3. Optimization Framework
In this section, we develop a mathematical model of this problem. Two important characteristics of employing BEVs, namely a reduced operating rang and the possibility to recharge at certain stations to increase this range, are considered in this model. Some mobile charging service characteristics are simplified, where parameters such as the number of BEVs requesting the charging service, the electricity demand of each car, and service time windows in each customer node are set in advance. The partial recharge strategy is adopted in this section. For the partial recharge strategy, the MCV departs from the depot fully charged, but may depart from a charging station with any state of charge. Furthermore, the MCV returns to the depot with an empty battery when it has been recharged once during its route.
Then, we follow common Vehicle Routing Problem (VRP) modeling technologies and simplify several real-world characteristics. Firstly, a flat terrain is assumed and the case that vehicles increase their speed to fulfill time windows requirements is not in considered. Therefore, the travel speeds between every node are constant and given. Then, the recharging rate of
charger is fixed and given as a constant,
, where
. In the real world, the charging time increases for last
–
of the battery capacity in the recharging processes [
50]; for simplification reasons, the last assumption is presented. Finally, the charging services are performed by a homogeneous fleet of mobile charging vehicles with a fixed battery capacity,
, and energy consumption rate,
. Neubauer et al. [
51] found that the travel ranges of BEVs can vary by up to
, depending on the weather, temperature, use of in-car heater, speed, and so on. Then, a detailed travel range report was presented by Tesla, a maker of BEV, which reported that the travel range might be reduced by half when the speed is doubled [
52]. Lee and Han [
10] has focused on the uncertain travel range by a probabilistic consideration, which is simplified in this paper.
The mathematical model can be formulated as a mixed integer framework as follows:
Objective: Minimize the total traveled distance, as follows:
where
, is a parameter that denotes the distance for each arc, it can be calculated by the locations of starting node and ending node of this arc.
is a binary decision variable, and takes the value of
if the arc
is traveled, and
otherwise.
The charging station constraints are as follows:
is a binary decision variable, which means whether a plug-in charging station of type, , should be built in the node, . When the type charging station is built in the node, , ; otherwise . denotes the cost of building a plug-in charging station of type, . Then, denotes the total budget limit for building the charging facilities. Constraint (2) is the budget limit, where denotes the set of candidate nodes where a plug-in charging station can be built, as the third type of charger has been built in the depot node . Because all of the cars start from the depot and will return to the starting point after completing all of the tasks, therefore, a third type of charging equipment with the lowest charging rate can be built at the starting point to reduce costs, and all of the vehicles have enough time to charge. The dummy charging stations corresponding to the initial point should be the third charger station, which is limited by Constraint (4). Constraint (3) ensures that only one type of charging device can be built for each dummy node. Constraint (5) ensures that the charging type of the dummy nodes is the same as that of the corresponding charging station. Constraint (6) indicates the binary constraint for the decision variable.
The flow constraints are as follows:
where
is a binary variable indicated by Constraint (11), it will equal one if an arc is traveled, and zero otherwise. Constraint (7) enforces the connectivity of the customer visits, which ensures that every customer could be visited only once by any mobile charging vehicle. Constraint (8) handles the connectivity of the visits to the dummy nodes, which guarantees the connectivity of visits to the dummy recharging stations and restricts each dummy node visited once at most by any mobile charging vehicle. Constraint (9), a flow conservation at intermediate nodes, enforces that the number of outgoing arcs equals the number of incoming arcs at each vertex. Constraint (10) controls the total number of vehicles by limiting the number outgoing from the depot
. In order to avoid the increase in the number of mobile charging vehicles due to an insufficient budget, we restrict the number of vehicles here.
The time constraints are as follows:
As the travel speeds between every node are constant and given, and the distance can be calculated easily by the locations of the starting node and ending node of this arc, therefore, travel time, , is a predetermined parameter. For each customer node, the positive service time, , and the time window is assumed to be predetermined also, and it could be noted as . As a result of the constraints of the time windows, the mobile charging service cannot begin before , and is not allowed to start after , but it might end later. Therefore, the waiting time could be caused by it. is the end time of charging service, which equals to . In order to track the arrival time of the mobile charging vehicles, a decision variable is defined for each node, . denotes the recharging amount in node , which is obviously less than or equal to the battery capacity, . represents the recharging rate of the charger.
