2.1. Research on the Time-Dependent Vehicle Routing Problem (TDVRP)

The urban transportation network exhibits time-varying dynamic nature, wherein vehicular velocities may fluctuate across varying time periods. This variation significantly affects the EV energy consumption and travel time cost of urban logistics distribution. Scholars initially focused on TDVRP, which involved studying the real-time speed impact of different weather and congestion levels on vehicle routing [19], identifying ways to evade traffic congestion by analyzing urban traffic’s time-varying characteristics [20], and addressing the transport of valuable goods by proposing a secure TDVRP with time windows including pickup and delivery with uncertain demands [21]. Researchers then discovered that time-varying vehicle speeds caused various carbon emissions due to different fuel consumption, leading to the development of the Time-Dependent Green Vehicle Routing Problem (TDGVRP). Soysal et al. [22] and Çimen et al. [23] established the TDGVRP model, which considered time-varying vehicle speeds, fuel consumption, and carbon emissions. TDGVRP was later focused on cold chain transportation by Guo et al. [24], who proposed a two-stage algorithm that enabled vehicles to wait in place after service to avoid harsh traffic conditions. Recently, with the popularity of EV fleets, the Time-Dependent Electric Vehicle Routing Problem (TDEVRP) is rising gradually. TDEVRP is, in fact, a subset of TDGVRP, which considers time-varying vehicle speeds’ impact on EVs’ energy consumption rather than FVs’ fuel consumption. Lu et al. [25] considered the impact of charging decisions and congestion conditions on the overall delivery process based on traditional constraints in TDEVRP. Bi and Tang [26] examined TDEVRP by incorporating time-varying random traffic conditions and employing the analytical battery model to determine EVs’ charging and discharging patterns. They aimed to minimize the overall service time and developed a hybrid rollout algorithm to tackle the problem. Zhang et al. [10] and Keskin et al. [27] approached TDEVRP from different viewpoints. The former incorporated charging during congestion periods into the objective function, while the latter considered the impact of queuing time at charging stations on overall distribution. The literature above provides a theoretical framework and solution methodology for TDEVRP.

2.2. Research on the Open Vehicle Routing Problem (OVRP)

OVRP has found practical application in a variety of scenarios, such as newspaper distribution [28], express service [29], campus shuttle [30], and home health care [31]. Scholars have examined the OVRP from diverse perspectives considering different constraints. Atefi et al. [32] developed a unique OEVRP model that utilized multiple carriers to deliver packages for a Canadian major producer of cookies and snacks. Azadeh et al. [33] introduced a closed-open hybrid VRP based on the cooperation model of self-operated fleets and 3PL fleets, allowing some vehicles to return to the depot, while the rest of the vehicles do not. Niu et al. [34] discussed the OVRP of outsourcing distribution to a 3PL fleet, and demonstrated the advantages of outsourcing logistics through cost composition analysis. Brandão [35] studied a multi-depot OVRP with the goal of minimizing the travel distance of vehicles. Xia and Fu [36] introduced customer satisfaction rate into the OVRP as a constraint, and constructed an optimization model to minimize the total distribution costs. In addition, many scholars have designed various algorithms for OVRP. Faiz et al. [37] presented two integer linear programming models for OVRP. The first one is an arc-based mixed integer linear programming model solved by a general-purpose solver, while the second one is based on a path-based formulation, which solved by a column generation framework. Chen et al. [38] developed an urgency level-based insertion heuristic to construct an initial solution, and employed a reinforcement learning based variable neighborhood search algorithm to solve the OVRP. Brandão [39] proposed an iterated local search algorithm for OVRP, and solved it with test sets including Solomon, Homberger and Gehring. With the rise of green logistics in recent years, the Green Open Vehicle Routing Problem (GOVRP) has gradually attracted attention. Niu et al. [40] considered vehicle fuel consumption and its environmental impact in GOVRP, conducted a simulation experiment based on the actual road conditions in Beijing, and found that compared to closed routes, open routes reduced total cost by 20%, with fuel consumption costs and carbon emission costs lowered by almost 30%. The above literature provides methodological guidance for OEVRP. However, there is a dearth of literature on OEVRP, indicating that related research has not received sufficient attention.

2.3. Research on energy replenishment strategies in EVRP

Efficient energy supplement strategies have always been a prominent research topic in EVRP, aiming to alleviate range anxiety and enhance delivery efficiency in EV distribution. Initially, scholars mainly focused on charging strategies, such as full charging, which has been utilized in the studies of Lin et al. [41], Granada-Echeverri et al. [42], and Kucukoglu et al. [43]. Full charging strategy involves completely charging the EV before departing from the charging station, which can effectively mitigate range anxiety. However, this strategy incurs high charging costs and long charging durations, may easily leading to a decrease in customer satisfaction. The strategy of partial charging, in its assessment of charging costs and service timeliness, affords greater flexibility in charging operations. Cortés-Murcia et al. [44] and Zhou et al. [45] presented partial charging strategies for EVs when designing EVRP charging schemes. The antecedent enables EVs to be dispatched for multiple distribution tasks, whereas the latter deftly employs the partial charging period to facilitate the delivery of parcels to clients closer to the charging station. In addition, there are different charging technologies in the charging process. Schiffer and Walther [46] and Dönmez et al. [47] proposed a linear charging function when designing the EVRP charging scheme. The former considers a single fast-charging mode at the charging station, while the latter considers a combination of three charging speeds: ordinary charging, fast charging, and super-fast charging. In recent years, battery swapping has emerged as a viable solution for energy replenishment in EV distribution. Jie et al. [48] and Zhou et al. [49] discussed the EVRP considering battery swapping strategy. The former developed a hybrid algorithm to tackle the model, while the latter proposed a multi-objective whale optimization algorithm for the model. Moreover, Raeesi and Zografos [50] devised a synchronized mobile battery swapping strategy for the EVRP, involving the replacement of depleted batteries with fully charged ones at designated times and locations by battery-swapping vans. The literature above can offer theoretical references and methodological support for EVs’ energy replenishment strategy in distribution.

2.4. Research on metaheuristics of solving VRP

The application of efficient meta-heuristic algorithms to optimize routing problems has been a focus of research in logistics transportation [51]. The TDOEVRP-HERS poses unique challenges due to the time-dependent nature of its constraints, alongside the specifics of 3PL fleets, such as non-mandatory return trips, flexible energy replenishment options, and uncertain battery level consumption. Two algorithmic strategies that have shown promise in addressing these complexities are the ACO [52] and ALNS [53]. ACO is a population-based metaheuristic that draws inspiration from the foraging behavior of ants [54]. Its utilization in complex VRPs has received extensive research attention. Su and Fan [55] used ACO to minimize penalty costs in the Green Vehicle Routing Problem (GVRP), incorporating fuel consumption, carbon emissions, and customer satisfaction constraints. Similarly, Zhang et al. [56] utilized ACO to cut down energy usage in EVRP. Despite its successes, the original ACO has shortcomings, leading researchers to develop the improved ACO. Zhang et al. [57] enhanced traditional ACO with three mutation operators for the multi-objective VRP with flexible time windows, improving local search and global exploration. Li et al. [58] adjusted pheromone update methods of ACO, focusing on optimal solutions for the multi-objective multi-depot GVRP. Jia et al. [59] created a bi-level ACO for EVRP with capacity constraints. The upper level, unconstrained by electricity, generates routes via ACO, while the lower level uses a heuristic to determine charging plans, using its solutions to update the upper level’s pheromone. ALNS is a well-regarded metaheuristic that has successfully addressed numerous COPs. It operates by iteratively exploring a set of candidate solutions in the neighborhood of the current solution and adaptively selecting the most promising ones [60]. Scholars have employed ALNS to solve many VRP variants. Sun et al. [61] addressed the time-dependent profitable pickup and delivery problem with time windows by ALNS. Chen et al. [62] discussed a VRP with time windows and delivery robots and developed ALNS to tackle it. Kuhn et al. [63] used the ALNS to integrate order picking and vehicle routing for micro-store supply. ACO is known for generating high-quality initial solutions that respect the problem’s constraints. ALNS is particularly effective when dealing with dynamic environments, given its capacity to continually adjust and refine solutions [64]. The combination of ACO’s global exploration and ALNS’s local exploitation may lead to better-quality solutions. The comprehensive search process increases the chance of finding optimal or near-optimal solutions for the TDOEVRP-HERS. However, the hybrid application of these two powerful algorithms in the field of EVRP is currently under-researched, suggesting a need for further exploration.

