2.1. Electric vehicle routing problem

The EVRP aims to determine the optimal route for EVs considering various constraints and operations [10]. Currently, several variations of EVRP have emerged, including incorporating time windows into the routing problem [11], considering energy consumption as a key factor in route optimization [12], and addressing chance-constrained multi-compartment scenarios [13]. As recharging processes typically entail significant time, various recharge strategies have garnered widespread attention. The primary energy supply modes for EVs consist of two types: recharging [14–16] and battery replacement. The former can be broadly categorized into three groups [17]: full charging, linear partial charging, and nonlinear partial charging [18–20]. While EVs offer a cleaner option for sustainable development, their limitations in mileage and electric power have prompted researchers to explore the integration of traditional fuel vehicles and EVs. One way is to assemble a mixed fleet of several vehicle types [21,22]. Another way is hybrid electric vehicles (HEVs) [23,24], consuming both electricity and gasoline.

2.2. Two-echelon vehicle routing problem

The early literature on 2EVRP primarily focused on the two-echelon capacitated vehicle routing problem [25–28], with the prevalent use of branch and cut algorithms [29,30]. The literature in recent years has witnessed a growing body of research focusing on diverse transportation options, with particular emphasis on the integration of conventional and contemporary tools. Liu et al. [31] proposed a combination of conventional trucks and automatic delivery robots to optimize transportation costs and reduce emissions. Ding et al. [32] introduced a 2EVRP involving electric and conventional vehicles for transporting common goods. Faiz et al. [33] employed fuel trucks for the first echelon distribution while utilizing drones for the second echelon distribution during post-disaster rescue operations. Further breakthroughs have also been made in heuristic algorithms. Enthoven et al. [34] used an adaptive large neighborhood search heuristic method to solve a 2EVRP with intermediate position options. Anderluh et al. [35] employed a metaheuristic that integrated large neighborhood search into a heuristic cuboid splitting approach. The study conducted by Sherif et al. [36] explored green transportation in the battery industry through the utilization of a simulated annealing algorithm and an exchange neighborhood-solving method. Lehmann and Winkenbach [37] employed exact methods for the first stage, followed by heuristic algorithms for the second stage. Despite the availability of multiple approaches, selecting the most optimal one remains challenging [38].

2.3. Vehicle routing problem for cold chain logistics

In the context of cold chain logistics for the vehicle routing problem (VRP), sustainability discussions primarily revolve around establishing indicators such as carbon emissions, carbon trading and carbon taxes as cost targets [39–41]. Additionally, comprehensive assessments encompassing social, economic and environmental aspects are conducted [42]. Research on two-echelon distribution in cold chain logistics has been relatively limited. Govindan et al. [43] introduced a 2EVRP model incorporating time windows for the perishable food supply chain and successfully addressed it using a hybrid metaheuristic algorithm. Wang et al. [44] examined a two-echelon logistics framework involving small vehicles and semi-trailer trucks, highlighting the cost advantages of an enhanced ant colony optimization algorithm. Yan et al. [45] conducted a comparative analysis of solutions for different objectives in a location-routing problem. Al Theeb et al. [46] discussed the 2EVRP of vaccines, where vehicles were divided into large and small sizes. While these studies considered a heterogeneous fleet, vehicles in each echelon mainly differed in capacity, rarely in energy type. The literature has examined the single-echelon path problem, which involves a mixed fleet of electric and conventional vehicles (with only one type per service [3,5,47]). In terms of the 2EVRP for cold chain logistics, Zhang et al. [48] proposed a simplified two-stage hybrid transportation scenario that does not incorporate charging strategy. The summary of previous review studies is given in Table 1.

