# Dynamic Multi-Compartment Vehicle Routing Problem for Smart Waste Collection +
## 3 Problem Descriptions  
Our model proposes a separate and smart waste management problem that considers real-time information on the filling level of fragmented bins; each part is dedicated to a specific waste type to define dynamic routes for each compartmentalized vehicle. The signal base station receives information on the amount of waste in each intelligent bin at each station. It then communicates this information to the depot, which then sends compartmentalized vehicles to collect the selected separate waste stations. When a new station appears, the main task of the dispatching center is to include the new station in the current routing plan. Scheduled routes must be rescheduled depending on the position and waste load of the alarmed stations. The waste types, can be transported on the same vehicle without mixing the types; this is done by dividing the capacity of each vehicle into a number of compartments where each compartment can accommodate a waste type from one or more stations. As shown in Figure 1, the dynamic event considered is the appearance of a station that contains at least one waste type having a filling level \(\hat{x_{wi}}\) such as \(\hat{x_{wi}}\in\ [\alpha\,\ 100]\) (\(\alpha\) is the threshold to trigger the alarm for each waste type) (Figure 2). Figure 1: Dynamic vehicle routing for separate and smart waste collection. Our problem can be defined as follows: given a set of n stations (each station contains \(\hat{m}\) bins, and each bin is designated for a specific type of waste), a group of \(v\) homogeneous compartmentalized vehicles, a transfer station \(n+1\) where vehicles unload different types of waste collected in the bins, and a depot (where all vehicles begin and complete their path), a complete undirected graph is defined over \(n+1\) nodes, with distances \(d_{ij}\) for each edge \((i,\,j)\) in the graph. Each bin of type w in station \(i\) has a maximum capacity \(Cp_{w}\) (in kg). Let \(C_{ij}\) be the travel cost per unit of distance travelled. Each compartment in a vehicle has a capacity \(Cq_{m}\) (in kg). The sensors inside the bins transmit information on the bins' fill levels in \(m^{3}\), which is transformed into kilograms. To model and solve this problem, we will adopt the approach proposed by [35], which consists of dividing the working day into time slices \(t_{i}\) (Figure 3); each time slice represents a partial static Multi-Compartment Vehicle Routing Problem (M-CVRP), which optimizes the route of each vehicle, where vehicles must serve all known stations at minimum cost while satisfying the capacity of the vehicle compartments, and the bins capacity. The discretization strategy we followed is different from that proposed in the literature because we have stations that can be visited more than once in the same working day. The vehicle visits the stations that triggered the alarm, and collects the alarm types first. Then it contains the other waste types in the corresponding compartments if the capacity of each case is sufficient after visiting all the stations that triggered the alarm for each waste type. At the start of the day, the compartmentalized vehicles leave the central depot with a route plan containing the stations that were not collected the day before until the alarms are triggered to redefine the routes. The problem of separate and intelligent waste collection is assessed here. Consider the input data, which can be divided into two main types: static and real-time inputs data (Figure 4):  
* The static inputs are the information on the waste types and the bin's positions in a station, the bin's capacity, and the distance between the stations. Secondly, concerning  
Figure 3: The division of the working day. Figure 2: Fill level of each type in each station. vehicles, the static inputs are the number of compartments, which are identified with the number of waste types and the capacity of each compartment.