2.5. The research gaps

Table 1 presents a structured categorization of the literature pertinent to TDOEVRP-HERS, drawing from the elements outlined previously. This table systematically delineates distinctions among energy replenishment strategies, routing modes, and solution methodologies within existing studies. Its design facilitates an efficient understanding of the prevailing state-of-the-art and highlights potential research gaps or avenues addressed by this research.

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Table 1. A summary of relevant works.

https://doi.org/10.1371/journal.pone.0291473.t001

The existing literature lays a solid groundwork for the in-depth investigation of EVRP in urban distribution. However, a review of the literature has identified several gaps in research that merit further exploration: (1) Many studies limit EVs to using charging stations as their sole energy replenishment way during distribution. However, in reality, EV fleets have started to adopt a variety of energy replenishment strategies in current urban distribution. As one of the world’s largest consumers of EVs, China explicitly outlined its intention in the “New Energy Vehicle Industry Development Plan (2021–2035)” to “accelerate the construction of battery swap infrastructures,” indicating that battery swapping will become a primary method of supplementing EV energy in the future. Despite some scholars have explored EVRP with battery swapping, there remains a significant shortage of EVRP research that considers hybrid energy replenishment strategies with both charging and battery swapping. (2) The time-varying traffic conditions of urban road networks lead to EVs traveling at varying speeds throughout the day, with speed being the most significant factor affecting both EV travel time and power consumption. The fluctuating vehicle speed makes calculating EV energy consumption and remaining mileage challenging, thereby complicating the planning of urban distribution routes. Currently, the academic community has not given sufficient attention to developing reasonable rules for predicting EV energy consumption during urban distribution. (3) To reduce operational costs, an increasing number of enterprises are outsourcing their logistics transportation business. Logistics outsourcing involves a 3PL fleet that departs from the enterprise’s depot to perform distribution tasks without having to return to the depot upon completing the tasks, known as open route distribution. Although this cooperation mode is gaining popularity in urban distribution, there is a dearth of literature on TDOEVRP, which warrants further exploration. To fill these research gaps, this study investigates TDOEVRP-HERS, considering customer demand, service time, EV capacity, time-varying speed, different energy supplement strategies, and open routing. A MIP model is constructed to minimize the total distribution cost of transport fleets, and a HALNS is designed to tackle this model. The study’s results provide decision-making references for companies interested in hiring environmental-friendly 3PL fleets and governments seeking to promote sustainable urban logistics.

4.1. Problem framework

This subsection introduces and analyzes the TDOEVRP-HERS in urban distribution. A retail company, lacking its own self-operated transportation fleets, delegates all package distribution tasks to a professional 3PL fleet. This 3PL fleet employs homogeneous EVs to deliver identical products to customers scattered all over a city. The objective is to minimize the total operating costs for the 3PL fleet, utilizing up to K vehicles. A complete graph G = {V,A} is defined where V = {0,1,…,n} is the set of all vertices, 0 stands for the depot, and A = {(i,j):i,j∈V and i≠j} is the set of arcs connecting nodes. Each arc (i,j)∈A has an associated distance d_ij. Unlike other studies, the infrastructure for EV energy replenishment in this study is bifurcated into two types: Charging stations (F) and battery swapping stations (B). So, is the set of customer nodes.

Each customer i∈C is assigned one and only one EV to fulfill their delivery requirement, with the customer possessing a non-negative demand D_i and a designated service time s_i. Specifically, before departing from the retail company’s depot, all EVs within the 3PL fleet load the maximum load capacity of W kilograms in goods for distribution. Concurrently, the EV fleet is fully charged and primed for action. During the transportation process, priority is accorded to EVs for customer service. However, should an EV find its energy insufficient to sustain the task, it must undertake an energy replenishment. According to the remaining delivery tasks, the EV can choose to replenish the battery level at either a charging station or a battery swapping station. Once the energy replenishment is completed, the EV resumes serving the remaining customers. Upon completion of all transportation tasks, the EV fleet can proceed to other enterprises for additional distribution tasks, without having to return to the retail company’s depot. The schematic of the TDOEVRP-HERS is depicted in Fig 1.

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Fig 1. Schematic of the TDOEVRP-HERS.

https://doi.org/10.1371/journal.pone.0291473.g001

In this problem, all the necessary parameters, such as EVs’ maximal load W and battery capacity Q, depot coordinates, customers’ location, delivery volume, and service time, are known. The study aims to optimize the 3PL fleet’s transportation routes, while meeting all customers’ distribution demands, and minimizing the total distribution costs that the retail enterprise has to pay. Distribution costs include EV fixed dispatch costs, travel costs, customer service costs, as well as charging and battery swapping costs. Please see S1 Appendix for all the parameters and decision variables used.

4.2. Assumptions

To clearly construct the MIP model, the ensuing assumptions are defined: (1) Each EV in the 3PL fleet is dispatched at most once, and the delivery volume of each customer is less than EV capacity. (2) Urban traffic network affects EV speed, which varies with time. (3) The energy expenditure of EVs is affected by a gamut of factors encompassing vehicular self-weight, travel velocity, real-time load, and traversed distance. (4) Once an EV enters a charging station or battery swapping station, it can directly replenish energy without having to wait in line. To ensure the timeliness of distribution, the logistics fleet limits each EV to enter a charging station only once, whereas the number of times it enters a battery swapping station is unlimited. (5) The charging station adopts a full charging strategy with a constant charging rate, while the battery swapping station has a fixed battery swapping time and the battery level of EVs becomes full upon completion of battery swapping.

4.3. TDOEVRP-HERS model

Based on the problem description and variables mentioned earlier, the optimization model of TDOEVRP-HERS is built below: (2) (3) (4) (5) (6)

The objective function (2) minimizes the total distribution cost for the 3PL fleet. This cost includes the fixed vehicle dispatching cost (P₁), travel cost (P₂), customer service cost (P₃), and energy cost (P₄). Notably, P₄ is the sum of charging and battery swapping costs. Expressions for these costs within the framework of TDOEVRP-HERS are presented in Formulas (3) to (6).

In tandem with objective function (2), the study addresses various constraints, organized into five distinct groups: arc and vertex constraints, vehicle capacity constraints, battery level constraints, time constraints, and binary constraints. The arc and vertex constraints are presented through constraints (7) to (10), while vehicle capacity constraints are articulated by constraints (11) and (12). The battery level constraints are captured by constraints (13) to (18), and time-related restrictions are laid out in constraints (19) to (22). Lastly, constraint (23) enforces a binary constraint.

(7)

(8)

(9)

(10)

Constraint (7) stipulates that each EV is dispatched at most once. By enforcing this restriction, the same EV is not redundantly assigned to multiple delivery routes, thereby exerting an indirect effect on the costs associated with vehicle dispatch (P₁). Constraint (8) necessitates that each client is visited uniquely once. In the context of the objective function, this constraint aids in minimizing the travel and service costs (P₂ and P₃), as it discourages unnecessary detours or repeat visits. Constraint (9) enforces flow conservation among customer locations, battery swapping stations, and charging stations. This constraint indirectly impacts energy costs (P₄) by influencing the distribution of EVs and their energy replenishment. Constraint (10) states that each EV is allowed to be recharged at most once on the route. Regarding the objective function, this constraint directly affects the energy cost component (P₄) by placing a cap on recharging events, which can influence the allocation of EVs and their corresponding energy consumption patterns.

