A company has to collect waste for different customers, at different locations, using two vehicles both with a certain capacity. The vehicles have to begin and end at the depot and when the capacity of a vehicle is exceeded it has to return to the disposal site, which is also the depot. But the vehicles are not limited to one route a day. I have to make a schedule for both vehicles in such a way that the total travel time of both vehicles is minimized.
In literature this is a variation of the Capacitated Vehices Routing Problem (CVRP), but instead of minimizing the travel distance or travel cost of the vehicles I have to minimize the total travel time.
The problem is defined on a graph $G=(V,A)$, with nodes $V$=$V^d \cup V^c$ where $V^d$={$0$} is the depot and disposal site and $V^c$={$1,2,..,n$} is the set of customers and a set of arcs $A$={$(i,j)|i,j$$\in$A, $i\neq j$}. Let $K$={1,2} the set of vehicles and $t_{ij}$ the time traveled between $i$ and $j$. Each node $i\in V$ has an associated service time and amount of waste, respectively $s_i$ and $q_i$. Both vehicles have capacity $Q$. Lastly we introduce two dummy variables:
$x_{ijl}=1$ if customer $j$ is served directly after customer $i$ by vehicle $l$ and $0$ elsewhere.
$y_{il}=1$ if customer $i$ is served by vehicle $l$ and $0$ elsewhere.
$d_{il}$ = the amount of waste in the vehicle up to costumer $i$.
If the capacity if exceeded the vehicles goes to the depot and there it takes $30$ min to dump the waste, so let $m$=the amount of routes the vehicles make.
Now I want to minimize the total travel time of the vehicles. Thusfar I have this:
Minimize the following function
$$ T=\sum_{l=1}^2 (t_{0il}+t_{j0l}+ \sum_{i=1}^{n-1} t_{ij}x_{ijl}) + \sum_{i=1}^n s_i + 30m$$
under the constraints:
$\sum_{l=1}^2 \sum_{i=1}^n x_{ijl}=1$ $\qquad$$\forall i\in A$
$\sum_{j=1}^n x_{ijl} - \sum_{j=1}^n x_{jil}=0$ $\qquad$$\forall i\in A$
$\sum_{j=1}^n x_{0jl}=1$ $\qquad$$\forall l\in K$
$\sum_{i=1}^n x_{i0l}=1$ $\qquad$$\forall l\in K$
$x_{ijl}=1 \Rightarrow d_{il}y_{il}-q_{i}=d_{jl}y_{jl}$ $\qquad$$\forall i,j\in A, l\in K$
$\sum_{i=1}^n q_{i}\le Q$
$x_{ijl}$={$0,1$} $\qquad$$\forall i,j\in A, l\in K$
But now I haven't made constraints about the amount of routes and I don't know how I'm supposed to do that. Also, I'm still not quite sure whether minimizing this particular function is the best way. Does anyone have any input? It will be much appreciated.