A Tailored Meta-Heuristic for the Autonomous Electric Vehicle Routing Problem Considering the Mixed Fleet

In recent years, some phenomena such as the COVID-19 pandemic have caused the autonomous vehicle (AV) to attract much attention in theoretical and applied research. This paper addresses the optimization problem of a heterogeneous fleet that consists of autonomous electric vehicles (AEVs) and conventional vehicles (CVs) in a Business-to-Consumer (B2C) distribution system. The absence of the driver in AEVs results in the necessity of studying two factors in modeling the problem, namely time windows in the routing plan and different compartments in the loading space of AEVs. We developed a mathematical model based on these properties, that was NP-hard. Then we proposed a hybrid algorithm, including variable neighborhood search (VNS) via neighborhood structure of large neighborhood search (LNS), namely the VLNS algorithm. The numerical results shed light on the proficiency of the algorithm in terms of solution time and solution quality. In addition, employing AEVs in the mixed fleet is considered to be desirable based on the operational cost of the fleet. The numerical results show the operational cost in the mixed fleet decreases on average by 57.22% compared with the homogeneous fleet.


I. INTRODUCTION
One of the most important causes of CO2 emissions and global energy consumption is transportation systems with a share of about 20-25%. 26% of total greenhouse gas (GHG) emissions belonged to fossil fuel usage in the transportation system of the United States in 2014 [1]. Additionally, trucks comprised 74% of domestic transportation in 2012, and scholars have predicted that the transportation volume will increase by 39% over the next 20 years. According to the statement of the European Union, a significant share of 23.2% of the total GHG emission was related to transportation in 2014 and this is expected to increase by about 80% in 2050 compared with 1990. Hence, in the future, transportation will become the main source of GHGs. Many countries have proceeded with new environmental standards The associate editor coordinating the review of this manuscript and approving it for publication was Kuo-Ching Ying . and targets to mitigate greenhouse gas emissions and rely on fossil fuels. Therefore, they start to set new regulations to encourage the use of alternative fuel vehicles, such as solar, electric, biodiesel, LNG (Liquefied Natural Gas), and CNG (Compressed Natural Gas) vehicles. In this regard, several governments and vehicle manufacturers have moved towards the production of alternative fuel vehicles [2]. In the scientific literature, this issue has also been attended to by researchers. Accordingly, one of the derivatives of the VRP area is the green vehicle routing problem (GVRP) which deals with minimizing pollution in routing problems. For example, Zhang et al. presented an optimization model for the capacitated green vehicle routing problem by considering alternative fuel-powered vehicles (AFVs) [3].
On the other hand, there is a highly competitive environment in the logistics industry. Therefore, emerging technologies play a significant role in promoting and developing logistics systems and the related companies need to attune VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ quickly to the expected changes and innovations. On account of the tough competition in the global market, an intelligent and efficient logistics system has become crucial for companies to be distinguished from their competitors, such as autonomous vehicles (AVs) or drones. Moreover, the distribution system comprises about 20% of the total logistics cost [4]. Therefore, deciding on the delivery process is a critical problem for logistics systems and efficient solutions are necessary to reduce costs. One solution is to decrease human resource costs. Bad behaviors of drivers, including careless and drowsy driving habits, cause probable risks to transportation safety. Companies also need to evaluate their effect on logistics and how to keep their position in the competitive market. Conclusively, supply chains need selfassessment to develop their business strategy for having a transportation method with less dependency on the human factor. Thus, the modern logistics industry has an emphatic claim of high efficiency. To this end, AVs, as an emerging technology, are a hot topic in theoretical and applied studies. Therefore, using them in logistics can decrease operational costs, improve transportation efficiency, and omit potential human-made risks [5]. Automation, particularly AVs, depicts one of the technological developments in the logistics industry. The future trends of AVs, particularly trucks, will not only affect employment, lead time, and reliability but also affects how the supply chain interacts with its actors. However, there is a long implementing this new technology will take a long time due to numerous obstacles. In addition to the high costs of research and development, regulatory requirements, ethical issues, and road standards, consumer compliance must be addressed. Even though there exist several obstacles, they can be overcome in the future by using some existing technologies. Compared with conventional transport models, it has been proven that AVs have significant improvements in efficiency and service quality. Human resource costs are high in dense urban cities and this type of vehicle may be advisable [6].
The advent of autonomous trucks (ATs) can mitigate the shortage of truck driver risks. The United States faced a shortage of 48,000 truck drivers in 2015, which is predicted to increase to 175,000 by 2024 [7]. The use of ATs also leads to an increase in overall traffic safety. According to a survey on traffic crashes, only 6% of all accidents are related to vehicles or the environment, and 94% of them are associated with driver errors [5].
Although some strategies are made to employ ATs, they appear to be limited and difficult to implement such as booking a line in the road [8] and allocating special infrastructure [9]. The least operational strategy for the future is the usage of the mentioned infrastructure for conventional vehicles (CVs) that need drivers [6].
Employing autonomous electric vehicles (AEVs) as the new technology in the logistic system can bring the aforementioned benefits, because of which the routing problem of a mixed fleet consisting of AEVs and conventional vehicles (CVs) is studied in this paper. To model this problem, some attributes of AEVs are considered. AEVs have several compartments that deliver the customers' orders in the absence of the driver. They arrive at the customers' locations in the planned time windows and the customers can take their orders using a unique PIN. In this paper, considering different compartments for AEVs and the time windows in the routing problem are introduced as the main contributions of the model that a new solution method is developed based on them. The following sections of this paper include: Section II provides the literature review on the related topics. Section III consists of the problem description, the model formulation, and the related definitions. The proposed solution method and computational results are presented in Sections IV and V, respectively. The conclusion and future research are presented in Section VI.

