## 2 Mathematical Formulation  
CVRP is usually stated as a graph of a set \(C=\{c_{i}\,|\,i=1,\,2,\,\ldots,\,n\}\) of \(n\) nodes, resembling customers, along with associated demands \(D=\{d_{i}\,|\,i=1,\,2,\,\ldots,\,n\}\) and required service times \(ST=\{st_{i}\,|\,i=1,\,2,\,\ldots,\,n\}\). If no service time is specified for the customers in a problem definition, then it is set to \(0\). Among the nodes, a special node, say \(c_{h}\), is called the depot, from where the demands of other nodes are met using some vehicles. For the depot \(c_{h}\), the service time \(st_{h}=0\). A vehicle starts from the depot, serves the demands of some nodes, and comes back to the depot. The objective is to serve all the customers using a set of vehicles, keeping the cost (usually calculated as the total distance travelled by the vehicles) as low as possible. A capacity constraint is associated with every vehicle, which specifies the maximum amount of total demands of the nodes that can be served by it. Maximum travel distance is another constraint that, if specified, puts a limit on the total distance traveled along with service time spent by a vehicle [43]. The required variables for the CVRP formulation are presented in Table 1.  
Some notations and variables related to the formulation of CVRP are stated in Table 1, based on which the problem can be formulated as a minimization problem as  
\[\textit{Minize}\ \sum_{j=1}^{n}\sum_{k=1}^{n}\sum_{v=1}^{v}x_{jk}^{v}\textit{dist} \big{(}c_{j},c_{k}\big{)} \tag{1}\]  
fulfilling the following constraints  
\[\sum_{v=1}^{m}\sum_{j=1}^{n}x_{jk}^{v}=1\hskip 56.905512 ptk=1,2,\ldots,n;k\
eq h \tag{2}\]  
\[\sum_{v=1}^{m}\sum_{k=1}^{n}x_{jk}^{v}=1\hskip 56.905512 ptj=1,2,\ldots,n;\ j\
eq h \tag{3}\]  
\[\sum_{j=1}^{n}x_{jo}^{v}-\sum_{k=1}^{n}x_{ok}^{v}=0\hskip 56.905512 ptv=1,2, \ldots,m;o=1,2,\ldots,n;\ o\
eq h \tag{4}\]  
\[\sum_{j=1}^{n}\sum_{k=1}^{n}x_{jk}^{v}d_{j}\ \leq Q\hskip 56.905512 ptv=1,2,\ldots,m \tag{5}\]  
\[\sum_{j=1}^{n}\sum_{k=1}^{n}x_{jk}^{v}\Big{(}\textit{dist}\big{(}c_{j},c_{k} \big{)}+\textit{st}_{c_{j}}\Big{)}\leq TT\hskip 56.905512 ptv=1,2,\ldots,m \tag{6}\]  
\[\sum_{k=1}^{n}x_{hk}^{v}=\sum_{k=1}^{n}x_{kh}^{v}\leq 1\hskip 56.905512 ptv=1,2, \ldots,m \tag{7}\]  
\[\sum_{j\in\mathbb{Z}}x_{jk}^{v}\leq|Z|-1\hskip 56.905512 ptZ\subseteq\{1,2,\ldots,n \};h\
otin Z;v=1,2,\ldots,m \tag{8}\]  
\[x_{jk}^{v}\in\{0,1\}\hskip 56.905512 ptj,k=1,2,\ldots n;v=1,2,\ldots,m \tag{9}\]  
Equations (2)-(8) are the constraints to ensure the validity of a solution to Equation (1). Equations (2) and (3) ensure that a customer is served by exactly one vehicle. Equation (4) ensures the direction of a vehicle to serve a node i.e., if a vehicle travels from node \(c_{j}\) to node \(c_{o}\) then it never returns to the node \(c_{o}\) from any other node \(c_{k}\) or depot \(c_{h}\). Equation (5) restricts the customers to be assigned to a vehicle exceeding the maximum load capacity of that vehicle. Equation (6) confirms the fulfillment of the maximum travel distance constraint of a vehicle. Equation (7) forces a vehicle to start the journey from and end at the depot \(c_{h}\) and never visit the depot any more. Equation (8) restricts sub-tour generated within a route. Equation (9) is the integrality constraint for the formulation. \begin{table}
\begin{tabular}{c c} \hline \hline \(N\) & Total number of nodes \\ \(m\) & Total number of required vehicles \\ \(\textit{dist}\big{(}c_{j},\ c_{k}\big{)}\) & Cost to visit node \(c_{k}\) from node \(c_{j}\) \\ \(\textit{st}_{i}\) & Service time required when a vehicle visits node \(i\) (for the depot, \(\textit{st}_{h}=0\)) \\ Q & Maximum capacity of a vehicle \\ TT & Maximum distance of a vehicle can travel to \\ \(d_{i}\) & Demand of a customer to be served by a vehicle \\ \(Z\) & Customer set served by a vehicle; \(|Z|\) is the number of customers served by a vehicle \\ \(x_{jk}^{v}\) & Binary decision variable set to 1 if vehicle \(v\) serves node \(k\) after serving node \(j\), or 0 otherwise \\ \hline \hline \end{tabular}
\end{table}