(11)

(12)

Constraint (11) asserts that the duty payload apportioned to an individual EV must not transgress its supreme capacity. From the perspective of the objective function, this constraint influences the travel and energy costs (P₂ and P₄) by constraining the goods that each EV can carry. Inefficient or overloaded routes that could potentially inflate costs are avoided. Constraint (12) states that the load of an EV after serving a client is determined by subtracting the client’s demand from the load before service. This constraint directly impacts the customer service cost (P₃), as it influences the load adjustments after each service operation.

(13)

(14)

(15)

(16)

(17)

(18)

Constraint (13) mandates every EV to be completely charged prior to its departure from the depot. This constraint directly affects the energy cost (P₄), as it ensures that the initial energy state of each EV is at its maximum capacity, thus minimizing the need for immediate recharging or swapping. Constraint (14) expounds that the EV battery level remains unchanged at customer points. This constraint impacts the energy cost (P₄) by maintaining a consistent energy level throughout the service process, which in turn affects the EV’s overall energy consumption. Constraint (15) upholds the energy equilibrium of each EV during its transit from the preceding node i to the subsequent node j, considering the energy consumption along the arc (i,j). This constraint directly influences the energy cost (P₄) by accounting for the energy consumed during travel and distribution tasks. Constraint (16) ensures that EV’s remaining energy is not negative when it reaches any node. This constraint affects the feasibility of the solution and indirectly impacts the EV dispatching cost (P₁) by preventing scenarios where an EV’s energy level becomes insufficient for completing the assigned tasks. Constraint (17) defines the amount of charging at a charging station, which directly affects the energy cost (P₄) by determining the energy replenishment needed during the route. Constraint (18) indicates that when the EV departs from the battery swapping station, its battery level becomes full. This constraint impacts the energy cost (P₄) by ensuring that the EV commences its journey with a fully charged battery after swapping.

(19)

(20)

(21)

(22)

Constraint (19) embodies the correlation amongst the EV’s arrival time, service duration, and departure time at a customer vertex. This constraint affects both the travel cost (P₂) and the customer service cost (P₃) in the objective function. Constraint (20) signifies the temporal interdependence between the EV’s exit time from the present node i and its entry time at the ensuing node j. This constraint impacts the travel cost (P₂) within the objective function, as it makes the EV’s route go through different traffic conditions. Constraint (21) calculates EV’s charging duration at a charging station. This constraint directly affects the energy cost (P₄) within the objective function, as it influences the charging-related expenses incurred during the distribution process. Constraint (22) signifies the time relationship before and after the EV replenishes energy at a charging station or battery swapping station. This constraint impacts the objective function by regulating the timing of energy replenishment, thereby affecting both the energy cost (P₄) and the overall distribution efficiency.

(23)

Constraint (23), a binary constraint, introduces a discrete decision-making element into the optimization framework. The binary constraint adds a layer of restriction to the optimization process, as the decision to activate or deactivate certain components directly affects the distribution costs captured by the objective function. Depending on the problem specifics, constraint (23) might influence the allocation of vehicles to particular routes, the utilization of charging or battery swapping stations, or the selection of specific distribution tasks.

5.1. General framework of HALNS

Algorithm 1. The pseudocode of HALNS

1: ACO generates initial solution initialSol

2: Global optimal solution bestSol ← initialSol

3: Current solution currentSol ← initialSol

4: while the termination condition is not met do

5: Use roulette rules to choose the destruction operator and repair operator.

6: Use the destruction operator on currentSol, and then use the repair operator to repair it, and get an updated solution updateSol.

7: if updateSol is better than bestSol then

8:  bestSol ← updateSol

9:  currentSol ← updateSol

10: else if updateSol is better than currentSol then

11:  currentSol ← updateSol

12: else

13:  if simulated annealing criterion acceptance then

14:   currentSol ← updateSol

15:  end if

16: end if

17: Update the scores of destruction operators and repair operators.

18: if the number of iterations to update operator weights has been reached then

19:  Update the weights of destruction operators and repair operators, and reset the score to zero.

20: end if

21: end while

The proposed HALNS ingeniously integrates the ALNS technique with the ACO method, synergizing their respective strengths. The ALNS, with its capability for intensive local search, is coupled with the ACO’s global exploration potential. This fusion is achieved by initializing the solution process using ACO’s probabilistic approach, generating diverse initial solutions that serve as a foundation for ALNS to perform more refined local searches. The adaptability of ALNS complements the ACO’s explorative prowess, enabling it to fine-tune solutions and guide the search towards promising regions of the solution space. This harmonized approach leverages the ALNS to enhance the solutions found by ACO, ultimately contributing to improved convergence and solution quality. The implementation framework of HALNS is meticulously outlined in Algorithm 1. The process begins by designing an ACO to generate a higher-quality initial solution. Next, five destruction operators—Random Removal (RdR), Basic Worst Removal (BWR), Related Removal (RlR), Single Point Removal (SPR), and Station-based Removal (SbR)—are introduced. In addition, three repair operators are included: Improved Random Insertion (IRdI), Improved Greedy Insertion (IGI), and Improved Regret Insertion (IRgI). During the ALNS iteration, these operators are randomly selected to work on the current solution. Simultaneously, an appropriate acceptance criterion is established to adaptively adjust the operators’ weights and update the current and optimal solutions. The termination conditions for the algorithm may include the maximum number of iterations or the longest runtime.

5.2. The construction of the initial solution

The quality of the initial solution significantly influences the ensuing optimization of the ALNS, and a superior initial solution increases the likelihood of achieving a satisfactory solution. Given the characteristics of the TDOEVRP-HERS, the study employs the ACO to procure an improved initial solution. First, all ants are allocated to depot 0. Each ant then embarks on its journey. Based on the transition probability Formula (24), ants employ a roulette-wheel selection process to choose the customer nodes they visit. These selected customer points are added sequentially to the current route until the ants are no longer eligible to continue customer visits. At this point, the ants are reset to depot 0 and initiate their journey anew. This operation is repeated until all customers in the area have been visited by the ants.

(24)

where is the transition probability for ant m to travel from node i to node j. unvisit_m is the set of customer nodes that ant m has yet to visit. visited_m is the set of customer nodes that ant m has already visited. τ_ij is the pheromone heuristic parameter, and ϑ_ij is the expectation heuristic parameter, ϑ_ij = 1/d_ij. β₁ and β₂ are the weights for pheromone and expectation heuristic factors, respectively.

Once all ants have completed visiting all customers, a global pheromone update is executed on the current travel plan, comprising all ants, as per Formula (25). (25) (26) where is the pheromone before update, while is the updated pheromone. rho is the pheromone evaporation rate with 0≤rho<1. Set rho = 0.2. signifies the pheromone increment on arc (i,j) for ant m. f is a constant that represents the amount of pheromone secreted by an ant in each travel. Set f = 5. Distance_m is the total distance traveled by ant m.

The specific steps of the ACO are as follows:

Step 1: Parameter initialization. Begin by inputting the necessary test data, which include the location coordinates of all nodes in the transport network, customer delivery volume, EV’s maximum load and battery capacity, congestion time, and time-dependent vehicle speed. Let Iter_Ant represent the current iteration, set Iter_Ant = 1. Let maxIter_Ant represent the maximum number of iterations, set maxIter_Ant = 200. Let minLen represent the shortest distance among all ants’ travel, set minLen = Inf. Let antNum represent the number of ants, set antNum = 30. Let initialRoute represent the initial optimal travel route.