II. LITERATURE REVIEW
In the real world, employing AEV in the logistic system requires considering some constraints based on the properties of AV. Since the AEV routing problem is new in the literature, we use a combination of similar areas in the literature. This section reviews some of the existing literature on AVs, multi-compartment vehicle routing problems (MCVRP), and Electric Vehicle Routing Problems with Time Windows (EVRPTW), and their similarities to the problem of this paper. The literature review structure is shown in Fig. 1 based on the combination of these areas, and the related literature is studied.

A. AUTONOMOUS VEHICLES
Most of the existing literature on autonomous vehicle routing takes the application of AVs and planning in particular places, such as indoor warehouses [10], into consideration. There are fewer studies concerning the use of AVs in public spaces. In this regard, De Almeida Correia and van Arem [11] surveyed the effect of AVs on urban transportation, and Nasri et al. [6] studied the route and speed optimization of ATs. Cai et al. [12] developed a mathematical model of VRP for AVs considering carbon emissions and time-varying speeds in grade paths. Chen et al. [13] introduced a vehicle routing problem with time windows and delivery robots (VRPTWDR). In their study, a driver served a customer while the robots were dispatched to serve nearby customers. They revealed that utilizing delivery robots contributes to significant operational time savings.

B. MULTI-COMPARTMENT VEHICLE ROUTING PROBLEM
The multi-compartment vehicle routing problem (MCVRP), which can transport various goods with particular properties in separate segments, has received extensive research attention in recent years. The size and number of compartments can be fixed or flexible. Access to these compartments is also needed when loading and unloading. In fact, the routing problem of MCV is CVRP extended by adding constraints on loading several compartments on each vehicle. There are various applications to use MCVs, such as food delivery, waste collection, and fuel distribution. Christofides et al. [14] were the first researcher to address the multicompartment vehicle routing problem (MCVRP). They studied vehicles with two types of compartments, refrigerated and non-refrigerated, in the food transportation industry. Moreover, the use of fixed compartments is more common than their flexible type in the literature, such as [15] for fuel distribution and [16] for waste collection. Derigs et al. [17] presented a general model for MCVRP application in food and fuel distribution and applied a construction heuristic method to solve the model. Ostermeier et al. [18] considered MCVs with compartments flexible in position and size. Henke et al. [19] considered MCVs with flexible compartments to collect glass waste but restricted their sizes to predefined values. They applied a variable neighborhood search (VNS) method to solve the proposed model. Henke et al. [20] also developed their previous study and applied a branchand-cut algorithm to solve it. Hübner and Ostermeier [21] extended MCVRP in grocery distribution by considering loading and unloading costs to evaluate the cost added by MCVs operations. In addition, Ostermeier and Hübner [22] evaluated the difference in cost between a homogenous fleet, including MCV or single compartment vehicle (SCV), and a mixed fleet, including both of them, indicating that the mixed fleet was more efficient. They solved the problem with a Large Neighborhood Search.