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Table 1. Summary of previous review studies.

https://doi.org/10.1371/journal.pone.0318765.t001

As discussed above, owing to current technical limitations, EVs still face challenges such as mileage anxiety and longer recharging times compared to refueling. These factors make them less suitable for long-distance distribution. Limited attention has been given to the integration of a mixed fleet of fuel vehicles and EVs. Additionally, the exploration of the partial recharge strategy remains scarce in the context of 2EVRP [49]. Therefore, this paper aims to introduce a novel concept: the 2EVRP with a Mixed Fleet (2EVRPMF) for cold chain logistics. The distinctive features of this research include:

1. The use of fuel vehicles and EVs in the first and second echelons, respectively, forming a mixed fleet.
2. The adoption of different recharge strategies, namely partial recharge and full recharge, in the 2EVRPMF. An algorithm for selecting a recharging upper limit is developed specifically for the partial recharge strategy.
3. The introduction of a customized and enhanced Ant Colony Optimization method for route planning within the 2EVRPMF.

3.1. Problem description

The paper introduces a two-echelon distribution network. The primary echelon comprises a solitary distribution center and multiple transfer stations (satellites). These stations are serviced by a fleet of fuel vehicles. In the secondary echelon, there are additional transfer stations, self-pick-up points (customers), and recharging stations. The total number of EVs is fixed, although the specific allocation for each EV is not predetermined. Following the policy that establishes a 1:1 ratio between the number of new public charging stations and newly promoted EVs in the public sector [50], EVs have the option to visit nearby recharging stations if their charge level becomes low (as depicted in Fig 1). The coordinates of the distribution center, transfer stations, self-pick-up points, and recharging stations are provided. The demand for self-pick-up points is specified. The demand for transfer stations is calculated as the sum of the demands for the served self-pick-up points. Several assumptions are made to facilitate the study, including:

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Fig 1. Two-echelon distribution network.

The definition of sets, parameters, and variables of this 2EVRPMF is shown in Table 2.

https://doi.org/10.1371/journal.pone.0318765.g001

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Table 2. Mathematical notations.

https://doi.org/10.1371/journal.pone.0318765.t002

1. (1). The distribution center is well-stocked with sufficient goods.
2. (2). The demand of transfer stations is divisible, allowing goods at a transfer station to be delivered by multiple vehicles if necessary.
3. (3). The demand of self-pick-up points is indivisible, meaning each self-pick-up point can only be served by one vehicle at a time.
4. (4). Fuel and battery discharge are independent of load and solely depend on the distance traveled.
5. (5). Each transfer station is equipped with a recharging station, and EVs leave the transfer station with a 100% State of Charge (SoC).
6. (6). Service times and time windows at the self-pick-up points are not considered in the model.

3.2. The optimization model

The objective of the mathematical model for 2EVRPMF is to minimize the overall cost, encompassing fixed costs, energy consumption costs, deterioration costs and carbon emission costs.

3.2.1. Fixed cost.

The fixed cost of a vehicle encompasses expenditures for depreciation, driver remuneration and other associated outlays. This can be expressed as

(1)

3.2.2. Energy consumption cost.

The energy consumption is derived from both the transportation and refrigeration processes, encompassing three distinct components: fuel consumption in the primary echelon, electricity consumption in the secondary echelon and electricity consumption for refrigeration at each echelon. The cost associated with energy consumption can be expressed as

(2)

3.2.3. Deterioration cost.

Deterioration of perishable goods occurs during both the transportation and unloading processes. For simplification purposes, it is assumed that the service time at the customer node is zero. Therefore, the cost of deterioration is considered as damage incurred during transportation.

The arrival time at transfer stations in the first echelon is influenced by both the serving sequence and assigned vehicles. This phenomenon can be mathematically represented as follows

(3)(4)

The arrival time at each node in the second echelon is considered under two scenarios when EVs are utilized, presented as

1) the nlformer node is a customer.

(5)

2) the former node is a recharging station

(6)

According to the literature [51,52], a commonly utilized function is introduced for assessing the quality of refrigerated goods, given as

(7)

To specify the amount of goods delivered by fuel vehicles in the first echelon, is introduced as it is related to the deterioration cost.

(8)

3.2.4. Carbon emission cost.

The combustion of fuel oil and the generation of electricity both result in direct or indirect carbon emissions. In light of the current government-led carbon trading system, it is imperative to establish a unit price for carbon emissions. Consequently, this unit price serves as an effective means to curb carbon emissions, presented as

(9)

The objective is to propose an optimization model for 2EVRPMF, to minimize the overall cost including fixed costs, energy consumption costs, deterioration costs, and carbon emission costs, given as.