Step 2: Determine whether the Iter_Ant satisfies the relationship of Iter_Ant≤maxIter_Ant. If it satisfies this condition, assign ant m with m = 1, and proceed to Step 3 (Here, tabu_m is the ant m‘s taboo list, and tabu_m = ∅); otherwise, advance to Step 11.

Step 3: Determine whether m satisfies m≤antNum, and if so, enter Step 4; otherwise, skip to Step 9.

Step 4: Place ant m in depot 0 and assign it the maximum allowable load.

Step 5: Based on the transition probability (Formula 24), select a customer node that fulfills the constraints and add it into the current solution. Then, remove this node from ant m‘s unvisit_m set, and place it in ant m‘s tabu_m set.

Step 6: Repeat the operation of Step 5 until ant m no longer meets the requirements to visit any of the remaining customer nodes.

Step 7: Reposition ant m to depot 0 and reset ant m‘s maximum load. Determine whether the unvisit_m is an empty set. If unvisit_m ≠ ∅, revert to Step 5; otherwise, save ant m‘s total travel distance antLen_m to the iterLen_iter set, and save ant m‘s travel routes antSol_m to the iterSol_iter set, proceeding subsequently to Step 8.

Step 8: Let m = m+1, clear the taboo list tabu_m, reset the unvisit_m set, and then proceed to Step 3.

Step 9: Analyze all travel distances within the iterLen_iter set to derive the optimal travel distance bestLen_iter and the optimal travel routes bestRoute_iter. If bestLen_iter≤minLen, set minLen = bestLen_iter and update the globally optimal routes as initialRoute = bestRoute_iter. Conversely, if bestLen_iter>minLen, then set bestLen_iter = minLen and bestRoute_iter = initialRoute. Subsequently, pheromones along the route are updated following Formula (25) before progressing to Step 10.

Step 10: Let Iter_Ant = Iter_Ant+1, return Step 2.

Step 11: Following the principle of greed, insert the charging station or battery swapping station into the optimal initial route, initialRoute, according to the battery level constraints. This process culminates in the acquisition of the initial solution, initialSol, tailored for ALNS.

5.3. Operator design

In this study, a comprehensive set of destruction and repair operators are employed within the HALNS. In addition to the fundamental RdR, BWR, and RlR operators, the algorithm introduces two novel operators, SPR and SbR, within the destruction phase, thereby contributing originality to the approach. Moreover, to enhance the algorithm’s performance, three improved repair operators—IRdI, IGI, and IRgI—are adapted from the existing literature. This judicious combination facilitates the efficient restoration of solutions. The sequence of operations unfolds as follows: five destruction operators—RdR, BWR, RlR, SPR, and SbR—disassemble the currentSol, which is then reconstructed using the repair operators—IRdI, IGI, and IRgI. Throughout this iterative process, redundant charging or battery swapping stations within currentSol are methodically removed, and new energy replenishment strategies are reintegrated during the repair phase to address route infeasibility. For a visual depiction, refer to Fig 2, where the leftmost image presents currentSol, the central image demonstrates the outcome of applying the destruction operator (resulting in the removal of six customers and three energy replenishment stations), and the rightmost image showcases updateSol following the repair operator’s intervention.

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Fig 2. Schematic of the route destruction and repair.

https://doi.org/10.1371/journal.pone.0291473.g002

5.3.1. Destruction operator.

The destruction operator selectively removes a predetermined number of customer nodes from the current solution. Given a total customer count, n, the quantity r designated for removal is determined by the equation r = [α×n]. In this equation, α represents the removal ratio, and the [] is the rounding symbol. Set α = 0.1. The customers removed through this operation are subsequently added to the unvisited customer set, C_unvisit, while any invalid routes and superfluous charging or battery swapping stations are discarded. The general structure of destruction operators is offered in Algorithm 2.

Algorithm 2. The general framework of destruction operators

input: A currentSol and the number of requests to be removed r

output: A partial_solution

1: Initialize removal list (C_unvisit ← ∅)

2: Use roulette rules to randomly choose a destroy heuristic based on their weights

3: while termination criteria are not met do

4: Apply the selected destruction operator to remove a request r

5: C_unvisit ← C_unvisit∪{r}

6: end while

1. (1) RdR: Randomly remove r customer nodes from the current solution. Algorithm 3 illustrates the implementation framework of the RdR operator.

Algorithm 3. The general framework of the RdR operator

1: Initialize i ← 0, C_unvisit ← ∅

2: partial_solution ← currentSol

3: while i<r do

4: Randomly choose a route path_D to destroy, and randomly remove a customer node o from path_D.

5:  C_unvisit ← C_unvisit∪{o}

6:  If path_D has no customer nodes, delete path_D.

7:  i++

8: end while

9: Update partial_solution, i.e., remove nodes within C_unvisit from currentSol.

10: Remove redundant charging or battery swapping stations within partial_solution.

1. (2) BWR: The cost of a customer node i in the current solution is denoted as cost(s,i) = f(s)−f_−i(s). Here, f(s) represents the objective value of the current solution, and f_−i(s) is the objective value after removing customer node i. Initially, BWR calculates the objective value of the current solution. Subsequently, it computes the objective values after the removal of different customer nodes. After that, it derives the cost associated with each customer node. Finally, it selects the customer node with the highest cost for removal and repeats above step until r customer nodes have been removed. This strategy was proposed by Hemmelmayr et al. [71]. Algorithm 4 illustrates the implementation framework of the BWR operator.

Algorithm 4. The general framework of the BWR operator

1: Initialize removal list (C_unvisit ← ∅)

2: partial_solution ← currentSol

3: Compute cost(s,i) for all customers i∈C.

4: while |C_unvisit|<r do

5: Select the customer node o with the highest cost(s,i) for removal.

6: C ← C\{o}, C_unvisit ← C_unvisit∪{o}

7: Compute cost(s,i) for all customers i∈C.

8: end while

9: Update partial_solution, i.e., remove nodes within C_unvisit from currentSol.

10: Remove redundant charging or battery swapping stations within partial_solution.

1. (3) RlR: Initially, a customer node i is randomly selected for removal and placed in the C_unvisit set. The similarity function is then used to calculate the similarity between each customer node in the current solution and the removed node. The node with the highest similarity in the current solution is selected for removal and added to the C_unvisit set. Subsequently, a customer node is randomly selected from C_unvisit, and the similarity calculation process is repeated, leading to the removal of the node with the highest similarity. This step is repeated until r customer nodes have been removed. The similarity function used in this paper is defined as follows:

(27)

where α₁ and α₂ represent weight coefficients satisfying the condition α₁+α₂ = 1. Set α₁ = 0.6, α₂ = 0.4. d_ij signifies the distance between two customer nodes, and |D_i−D_j| represents the difference in delivery volume between two nodes. If nodes i and j are on the same route, then η_ij = 5; otherwise, η_ij = 0.

1. (4) SPR: SPR is an approach that involves the defined areas between two energy replenishment stations, or between a replenishment station and a depot, known as a station service area. This process initiates by randomly selecting a service area and a customer node within that area as an operational point. Subsequently, all customer nodes either to the left or right of the operational point within the service area are deleted. The number of removed nodes is counted as r_d, leaving a remaining removal capacity, denoted as r_a, where r_d≤r and r_a = r−r_d. If r_a>0, another station service area is randomly selected and the same removal operation is performed, with a simultaneous update of r_a. In the final round of the removal operation, if a situation where r_d>r_a arises, r_a nodes are randomly chosen for removal from the set of r_d nodes. Algorithm 5 illustrates the implementation framework of the SPR operator.

Algorithm 5. The general framework of the SPR operator

1: Initialize removal list (C_unvisit ← ∅)

2: partial_solution ← currentSol, r_a ← r

3: Count all service areas and put them into the set Serve.

4: while r_a>0 do

5: Randomly select a station service area i∈Serve.