C. ELECTRIC VEHICLE ROUTING PROBLEM WITH TIME WINDOWS
Considering the green supply chain and clean routing operations, one of the main problems is the Electric Vehicle Routing Problem (EVRP) which uses electric vehicles (EV) instead of internal combustion engine vehicles (ICEV) [23].
Additionally, the EVRPTW is also a generic extension of Vehicle Routing Problem with Time Windows (VRPTW) that considers the charging process in the model [24].
In the literature, there are two strategies to recharge electric vehicles, including full charge and partial charge. In the full charge strategy, the EVs are recharged up to full capacity whereas in the partial charge strategy, the EVs are partially recharged according to their needs. Schneider et al. [25] studied EVRPTW as an extension of the green vehicle routing problem (GVRP). They supposed EVs were full charge in the charging stations. The objective function of their model minimized the distance traveled by applying the least number of vehicles. Unlike short refueling times in the GVRP, the recharging time may be significantly longer in the EVRPTW. They presented a hybrid heuristic as a solution method, that combines a variable neighborhood search algorithm with a tabu search heuristic.
Keskin and Çatay [26] relaxed the full recharge assumption and permitted EVs to be partially recharged at any level. They extended an EVRPTW with Partial Recharges (EVRPTW-PR) and utilized an ALNS approach for solving it. Desaulniers et al. [27] proposed a similar problem by studying various charging strategies and solving them optimally via branch-price-and-cut algorithms. Goeke and Schneider [28] studied a routing problem for a heterogeneous fleet, including electric and conventional vehicles. In their model, the total cost was considered as a function of loading times, vehicle speed, and route slope. They developed an ALNS algorithm that is improved by a local search. Keskin and Çatay [1] developed a mathematical model for the EVRPTW problem. They applied a partial recharging strategy with three types of normal, fast, and super-fast chargers. These chargers differed only in the charging time and cost, but the amount of charged energy was the same. They minimized the charging cost and the number of vehicles used in the proposed model. Hiermann et al. [29] also addressed the EVRPTW and fleet size for an electric vehicle fleet simultaneously. Their mathematical model consisted of vehicle purchase and routing costs. They solved the model by means of branch-and-price and proposed a hybrid heuristic that combines an ALNS and labeling procedure for intensification. Chen et al. [30] investigated the tour planning problem for electric vehicles. They aim to derive the optimal route considering maximum profitability and minimum range anxiety within the time horizon. Then, they used an interactive branch and bound algorithm to solve the model. Pelletier et al. [31] comprehensively reviewed the distribution problem by taking EVs into consideration. TABLE 1 structures the related literature and shows the research gap in the literature.
According to a study of Eshtehadi et al. [32], the routing problem for electric vehicles with modular compartments and a heterogeneous fleet, including both diesel and electric vehicles, are the possible research areas in the VRP literature. In this regard, this paper addresses • A mixed fleet, including AEVs and CVs, both of which have different operational costs.
• Due to the absence of the driver in AEVs, their loading spaces are divided into separate compartments in a way that each customer accesses only his/her orders. VOLUME 11, 2023 • AEVs and CVs should meet the customers in the announced time windows.
• AEVs can also be recharged partially at any charging station, but cannot enter the paths that are not equipped with the relevant infrastructure to pass them.
• A novel solution method was proposed for the problem by combining the Variable Neighborhood Search (VNS) algorithm and Large Neighborhood Search (LNS) algorithm with different strategies for neighborhood searches.
To the best of our knowledge, this is the first research considering a mixed fleet, including CVs and AEVs, with the mentioned attributes of AEVs. Our analysis showed that employing AEVs in the fleet has a significant impact on reducing the operational cost of the fleet. The computational results implied that the proposed solution method works very well.

III. MODEL FORMULATION
In this section, we first present the problem description and then formulate a mixed integer programming (MIP) mathematical model. The related parameters and decision variables are respectively summarized in Tables 2 and 3.

