| # Original Formulation: Capacitated Arc-Routing Problem (CARP) – Two-Index Formulation |
|
|
| *Source: “Cut-First Branch-and-Price-Second for the Capacitated Arc-Routing Problem,” Bode and Irnich, Operations Research 60(5):1167–1182, 2012.* |
|
|
| ## Sets and Parameters |
|
|
| - $G = (V, E)$: undirected graph with node set $V$ and edge set $E$. |
|
|
| - $E_R \subseteq E$: subset of required edges (those with positive demand). |
| |
| - $K$: fleet of $|K|$ homogeneous vehicles, stationed at depot $d \in V$. |
| |
| - $q_e \ge 0$: integer demand on edge $e \in E$; $Q$: vehicle capacity. |
|
|
| - $c^{\mathrm{serv}}_e$: service cost for required edge $e \in E_R$; $c_e$: deadheading cost for edge $e \in E$. |
| |
| - Notation: for $S \subseteq V$, $\delta(S)$ is the set of edges with exactly one endpoint in $S$, $\delta_R(S) = \delta(S) \cap E_R$, $E_R(S) = E(S) \cap E_R$; $\delta(i) := \delta(\{i\})$; for variable $y$ and subset $F$, $y(F) = \sum_{e \in F} y_e$. |
| |
| ## Decision Variables |
| |
| - $x^k_e \in \{0,1\}$ for $e \in E_R$, $k \in K$: $1$ if vehicle $k$ services edge $e$. |
| |
| - $y^k_e \in \mathbb{Z}_+$ for $e \in E$, $k \in K$: number of times vehicle $k$ traverses edge $e$ without servicing it. |
| |
| - $p^k_i \in \mathbb{Z}_+$ for $i \in V$, $k \in K$: auxiliary parity variable at node $i$ for vehicle $k$. |
| |
| ## Objective |
| |
| $$\begin{equation} |
| \min \sum_{k \in K} {c^{\mathrm{serv}}}^{\!\top} x^k + \sum_{k \in K} c^{\top} y^k \tag{1} |
| \end{equation}$$ |
| |
| ## Constraints |
| |
| $$\begin{align} |
| \sum_{k \in K} x^k_e &= 1, & \forall\, e \in E_R \tag{2} \\ |
| x^k(\delta_R(S)) + y^k(\delta(S)) &\ge 2\, x^k_f, & \forall\, S \subseteq V \setminus \{d\},\; f \in E_R(S),\; k \in K \tag{3} \\ |
| x^k(\delta_R(i)) + y^k(\delta(i)) &= 2\, p^k_i, & \forall\, i \in V,\; k \in K \tag{4} \\ |
| q^{\top} x^k &\le Q, & \forall\, k \in K \tag{5} \\ |
| p^k \in \mathbb{Z}_+^{|V|},\; x^k &\in \{0,1\}^{|E_R|},\; y^k \in \mathbb{Z}_+^{|E|}, & \forall\, k \in K \tag{6} |
| \end{align}$$ |
|
|
