# The Electric Vehicle Routing Problem with Time Windows, Partial Readers, and Parcel Lockers
## 3 Problem Description and Mathematical Model  
With the presence of parcel lockers, there are two delivery methods offered, including (1) home delivery and (2) self-pickup at parcel lockers. With these delivery methods, we propose three types of customers for our problem. The first type is home delivery customers (H), who are required to directly receive parcels at their home locations. The second type is self-pickup customers (S), who come to the designated parcel lockers to pick up parcels. Each self-pickup customer is allowed to select a preferred parcel locker, which is convenient for collecting parcels. The last type is flexible customers (F), who are willing to be served by either home delivery or self-pickup methods. Note that the preferred delivery method for each customer is known in advance. Let \(C=\{C_{H}\cup C_{S}\cup C_{F}\}\) be a set of all customers (i.e., consisting of three subsets of home delivery, self-pickup, and flexible customers, respectively), \(P\) denotes a set of parcel lockers, \(F\) denotes the set of recharging stations (CSs), and \(\{0,0^{-}\}\) is the origin and destination depots. We define EVRPTW-PR-PL as a directed graph \(G=(N,A)\), where the set of nodes \(N\) can be separated into subsets as \(N=\{0,0^{\prime}\}\cup C\cup P\cup F^{\prime}\). Note that set \(F^{\prime}\) includes set \(F\) and its copies of each CS to allow multiple visits to CSs. For later formal expressions, we denote subsets \(N^{{}^{\prime}}=N\setminus\{0,0^{\prime}\}\), \(N^{(o)}=N\setminus\{0^{\prime}\}\), \(N^{(d)}=N\setminus\{0\}\), and \(F^{+}=F^{\prime}\cup\{0\}\). The set of arcs \(A\) is defined by \(A=\{(i,j)|i\in N^{(s)},j\in N^{(d)},i\
eq j\}\). For each arc \((i,j)\in A\), travel cost and travel time are defined by \(c_{ij}\) and \(t_{ij}\), respectively. All customers are served by a homogeneous fleet of EVs (denoted by set \(K\)) with a load capacity of \(Q\), battery capacity of \(B\), and electric consumption rate of \(r\). Each utilized EV starts at the depot with a fully charged battery and ends at the depot. During transportation, each EV is allowed to recharge its battery at CSs with a charging rate of \(g\). Note that each CS may serve EVs multiple times. Furthermore, each parcel locker with a capacity limited by the number of customers (denoted by \(\mathcal{E}\)) may serve several self-pickup and flexible customers and can be visited by one or more vehicles. Let \(\xi\) denote the designated parcel locker of customer \(i\in C_{S}\cup C_{F}\). Consequently, each parcel locker \(i\in P\) has a subset of associated customers \(\mathcal{C}_{i}\subseteq C_{S}\cup C_{F}\), which represents the possible connection between the parcel locker and customers in the subset. Each customer \(i\in C\) has a predefined time window \([e_{i},l_{i}]\), a non-negative demand \(d_{i}\), and a service time \(s_{i}\). Demand and service time of the remaining vertices (i.e., parcel lockers and CSs) are set to 0, while their time windows are denoted by \([0,T_{max}]\), with \(T_{max}\) as the maximum duration of the routing plan. Figure 1 presents an illustrative network including 4 home delivery customers (1-4), 4 self-pickup customers (5-8), 4 flexible customers (9-12), 4 parcel lockers (P1-P4), 3 recharging stations (CS1-CS3), and depot 0. Regarding the self-pickup method, the sets of possible connections for parcel lockers are \(\mathcal{C}_{1}=\{5,8,10\}\), \(\mathcal{C}_{2}=\{9,11\}\), \(\mathcal{C}_{3}=\{6,12\}\), and \(\mathcal{C}_{4}=\{7\}\) (see hash lines in Figure 1). The figure depicts a routing plan for serving all customers along with their corresponding delivery methods by using three EVs. All home delivery customers have their parcels delivered to their homes, while self-pickup customers receive their parcels at parcel lockers (see red hashed lines in Figure 1). Flexible customers 9 and 11 are not assigned to the self-pickup method (see black hashed lines in Figure 1), while customers 10 and 12 take their orders from the parcel lockers.