6: Randomly select a customer node o within service area i.

7: All m customer nodes either to the left or right of node o within service area i are deleted.

8: r_d ← m, C_unvisit ← C_unvisit∪m, Serve ← Serve\{i}

9: if r_d>r_a

10:  r_a nodes are randomly chosen for removal from the set of r_d nodes.

11:  r_a ← 0

12: end if

13: r_a ← r_a−m

14: end while

15: Update partial_solution, i.e., remove nodes within C_unvisit from currentSol.

16: Remove redundant charging or battery swapping stations within partial_solution.

1. (5) SbR: SbR begins with the random selection of either a charging station or a battery-swapping station. From the route associated with this selected station, the nearest m customer nodes are removed, where m can be divisible by r. This procedure is repeated until r customer nodes have been removed. The study sets m = 10. Algorithm 6 illustrates the implementation framework of the SbR operator.

Algorithm 6. The general framework of the SbR operator

1: Initialize m ← 10, C_unvisit ← ∅

2: partial_solution ← currentSol

3: while |C_unvisit|<r do

4: Randomly select a station i∈F∪B.

5: Get routes associated with this selected station i.

6: Find the nearest m customer nodes on associated routes to station i and remove them.

7: C_unvisit ← C_unvisit∪m

8: end while

9: Update partial_solution, i.e., remove nodes within C_unvisit from currentSol.

10: Remove redundant charging or battery swapping stations within partial_solution.

5.4. Acceptance criteria

The acceptance criteria for this research utilize the simulated annealing approach as proposed by Adulyasak et al. [72]. During the iterative process, if the updated solution updateSol demonstrates better performance than the current solution currentSol, the updated solution is retained. Otherwise, the updated solution is maintained with a probability of . The initial temperature is set to T₀ = 10000 and follows a cooling schedule defined by T_n = cT_n−1, where the cooling rate c = 0.995. The entire search process concludes once the number of iterations reaches a predefined maximum limit.

5.5. The specific realization of the adaptive process

The fundamental principle of ALNS involves the adaptive adjustment of an operator’s weight based on its historical performance. Initially, all operators are assigned equal weight and score, with this study setting the initial operator weight to 10 and the score to 0. Throughout the algorithm’s iterative process, the destruction and repair operators are first selected following the roulette wheel rule. Subsequently, these operators are scored differently based on the quality of the updated solution (updateSol) after each iteration. When the number of iterations attains the pre-set threshold, the weights of the operators are updated. As a result, operators with greater weights are more likely to be selected in subsequent iterations.

The search process of ALNS employed in this study consists of N_a stages, each of which must execute N_b iterations. Thus, the maximum number of iterations, maxIter = N_a⋅N_b. Set N_a = 4, N_b = 50, i.e., maxIter = 200. Assume the weight of operator i in stage j as ω_ij and the usage probability of operator i in stage j as p_ij. Then, under the current weight, p_ij can be calculated by the formula , where H represents the set of operators to which operator i belongs. Define the score of operator i in stage j as ε_ij. In this study, four levels of scores are established, and the update rules for ε_ij are as follows: (28)

In the scoring scheme, the corresponding destruction and repair operators accrue points based on the quality of the updated solution relative to both the global optimal and the current solution. Specifically, if the updated solution surpasses the global optimal solution, the operators involved receive an additional 50 points. If the updated solution, while not surpassing the global optimal, improves upon the current solution, the operators gain 20 points. In cases where the updated solution is not an improvement over the current one but is still accepted, the operators are credited with 10 points. Lastly, if the updated solution neither improves upon the current solution nor is accepted, no points are added to the corresponding destruction and repair operators.

The update rules of the operator weights are as follows: (29) where ω_i(j+1) denotes the weight of operator i in stage (j+1), θ is the weight adjustment coefficient, indicating the importance of historical weight and operator performance when the operator weight is updated. Set θ = 0.3. Additionally, π_ij represents the number of times operator i is selected during the N_b iterations of stage j. Should operator i not be utilized during the current stage j, its weight remains unchanged in the subsequent stage. Following a weight update, both ε_ij and π_ij are reset to zero.

6.1. Data description

Currently, there is no standardized test dataset specifically tailored for TDOEVRP-HERS, and the geographical distribution of customers varies when 3PL fleets undertake transportation tasks for different enterprises. Consequently, this study refines Solomon’s VRP test sets [73], encompassing clustered distribution test sets (C-type), random distribution test sets (R-type), and randomly clustered distribution test sets (RC-type), utilizing these improved test instances as the experimental dataset. Notably, numerous scholars have generated new test instances based on Solomon’s test sets. For example, Schneider et al. [74] employed the Solomon test sets as a benchmark and introduced 21 charging stations into each test instance, thereby creating the EVRP experimental dataset they necessitated.

Building upon the work of Schneider et al., this study further improves the test instances by converting some of the charging station nodes into battery swapping station nodes. Therefore, new test sets tailored for TDOEVRP-HERS are constructed. Each test instance consists of one depot, 100 customers, five charging stations, and five battery swapping stations. The data provided in these test instances include the coordinates of all vertices, as well as the delivery volumes and service durations for each customer.

To meet the test requirements of this paper, the following data are supplemented: (1) Referring to the update frequency of the traffic congestion index in Beijing, set the duration of each time period as H = 15 minutes, thereby dividing the whole day into 96 time periods. (2) The earliest working time for the enterprise’s depot is set as 7:00 a.m. During the traffic peak hours of morning (8:00–9:00) and evening (17:30–19:00), the third-party EV fleet operated at a speed of 20 km/h on the road. (3) The EV fleet is assumed to operate with three time-varying speeds during normal time periods. Utilize the remainder function χ = mod(R,3) to determine these three different velocity values, where R represents the R-th time period. When χ takes the value 1, 2, or 0, respectively, it corresponds to the respective time-varying travel speeds of the EV as 54 km/h, 72 km/h, or 42 km/h. (4) The remaining parameters of the numerical experiment in this paper are set as follows: ϕ^d = 1.184692, φ^d = 1.112434, g = 9.8 m/s², θ_ij = 0°, C_r = 0.012, L = 3000 kg, R_c = 0.7, A_w = 3.8 m², ρ = 1.2041 kg/m³, W = 800 kg, Q = 50 kWh, η = 90%, p_e = 60 kW, φ = 10 min, c₁ = 120 yuan/vehicle, c₂ = 0.5 yuan/min, c₃ = 0.4 yuan/min, c₄ = 0.7 yuan/min, c₅ = 50 yuan/time.

The prescribed methodologies within the HALNS framework are implemented using the MATLAB R2020b software and performed on a microcomputer, furnished with a 3.60 GHz processor and a memory capacity of 16 GB of RAM.

6.2. Parameter tuning

An initial series of experiments is undertaken to fine-tune the parameters of the ALNS metaheuristic, as detailed in Table 2. For this purpose, 27 representative instances, as proposed by Ticha et al. [75], are chosen. These include the SOL instances r101, r105, c103, c104, rc101, and rc105, each with 50 customers, and the NEWLET instances 1, 2, and 3, each comprising 50 customers and 100 nodes. These nine instances are considered for three correlation tiers: No-Correlation (NC), Weak Correlation (WC), and Strong Correlation (SC).

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Table 2. Parameter values.

https://doi.org/10.1371/journal.pone.0291473.t002

Utilizing these instances, the following procedure is embraced. Initially, the parameters of the RlR heuristic are tuned. The ALNS scheme is then implemented, limited to this destruction operator and the IGI operator. Successively, attention is directed to one of the parameters, testing several values for it. For each value, the tuning instances are solved five times; the value that consistently manifests the best average solution quality is ultimately adopted.

For the SbR heuristic, an identical methodology is applied. Other parameters are systematically adjusted in a similar manner, but the comprehensive ALNS scheme is employed rather than relying on individual destruction and repair heuristics.

