“That’s (not) Rubbish!” Planning the Reverse Logistic for Recyclable Solid Waste Using a Vehicle Routing Problem: A Case Study in Vitória/ES

In this section, some vehicle routing concepts are presented, in addition to a taxonomy of solid waste collection route optimization problems.

The Vehicle Routing Problem (VRP) has been widely studied in the last years and during this time, many mathematical models and heuristics have been suggested for many cases. To address these cases, different extensions of the VRP have been proposed adding different constraints for each type of problem. One of these extensions is the Vehicle Routing with Multiple Trips (VRPMT), which is important in the context of transport companies that perform many trips during the day. Despite its importance, VRPMT has been only investigated in the last two decades, and few papers were published so far [8].

Optimization models are commonly used in order to reduce waste collection costs [9]. The first study on economic optimization was carried out by Anderson and Nigam [10], and since then, several authors have worked with cost minimization in the objective function.

The traditional MSW collection method consists of collecting household waste (door-to-door collection) and dry waste from Voluntary Delivery Points (VDSs) with predefined stops and routes and transporting the waste to the disposal station in trucks along the stipulated routes. This process generates labor costs, fuel costs, maintenance, among others, being considered a very high cost and responsible for most expenses with urban waste [11].

Using dynamic routing model, Abdallah et al. [11] developed a model to reduce costs and carbon dioxide emissions. In the study, it was possible to reduce 10% of the costs and between 5 and 22% of emission, using a model associated with IoT based bins.

Using a multi-objective optimization model to design a cost-effective waste management supply chain, Olapiriyakul et al. [5] considered sustainability issues such as land-use and public health impact. The model presented 20% cost reduction and above 50% satisfaction level.

Rizvanoğlu et al. [12] applyed more than one technique to develop ideal routing for the collection and transport of solid waste, aiming to reduce costs to a minimum, along with the geographic information system to determine the best route it was possible to save 28% of the route with GIS analysis and more than 30% with linear programming analysis according to existing routes for collecting and transporting urban solid waste.

Generally, collection and transport are the most important and expensive aspects of the process due to the labor intensity of the work and the massive use of vehicles in the collection and transport process [13].

Marseglia et al. [14] developed a mathematical model, based on the Bin Packing and VR schemes, for the deployment of routes of mobile containers in the selective collection of urban solid waste. The authors obtained above 19%, on average, of improvement. In the number of vehicle routes dispatched from the depot, the improvement remains inconclusive.

Jorge et al. [15] used a hybrid metaheuristic to solve the smart waste collection problem with workload concerns. The authors developed a look-ahead heuristic to decide the days on which collection is necessary and which bins need to be collected and also a simulated annealing/neighborhood search algorithm to choose the bins that are profitable to collect and the best routes to visit the bins. The results achieved by the hybrid metaheuristic were at least 45% higher than the profit obtained by the company.

Rathore and Sarmah [16] published a study that examines the economic, environmental and social viability of circular economy in MSW management. A concept is proposed for utilizing collected organic MSW, converting it into biogas and then using it as fuel in a thermal power plant to reduce the load on coal mines. For this, a nonlinear mixed integer program (MINLP) model is formulated for the minimization of the total cost which is the sum of (i) running cost, (ii) transportation cost, (iii) rental cost, (iv) environmental cost, (v) social cost, and (vi) penalty cost. With the application of the model in a case study in the city of Bilaspur, India, it was possible to verify the reductions in carbon emissions, reductions in social costs and even a 60% reduction in the cost of transport due to the optimization of routes and reduction of the amount of vehicles. The authors conclude that it is possible to apply the model globally, however, according to the analysis of collection costs, the model performs better in high-income countries compared to low-income countries for the same scenario.

Using a novel robust bi-objective multi-trip periodic capacitated arc routing problem under demand uncertainty to treat the urban waste collection problem, Tirkolaee et al. [17] aimed to minimize the total cost and minimize the longest tour distance of vehicles. The largest change in the total cost was the reduction of 11.97% which occurred for 20% increase in the longest allowable tour, and the largest change in the longest tour was the reduction of 25.29% for a decrease of the parameter.

