Considering the economic factors of logistics distribution costs and the influence of logistics distribution customer service time window on customer waiting time, the paper formulates a multi-objective optimization model which aims at maximizing the overall satisfaction of the system and minimizing the delivery time and total cost of the system.

(1) Minimizing the logistics distribution services cost

Considering the logistics integrated transportation service of electric vehicles in logistics distribution services and the site selection and construction of the replacement station, the logistics distribution cost of this paper mainly includes the total cost of electric vehicle power exchange cost, electric vehicle fixed cost, replacement station construction cost, and time window penalty cost. The cost of electric vehicle power exchange is related to the electricity exchange price and power exchange; the higher the electricity exchange price and the more the electricity exchange, the higher the power exchange cost; for this reason, it is necessary to optimize the power exchange of electric vehicles into the power exchange station. The length of the driving path of electric vehicles determines the cost of their logistics distribution travel time, and choosing a suitable location to build a replacement station can make the electric vehicle exchange power in its suitable power exchange during the driving distance while reducing the time cost; the longer the route travel time corresponds to the higher the route travel time cost, so the path should be reasonably selected during the route driving process so that the electric vehicle can be replenished. The specific mathematical expression is as follows: min ⁡ F 1 = ⁡ min Γ 1 + Γ 2 + Γ 3 + Γ 4 . (1)

The cost of electric vehicle replacement is related to the remaining electricity and power exchange price of electric vehicles driving into the power exchange station during distribution; taking into account the prevention of electric vehicles from re-entering the power exchange station multiple times and affecting the power exchange of other electric vehicles resulting in resource occupation, the replacement price will be set according to the remaining power level of electric vehicles; if the remaining electricity is more, the corresponding purchased electricity price is correspondingly higher. The expense linked to power exchange is associated with the frequency of power changes in electric vehicles during distribution, the remaining power to the power exchange station, and the electricity exchange price, and the calculation method is the sum of the purchase cost of the new replacement battery and the cost of a single battery rental. Γ 1 = ∑ i ∈ I ∑ s ∈ S α ∑ k ∈ K T s + γ s k , n − γ s k λ r t ⋅ x i j k . (2)

The construction cost of the substation is related to the construction cost of a single substation and the number of substations under construction, as shown in the following equation. Γ 2 = ∑ s ∈ S L i B s . (3)

Electric vehicle distribution costs are divided into variable costs and fixed costs. Variable costs are related to the length of the driving path of electric vehicles; the farther the driving distance, the greater the variable cost. Fixed cost is the total cost of the vehicle paid by the enterprise to purchase an electric vehicle that is put into use. Γ 3 = ∑ k ∈ K ∑ j ∈ J h ⋅ x o j k + g ⋅ ∑ i ∈ I ∑ j ∈ J ∑ k ∈ K d i j ⋅ x i j k . (4)

Considering the impact of the service time of delivery on customer delivery service satisfaction, the cost caused by the delivery time in violation of the merchant’s requirements in the objective function also takes into account the total cost, and this paper establishes a mathematical model based on the soft time window constraint, describing it as follows: if the delivery vehicle a i delivers the goods before and the unit opportunity cost is w 1 , if the delivery vehicle delivers the goods afterward b j , resulting in a decrease in satisfaction, and the unit penalty cost is w 2 . The cost calculation is shown in Figure 2.

The total time cost is shown below: S r i k = w 1 a i − r i k , r i k < a i , 0 , a i ≤ r i k ≤ b i w 2 r i k − b i , r i k > b i , i ∈ I , k ∈ K , (5) Γ 4 = ∑ i ∈ I ∑ k ∈ K S r i k . (6)

(2) Minimizing the total distribution distance of the logistics system.

Considering the reasonable planning of the site selection of the layout of the substation and the distribution path of the logistics enterprise, the logistics distribution distance can be reduced, and the expression of the logistics distribution distance F ₂ is given as follows: F 2 = ∑ i ∈ I ∑ j ∈ J ∑ k ∈ K d i j x i j k , i ≠ j . (7)

(3) Maximum customer satisfaction.

