to express the customer’s satisfaction. The fuzzification rules refer to the method of Wang et al. [26]; in the paper by Wang, if the time windows is extended too much, such as extending 100 to both sides, the average customer satisfaction will be higher. At this time, the problem is close to the problem of no time windows requirement, and it is not easy to distinguish various routes. If the expansion is too small, such as 30 for each expansion, the satisfaction degree will be too large, and the average satisfaction degree will be too strongly correlated with the time windows. At this time, the problem is close to the hard time windows requirement, and the expected effect is also not achieved.

Based on the research of Wang et al., extend the time windows by 50 on both sides, and the time windows is fuzzified according to equations (16) and (17):

$$ (16)

$$ (17)

Assuming that the time the vehicle arrives at the customer i is t_i, the membership function expression used in this paper to deal with the customer time windows is

$$ (18)

The meaning of equation (18) is that if the customer arrives before the earliest time EET that the customer can tolerate or arrives after the latest time ELT that can be tolerated, the customer is no longer in working hours and cannot deliver the goods, so customer satisfaction is 0. If after the earliest tolerated time EET but before the normal work start time ET or after the normal working time LT but before the latest tolerated time ELT, customer satisfaction can hand over the goods because it is not part of the normal working hours and satisfaction is not high, but the satisfaction level is higher when the arrival time is about close to normal working hours. If the vehicle arrives during normal working hours [ET, LT], the customer is most satisfied with 100% satisfaction (here, marked as 1).

In order to express it more intuitively, use the polyline expression (18) in the rectangular coordinate system, as shown in Figure 1. It can be seen from Figure 1 that

• (1)

  If ⊗t_i is before the earliest time, the customer can tolerate, which means ⊗t_i ≤ EET_i; then, the waiting time windows opens, and the customer satisfaction is 0

• (2)

  If ⊗t_i exceeds the time windows expected by the customer, but is after the earliest tolerable time, ⊗t_i ∈ [EET_i, ET_i), the customer satisfaction value is between [0, 1)

• (3)

  If ⊗t_i is within the service time windows[ET_i, LT_i], then the customer satisfaction is 1

• (4)

  If ⊗t_i exceeds the time windows expected by the customer, but is before the latest tolerable time, ⊗t_i ∈ (LT_i, ELT_i]; then, the customer satisfaction is[0, 1)

• (5)

  If ⊗t_i exceeds ELT_i, which means ⊗t_i > ELT_i, then the customer satisfaction is 0

5.2. Satisfaction Threshold

In the actual distribution, the demands of different customer satisfaction are generated according to different products or important levels of customers. In order to save cost and provide personalized service, customers are classified and the minimum satisfaction threshold is set. As shown in Figure 2, for a certain customerj, the minimum satisfaction threshold is l_j. Then, corresponding to this threshold, the time interval [t_js, t_je] for which the vehicle can serve the customer is the part of the broken line in bold, as shown in Figure 2, which is the part above the line y = l_j (including the equal).

6. Experiment and Analysis

In order to solve the model, the author expands the hard time windows into a soft time windows in Solomon’s example and conducts numerical experiments in two groups based on the improved quantum evolution algorithm [28]. A new idea is provided for solving the problem of serious loss of effective information in the quantum evolutionary algorithm by defining the quantum cell body.

There are 3 types of Solomon’s calculation examples: Rdp, Cdp, and Rcdp. The geographic location of customers in the Rdp calculation example meets a random distribution, the geographic location of customers in the Cdp calculation example meets the clustering distribution, and the geographic location of the customer in the Rcdp calculation example is generated by a mixture of random distribution and cluster distribution. Except for the 9 small- and medium-sized customers, the customer size of the remaining 56 examples is 100. Each type of example can be divided into two categories: the first type of vehicle has a short planning time and weak installation capacity. The second type of vehicles has a long planning time and strong installation capacity; among them, Rdp1, Cdp1, and Rcdp1 have a short planning time and a low vehicle loading capacity (both 200), so the customers that each delivery vehicle can serve are relatively limited (about 5–10 customers), while the planning time for Rdp2, Cdp2, and Rcdp2 is longer, the loading capacity of the vehicle is larger (1000, 700, and 1000, respectively), and each delivery vehicle can serve more customers (some routes can exceed 30).

