A new procedure to plan routing and scheduling of vehicles for solid waste collection at a metropolitan scale

In this paper a new decision procedure for urban solid waste collection problem is introduced. The problem objective is to minimize the overall cost, which is essentially based on the time spent by the vehicles in the network. In the last years such a problem has been considered by several authors in the framework of the more general problem of planning a waste management system at a municipal scale. In particular, the vehicle scheduling and routing problem for solid waste collection has been addressed only as a sub-problem of the higher-level problem concerning municipal solid waste management. Some examples can be found in [1], [2], [3], [4], even if in none of them the waste collection problem has been considered at a district level, i.e., taking into account the whole road network in detail. This is mostly due to the intrinsic complexity of the problem, and to the high dimensionality of any real case study. In this connection, the purpose of this work is that of proposing a computationally sustainable approach to automatically determine “reasonably good” solutions for real case studies concerning solid waste collection in urban areas, represented in full detail as regards the road network. The development of such an approach is heavily based on heuristics, and any effort is made to take into account all features characterizing the considered real problem. Let us define the basic waste collection problem. A road network is defined, in which links are characterized by the service demand (i.e., the quantity of refuses to be collected for any link in the network), which is null for a non-empty subset of links, and by some set of parameters, such as the time required to traverse the link, and the time required to cover the link collecting refuses, again for any link in the network. A limited number of identical vehicles is available, and each of them is characterized by a finite capacity. The service of each vehicle starts and ends at a predefined node (depot), and it is named “working shift”; the duration of each working shift has an upper bound. A generic working shift is composed by several routes, i.e., paths starting and ending at the depot; during each route the vehicle can collect at most a quantity of refuses equal to its capacity. The objective is that of collecting all the refuses (i.e., servicing all the links with positive demand) while minimizing the overall time spent, and taking into account both time (related to each working shift) and capacity (related to each route) constraints. As a matter of fact, in the previously mentioned works waste collection problems have been modelled within a vehicle routing problem framework, in which demand has been associated with nodes. Differently, the model proposed in this paper associates the demand with the links, due to the fully detailed representation of the road network. Thus, the problem falls in the class of arc routing problems, which is usually utilized to represent problems as mail delivery, street sweeping, etc. For a survey about such problems and the possible approach to solve them, see [5]. A particularly interesting problem in this class is the so called CARP (Capacitated Arc Routing Problem), which is characterized by the presence of a constraint on the payload of the vehicles, and by the presence of links with null demand; CARP has a very general formulation, and it is NP-hard. A number of heuristic algorithms for CARP has been developed by several authors, and their performances are reported in the literature (see again [5]). In any case, the basic waste collection problem represents a non trivial generalization of the CARP, because of the addition of two sets of constraints, namely those related to the maximum duration of a working shift, and those imposing that each route must start and end at the depot. For these reasons, it is not possible to follow already existing approaches, and a heuristic approach, purposely devised to cope with the features of the considered problem, must be developed. The algorithm developed to solve the basic waste collection problem builds a working shift by incrementally generating the routes, whose construction consists of two phases. In the first phase forward phase – at the current node, the outgoing link is chosen which provides the maximum “attractiveness coefficient”, which is defined as the ratio between the amount of waste to be collected and the time necessary to cover the link (with service). In the second phase backward phase the best path to the depot, computed on the basis of the link attractiveness coefficient, is built, checking whether it fulfils capacity and time constraints. If such constraints are satisfied, the link is added to the route under construction and the procedure is repeated at the new node reached; if constraint concerning the vehicle capacity is violated, the link is not added and the route under construction is taken as complete; finally, if the time constraint is violated, the whole considered working shift, as well the route under construction, is taken as complete. In order to make the procedure capable of representing real waste collection problems, a different procedure has been developed for the case of a more complicated kind of working shifts, in which some routes do not start or end at the depot. The model is inspired by the real case study relevant to the city of Genova, located in the northwestern coast of Italy. Present solution to the above problem relies on human expertise and on a tradition of hand-made solutions, which are reasonably updated whenever a significant change in operational conditions occurs. In order to test the procedure, a large district belonging to the city of Genova has been selected, namely Sampierdarena. This district has been selected as it is quite representative of the whole city. Referring to the characteristics of the district, six different classes of vehicles have been identified. The depots of the vehicles are different: for trucks the depot is located outside of the center of Sampierdarena, and for three-wheeler the depot is situated in the center of the district. Nowadays waste collection in the considered district is performed through 9 different working shifts (4 in the morning, 2 in the afternoon and 3 during the night), with a total daily cost of 5,825 €. The implementation of the procedure presented in the previous sections has been made using C++ language, and the best solution found shows a total daily cost of 5,228 €, thus allowing a 5% reduction in the overall cost. Moreover, the number of vehicle classes required to complete the service decreases from 6 to 4.