A. PROBLEM DESCRIPTION
In this paper, we assumed there is a mixed fleet, including AEVs and CVs. An instance of the AEV is designed based on https://b2n.ir/autonomous in Fig. 2.
Customers with different demands are on the nodes of the network. The operational costs of these vehicles are distinct. The problem assumptions are as follows: 1-The mixed fleet comprises a sufficiently large number (or infinite) of AEVs and CVs; 2-There are N nodes in the logistic network, including one depot and F charging stations and N − F − 1 number of customers; 3-Decision variables x ijv and z ivc obtains the routing and loading of vehicles, respectively. As well, During the tour, when should the AVs go to the charging stations and how much should they be charged? these decisions are made with the variables y iv and q iv . For scheduling the tours according to time widows the variable τ i is used in the model. 4-AEVs leave the depot and node i with the full charge state of Q v and q iv , respectively. They partially recharge in the charging stations as much as y iv − q iv and return to the depot with the amount of y iv − H · d i0 charge. Since AEVs have no drivers, the fleet needs on-time event planning services. Therefore, considering the time windows [e i , l i ] and the service time s i for node i in the delivery process is crucial; 5-AEVs have several fixed compartments with different capacities of Ccap cv, each of which is only assigned to one customer. Their different capacities are on account of the coverage of customers with different demand sizes; 6-Due to the road's lack of infrastructure, there are no guided lines or charging stations, and AEVs are not allowed to enter some of the routes in which a ij is equal to 0; 7-The transportation cost of each vehicle TC v includes the insurance, taxes, maintenance, depreciation costs, and air pollution fines.

B. MATHEMATICAL MODEL
The mathematical model can be expressed as (1)- (19), shown at the bottom of the next page.
The objective function includes the following cost terms in Equation (1): 1-1-the overloading costs, including insurance, depreciation, pollution penalty, tax, and maintenance; 1-2-the fuel consumption cost for CVs; 1-3-the driver cost of CVs, 1-4-the full charging cost of AEVs in the depot; 1-5-the recharging cost of AEVs in the charging station; 1-6-the remaining charge of AEVs when returning to the depot is used to calculate the pure charge cost of each route.
Constraints (2) enforce that each vehicle only takes one tour. Constraints (3) guarantee that every customer is visited only once and by one vehicle. Constraints (4) for each node and vehicle ensure that the number of input arcs is equal to the number of output arcs. Constraints (5) denote that if a path lacks the incumbent infrastructures for AEVs, this type of vehicle cannot pass. Constraints (6) prevent the formation of sub-tours by indicating the location of i in the tour of vehicle v and the depot as the start and end nodes of each tour. Constraints (7) and (8) are loading capacity constraints for conventional vehicles and each compartment of AEVs, respectively. Constraints (9) ensure that if the order of customer i is loaded on a compartment of vehicle v, customer i must be visited. Constraints (10) show that each compartment is assigned to only one customer. The compatibility of arrival times is assured with constraints (11). If vehicle v travels from node i to node j, the arrival time at node j is greater than the arrival time at node i plus service time and the time interval between the two nodes. Constraints (12) satisfy the problem of recharging visits. These set the arrival times of related nodes between the charging stations and other nodes. In fact, since recharging time is dependent on the remaining charge level q iv when arriving at station F, this time is equal to (y ivq iv )g. Constraints (13) check the compliance of the arrival time at a node with its time windows.
Constraints (14 -17) control the charge level of AEVs at every node and result in AEVs being charged at the charging station to the required amount to continue the route. Constraints (14) and (15) check the battery charge level at each vertex. Constraints (16) ensure that an AEV has adequate energy to reach its next destination in its route. Constraints (17) ensure that the battery charge level of an AEV is Q when leaving the depot. Lastly, Equations (18) and (19) show the type and domains of the decision variables.
The proposed model is nonlinear because of the last two terms in the objective function. Therefore, we use the following method to linearize it. If w ijv = q iv ·x ijv , p ijv = y iv · x ijv and M is a large number, then Thus, the mathematical model is modified by adding constraints (20 -26) and the objective function (1) is changed as follows: v∈V i∈I x ijv ≤ a ij , ∀i, j ∈ I , v ∈ AV (5) x ijv , z ivc ∈ {0, 1} , ∀i, j ∈ I , v ∈ V , c ∈ C (18) y iv ≥ 0, q iv ≥ 0, τ i ≥ 0, u iv ∈ {0, . . . , N }, ∀i ∈ I , v ∈ V