(10)

s.t.

(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)

Eqs. (11) and (12) ensure that the number of vehicles utilized in each echelon does not exceed the total available vehicle count. Eqs. (13) and (14) guarantee a balanced outflow for all nodes within each echelon. Eq. (15)–(18) represent the regulations governing each echelon. Eq. (15) stipulates that one transfer station is visited by vehicles from the first echelon, with each vehicle serving it no more than once. Eq. (16) indicates that one customer or recharging station is visited by vehicles from the second echelon and served only once. Eq. (17) implies that it is not permissible to directly visit a recharging station from transfer stations. Eq. (18) indicates that consecutive visits to two recharging stations are prohibited. Eq. (19) signifies that the quantity of goods required to be transferred by each transfer station equals the total demand of customers to be served. Eq. (20) elucidates that the amount of goods needed to be transferred by each transfer station is determined by the vehicles in the first echelon. Eq. (21) elucidates that the amount of goods needed to be transferred by each transfer station is determined by the vehicles in the first echelon. Eqs. (22) and (23) demonstrate the balanced outflow of each node, excluding the depot where the outflow equals the total demand of transfer stations. Additionally, for second echelon transfer stations, the outflow corresponds to their assigned unknown demand. Eqs. (22) and (23) also serve to prevent sub tours. The capacity constraints are formulated in Eqs. (24) and (25) respectively. Eqs. (26) and (27) indicate vehicles returning to depots or transfer stations without any load. Different states of charge (SoC) of battery upon leaving a node are illustrated in Eqs. (28)–(30): when departing from transfer stations, the SoC is full; when departing from recharging stations, it is limited by EVs’ recharging limit. For customer nodes, the SoC status remains unchanged from when the vehicle arrives. Eq. (31) represents the constraint on electricity consumption. Eq. (32) indicates that the safe SoC for EVs upon reaching a customer is . Eq. (33) calculates the amount of recharge required for an EV at a recharging station. Eq. (34) states that the transfer station can only serve nodes in the second echelon if it is served by vehicles from the first echelon. Furthermore, Eq. (35) ensures that transfer station k serve for customer or recharging station j only when vehicle e serves j and is used by k. Eq. (36) presents the range of recharging limits for two different recharge strategies: partial recharge and full recharge. Finally, Eqs. (37)–(42) specify the domain of variables.

4.1. Vehicle allocation

The proposed paper presents a vehicle allocation algorithm based on distance to determine the number of EVs assigned to each transfer station, thereby dividing the MDEVRP problem. The fundamental concept is outlined as follows:

1. 1). The demand of each transfer station is calculated. The demand for each transfer station is calculated using , where d represents the demand and represents the capacity of EVs. Here, n denotes the number of EVs allocated in the first-round plan.
2. 2). Any remaining unsatisfied demand is determined. A loop statement is employed to compare and incrementally assign one additional EV to the transfer station with the highest remaining demand. This iterative process persists until all EVs are assigned or there are no unfulfilled demands left.

4.2. Distribution route selection

Due to inherent characteristics of positive feedback, heuristic search, and enhanced solving efficiency, Ant colony optimization algorithm (ACO) proves to be an invaluable tool in addressing 2EVRP [53,54]. Consequently, this present study employs ACO as a methodology for determining distribution paths in two echelons.

4.2.1. Transfer probability.

The transfer probability serves as a crucial determinant for ants in selecting the next node. which is established based on the pheromone levels and heuristic function of the path connecting the previous node and the candidate node, depicted as

(43)

where is pheromone. Initially, each path has a pheromone value of 1. This value is subsequently updated through the cumulative selection of ants and the volatilization process. The heuristic function represents the expectation of ant transfer from node i to node j. A smaller distance implies a higher expectation. The weight coefficients α and β determine the influence of pheromone and heuristic function respectively. Finally, denotes a collection of nodes that an ant k needs to visit.