Customer satisfaction is used to evaluate the service level of logistics distribution enterprises based on the logistics distribution time window. The length of its delivery time will directly affect the customer’s evaluation of its satisfaction. Its linear function expression F′ ₃ is F 3 ′ = 0 , r i k ≤ a i . ⁡ min a i − r i k a i − a i . ⁡ min , a i . ⁡ min < r i k ≤ a i 1 , r i k ∈ a i , b j r i k − b j b j , ⁡ max − b j , b j , ⁡ max > r i k ≥ b j 0 , r i k > b j , ⁡ max . (8)

For ease of calculation, customer dissatisfaction is considered, and customer dissatisfaction is represented by F ₃, and the F ₃ expression is expressed as follows: F 3 = 1 − F 3 ′ . (9)

The planning and replacement management of electric vehicle logistics distribution routes should meet the following constraints: 1) Logistics distribution constraints: the constraints of logistics distribution mainly include path constraints, load constraints, arrival/departure time constraints, and remaining power constraints.

(1) Path constraints

Path constraints will restrict vehicle movement, considering the number of vehicles entering and leaving a node. ∑ i ∈ I , i ≠ j x i j k = ∑ i ∈ I , i ≠ j x j i k , ∀ i ∈ I , k = 1 , 2 ⋯ K , (10) ∑ k ∈ K x i j k = 1 , ∀ i ∈ I , ∀ j ∈ J , i ≠ j , (11) ∑ i ∈ S α , j ∈ S α ∑ k ∈ K x i j k ≥ 1 , ∀ i ∈ S α , j ∈ S α , k ∈ K , (12) S = j ∑ i ∈ I x i j k = 1 , j ∈ J , k ∈ K , (13) ∑ k ∈ K ∑ j ∈ J x o j k = ∑ k ∈ K ∑ j ∈ J x j o k , (14) x i j k ∈ 0 , 1 , ∀ i ∈ I , j ∈ J , k ∈ K , (15) ∑ i ∈ I , j ∈ J ∑ k ∈ K x i j k ≤ M ⋅ B s . (16)

Eq. 10 indicates that each vehicle enters a node and leaves a node an equal number of times. Eq. 11 indicates that each customer can receive service from at most one electric vehicle. Eqs 12, 13 indicate that the distribution path of the electric vehicle forms a closed loop connecting end to end. Eq. 14 indicates that each electric vehicle departs from the distribution center and ends up in the distribution center. Eq. 15 defines that the value of a decision variable can only be 0 or 1. Eq. 16 indicates that the station can be visited many times.

(2) Load constraints

Formula 17 indicates that the total customer service of each electric vehicle cannot exceed the maximum cargo capacity of electric vehicles. Eq. (18) represents the overall count of electric vehicle transfers.

(3) Arrival/departure time constraints

When a customer provides delivery within an acceptable timeframe, there is zero opportunity cost and penalty cost. If the time window is exceeded, the penalty cost will be paid according to the length of the violation. a i . ⁡ min ≤ r i k ≤ b j , ⁡ max , i ∈ I , ∀ k ∈ K , (19) β o j a = 0 , ∀ j ∈ J , (20) β j k b = β o k a + d i j v k + q i α i k + α s k , i ∈ C , j ∈ J , k ∈ K , s ∈ S , (21) β j o k b = β o j a + ∑ i ∈ I ∑ j ∈ J d i j x i j k v k + ∑ i ∈ I , s ∈ S α α s k + α i k ∑ i ∈ C q i , k y i k , x i j k = 1 , y i , k = 1 . (22)

Eq. 19 means that the electric vehicle delivery time cannot exceed the customer’s maximum tolerable time window, Formula 20 and Formula 21, respectively, represent the time when the electric vehicle leaves the logistics distribution center and arrives at the customer, and Formula 22 represents the total time of the entire logistics distribution network electric vehicle to complete the logistics distribution use.

(4)Power constraints

Formula 23 indicates that the remaining power of the electric vehicle cannot exceed its power limit. Eqs 24–26 represent the electric vehicle leaving the distribution center and the customer node and the power level of the replacement station. Formula 27 represents the relationship between the remaining power of the electric vehicle leaving the previous customer point and the next customer point. Formula 28 represents the relationship between the power of two customer nodes. In summary, the multi-objective optimization model of electric vehicle logistics distribution path optimization and power exchange strategy considering customer satisfaction is represented as follows: min ⁡ F = F 1 , F 2 , F 3 , s . t . 10 ∼ 28 . (29)

The A* (A-Star) algorithm is the most efficient direct search method for solving the shortest path and is a common heuristic for many other problems. Its heuristic function is f n = g n + h n . (34)

The above equation f n when each node is searched, its corresponding heuristic function, in this article, represents the valuation function that reaches the customer point C; f n consists of two parts, of which the first part g n represents the actual cost of customer n to the distribution center current customer; the second part h n is to estimate the cost of the current square to the destination, that is, the distance between the previous customer point and the next customer point when the electric vehicle is delivered. Each time the algorithm scales up, it picks the node with f n having the lowest value as the next node on the optimal path.

