# Assessing the Opportunity Offered by Electric Vehicles in Performing Service Trips to End Consumers
## 3 Materials and Methods
### Routing and Scheduling  
The problems described by _service patterns 1-3_ can be formulated as a vehicle-routing problem (VRP), taking into account the availability of multiple depots and time window constraints. The list of the symbols used in this section is as follows:  
* \(i,j\) are the indexes for end consumers;
* \(v\) is the index for LCVs;
* \(b\) is the index for cargo e-bikes;
* \(d\) is the index for depots;
* \(C\) is the set of end consumers;
* \(D_{1}\) is the set of depots for _service patterns 1_ and \(2\);
* \(D_{2}\) is set of depots for _service pattern 3_;
* \(U_{1}\) is the union between set \(C\) and set \(D_{1}\) (\(U_{1}\) = \(C\cup D_{1}\));
* \(U_{2}\) is the union between set \(C\) and set \(D_{2}\) (\(U_{2}\) = \(C\cup D_{2}\));
* \(V\) is the set of vehicles;
* \(B\) is the set of cargo e-bikes; \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Mechanical**} \\ \hline Length & 4.1 m \\ Width & 2.15 m \\ Height & 2.2 m \\ Cargo Volume & 19.6 m\({}^{3}\) \\ Curb weight & 2114 kg \\ Frontal Area & 4.26 m\({}^{2}\) \\ Drag coefficient & 0.3 \\ Rolling friction coefficient & 0.013 \\ Tire type & 225/65R16 \\ \hline
**ICE** & \\ \hline Engine Power & 81 kW \\ Engine Torque & 330 Nm \\ Fuel consumption & 9.8 L/100 km \\ CO\({}_{2}\) emissions & 256 g/km \\ \hline \multicolumn{2}{c}{**Electric propulsion systems**} \\ \hline Motor type & SSM—Magnet-less \\ Motor Power & 57 kW \\ Motor Torque & 225 Nm \\ Battery type & NMC \\ Battery Energy & 52 kWh \\ _Cell Type_ & _Lithum-Ion_ \\ Chemistry Cathode & NMC \\ Chemistry Anode & Graphite \\ Capacity & 63 Ah \\ Nominal Voltage & 3.75 V \\ Operating voltage & 2.7-4.12 V \\ Battery temperature range & \(-\)40–65 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The LCVs’ technical specifications. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Mechanical/Electric Propulsion Systems**} \\ \hline Motor type & Electric \\ Weight & 49 kg \\ Consumption & 0.009 kWh/km \\ Cargo volume & 2 m\({}^{3}\) \\ Length & 1 m \\ Width & 2.5 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cargo e-bike technical specifications. \(B_{d}\) is the subset of cargo e-bikes belonging to depot \(d\);  
\(c_{i\bar{i}v}\) is the cost to move from \(i\) to \(j\) by vehicle \(v\) (\(c_{i\bar{i}v}=TT_{i\bar{i}v}+ST_{j\bar{v}}\));  
\(TT_{i\bar{i}v}\) is the travel time to move from \(i\) to \(j\) by vehicle \(v\);  
\(ST_{j\bar{v}}\) is the service time at node \(j\) of vehicle \(v\);  
\(g_{i\bar{j}b}\) is the cost to move from \(i\) to \(j\) via vehicle \(b\) (\(g_{i\bar{j}b}=TT_{i\bar{j}b}+ST_{j\bar{v}}\));  
\(TT_{i\bar{j}b}\) is the travel time to move from \(i\) to \(j\) via cargo e-bike \(b\);  
\(ST_{j\bar{b}}\) is the service time at node \(j\) of vehicle \(b\);  
\(q_{j}\) is the quantity to be delivered to customer \(j\);  
\(p_{j}\) is the quantity to be picked up from customer \(j\);  
\(w_{v}\) is the LCV \(v\)'s capacity;  
\(w_{\bar{v}}\) is the cargo e-bike \(b\)'s capacity;  
\(t_{lim}\) is the threshold value for route duration;  
\(\ell_{v}\) is the vehicle \(v\)'s range;  
\(\ell_{\bar{b}}\) is the vehicle \(b\)'s range;  
\(t_{i}\) is the arrival time at customer \(i\);  
\(\left[a_{i},b_{i}\right]\) is the time window at customer \(i\);  
\(x_{i\bar{j}v},y_{i\bar{j}v}\) are the problem variables. The problem formulation collapses into a single depot VRP [55; 56] for _service patterns_\(1\) and \(2\) and, for these service types, the formulation is essentially the same. In fact, the only difference is in the type of vehicle engines. The formulation is as follows:  
\[\text{Minimize }Z_{1}=\sum_{i\in\mathcal{U}_{1}}\sum_{j\in\mathcal{U}_{1}} \sum_{v\in V}c_{i\bar{i}v}\cdot x_{i\bar{i}v}=\sum_{i\in\mathcal{U}_{1}}\sum _{j\in\mathcal{U}_{1}}\sum_{v\in V}\left(TT_{i\bar{i}v}+ST_{j\bar{v}}\right) \cdot x_{i\bar{i}v} \tag{1}\]  
s.t.