Braier et al. [18] used mathematical programming techniques to optimize the routes of a recyclable waste collection system. The study was applied to the city of Morón, a large municipality outside Buenos Aires, Argentina and it was possible to achieve improvement while in some cases actually reducing total vehicle distance compared with the old manual routes. In a few cases, distance did increase, but only slightly.

Jatinkumar Shah et al. [19] developed a stochastic optimization model based on chance-constrained programming to optimize the planning of waste collection. The study aimed to minimize the total transportation cost while maximizing the recovery of value still embedded in waste bins. The authors considered High and Low-capacity trucks (LCT), and as the low-capacity vehicles increase, the value of the objective function or the transportation cost decreases. This conclusion was based on the point that the fixed cost of adding new LCT.

The VRPMT is an important mathematical model for real situations occurring in daily logistics and yet it has few published papers. Most of the published papers deal with new methods to solve it. Very few papers deal with real cases.

The trucks, for collection of selective waste, start the trip from the central depot at the beginning of the shift work. From the depot, the trucks must go through all VDS in one shift and one trip finishes when the truck’s capacity is reached. After reaching full capacity, the truck must return to the depot to unload. After the truck is unloaded, another trip is started in order to collect the VDS that were not collected yet. This process continues until the end of the work shift of the driver and the assistants.

The VRPMT applied to waste collection consists of routing vehicles from a fleet to collect solid waste while minimizing travel costs and distances. The VRPMT differs from the traditional VRP because the collecting trucks must empty the load at disposal sites. The problem is illustrated in Figure 2, for multiple vehicles and a single disposal site.

2.png

Figure 2. Schematic vehicle routing with multiple trips [24]

The proposed mathematical model is presented in five parts: the parameters, the sets, the decision variables, the objective function and the constraints. The VRPMT considers np the number of VDS; nv the number of vehicles in the fleet; nt the number of trips.

Mathematical model sets are:

Mathematical model parameters are:

Mathematical model decision variables are:

Objective Function:

$\begin{gathered}\text { Minimize } \sum_{k \in K} c f_k z_k+\sum_{i \in C T} \sum_{j \in C T} \sum_{k \in K} \sum_{t \in T} d s_{i j} c v_k x_{i j k t}\end{gathered}$        (5)

Subject to:

$\begin{gathered}\sum_{k \in K} \sum_{j \in C V} \sum_{t \in T} x_{i j k t}=1 \\ \forall i \in C\end{gathered}$        (6)

$\begin{gathered}\sum_{j \in C V} x_{0 j k t}=1 \\ \forall k \in K, t \in T\end{gathered}$                      (7)

$\begin{gathered}\sum_{k \in K} \sum_{t \in T} x_{i 0 k t}=0 \\ \quad \forall k \in K, i \in C T\end{gathered}$                         (8)

$\begin{gathered}\sum_{k \in K} \sum_{j \in C T} \sum_{t \in T} x_{n p+1 j k t}=0 \\ \forall k \in K\end{gathered}$                (9)

$\begin{gathered}\sum_{i \in C T} x_{i h k t}-\sum_{j \in C T} x_{h j k t}=0 \\ \forall h \in C, k \in K, t \in T\end{gathered}$                  (10)

$\begin{gathered}\sum_{i \in C} d m_i \sum_{j \in C V} x_{i j k t} \leq Q_k \\ \forall k \in K, t \in T\end{gathered}$                  (11)

$\begin{gathered}\sum_{i \in C 0} \sum_{j \in C V} \sum_{t \in T} \frac{d s_{i j}}{v v_k} x_{i j k t} \leq t u_k \\ \forall k \in K\end{gathered}$                   (12)

$\begin{gathered}x_{i i k t}=0 \\ \forall i \in C T, k \in K, t \in T\end{gathered}$                   (13)

$\begin{gathered}x_{i j t} \in\{0,1\} \\ \forall i \in C T, j \in C T, k \in K, t \in T\end{gathered}$                  (14)

$\begin{gathered}t c_{j k t} \geq t c_{i k t}+\frac{d s_{i j}}{v v_k}-\left(1-x_{i j k t}\right) M \\ \quad \forall i \in C 0, j \in C V, k \in K, t \in T\end{gathered}$                             (15)