This article assumes that the distance between customer points is the Euclidean distance d i j . g n represents the cost of an electric vehicle from customer point C to the logistics distribution center. h n is the cost value between any customer and the next customer. In this article, we select logistics distribution center O as the starting point and add all customer points to the open list. At this time, the minimum value in the opening list is taken, and only one node in the logistics distribution center O is opened in the initial stage. So, remove the O-points from the open list and add the O-points to the off-list. Take the adjacent customer points of the O point and add the customer points with the smaller valuation function to the open list. At this point, these adjacent customer points are the parent nodes of the adjacent points; delete these parent nodes in the open list, and then, center on the parent node; look for the customer point with the smallest neighbor valuation function, and cycle through the above steps until all customer delivery needs are met. When exploring the path, consider the load limit of the electric vehicle and the power level of the electric vehicle, and return to the distribution center if its load limit is exceeded. Set open list to V 1 , and the closed list is V 2 . At this point, the vehicles are connected from front to back in the set of nodes in the v table. The resulting path is the optimized solution. Suppose the initial pheromone it generates is τ i j R = λ τ c , τ c is a pheromone for other paths. λ is greater than 1.

The specific steps are as follows.

1) Build the initial function, and initialize the start list V 1 and the closed list V 2 , which calculates the valuation function for adjacent customer points C 1 of logistics distribution center O. Substitute logistics distribution center O and its neighbors into V 1 .

After the A* algorithm is calculated, the initial optimal solution is obtained. Ants transfer from customer point i to select the next customer point j through certain probability selection rules. In the traditional ant colony algorithm, the state transition probability of ant m from node i to node j is expressed as shown in Eq. 31: P i j m = τ i j α η i j β ∑ s ∈ J k i τ i s α η i j β , j ∈ a l l o w e d m , 0 , o t h e r w i s e . (35)

In Eq. 31, a l l o w e d m represents all nodes that ant m can select next, C represents a collection of customer points that can be selected after ant m passes through customer point i, and α is a pheromone heuristic factor and reflects the factors that affect the path of pheromones on the ant’s selection path.β is the desired heuristic factor, and the relative importance of visibility is expressed in the path.

The heuristic factor n i j is the expectation of the ant from the customer point i to the customer point j, which is the key to the ant choosing the next node. This paper studies the distribution strategy with the lowest cost of logistics and distribution services, the smallest logistics distribution distance, and the greatest customer satisfaction. Combining the above factors as heuristic factors affects the optimal distribution strategy. Therefore, this study will design the heuristic factorial as follows: η ij = 1 min ⁡ F . (36)

The cost of logistics distribution services, logistics distribution distance, and customer satisfaction are used as the denominator of the heuristic factor in order for the vehicle to select the next customer demand point j by the customer point i, and the expectation is that the total cost is the smallest, the logistics distribution distance is the smallest, and the customer satisfaction is the largest. The transfer probability of customer points that meet the conditions of small total cost, short distance, and high satisfaction is increased so that vehicles are prioritized for customer points with small total distribution costs.

The pheromone volatility factor pertains to the rate at which pheromones dissipate. Its value intricately influences both the algorithm’s global search capacity and convergence speed. If set too high, pheromones evaporate rapidly, causing the exclusion of potentially superior paths. Conversely, a value set too low results in excessive residual pheromones along the path, thereby impacting the algorithm’s efficiency.

The size of the pheromone volatilization factor ρ-value in the ant colony algorithm determines the persistence of the above pheromone retention in the optimization path. Therefore, this paper selects the size of ρ for segmentation and adjusts the size of the pheromone volatilization factor as the number of iterations increases. ρ = 0.1 , 0.25 N ≤ n ≤ 0.5 N , 0.3 , 0.5 N ≤ n ≤ 0.75 N , 0.6 , 0.75 N ≤ n ≤ N , (37)

where n represents the current number of iterations and N represents the total number of iterations of the algorithm. Start setting ρ to a smaller value, guided by pheromones, to find the optimal path. After 0.5 N, the pheromone accumulation on the path is too high, and ρ is set to 0.3 to improve the pheromone volatilization effect and avoid the risk of falling into local optimization. When the number of iterations is more than 0.75 N, the pheromone concentration on the path reaches a large value, resulting in the corresponding increase of the ρ-value.