IV. THE SOLUTION METHOD
The presented model (2) - (27) can be relaxed to the well-known VRP by removing the compartment capacity, electrical charging, and time window terms. Since the conventional VRP is NP-hard [33], our problem is also NP-hard. Solving large sized problems to optimality in reasonable time is usually unrealistic. Therefore, developing efficient heuristic or meta-heuristic algorithms to quickly obtain good enough solutions would be a better alternative [34]. According to [35] large neighborhood search/adaptive large neighborhood search (LNS/ALNS), by 28% and variable neighborhood search by 18% are the most usage rates of the ten principal algorithms in the EVRP literature. Shaw [36] introduced the LNS algorithm and Ropke and Pisinger [37] presented the ALNS algorithm as an extension of LNS. Using large neighborhood search in solving various transportation and scheduling problems have recently shown outstanding results [35]. This method explores a complex neighborhood by using some heuristics. Employing these large neighborhoods lead to finding better candidate solutions in each iteration and traversing search path with Higher reliability.
The ALNS algorithm includes different destroy and repair structures to achieve a quality solution. It is worth noting that the ALNS algorithm is relatively fast compared to other popular metaheuristics to solve the VRP. In this method, some heuristics are recalled to determine the sequence of different neighborhoods to use each time.
Mladenovic and Hansen [38] proposed the Variable Neighborhood Search (VNS) metaheuristic initially. The main components of this algorithm are the shake and local search. The shaking procedure changes the neighborhood structure systematically to escape from the local optimum valley and in the local search procedure, the search continues among the current neighborhood.
Herein, the sequential Clarke and wright (C&W) algorithm [39] was utilized to construct a feasible initial solution, and a hybrid VLNS algorithm was employed to improve the procedure. The VLNS algorithm employs VNS to select neighborhood structures used in the LNS algorithm. In the following, the neighborhood structures of this algorithm are described.

A. NEIGHBORHOOD STRUCTURES
Neighborhood structures in an ALNS algorithm include removal and insertion operators. Since the problem has two types of arcs, customer to customer and customer to charging station, their operators will have various impacts on the answer. Thus, we used separate customer removal/insertion (CR/ CI) and recharging station removal/ insertion (SR/ SI) according to Fig. 3.

1) CUSTOMER REMOVAL OPERATORS
This paper employs the following five customer removal operators in the VLNS algorithm widely applied in the literature.
1-Random removal: In this method, the number of customers is selected randomly and removed from the current solution. This method causes diversities in the solution space. 2-The worst energy consumption removal: This operator removes customers with higher energy consumption (E ij )  Shaw [36]. This operator calculates the weighted sum of difference for distance, lower bound of the time window, and the demand quantity between the two nodes i and j with the formula obtained in Shaw's study {w 1 d ij + w 2 |e ie j | + w 3 |d i -d j | + w 4 R ij }. If nodes i and j are in the same VOLUME 11, 2023 route, then R ij is equal to 1 and w 1 , .., w 4 are the related weights.
It then removes the nodes with the maximum amount of the mentioned formula. By defining different weighting coefficients for this expressed formula, other operators are developed as follows: -Proximity-based: the nodes with the longest distance are removed when w 1 = 1 and w 2 , w 3 , w 4 = 0; -Time-based: the nodes with the maximum earliest service start time difference are removed when w 2 = 1 and w 1 , w 3 , w 4 = 0; -Demand-based: the nodes with the highest demand quantity difference are removed when w 3 = 1 and w 1 , w 2 , w 4 = 0.

2) STATION REMOVAL OPERATORS
We used the Worst-Distance and a new station removal operator as the station removal operators.
1-Worst-Distance station: This operator selects and removes a station with the longest distance from the last node before it. 2-Unnecessary Station Removal: This operator tries to eliminate the visiting station with the minimum necessity. For this purpose, the operator checks the condition y i − H · d ij > 0 for every station where nodes i and j respectively are the nodes before and after that station. Afterward, it lists the feasible stations in the non-decreasing order of the quantity of energy, they charge and remove a random number of stations from the top of the list.