$\begin{aligned} t c_{i k t} & \geq 0 \\ \forall i \in C T, k & \in K, t \in T\end{aligned}$                      (16)

$\begin{gathered}t c_{i k t} \leq t u_k \\ \forall i \in C T, k \in K, t \in T\end{gathered}$                       (17)

$\begin{gathered}\sum_{j \in C V} \sum_{t \in T} x_{0 j k t} \geq z_k m \\ \forall k \in K\end{gathered}$                     (18)

$\begin{gathered}\sum_{j \in C V} \sum_{t \in T} x_{0 j k t} \leq z_k M \\ \forall k \in K\end{gathered}$                    (19)

$\begin{gathered}z_k \in\{0,1\} \\ \forall k \in K\end{gathered}$                          (20)

$\begin{gathered}l k \leq z_k \leq n k \\ \forall k \in K\end{gathered}$                (21)

The Objective Function (OF), Eq. (5), aims to minimize the total distance of all trips during the collection. The set of constraints (6) ensures that each VDS i is visited by a single vehicle k on a single trip t. Constraints (7) ensure that all vehicles must leave the depot, including to the virtual depot. Constraints (8) ensure that no vehicle return to the depot. Constraint (9) ensure that no vehicle can leave from the virtual depot. Constraints (10) represent flow networks, every vehicle arriving at node i must leave this node. Constraints (11) are capacity constraints, the total loaded on each vehicle k in a trip t cannot exceed the capacity Q_k of the vehicle. Constraints (12) guarantee that all vehicles must have a maximum time of travel and it must be inferior than the work shift. Constraints (13) guarantee that arches from a node to itself cannot exist. Constraints (14) ensure that variables x_ijkt are binary. The set of constraints (15) ensure that sub-tours are not created. Constraints (16) and (17) ensure that time must be inferior than the work shift and it cannot be negative. Constraints (18) and (19) were created for the logic in the model, variable z_k receives value 1 in case vehicle k performs a trip t and 0 otherwise. Constraints (20) ensure that variables z_k are binary. Constraints (21) guarantee the minimum and the maximum number of vehicles used to collect recyclable waste.

There is an accelerated and constant increase in the urban population worldwide, and it has led to an exorbitant rise in municipal solid waste generation. Nowadays, there is a growing preoccupation concerning the disposal of the waste collected for the environment and society as well as the expenses associated with it.

This study developed a mathematical model to plan the reverse logistics of RSW collection, employing linear programming, to a medium-sized city in Brazil. First, the methodology developed by FUNASA [14] was applied to dimension the fleet of trucks, and then a mathematical model based on the VRPMT was proposed to optimize the routes to collect recyclable waste in the city of Vitória, Brazil. Simulating one scenario, where the fleet of trucks must collect the recyclable waste in VDSs geographically in an 8-hour work shift, the model presents economic benefits by promoting a reduction of costs associated with the collection. The results showed that it is possible to reduce the costs by 48.70%, compared to the manual planning currently performed by the city council, therefore, presenting a more efficient collection. The model can be used as a tool to support the planning of the reverse logistics of the RSW, and it may help the city council to reduce the logistics cost. In addition, the trips no longer need to be made manually, it is possible to automatically plan the routes in a standard way, requiring only input data. The introduction of multiple routes during one day proved to be a paradigm shift, and it has not been applied to the selective waste collection of a medium-sized city. A strong waste collection plan can aid the decision-maker choose the best alternatives, reducing fuel consumption and costs associated with the travel distances of the fleet of trucks. It should be stressed that the model is limited by small instances, since it cannot plan the routes of collection for a large number of VDSs with a small GAP.

Therefore, for future directions, a SA metaheuristic can be considered. Considering that exact methods are not able to solve large instances of the problem due to the combinatorial aspects of the mathematical model, the SA metaheuristic developed for VRPMT can solve bigger instances, since the problem is different from the classical VRP. The use of VRPMT and SA metaheuristic will allow to plan future expansions of the selective collection system with geographically dispersed points to large instances.