6.3. Experimental evaluation of the proposed model and algorithm

This section aims to validate the superiority of the proposed model and algorithm through five distinct experiments. Each experiment corresponds to a separate subsection and focuses on a specific aspect of the model or algorithm. Two experiments, found in Subsections 6.3.1 and 6.3.5, are dedicated to examining the effectiveness of the HALNS algorithm. Subsection 6.3.1 evaluates the efficiency of the HALNS in solving test sets featuring diverse geographic distributions of customers. Subsection 6.3.5 assesses the HALNS’s robustness to environmental disturbances, with a specific emphasis on its ability to achieve high-quality solutions amidst varying levels of traffic congestion. The other three experiments, detailed in Subsections 6.3.2, 6.3.3, and 6.3.4, are designed to validate the advantages of the TDOEVRP-HERS model. Subsection 6.3.2 investigates whether switching from FVs to EVs indeed leads to a reduction in the overall transportation costs of logistics fleets while also promoting a sustainable urban environment. Subsection 6.3.3 analyzes whether a hybrid energy replenishment strategy outperforms a single replenishment strategy. Finally, Subsection 6.3.4 examines whether employing a 3PL fleet is more economical and environmentally friendly than utilizing a self-operated fleet. In this way, the five experiments comprehensively evaluate the performance of the HALNS algorithm and the TDOEVRP-HERS model, providing valuable insights for practitioners and policymakers.

6.3.1. EV route planning for different test instances.

Various test instances, featuring diverse customer geographic distributions, serve to ascertain the feasibility of the HALNS proposed herein. Table 3 shows the test results, where TN signifies the test set name, TC signifies the total distribution cost incurred by the 3PL fleet (in yuan), FC signifies the fixed EV dispatching cost (in yuan), SC signifies the customer service cost (in yuan), DC signifies the vehicle travel cost (in yuan), EC signifies the EV energy replenishment cost (in yuan), EN signifies the number of energy replenishment times for all EVs, DN signifies the number of EVs enabled for distribution, RT signifies the running time of HALNS (in second), and AVE denotes the average value.

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Table 3. Test results for different test sets.

https://doi.org/10.1371/journal.pone.0291473.t003

The test results depicted in Table 3 reveal that: (1) The fixed EV dispatching costs, travel costs, and customer service costs account for an average of 29.27%, 39.48%, and 18.30% of the total distribution costs, respectively, totaling 87.04%. This finding suggests that vehicle fixed dispatching costs, travel costs, and customer service costs remain the primary factors affecting distribution costs in urban distribution. In practice, when enterprises engage 3PL fleets for urban distribution, they should require the fleet to use EVs with larger capacities to minimize the number of EVs dispatched. At the same time, to avoid increased travel costs due to traffic congestion, logistics fleets should consider the effect of time-varying vehicle speeds on travel time when performing distribution tasks. Furthermore, it is recommended that fleet managers train fleet members to reduce customer service time and improve service efficiency, which can not only reduce the service cost that enterprises have to pay for the fleet, but also improve the competitiveness of the transport fleet. These efforts can ultimately reduce the total distribution cost of the fleet and foster stronger collaborations between enterprises and 3PL fleets. (2) EV energy replenishment costs during distribution account for a range of 6.91% to 19.20% of the total distribution cost, with an average of 12.82%. It is noteworthy that the majority of EVs on distribution routes only require a single energy supplement at most to complete transportation tasks. Thus, using EVs for urban distribution has a negligible effect on distribution efficiency. Compared to traditional FVs, EVs incur lower energy consumption and offer additional advantages such as zero emissions and reduced noise levels. Embracing EVs for urban distribution can aid logistics fleets in reducing energy consumption and gaining cost edges, all while decreasing the environmental impact of transport activities. This aligns with the vision of policymakers who seek to promote green and sustainable development of urban distribution. (3) The HALNS’s running time ranged from a minimum of 211.47 seconds to a maximum of 281.32 seconds, with an average of 255.23 seconds. This result indicates that the HALNS proposed in this study can provide high-quality EV distribution and energy replenishment route planning solutions that meet decision objectives in a reasonable period, demonstrating both high efficiency and feasibility.

Fig 3 illustrates the EV distribution and energy replenishment route planning solutions for the C103, C203, R103, and RC203 test instances. The solutions for each test instance involving 100 customers are shown clearly and distinctly in the figure, with few instances of route detours and intersections. Based on test cases reflecting a realistic delivery scale, the proposed algorithm demonstrates its potential to consider various practical factors and offer valuable guidance for the transportation route optimization of logistics fleets.

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Fig 3. EV route planning solutions for different test instances.

https://doi.org/10.1371/journal.pone.0291473.g003

6.3.2. Comparative tests of TDOEVRP and TDOFVRP.

A comparative experiment is conducted to analyze the differences in urban distribution costs between the utilization of EVs and FVs. Multi-type test sets are employed while keeping relevant parameters unchanged. While FVs do not account for energy supplement time during distribution, they generate fuel consumption and carbon emissions. Following the methodology and experimental parameters employed by Liu et al. [20] for calculating fuel consumption and carbon emissions of FVs in their VRP study, the associated costs are incorporated into the distribution expenses of the logistics fleet. Table 4 presents the results of comparative experiments between TDOEVRP and TDOFVRP (Time-Dependent Open Fuel Vehicle Routing Problem), where TD signifies the total EV travel distance, GC signifies fuel consumption costs, and CC signifies carbon emission costs incurred by FVs. Remaining symbols maintain their previously established definitions.

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Table 4. Test results of EV urban distribution and FV urban distribution.

https://doi.org/10.1371/journal.pone.0291473.t004

The test results depicted in Table 4 reveal that: (1) Using EVs instead of FVs for urban distribution offers significant cost savings for logistics fleets. Specifically, substituting EVs for FVs leads to an average reduction of 31.03% in distribution costs, with the energy consumption cost for the logistics fleet decreasing from an average of 36.56% to 12.36% of the total distribution cost—a remarkable decline of 66.20%. This finding highlights the advantages of EV fleets in urban distribution, including energy savings and reduced operational expenses. To enhance competitiveness, logistics fleets should adopt EVs for their distribution operations, which aligns with society’s call for sustainable development within the logistics industry. Government departments should promote EV distribution by strengthening the construction and management of EV energy replenishment infrastructures and enhancing the convenience and safety of EV fleet use. (2) Using EVs for urban distribution leads to a mere 4.69% average increase in total distance traveled compared to using FVs. This finding can be attributed to the well-developed energy replenishment infrastructure in urban areas, which makes it easy for EVs to locate nearby charging stations or battery swapping stations. Additionally, using FVs for urban distribution generates an average carbon emission cost of 14.60 yuan, while EVs produce no carbon emissions, improving urban air quality and promoting sustainable logistics transportation. However, it is worth noting that the carbon emission cost of FV distribution accounts for only 0.92% of the total distribution cost, which is less than 1% of the proportion, indicating that logistics fleets do not prioritize reducing emissions when using FVs for urban distribution. This result suggests that China’s current carbon price is too low to incentivize logistics fleets to curtail emissions actively. Therefore, Chinese policymakers should formulate more reasonable carbon trading-related policies to promote sustainable logistics transportation.

6.3.3. Comparative tests of different energy replenishment strategies.

A comparative experiment is conducted to assess the effects of various energy replenishment strategies on EV distribution costs, including Hybrid Energy Replenishment Strategy (HERS), Pure Charging Strategy (PCS), and Pure Battery Swapping Strategy (PBSS). The experiment uses multiple types of test sets while keeping the other parameters unchanged. The comparative test results of the three energy replenishment strategies are presented in Table 5, where TT is the total distribution time for all EVs (in minutes), which encompasses EVs’ travel time on the route, customer service time, and energy replenishment time, and ET is the total energy replenishment time for all EVs (in minutes). Remaining symbols maintain their previously established definitions.