3) CUSTOMER INSERTION OPERATORS
In the following, the five insertion operators we employed to reinsert the removed nodes in the last sub-section are as follows: 1-Random Insertion: This operator reinserts the removed nodes in existing routes according to the remaining CVs' capacity and unloaded AVs' compartments randomly; 2-Greedy insertion: This operator selects the best position to reinsert the removed nodes according to the energy consumption cost is calculated as E i = E ji + E ik − E jk for j = 1, . . . , n and i = 1, . . . , n; 3-Greedy with noise insertion: This operator generates noise in the greedy insertion by defining new cost for node i with the formula C = C + dµε, where d is the maximum distance, µ is the noise parameter, and ε is the random number between [0, 1]. Indeed, this operator with noise diversifies the greedy operator; 4-Regret insertion: In the literature, E i is the alteration in objective function value by reinserting a node i into the best and second-best positions by calculating energy is the lowest energy consumption increase and E i2 refers to the second-lowest fuel consumption increase; 5-Regret with noise insertion: This operator, similar to insertion operator 2, causes a diversity of regret insertion by generating noise.

4) STATION INSERTION OPERATORS
We used the greedy station insertion operator to reinsert the charging station as follows: -The nearest station with a minimum cost: This operator detects the first unfeasible node (y i < 0) on the route starting from the depot and selects the best position for the station with the least cost by putting the nearest station before that or other previous nodes.

B. VEHICLE TYPE ALTERATION OF THE ROUTES
In addition to the mentioned removal and insertion operators as the neighborhood structures, we defined a new method, namely vehicle type alteration of the routes (VTA), since our model's fleet type is mixed. In this structure, the vehicle type of a route is converted to another type of vehicle. Therefore, the AEVs route changes to the CVs route and vice versa as Fig. 3. E.

C. REPAIR OPERATOR FOR LOADING CAPACITY FEASIBILITY
Regarding the proposed model, including capacity constraints for CVs and AEVs compartments, a mechanism is needed to check its feasibility. In our solution method, the algorithm first finds the best candidate solution using CR/CI and VTA structures, and before setting that as the best solution, it calls the repair mechanism to check the feasibility of loading capacity. In this section, we define the two sets, D − and EC. D − , consisting of the removed demands to ensure the feasibility of the solution and EC to determine empty capacity in vehicles. The repair mechanism includes removing and inserting the identified demand as follows:

2-The loaded orders of the AEVs removal operator
On account of the multi-compartment feature of AEV that each compartment has a specific volume and weight capacity, the capacity constraints must be checked for all the compartments. Therefore, follow the steps below to remove inconvenient or extra demands:  it is recognized that d 11 < Ccap 1,4 , d 11 is then assigned to compartment 4 in AEV 1 and the EC and D − are updated. Afterward, the rerouting operation is carried out in Fig. 3. F. In addition, the flowchart of the proposed VLNS algorithm is depicted in Fig. 5. According to Fig.5, VLNS start with a sequence of neighborhoods k 1 , . . . , k max including the customer removal and insertion operators, and an initial feasible solution obtained from the C&W algorithm. Then, the model is solved with k1 and the obtained solution is compared with the best solution. If the obtained solution is better, the algorithm uses k1 otherwise, it employs the next k. This step is repeated until k=k max .

V. COMPUTATIONAL RESULTS
In this section, we aim to verify the performance of the proposed algorithm. For this purpose, we used data that was reported previously by Schneider et al. [25] and the model parameters published in a study by Goeke and Schneider [28], as shown in TABLE 4. The costs related to the two types of vehicles are considered in [28]. Because in this paper, the fleet includes CVs and EVs, and except for the driver's fee, AEVs have all related costs. We generate the capacity of CVs and the compartments of AEVs based on the datasets studied in [32]. The proposed VLNS algorithm components are described in TABLE 5.