Considering the generality and time economy, take one from the middle position for each small category in the 56 examples of large-scale customers for numerical experiments. The data characteristics of the six calculation examples are shown in Table 1. The table lists the number, number of customers, total number of items, and number of available vehicles for each test case in Solomon’s example.

6.2. Experiment and Analysis under Different Satisfaction Thresholds

In order to provide customers with more personalized services while saving business costs, different satisfaction thresholds (the minimum satisfaction of a single customer) can be set for different customers when providing delivery services to customers.

Taking equation (1) as the objective function, without considering the vehicle startup cost, the experimental environment and experimental parameters are the same as 6.1, the customer satisfaction threshold S_lim is from 0.1 to 0.9, the step size is 0.1, and the maximum number of iterations for each threshold is set to 100. Calculate the grey distribution time windows, the distribution cost C_f of different threshold constraints, and the average customer satisfaction S_f. The results are shown in Table 3.

Table 3. Distribution costs and average satisfaction under different satisfaction thresholds.

	Slim	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
C105	Cf	363.25	363.25	369.74	372.32	392.63	401.88	406.18	412.96	417.39	445.12
	Sf	0.67	0.75	0.77	0.78	0.81	0.94	0.96	0.97	0.99	1.00
C205	Cf	141.54	245.98	249.35	250.15	269.00	326.52	380.49	443.23	461.85	479.11
	Sf	0.42	0.62	0.63	0.69	0.77	0.80	0.84	0.87	0.90	1.00
R105	Cf	609.70	646.43	654.70	692.30	727.23	750.84	796.38	800.95	814.62	905.56
	Sf	0.47	0.55	0.57	0.66	0.69	0.74	0.80	0.82	0.88	1.00
R205	Cf	745.62	771.72	772.37	774.22	789.13	792.56	796.22	824.96	842.06	864.15
	Sf	0.55	0.73	0.74	0.76	0.80	0.84	0.87	0.93	0.97	1.00
RC105	Cf	558.88	597.58	606.66	611.61	620.05	663.01	736.14	762.89	784.18	805.57
	Sf	0.63	0.65	0.69	0.74	0.80	0.84	0.88	0.89	0.90	1.00
RC205	Cf	621.15	627.88	632.60	635.74	637.57	640.39	647.19	664.47	677.83	681.73
	Sf	0.59	0.61	0.71	0.72	0.72	0.83	0.86	0.88	0.92	1.00

In order to facilitate understanding and more intuitive performance of the data in Table 3, using Figures 3–8, respectively, the five examples of distribution costs and average under different satisfaction thresholds are shown.

In the six figures, the 10 small circles on the right of each picture correspond to the 10 sets of data in Table 3. The last small blue glowing circle corresponds to the data with a satisfaction degree of 1. When the customer satisfaction threshold is set to 1, it means that the vehicle must arrive within the time windows [ET, LT] expected by the customer. At this time, the problem is transformed into a hard time windows constrained vehicle routing optimization problem (CVRPTW).

This experiment shows that taking into account customer satisfaction, the distribution cost is significantly increased; the distribution cost reaches the highest when the customer satisfaction is 1, that is, under the hard time windows; the distribution cost increases as the satisfaction threshold increases, and the distribution cost gradually increases. In actual production, distribution companies can classify customers and provide personalized distribution services for different customers, so as to achieve the goal of reducing business operating costs without affecting service quality.

In addition, it can be seen from Figures 3 to 8 that the distribution cost gradually increases with the relatively smooth customer satisfaction, indicating that the grey time windows has certain advantages in solving this kind of random travel time vehicle routing problem.

7. Conclusion

The paper abstracts the common problem of uncertain distribution of road conditions into the problem of uncertain distribution time. Combining the grey time windows of delivery with the customer time windows, the choice of vehicle delivery route is decided and customer satisfaction is determined, new grey time windows and multiobjective vehicle routing problem are constructed, and the construction method and whitening of the grey time windows method are given. Finally, based on the improved quantum evolution algorithm, the author designed two sets of numerical experiments on the modified Solomon’s examples. Both sets of experimental results show that the grey time windows has certain advantages in solving the dynamic travel time of the vehicle path and can provide a reference for logistics distribution enterprises.

The grey delivery time windows proposed in this paper expands the research scope of the vehicle routing problem. Since no actual logistics data was used for experiments, this study also has limitations. Simulation experiments based on the operating data of large logistics companies are also the direction of the authors’ future research.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.