4.2.2. Pheromone update.

The pheromone release by each ant occurs along the path it traverses. Over time, the accumulation of pheromone concentration on the optimal route gradually intensifies. Consequently, upon completion of a full cycle by all ants, the total sum of released pheromone concentrations on each path can be determined.

(44)(45)

where represents the concentration of pheromones released by ant k on the path between node i and node j. When ant k travels from node to node j, this concentration is calculated as the total amount of pheromones Q released by the ant in a cycle divided by the length of all possible paths taken, with any remaining value equal to 0.

The pheromone concentration in the original path gradually diminishes over time, while simultaneously, all ants complete a cycle. Consequently, the new generation exhibits a reduced level of pheromone concentration, presented as

(46)

where ρ represents the factor of pheromone volatilization, and t denotes the number of generations.

4.2.4. Measures improvement.

In order to enhance the global convergence of ACO, this paper introduces a genetic mutation process and a pheromone rollback mechanism. The genetic mutation process is executed at an end of each iteration [55], where two nodes from the best routes found in that iteration are exchanged to achieve better results. The number of mutations is correlated with the frequency of pheromone rollbacks, resulting in faster problem-solving speed. Additionally, the pheromone rollback mechanism occurs when consecutive iterations yield identical optimal routes. It rolls back the pheromone levels of these routes to their earliest occurrence while simultaneously adjusting the volatilization factor. This improvement can be expressed as

(47)(48)

where c represents the count of roll back processes, denotes the increment of volatilization factor, signifies the maximum value of volatilization factor, and n indicates the number of consecutive iterations that remain unchanged.

4.3. Recharging limit determination

The determination of a recharging limit method should be investigated when implementing the partial recharge (PR) strategy. To simplify this determination process, a candidate matrix for the recharging limit is set as within an optional range. The steps for this determination process are outlined as follows:

1. establish a pheromone matrix that defines a recharge limit, records the pheromone trail of ants from a customer depot to a specific recharging station and ensures recharging up to a predetermined upper threshold;
2. employ the recharging limit pheromone matrix and the roulette method to determine the specific recharging station and its corresponding upper limit when approaching a recharging point;
3. determine the SoC of the EVs upon arrival at station i, and calculate the required amount of electricity for recharging;
4. if is negative, continue with step 2)-3) until becomes positive;
5. the recharging limit pheromone matrix is updated after each iteration by incorporating the route of each ant that generates a feasible solution.

5.1. Benchmark instances

The benchmark sets utilized in this experiments are derived from the 2EVRP instances proposed by Perboli et al. [27], Hemmelmayr et al. [56] and Baldacci et al. [57] To accommodate the distinctive characteristics of EVs employed as city freighters, including the distribution and quantity of recharging stations, as well as the nominal battery capacity, this study adopts the approach suggested by Schneider et al. [58] and Breunig et al. [6] in EVRP. Specifically, the number of charging stations is determined by rounding one-fifth of the total customer count. The placement of recharging stations prioritizes transfer stations with recharging capabilities. Any remaining stations are then randomly distributed within a range that does not exceed the locations of customers. The nominal battery capacity is determined as , where D represents the maximum distance between customers and their nearest recharging station.

The paper investigates the establishment of a self-established logistics network by company A in Beijing [59]. The determination of carbon price is based on data sourced from China’s official carbon trading website [60], combined with an evaluation of carbon emissions per unit energy consumption across various energy sources [61]. Furthermore, the parameters for fuel vehicles in the first echelon and EVs in the second echelon are defined based on commonly employed types of cold chain distribution vehicles in China. The parameters valued are given in Table 3.

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Table 3. Parameters valued.

https://doi.org/10.1371/journal.pone.0318765.t003

5.2. Parameter tuning

The results can be significantly influenced by the parameters of the heuristic algorithm. Therefore, in line with the tuning approach employed by Keskin and Catay [62] as well as Jie et al. [63], this paper selects three instances from benchmark sets with varying sizes (Set2a_E-n22-k4-s6-17, Set2c_E-n51-k5-s6-12-32-37, Set5_100-5-1b) for parameter optimization. The experiment is conducted ten times for each instance. The average percentage deviation from the optimal solution is then calculated across all solutions for each parameter. Subsequently, adjustments are made to minimize this average percentage deviation of the parameter. This iterative process continues until calibration is completed for all parameters. The initial values of each parameter were determined based on previous literature [64].