A. PERFORMANCE OF SOLUTION ALGORITHM
In this section, we analyzed the proposed solution method performance. Note that the algorithms were implemented with python 3.8 programming language and run on a personal computer with a Core i7 8550u 1.80GHz CPU and 12GB of RAM.
As mentioned in the last section, we proposed a VLNS algorithm as a solution method for the model. The algorithm was evaluated in two steps, a comparison with the exact method using the CPLEX solver in GAMS 26.1 software and the ALNS algorithm proposed in research by Eshtehadi et al. [32] without applying VTA heuristics. In the first step, we employed a small class of data, including 5, 10, and 15 customers, to compare our VLNS algorithm with the CPLEX solver. The results are presented in TABLE 6. The first column is the original name of the instances in the mentioned data set [25]. The column of ''Gap'' is the percentage difference between the best objective function VOLUME 11, 2023   values obtained with VLNS and CPLEX with the formula (Z VLNS -Z CPLEX ) / Z VLNS .
Note the relative MIP gap tolerance shown in the column of ''Relative Gap'' is 1e-4. This default value indicates to CPLEX to stop when an integer feasible solution has been proved to be within 0.01% of optimality. The value of 1 in this column shows a feasible not optimal solution in the defined time for the related sample. The value of 1 in this column shows a feasible (not optimal) solution in the ''Obj. Value'' column for the related sample. Based on the results, for the small class with five customers, CPLEX obtained the optimal solution for five instances. Due to this, for other instances, the time limit of the optimization solver was set to 3000 and 7200 seconds in CPLEX.
According to TABLE 6, the worst gap between the VLNS algorithm and CPLEX is related to instance RC108C10 with the amount of 0.62%. It is worth noting that the largest solvable problems using CPLEX are nine-node instances (five customers and four stations) according to the numerical experiments.
It is due to the assignment of each compartment of AEV according to its capacity to only one customer. In fact, for each AEV with |C| compartments |C| capacity constraints must be met. The negative amount of gap means the best solution for VLNS was better than that of CPLEX.
The average amount of Gap column was -6.59%, the run time of VLNS was 18.70, and CPLEX was 5022.93 seconds. Therefore, considering the best solution and the run time, the performance of VLNS on average was better than that of the obtained feasible solution of CPLEX.
In the second step, we validated the presented VLNS algorithm by running the medium and large instances and comparing the result with the ALNS algorithm without using ''vehicle type alteration of the routes'' which is shown as ''ALNS without VTA Alg.'' in TABLE 7. To this end, we applied the instances with 30, 50, and 100 customers. Both  of the algorithms ran four times for each group and the best result is reported in TABLE 7.
In TABLE 7, the column of Gap demonstrates the percentage difference between the best solution of the two algorithms ((Z VLNS -Z ALNS ) / Z ALNS ) and the total average of the three groups, including 30, 50, and 100 customer nodes, was -40.3% with the average run time -10.12476 %. Hence, the performance of VLNS with less time difference for the best result is considered to be better. This result also indicates that VTA is effective.

B. THE IMPACT OF CONSIDERING A MIXED FLEET
Herein, a mixed fleet, including AEVs and CVs, was examined and two scenarios were defined to investigate the impact of this problem definition. Scenario 1. Considering the homogeneous fleet: This fleet includes one type of vehicle, CVs. This problem was defined as the capacitated vehicle routing problem with the time window (CVRPTW) obtained by removing the constraints related to AEVs from the studied model.  Fig. 6.
According to Fig. 6, the average cost of the mixed fleet is lower than that of the homogeneous fleet. Furthermore, with the increase in the number of customers, the difference cost between the two fleets also increased (50 and 100 customers). The difference in cost of the homogeneous fleet by increasing the number of customers (the difference between the number of customers 50 and 100) was greater than this difference in the mixed fleet. Therefore, it could be concluded that using AEVs in the fleet is beneficial in terms of the operation cost.

VI. CONCLUSION
In this work, we developed a mathematical model for a mixed fleet, including autonomous and conventional vehicles, according to the properties of AEVs. Because of the driver's absence, the loading area of AEV is separated into several compartments with different capacities each of which is assigned to one customer. As well as, due to the accurate timetabling of this type of vehicle, time window constraints are considered. Then, the mathematical model was developed based on these properties for the first time in this study. After determining whether the model is NP-hard, we developed a hybrid algorithm based on LNS and VNS, called VLNS, to solve the model. This algorithm employed different neighborhood structures, such as customer removal/ insertion, charging station removal/insertion, vehicle type alteration of the routes, and a repair mechanism for loading constraints that some of which were new and improved the solution method. This was proven by comparing the proposed VLNS with the presented ALNS in the literature. The numerical results indicated that considering a mixed fleet reduces the operational costs in the vehicle routing problem. One of the main limitations of this study is that autonomous vehicle was not employed widely, and it is possible we faced other constraints and costs during employing them in the real world.
Future research could be conducted to analyze this problem in the stochastic environment as a robust model for the time window. Because considering of time window is necessary for AEV routing problems due to the need for precise timetabling in the absence of the driver. Therefore, considering this parameter in the stochastic area leads to the mathematical model being usable in the real world.