**2. General sector arc-routing problem and its related problems**

For better understanding the problems related to routing waste collection vehicles, we indicate below three of them:

Let us consider a strong-connected mixed graph G(V-VR, VR,L-LR, LR), where V is the set of vertices, VR are the required vertices, and L is the set of links (L ⊆ E ∪ A, E —not oriented set of edges, A—oriented set of arcs, and LR ⊆ ER∪ AR, ER— set of required edges and AR—set of required oriented arcs) that must be covered for the known garbage offer (qR > 0). With this basic statement, the following problems consider the elementary needs to be solved by specialized methods [5].

*Optimized Planning and Management of Domiciliary and Selective Solid Waste... DOI: http://dx.doi.org/10.5772/intechopen.111432*

There are three different methodologies of designing routes for waste collection: Chinese Postman Problem or CCP-based sectoring and routing, CARP-based solutions, and SARP-based solutions.

The CCP-based solutions consider that there are areas of collection that must be covered at certain times and on certain days. These areas can be covered in less than one workday through route design. The goal is to minimize the number of trucks required to cover the areas while joining (sectoring) different coverage areas in multiple trips to the landfill or transfer station. The routes are partially built by the Chinese Postman method, and a truck scheduling problem is performed to join the trips and compose sectors using a given fleet [6].

In the CARP-based solutions, the methods used consider a strongly connected positive-weighted graph as indicated previously. A fleet (homogeneous or heterogeneous) starts and ends at the garage (v0∈V) and passes through one or more intermediate facilities (vI∈V), that is, a landfill or transfer station. This problem was first proposed as a symmetric graph G(V, E-ER, ER) by Golden and Wong [7].

Using this approach, residential refuse collection applications and software for different cities worldwide were investigated by Ghiani et al. [8], and the most recent surveys on methods and their applications in garbage collection were conducted by Ghiani et al. [9], Han and Ponce-Cueto [10], and Mourão and Pinto [11]. With CARP solutions, the coverage of the area is not coordinated in a specific region, that is, shift contiguities are not guaranteed, or the same vehicle may collect in opposite places in a city, making managing the execution and measurement of daily processes complex [12]. Most recently, advanced methods have been developed by Boyaci et al. [13], considering fast lower bounds for CARP with intermediate facilities, and Janela et al. [14], considering balanced routes heuristics for MCARP.

The SARP was proposed by Mourão et al. [2]; it is considered an arc-routing problem (ARP), although it can be extended to node-routing applications for commercial and health waste collection. The SARP is, however, a more general problem than CARP as it includes all the problems of routing in arcs and nodes either with or without intermediate facilities [15]. In the SARP, trip contiguity, connectivity, and network compactness in a sector are considered objectives to be achieved by the final sector circuits covered by the routes, considering workload and fleet capacity constraints [16].

SARP resolutions are found by using clustering-based heuristic algorithms from generalized heterogeneous group assignments (sectors and circuits), and the Mixed Rural Postman Problem (MRPP) was applied to the circuits by Batista et al. [17] and Corberán et al. [18]. CARP heuristic method modifications have also been used by Corberán et al. [18]; they have been used by Ghiani et al. [19, 20] in zoning. Wølk and Laporte [21] used CARP with districting for waste collection in Denmark, and Boyaci et al. [13] applied fast upper and lower bounds for CARP with intermediate facilities and deadline methodology to large-scale world city instances. SARP was investigated by Mourão et al. [2], Rodrigues [15], and Cartinhal et al. [22].

In **Figure 1**, we can see an example of two circuits (trips) in the same collection sector: the chart on the left indicates the first trip (first circuit), starting at the garage, passing through the sector to collect, and ending at the disposal site. The chart on the right shows the second trip, starting from the end of the previous trip (transfer station), passing through the collection sector, and returning to the disposal site for unloading. The routes between garage and sector, sector and transfer station, and then transfer station and sector can be optimized by the "shortest path" or the "fastest path."

**Figure 1.**

*Sector 220 and its circuits with routes for each circuit. (a) Sector 220 – Petropolis/RJ. (b) Circuits of the sector 220, (c) Sector 220 first trip, (d) Sector 220 second trip.*