The parameter tuning results are presented in Table 4. It is important to note that the weight coefficients of pheromone and heuristic function, denoted as α and β respectively, are adjusted simultaneously as a pair.

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Table 4. Final values of parameter.

https://doi.org/10.1371/journal.pone.0318765.t004

5.3. Results discussion

The routing method is utilized to calculate two distinct distribution strategies based on the case of company A’s self-established logistics network in Beijing. Each recharge scheme is executed five times. Table 5 presents the outcomes of the full/partial recharge scheme, with recharging station nodes highlighted in bold font. RUL represents the abbreviation for recharging the upper limit. The optimal outcome’s route map is illustrated in Fig 2. The analysis reveals that both the full and partial recharge schemes require a fleet of 3 vehicles in the first echelon and 10 vehicles in the second echelon to ensure efficient distribution. Additionally, enroute recharging is required for 5 EVs in the second echelon. In comparison, the full recharge strategy holds a slight advantage over the partial recharge strategy due to its lower overall cost. Furthermore, it is noteworthy that each station’s optimal solution under the partial recharge scheme indicates an approximate 100% recharge amount.

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Table 5. The optimal result.

https://doi.org/10.1371/journal.pone.0318765.t005

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Fig 2. The optimal route of FR (above) and PR (below) scheme.

https://doi.org/10.1371/journal.pone.0318765.g002

In order to determine the optimal scheme for this specific case, the average costs of five runs in each scheme are compared and presented in Table 6. The limited literature on mixed fleet scenarios in a two-echelon cold chain network necessitates an examination of the effectiveness of the 2EVRPMF model through analysis of a case involving a diverse fuel vehicle fleet. In Table 6, ‘MF’ refers to the full recharge scheme for a mixed fleet, while ‘HF’ represents a fuel heterogeneous fleet.

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Table 6. Average result of 5 runs with MF and HF.

https://doi.org/10.1371/journal.pone.0318765.t006

The results in echelon 1 demonstrate comparable performance between MF and HF, as both employ fuel vehicles at this tier. In echelon 2, the energy consumption cost of MF is significantly lower by 42.20% compared to HF due to the considerably higher unit price of fuel relative to electricity. Furthermore, the utilization of EVs in echelon 2 leads to a remarkable reduction of 69.16% in carbon emission costs. Although HF holds a slight advantage concerning deterioration costs and distribution time, there remains a noticeable disparity in total costs when compared with MF. Hence, it becomes apparent that this proposed model for 2EVRPMF exhibits superior performance, as it not only reduces overall expenses but also contributes towards sustainable development through decreased carbon emissions.

5.4. Sensitivity analysis

To establish an appropriate policy for the 2EVRPMF model, it is crucial to investigate the performance of each scheme under various scenarios. Considering the inherent connection between recharge schemes with EVs and recharging stations, factors such as battery capacity and recharging rate must also be taken into consideration. Consequently, a sensitivity analysis has been conducted specifically focusing on evaluating the effectiveness of FR and PR schemes within three instances (E-n51-k5-s6-12-32-37, E-n51-k5-s2-4-17-46 and E-n51-k5-s2-17) selected from benchmark instance Set 2c.

5.5. Managerial insights

This paper presents a two-stage hybrid fleet routing model for perishable goods, to optimize fixed delivery costs, energy consumption costs and deterioration costs to the greatest extent possible. The study also takes into account the impact of both full and partial charging on the model’s outcomes. It demonstrates that utilizing fuel vehicles in suburban areas and electric vehicles in urban areas is not only cost-effective but also environmentally friendly. The number of vehicles required for different charging strategies is comparable, with only minor variations in overall cost performance observed. The capacity of charging stations and battery capacity of electric vehicles impose limitations on total costs. These findings can assist enterprises in effectively managing their fleets based on local charging station infrastructure and EVs technology, ultimately leading to long-term cost optimization.