Tactical waste collection: column generation and mixed integer programming based heuristics

Abstract

Environmental considerations and corresponding legislation cause a shift from waste management to materials management, requiring efficient collection of these flows. This paper develops a model for building tactical waste collection schemes in which a set of capacitated vehicles visits a set of customers during a given time period. Each vehicle must visit the disposal facility to discharge the waste after each customer visit. This is motivated by the fact that the waste of each customer has to be weighed at the disposal facility. The goal is to find a set of routes for each vehicle that satisfy both the demand and the frequency constraints and minimize the total cost. Since a state-of-the-art solver could not find a solution with a reasonable gap within an acceptable time limit, a column generation and a mixed integer programming-based heuristic are proposed. While the mixed integer programming-based heuristic outperforms the column generation heuristic in terms of solution quality, the lower bound provided by column generation allows to prove the small optimality gaps of the solutions obtained. Moreover, by applying both heuristics on instances derived from real-life data, they proved to be capable of finding good quality solutions in small computation times.

Appendices

Appendix A: IP model to find fleet size

The following IP is solved to obtain the maximal fleet size |V| used throughout the paper. It minimizes the number of vehicles needed, if collection was performed using only type 1 trips. The model was able to find an optimal solution for all test instances within seconds.

$$\begin{aligned} ({\text {F2}})\quad {\textsf {minimize}}\quad z \end{aligned}$$

subject to

$$\begin{aligned}&x^{1}_{vmd} \le L \cdot y^{1}_{vmd} \quad v \in V, m \in M, d \in D \\&\sum _{v\in V} \sum _{d\in D} x^{1}_{vmd} = Q_m \quad m \in M\\&\sum _{m \in M}\left( s_{m} \left( x^{1}_{vmd}\right) + t^1_{m}y^{1}_{vmd}\right) \le T \quad v \in V, d \in D \\&\sum _{m \in M} y^{1}_{vmd} \le 1 \quad v \in V, d \in D \\&z \ge \sum _{v \in V} \sum _{m \in M}y^1_{vmd} \quad d \in D \\&w_{md} \ge y^{1}_{vmd} \quad v \in V, m\in M, d \in D \\&\sum _{d \in D} w_{md} \le W_m \quad m\in M \\&y^{1}_{vmd} \in \{0, 1\} \quad v \in V, m\in M, d \in D \\&x^{1}_{vmd} \ge 0 \quad v \in V, m\in M, d \in D \\&w_{md} \in \{0,1\} \qquad m\in M, d \in D \\&z \in \{0, 1, 2, \ldots \} \end{aligned}$$

For the interpretation of the constraints, we refer to Sect. 3.

Appendix B: Initial instances

This appendix presents the data on the initial instances (Tables 6, 7). Note that $t^1_m$ represents time for a type 1 trip to a given customer, excluding the time to perform collection at this customer. This time thus represents the aggregated time to drive from the depot to the customer, from the customer to the disposal facility, the time to unload the waste and the time to drive from the disposal facility to the depot (at the end of the day). This should be multiplied with $c^{h}$ to obtain the cost of a type 1 trip. Similarly, $t^2_m$ represents the time needed for a type 2 trip to a given customer. The total time of such a type 2 trip is the time to drive from the disposal facility to the customer and back, and the time to unload the waste.

Table 6 Instance 1 (PMDa): data

Table 7 Instance 2 (PCa): data

Appendix C: Other test instances

This section presents the data on the other test instances (Tables 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21). Note that $T^1_i$ represents the time for a type 1 trip to a given customer, excluding the time to perform collection at this customer. This time thus represents the aggregated time to drive from the depot to the customer, from the customer to the disposal facility, the time to unload the waste and the time to drive from the disposal facility to the depot (at the end of the day). This should be multiplied with $C^{\text {hour}}$ to obtain the cost of a type 1 trip. Similarly, $T^2_i$ represents the time needed for a type 2 trip to a given customer. The total time of such a type 2 trip is the time to drive from the disposal facility to the customer and back, and the time to unload the waste.

Additional instances (suffixes .a, b, .b, c, .c, d and .d) were generated based on the initial cases (suffixes a). We varied the waste volumes $Q_i$ (instances indicated with b, c and d) and travel times $T^{\text {dep}}_i$, $T^{\text {disp}}_i$ and $T^{\text {disp,dep}}$ (instances indicated with .a, .b, .c and .d).

Table 8 Instance PMDb: data

Table 9 Instance PMDc: data

Table 10 Instance PMDd: data

Table 11 Instance PMD.a: data

Table 12 Instance PMD.b: data

Table 13 Instance PMD.c: data

Table 14 Instance PMD.d: data

Table 15 Instance PCb: data

Table 16 Instance PCc: data

Table 17 Instance PCd: data

Table 18 Instance PC.a: data

Table 19 Instance PC.b: data

Table 20 Instance PC.c: data

Table 21 Instance PC.d: data