To make the search process more instructive, after all ants have formed their paths, the established paths are updated globally, and only the path of the ants that find the globally optimal path is updated with pheromones. The update rules are Δ τ i j b e s t = ∑ m = 1 M Δ τ i j k , (38) Δ τ i j m t = G l b e s t , 0 (39)

where G is the total amount of pheromones left by ants passing through the optimal path and l b e s t is the path length corresponding to the current total cost of the smallest. When information is flooded, the residual information needs to be updated after each ant traversal is completed. Thus, at the time t + n, the information update rules on the optimal path (i, j) are as follows: τ i j t + n = 1 − ρ τ i j t + Δ τ i j b e s t . (40)

For edges (i, j) that are not optimal paths, the update rules are τ i j t + n = 1 − ρ τ i j t , (41)

where ρ represents a pheromone volatile factor.

According to the experimental data given in Table 1, this paper uses the method of combining the A* algorithm and the ant colony algorithm to generate the optimal route based on the pheromone iteration, and the optimal distribution path and optimal roadmap are obtained, as shown in Table 3 and Figure 3.

When there is no power exchange behavior, the improved ant colony algorithm in the first stage of this paper design is used to solve the path planning model and obtain the optimal distribution route, and the total cost of distribution generated under the path is 1942.85 yuan. From the path optimization results, it can be seen that subjected to the constraints of the customer’s time window, the first, second, and third paths require multiple electric vehicles for joint distribution to meet the customer’s time window needs, and the total number of delivery vehicles required for distribution is 5. The resulting fixed cost of electric vehicles is 1000 yuan, the distance cost of electric vehicles is 698.67 yuan, and the penalty cost and opportunity cost of the time window are the smallest, 244.16 yuan. It can be seen from this that the distribution route should be reduced as much as possible under the condition of meeting the constraints of the customer’s time window, and the number of vehicles used, that is, the fixed cost expenditure. The optimal delivery route diagram in this article is shown in Figure 4.

Based on the optimal path optimization map of logistics and distribution obtained above, this paper will next solve the site selection problem of the replacement station, considering the cost of the electric vehicle power exchange and the construction cost of the power exchange station. The difference between model solving in the power-swap mode and no swapping behavior is when the car has less power left, and it will enter the designated substation for power exchange. The entire power exchange process takes a shorter and fixed time than the charging time. The penalty cost and opportunity cost of the time window in this mode will have an impact, as well as the cost of replacing the electricity. Based on the optimal path optimization map, the location coordinates of the candidate points of the alternative station are obtained in this paper, as shown in Table 4.

Since the vehicle enters the power exchange station for power exchange will delay a certain amount of time, resulting in a change in the time when the electric vehicle arrives at each customer point, the corresponding time window penalty cost and opportunity cost will also change. According to the optimization results of the logistics distribution path obtained above considering the time window, the relevant parameters of the electric vehicle and those of the substation are combined. Considering the load capacity and power constraints of electric vehicles, the second stage of the two-stage hybrid algorithm–genetic algorithm solution is used to minimize the cost of power exchange, the construction cost of the power exchange station, and the total distribution cost. Thus, obtaining the total cost of power exchange for electric vehicles to reach each substation for power exchange, the total cost of each path is shown in Table 5 below.

From the results in Table 6, it can be seen that in the power exchange mode, the electric vehicle has insufficient endurance of paths 2 and 3, the customer point 11 of path 1 has an insufficient power problem, the reachable candidate points are 1 and 5, and the total cost of the path to reach candidate point 1 is the smallest, so customer 12 chooses to change power at candidate point 1. In the same way, it can be known that customer points 5, 8, 13, and 9 are replaced at candidate points 5, 5, 1, and 1, respectively. Vehicle 1 in paths 1, 4, and 5 selects candidate point 1, while paths 2 and 3 select candidate point 5, mainly because the customer point in path 1 requires a later delivery time, and the power change mode of the electric vehicle reduces the power replenishment time. As a result, vehicles have plenty of time to travel to distant substations, reducing their opportunity costs. Based on the above results, the best candidate addresses for the replacement station in the power exchange mode are candidate points 1 and 5, and the single power exchange cost is 1139.2 yuan, of which the electricity cost is 338.3 yuan, and the opportunity cost and penalty cost are 445.5 yuan. In the following analysis, this article will discuss the path optimization and site selection of different time windows and power exchange rates.