Constraint (12) is the time feasibility constraints for the customer nodes and depot node with an instance of . Constraint (13) considers the recharge time instead of the service time when the previous node, , is a recharging station. Constraint (14) enforces that every vertex is visited within the predetermined time window .
The electricity constraints are as follows:
For the mobile charging vehicle, the battery consumption rate is assumed to be a constant, , so every traveled link consumes of the remaining battery. Let denote the remaining charge level on arrival at vertex . Each vertex, , is assigned a positive number of requests, , and the demand of charging for each request, , but both are zero when node does not belong to set . Hence, it is obvious that the total amount of charging could be computed as at node . is an infinity value.
Constraint (15), Constraint (16), and Constraint (17) keep track of the state of charge of the battery for the customer nodes, charging the dummy nodes and deport, respectively, and ensure that the battery state of charge never falls below the electric demand of the next customer node. Because, in the early assumptions, all of the vehicles are full when they leave the depot , therefore Constraint (16) and Constraint (17) are different from the power when leaving the node where the electricity leaving node is equal to the sum of the remaining charge level, , and the recharging amount, , in Constraint (16), while equaling the capacity of battery in Constraint (17). Constraint (18) suggests that the MCVs can only recharge at the nodes with a charging station. Constraint (19) sets the upper and lower bounds of the battery state, which makes sure that the battery state of charge could not exceed its capacity and that the MCVs do not run out of charge. Constraint (20) represents the constraint for the demand of the customer nodes, which sets the upper and lower bounds of the demand.
However, there is a common bilinear term,
, in Constraint (13), which is nonlinear and leads to the nonconvex property of the whole problem. To simplify the solution, the nonconvex problem here is transformed into an equivalent MILP by the reformulation-linearization technique (RLT) [
53]. Therefore, commercial solvers can solve the latter problem directly and the optimal solution can be guaranteed.
Let
, for each
and
, to linearize the bilinear term. Thus, Constraint (13) can be rewritten as follows:
Following the rules of RLT,
is equivalent to the following linear constraints:
We can separately let equal to one, where the type of charger level is built in node , or zero, where the charger level is not built in node to prove the equivalence between these two. The plugged results are shown in following equations:
This time let
, then, Constraints (22)–(25) can be written as follows:
From Constraints (26)–(29), we can gain the result that
, which means when the
charger level is built in node
,
equals to the charging amount
. Then, as
, Constraints (22)–(25) can be written as follows:
From Constraints (30)–(33), we can get , which means there is no a charging activity when the charger level is not built in node . From the equivalence proof, is equivalent to the linear Constraints (22)–(25).
Next, we will further deal with the muti-equality-based constraint, Constraint (5): , , .
In air traffic control, Constraint (5) is referred to as coupling constraints, while in stochastic programming, it is referred as non-anticipativity. To reduce the complexity of the solution constraint, the following three propositions are proposed and verified.
Proposition 1. Constraint (5) equals essentially the following constraints: Proposition 2. Constraint (5) is equivalent to the following constraints: Proposition 3. Constraint (5) is equivalent to the following inequality constraints: Proof. As it is obvious for the necessary condition, therefore, the sufficient condition is proved in this part using contradiction. Firstly, we assume that Constraint (35) holds, while Constraint (5) is not established. Then, there must exist two nodes not equal, where is assumed. Therefore, one of two variables, and , has a value of zero. By Constraint (36), we can get is less than one strictly, and . Let , and without the loss of the generality. Through simple processing, we can obtain . That is all, the sufficient condition proved. □
Propositions 1 and 2 are easy to verify, so the certification processes are not stated here. Propositions 2 and 3 are simple forms of Constraint (5), and have equivalence. The corresponding constraints can be directly used in the model. So far, by applying the RLT technique and simple formula conversion, the model can be transferred into the following equivalent MILP form: Constraint (1) subject to Constraints (2)–(4), (6)–(12), (14)–(25), (35) or (37), (36).
Proposition 4. In an optimal solution, if a mobile charging vehicle has been recharged at once and returns to the depot at the end of its route with a positive battery state, that is, , then, when the same cars return to the depot with empty battery, that is, , the solution is also optimal.