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Table 5. Test results for different energy replenishment strategies.

https://doi.org/10.1371/journal.pone.0291473.t005

The test results depicted in Table 5 reveal that: (1) In each test instance, the EV energy replenishment time achieved with PBSS is significantly lower than the other two strategies. Compared to HERS, PBSS reduces EV energy supplement time by a range of 51.30% to 69.25%, with an average saving of 58.20%. Compared to PCS, PBSS exhibits even more substantial savings in EV energy supplement time, with an impressive average time-saving of 79.80%. However, it should be noted that PBSS comes with higher distribution costs for logistics fleets. The total distribution costs incurred by PBSS are the highest among all test instances. On average, the use of PBSS increases the distribution cost of the logistics fleet by 4.74% compared to PCS. Therefore, when the 3PL fleet undertakes distribution tasks for enterprises, it is crucial to assess the urgency of customer package delivery. If there are time-sensitive customers, opting for battery swapping stations for EV energy replenishment can enhance delivery efficiency. On the other hand, if customers do not prioritize timeliness, charging stations can be utilized to reduce distribution costs. (2) Although HERS does not offer the optimal solution in terms of both total distribution cost and total distribution time, it exhibits a modest increase in distribution cost of only 2.65% on average compared to PCS, which has the lowest distribution cost. Similarly, compared to PBSS, which achieves the shortest distribution time, HERS shows a slight increase in distribution time by an average of 2.94%. Despite PCS being cost-effective, it requires an average of 5.69% more distribution time than HERS. Likewise, although PBSS reduces distribution time, it incurs an average of 1.90% higher distribution costs than HERS. These findings highlight the ability of HERS to strike a balance between total distribution costs and distribution efficiency, providing flexibility in energy replenishment operations during transportation tasks. Therefore, using HERS to support the urban distribution of EV fleets is a more scientific and reasonable approach.

6.3.4. Comparative tests of open routing and closed routing.

A comparative experiment is conducted to analyze the differences in urban distribution route planning between self-operated transport fleets and 3PL fleets. Multi-type test sets are utilized while keeping the relevant parameters unchanged. It should be noted that the self-operated transport fleet follows a closed routing, which requires returning to the enterprise’s depot upon completing the distribution task. Conversely, the 3PL fleet adopts an open routing, eliminating the need to return to the depot after task completion. The test results are presented in Table 6, where TCSR denotes the proportion of total distribution cost saved by hiring a 3PL fleet instead of a self-operated transport fleet (in %), and TDSR denotes the proportion of total distribution distance saved by hiring a 3PL fleet (in %). Remaining symbols maintain their previously established definitions.

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Table 6. Test results of open routing and closed routing.

https://doi.org/10.1371/journal.pone.0291473.t006

The test results depicted in Table 6 reveal that: (1) Engaging a 3PL fleet leads to an average savings of 6.20% and 5.61% in total distribution cost and time, respectively, compared to deploying a self-operated transport fleet. This advantage stems from the fact that when a third-party EV fleet is hired, EVs are not required to return to the company’s depot after completing the distribution task. As a result, the empty travel time of EVs is reduced, leading to decreased travel costs in logistics transportation. Therefore, for companies with an imperfect warehouse network layout, specifically lacking multiple depots, opting for a third-party EV fleet instead of a self-operated fleet can effectively reduce their operational expenses. (2) Hiring a 3PL fleet leads to a considerable reduction in EV travel distance compared to dispatching a self-operated transport fleet, with an average decrease of 9.35%. This finding highlights the potential for further optimizing EV distribution-energy supplement route planning when employing 3PL fleets, resulting in more efficient distribution solutions for EVs. Therefore, promoting this collaboration model is recommended for practical logistics distribution.

6.3.5. Robustness analysis of HALNS under different traffic conditions.

While keeping other experimental parameters constant, five distinct combinations of traffic conditions (Ⅰ, Ⅱ, Ⅲ, Ⅳ, Ⅴ) are designed. These conditions are outlined in detail in Table 7, where STC represents severe traffic congestion during the current time period, MTC represents mild traffic congestion during the current time period, TFS indicates that the traffic flows smoothly during the current time period, and the values provided represent the average travel speeds of EVs under the respective traffic scenarios (in km/h). Employing the RC107 test set for experimentation, the test results are presented in Table 8, where DT denotes the travel time on the routes (in min), and PU denotes the total energy consumption of all transport EVs (in kWh). Remaining symbols maintain their previously established definitions.

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Table 7. Average speeds of EVs in different traffic conditions.

https://doi.org/10.1371/journal.pone.0291473.t007

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Table 8. Test results of the TDOEVRP-HERS in different traffic scenarios.

https://doi.org/10.1371/journal.pone.0291473.t008

The test results depicted in Table 8 reveal that: (1) The enhancement of traffic conditions yields a gradual reduction in the travel time of EVs during transit. Transitioning from traffic scenario Ⅰ to scenario Ⅴ, the en-route travel time of EVs diminishes by 42.62%. This finding emphasizes the affirmative significance of ameliorated urban traffic on transport efficiency. (2) While the amelioration of traffic conditions boosts the transportation efficiency of 3PL fleets, it also leads to a notable escalation in energy consumption. Shifting from traffic scenario Ⅰ to scenario Ⅴ, the energy consumption of EVs rises by 190.26%. Heightened energy consumption necessitates more frequent energy supplements. Consequently, EVs fail to achieve the lowest total distribution cost in traffic transport scenario Ⅴ, but instead achieve it in transportation scenario Ⅲ. This is attributed to the fact that the transition from transportation scenario Ⅲ to scenario Ⅴ results in a 75.87% increase in energy replenishment costs, whereas the reduction in travel costs amounts to only 16.81%. The rate of increase in energy costs surpasses the decrease in travel costs. Thus, it becomes apparent that when planning the transportation route for the urban distribution fleet, fleet managers must establish a reasonable travel speed range to strike the optimal balance between efficiency and total cost. (3) In comparison to traffic scenario Ⅲ, which boasts the lowest total distribution cost, the average disparity between the total distribution cost of other traffic scenarios and scenario Ⅲ is 3.46%. This minor gap in distribution costs showcases the adaptability and route optimization capabilities of the HALNS based on real-time traffic conditions and time-dependent constraints. When traffic conditions deteriorate, the proposed HALNS prioritizes the delivery of customers in closest proximity to minimize travel time. Conversely, when traffic conditions improve, the HALNS adeptly schedules the insertion of energy-supplementary stations, effectively striking a balance between energy costs and congestion impacts.

6.4. Algorithm comparative test

To assess the efficacy of the proposed HALNS, a series of comparative analyses are performed, spanning two experimental categories: comparisons with an exact solver and other metaheuristics. For small-scale computations, outcomes from the HALNS are juxtaposed with those from the widely-recognized CPLEX solver, while for large-scale computations, HALNS’s optimized results are set against those from two renowned metaheuristics. These benchmarks facilitate a thorough appraisal of the HALNS’s performance across varying scales, underscoring its aptitude and preeminence in addressing large-scale optimization challenges.

6.4.1. Comparison with exact solver.

In order to validate the proposed mathematical model and assess the performance of the HALNS algorithm, the CPLEX solver version 12.6 is used to address small-scale test instances. These results are then juxtaposed against those produced by the HALNS. To ensure the generality of the experiment, selected subsets from Goeke’s dataset [76] are utilized, with certain charging station coordinates adjusted to battery-swapping station coordinates to accommodate experimental requirements. Each test instance is run through HALNS 10 times to discern the optimal result. The findings are tabulated in Table 9. Within the table, NC denotes the number of customers, NS indicates the number of charging stations, NB signifies the number of battery swapping stations, and ‘gap’ highlights the gap between the ALNS-derived solution and that from CPLEX. Remaining symbols maintain their previously established definitions.