Based on the definition of the above scenario, the logistics distribution path of the logistics distribution center in scenarios 1, 2, 3, and 4 is shown in Figure 5, and the corresponding logistics distribution journey cost, upper time window opportunity cost, lower time window penalty cost, fixed cost, power replacement cost, and customer satisfaction results are given in Table 7.

Based on Figure 5, it can be seen that the electric vehicle logistics distribution path in scenario 1 has fewer crossovers, scenario 2 has less, scenario 3 has more crossover paths, and scenario 4 has the most crossovers. According to Table 4, with the increase in the number of path crossings, in order to meet the goal of maximum customer satisfaction, the corresponding logistics distribution costs and power exchange costs will increase. Based on scenario 2, it can be seen that the total cost of logistics distribution is lower when the goal of maximum customer satisfaction is not considered. At this time, the total cost of logistics distribution is 1995.27, which will only be distributed under the premise of meeting the time window with the shortest path as the goal, although the distance distribution cost and power replacement cost are reduced, but due to the lack of consideration of the customer’s time window factor, the customer’s satisfaction level decreases to 0.67. For scenario 3, the customer satisfaction level is the largest, 0.87; compared with scenario 2, the satisfaction level increased by 20%, and the total cost of logistics distribution under this scenario is 2012.54, mainly because the logistics distribution center delivers the goods within the specified time window, and the electric vehicle driving route needs to be adjusted in the logistics distribution process, which brings more logistics distribution path crossover and power exchange costs. Scenario 4 considers the goal of the minimum power exchange cost and the maximum customer satisfaction level of the electric vehicle; it can be seen from Table 4 that the customer satisfaction level of scenario 4 has increased by 15% compared with scenario 2, and the power exchange cost is 208.11, which is 97.12 yuan lower than the replacement cost of the electric vehicle and the loss cost of the electric vehicle.

In order to prevent electric vehicles from re-entering the power exchange station for power exchange due to more remaining electricity, it will affect the normal power exchange order of the power exchange station. This paper assumes that when the residual power of the electric vehicle is less than 20%, the profit factor of the replacement station is 1.2, and it is ascending in steps, and the profit factor of the replacement station increases by 0.1 for every 20% increase in the remaining electricity. This section discusses the impact of the remaining power exchange on the site selection decision, logistics and distribution costs, and power exchange costs of the replacement station. The relationship between the amount of electricity exchanged and the cost of exchanging electricity is shown in Figure 6.

As can be seen from Figure 6, the remaining power of electric vehicles is 0%–100%, and with the reduction of the remaining power of electric vehicle batteries, the total cost of logistics and distribution of electric vehicles has dropped from 2019.37 yuan to 1942.34 yuan, and the cost of power replacement has dropped from 354 yuan to 40 yuan. At the same time, the opportunity cost and penalty cost on the time window are also slowly increasing, from 293.27 yuan to 300.37. From the perspective of the degree of change in the cost of power exchange, the main reason is that with the reduction of the remaining power of electric vehicles, the profit factor of the replacement station is reduced, so when the remaining electricity is closer to 0, the unit replacement cost is smaller. From the perspective of the opportunity cost and penalty cost of the time window, the more the remaining power of the electric vehicle, the more it can ensure that the electric vehicle meets the distribution needs of the remaining customers, and there will be no need to replace the electricity in the middle, which will make the electric vehicle better meet the needs of customers, and the opportunity cost and penalty cost of the time window will be reduced.

Therefore, in order to meet the goal of the minimum total cost and the greatest customer satisfaction of logistics distribution, the gradient electricity price can be set according to the remaining electricity of the electric vehicle to reduce the unit replacement cost. On one hand, it can motivate electric vehicles to choose a power exchange station for power exchange, improve the income of the power exchange station, reduce the number of times the electric vehicle re-enters the replacement station, and effectively improve the battery utilization rate. On the other hand, due to the fixed power change time, electric vehicles can decide whether to change electricity according to their own remaining electricity and the time window needs of customer orders, effectively improving the efficiency of electric vehicle distribution.