Proof. Because there is plenty more time to charge the car, it therefore reduces the charging time so that the amount of electricity at the end of the route is exactly zero, that is, , and the solution is also an optimal solution. □
Corollary 1. If one car does not consume all of the electricity at the end of the route, that is, , and the solution is optimal, then the problem has infinite multiple optimal solutions.
Proof. By contradiction, we assume, in the optimal solution, that one car does not consume all of the electricity at the end of the route, that is, , while, when the car’s charge is , the solution will not be optimal, where is a small positive scalar. By Proposition 4, we can find in this case when the car’s electricity is full consumed the solution is also optimal, while , so the multiple optima exist. □
5. Conclusions and Discussions
In this paper, a novel approach to providing a service for the battery charge replenishment of BEVs is presented. Facility location and vehicle routing are two of the most crucial decisions in reducing the cost of many companies. Because the strategy of plug-in charging station location directly determines the route of each vehicle, and meanwhile, the station location strongly depends on the vehicle routing plan, when these two interrelated components are tackled separately, the solution could be suboptimal, which has been proven by many studies. Therefore, we have built our problem on LRP, and present the multiple types of plug-in charging facilities’ location-routing problem with time windows for mobile charging vehicles (MTPCF-LRPwTW-MCVs) in this paper. MTPCF-LRPwTW-MCVs can determine the station location and vehicle routing plan simultaneously.
Multiple types of charging stations with different charging speeds have been discussed by Wang and Lin [
36]. Hence, in this paper, we not only determine the location of the charging station, but also the type of chargers at each charging station. However, when the multiple types of charging stations are considered in this problem, a common bilinear term will be generated, which is nonlinear and leads to the nonconvex property of the whole problem. To simplify the solution, the nonconvex problem here is transformed into an equivalent MILP by the reformulation-linearization technique (RLT) [
53].
From RLT, we formulate the MTPCF-LRPwTW-MCVs as an MILP, which can be solved directly using the GAMS commercial solver. To demonstrate the model, 36 small instances are designed based on the benchmark instances for VRPTW, proposed by Schneider et al. [
43]. All of the instances can be solved exactly, and the sensitivity analyses, such as the battery capacity, the recharging rate, and so on, are also conducted. It could be concluded that the larger battery capacity or quicker charging rate could decrease the number of mobile charging vehicles and the total traveled distances, respectively. With more budgets, the number of fast charging stations can increase, so the number of mobile charging vehicles and total traveled distances will decrease, respectively. In the latter, we improve the number of corresponding dummy nodes of one candidate charging station,
, to allow a charging station to be used up to two times at most. This method improves the using frequency of the fast charging station as well as the efficiency, so the result is the same as raising the budget. In the above-mentioned four sensitivity analysis cases, we will find that the difference in the total distance of the optimization target increases when the number of customers increases. These are reasonable results due to more candidate customer nodes and more vehicle routing options.
Furthermore, the model can be extended in several ways. Firstly, it is interesting to investigate how problems with real-life road networks will affect the performance of the commercial solvers. The solution rate of the solver GAMS has a certain limit. When the total number of nodes is 100, the model solution time will reach more than 20 hours. Therefore, more efficient algorithms can be expected in future work. Moreover, the heterogeneity of both the mobile charging vehicles and the requests should be further discussed. It can be seen from the test results that when the number of customer nodes is small, as the vehicle battery capacity increases or decreases, the total running distance of all of the vehicles does not change much, indicating that the power of all of the vehicles used to provide services and operate by themselves is greater than the total demand. There is a certain waste of resources, so the combination of heterogeneous vehicles will be a worthy research direction, which can reduce the waste of resources. Then, it should be pointed out that the mobile charging vehicle routing will be more complex when the charging requests are dynamically considered. Because, on the way to providing services for the next customer, the drivers of the MCVs may receive new service requests. Therefore, under the previous customer’s service time window, how to re-plan the path, leading to maximum profit, has a certain research significance. Furthermore, the maximum travel distance can vary according to environmental factors, including weather, temperature, and so on. Therefore, these various factors should be reflected in future studies.