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Table 9. Small-scale experimental results.

https://doi.org/10.1371/journal.pone.0291473.t009

Table 9 delineates that the HALNS is adept at efficiently tackling small-scale TDOEVRPs-HERS. Given the lower number of vertices, HALNS consistently identifies exact solutions that coincide with those derived from the CPLEX solver. However, considering the NP-hard nature of TDOEVRP-HERS, the computational duration for CPLEX escalates dramatically with increasing instance size. As a result, CPLEX’s average execution time for all test instances clocks in at 105.83 seconds, in stark contrast to HALNS’s mere average of 2.26 seconds. This observation accentuates HALNS’s proficiency in navigating towards optimal solutions while preserving exceptional efficiency.

6.4.2. Comparison with other metaheuristics.

To further validate the superiority of HALNS, an Ant Colony System (ACS) and a Genetic Algorithm (GA) are developed to tackle the TDOEVRP-HERS. The parameter settings of ACS align with the work studied by Li et al. [77], while the parameter settings of GA follow the experimental setup of Park et al. [78]. Multi-type test sets are used to compare the performance of HALNS, ACS, and GA. Table 10 reports the test results, with the symbols retaining their previously defined meanings.

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Table 10. The results of algorithm comparative test.

https://doi.org/10.1371/journal.pone.0291473.t010

The test results depicted in Table 10 reveal that: (1) In each test instance, HALNS consistently outperforms ACS and GA in terms of total distribution cost and travel distance. On average, HALNS achieves savings of 3.37% in distribution cost and 3.18% in travel distance compared to ACS, and 5.09% in distribution cost and 4.31% in travel distance against GA, respectively. Fig 4 visually illustrates the gaps in the objective function values among the three algorithms, clearly showing HALNS routinely yielding the most favorable objective function values. Meanwhile, Fig 5 showcases the EV distribution route schemes generated by HALNS, ACS, and GA for test set C208. Unmistakably, the HALNS-generated solution delineates the most coherent distribution routes with minimal situations of route detours and intersections. In addition, the energy replenishment stations selected in the HALNS solution are conveniently located near the served customers, while the solutions generated by ACS and GA allocate some energy supplement stations far away from customers to the transport fleet, suggesting that ACS and GA may not reasonably optimize the energy supplement route in transportation, which will affect distribution timeliness and lower customer satisfaction. These findings exhibit the effectiveness of HALNS in optimizing logistics distribution routes, improving logistics distribution efficiency, and reducing logistics distribution costs. (2) Despite the strong optimization capabilities of the HALNS, it demands an average of 17.58% more running time than the ACS and 15.52% more than the GA. Fig 6 highlights the time-consuming calculation burden of the three algorithms through a line graph when processing different test instances. This result can be attributed to three key aspects of the HALNS: operational complexity, depth of search, and adaptivity overhead. Firstly, HALNS integrates multiple destroy and repair heuristics, enabling it to explore a broader solution space and thereby increasing its potential to uncover high-quality solutions. However, the utilization of these operators tends to be computationally taxing, and thus extends the overall processing time. Secondly, the adaptive nature of the HALNS means that it will often perform a more thorough and exhaustive search of the solution space, as compared to ACS or GA. This deeper exploration can yield better results but requires more processing time. Finally, unlike simpler algorithms like ACS or GA, HALNS incorporates an adaptivity mechanism where it modifies the use of different operators based on their performance. This feature enhances the flexibility and robustness of HALNS, but it contributes to the increased running time as the algorithm needs to constantly evaluate and adjust the weights assigned to various heuristics. Consequently, optimizing the solving efficiency of HALNS, without compromising solution quality, will significantly influence its broader application in future contexts.

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Fig 4. Objective function values of each algorithm for solving large-scale problems.

https://doi.org/10.1371/journal.pone.0291473.g004

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Fig 5. The EV distribution solution of each algorithm.

https://doi.org/10.1371/journal.pone.0291473.g005

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Fig 6. The processing time of each algorithm for solving large-scale problems.

https://doi.org/10.1371/journal.pone.0291473.g006

6.5. Sensitivity analysis

Upon validating the superior performance of the proposed HALNS, sensitivity analyses are performed on the destruction and repair operators to assess their efficiency and influence on computational costs. For these evaluations, test set R101 is specifically chosen, and the average changes between the initial and final solutions, as determined by the HALNS algorithm, are calculated. These findings are tabulated in Table 11. To scrutinize the efficacy of each operator, the HALNS is executed using a singular pair of destruction-repair operators, and improvements in the solutions are documented. A subsequent analysis, focusing on the individual performance of the destruction and repair operators, processes the average changes noted in Table 11 for each operator and visually represents them in Fig 7.

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Fig 7.

Performance of destruction (a) and repair (b) operators.

https://doi.org/10.1371/journal.pone.0291473.g007

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Table 11. Sensitivity analysis on the performance of destruction and repair operators.

https://doi.org/10.1371/journal.pone.0291473.t011

Results from Table 11 indicate that the optimal pair of heuristics is the combination of RlR with IGI and IRgI. These two pairings yield an average enhancement of approximately 120% from the initial to the final solution. Upon examining Fig 7, it is evident that RlR serves as the most effective destruction operator (Fig 7(A)), while IGI stands out as the superior repair operator (Fig 7(B)), showcasing a 12.20% overall improvement when compared to IRdI.

In conclusion, the operators employed in the proposed HALNS are pivotal in shaping computational costs. They enhance search ability by adeptly selecting and integrating elements during the solution formulation. By carefully choosing the most fitting destruction and repair operators, the HALNS can traverse the solution space with greater efficacy and attain marked enhancements in solution quality with diminished computational demands.

6.6. Discussion

Optimization algorithms have evolved remarkably, fostering advancements across various domains, ranging from online learning, scheduling, and transportation to medicine and data classification. Given the increasing complexity of contemporary decision problems, incorporating advanced algorithms will be instrumental in deriving more effective solutions. The decision problem tackled in this research, the TDOEVRP-HERS, predominantly relied on the HALNS algorithm. While HALNS offers significant merits, it becomes imperative to juxtapose it with other advanced optimization methods to foster comprehensive solution frameworks.

Advanced optimization techniques, such as hybrid heuristics, metaheuristics, hyperheuristics, self-adaptive algorithms, island algorithms, and polyploid algorithms, bring forth specialized capabilities, enabling researchers to address the nuanced facets of complex decision problems. Their adaptability and customizability render them suitable for a plethora of applications. For instance, an online-learning-based evolutionary many-objective algorithm has been manifested to optimize decision-making in dynamic environments [79]. Adaptive polyploid memetic algorithms have demonstrated effectiveness in scheduling applications, such as optimizing truck schedules at cross-docking terminals [80]. In the realm of VRP, algorithms that accommodate both exact and metaheuristic methodologies have been applied, ensuring robust solutions even in multi-objective settings [81].

Given the myriad applications of these algorithms, their relevance to TDOEVRP-HERS cannot be understated. Hybrid meta-heuristic algorithms, for instance, have shown considerable promise in supply chain network designs under uncertain conditions [82], suggesting potential adaptability for electric vehicle routing under time-varying traffic networks. Similarly, the customizability offered by hyper-heuristics can be of paramount importance, especially when devising metaheuristics for continuous optimization [83]. This points towards their potential utility in fine-tuning the solutions for TDOEVRP-HERS.

In conclusion, while the HALNS has proven its merit in this research, the vast landscape of advanced optimization algorithms holds significant promise. By embracing a multifaceted approach that incorporates insights from various algorithms and domains, the research community can pave the way for more affluent, nuanced solutions to intricate decision problems like TDOEVRP-HERS.