Waste Collection Vehicle Route Optimization: A Case Study at the Hellenic Military Academy ^†

Abstract

In this article, we present a case study of the waste collection problem at the Hellenic Military Academy. The waste is sorted by type and collected by a garbage truck. To minimize the travel cost of the waste collection vehicle, we apply the Markov Decision Process methodology. This approach enables the development of more efficient algorithms.

1. Introduction

In modern times, waste management is a critical area of study aimed at reducing transportation and storage costs. One effective approach to lowering waste disposal costs is converting waste into biofuels. The transformation of waste into biomass is a complex process with significant limitations and high costs [1]. Notably, a substantial portion of the total cost is associated with waste collection and garbage truck routing. In this paper, we develop a model using Markov Decision Processes to optimize the routing of a garbage truck at the Hellenic Military Academy.

2. The Description of the Problem

At the Hellenic Military Academy, bins of different colors have been placed for the purpose of collecting waste according to its type. The waste disposal areas are predetermined based on the building zoning of the Hellenic Military Academy. Waste collection is mandatory at these designated points to ensure the safety and operational efficiency of the camp [2]. The diagram below illustrates the spatial layout of the points where the waste collection bins have been placed.

The route described in Figure 1 is determined by the existing road network of the Hellenic Military Academy and the location of the garbage truck parking lot. At each collection point, waste is separated into three categories—food waste, plastic, and paper—and placed into brown, red, and yellow bins, each with a capacity of 1.1 m³. Given the importance of source sorting, clear instructions and guidelines are provided to facilitate proper waste disposal. The qualitative and quantitative assurance of organic waste is maintained for two key reasons: (a) the largest volume of organic waste is generated after meals (breakfast, lunch, and dinner) served to the Cadet Regiment, and (b) the operation of the Hellenic Military Academy requires waste disposal at specific times of the day. The collection and placement of waste into the bins are carried out by designated personnel assigned to this service, rather than individually by each cadet. The garbage truck has a capacity of Q. The vehicle begins its journey at Point 1 (Restaurant 1), collects waste from each designated point, and then returns to its parking spot. If the waste collected from any point exceeds the remaining capacity of the vehicle, the truck must return to the waste collection facility of the Municipality of Varis–Voulas–Vouliagmenis to empty its load. A key requirement is that all bins must be emptied from their respective collection points. Travel costs are proportional to the distance traveled by the vehicle, and the goal is to minimize the total travel cost while ensuring that all bins are emptied.

This problem is a variation of the well-known Vehicle Routing Problem (VRP), which has been extensively studied over recent decades [3]. In the specific problem being solved, a key stochastic parameter is introduced: the volume of waste in each type of bin (brown, red, and yellow) located at the eight collection points. Due to this stochastic element, the problem falls under the category of the Stochastic Vehicle Routing Problem (SVRP) [4]. To solve this problem, Markov Decision Processes (MDPs) are employed to determine the optimal route policy for the vehicle [5]. Once the optimal policy is established, algorithms with lower computational costs can be developed based on the structure of the policy.

3. The Mathematical Model of the Optimal Route

To simplify the mathematical expressions presented below, we will use the following notation: 0 represents the waste collection facility of the Municipality of Vari–Voula–Vouliagmeni, 1 represents the bin location at Restaurant 1, 2 represents the bin location at Restaurant 2, 3 represents the bin location at the first parking lot of the Hellenic Military Academy (Parking 1), 4 represents the bin location at the second parking lot of the Hellenic Military Academy (Parking 2), 5 represents the bin location at the first command building of the Hellenic Military Academy (Commanding Office 1), 6 represents the bin location at the second command building of the Hellenic Military Academy (Commanding Office 2), 7 represents the location of the bins in the Cadet Dormitories (Cadet Dormitories), and 8 represents the location of the bins in the Cadet Classrooms. Also, we classify food waste as Category 1, plastic waste as Category 2, and paper waste as Category 3. The vehicle can transport Category 1 waste (food waste) up to a maximum volume of $Q_1$ (m³), Category 2 waste up to a maximum volume of $Q_2$ (m³), and Category 3 waste up to a maximum volume of $Q_3$ (m³), subject to $Q_1+Q_2+Q_3\le Q$, where $Q$ is the capacity of the vehicle. We denote the cost of traveling from point $j$ to point $j+1$ as $c_{j,j+1}$ the cost of traveling between each point and the waste collection site of the Municipality of Vari–Voula–Vouliagmeni as $c_{j,0}$ where 0 represents the waste collection site. The vehicle visits each location where the bins are placed, and at that point, the volume of waste for each category to be collected becomes known. The garbage truck collects the maximum possible amount of waste for each type (category) at every location.

To solve the problem, we will use a variation of the model that has been proposed in [6]. We assume that the state of the vehicle after its first visit to each collection point, once it has reached its maximum possible load, is represented by the vector $\left(z_1,z_2,z_3\right)$, where $z_1$ is the available space in the garbage truck for food waste, $z_2$ is the available space for plastic waste, and $z_3$ is the available space for paper waste. Negative values of $z_1,z_2$, and indicate the volume of waste in category $i, i=1,2,3$ that cannot be collected due to the truck’s capacity limitations. Let $z=\sum _{i=1}^3{z}_i^{-}<0$ where $z_i^{-}=\mathrm{m}\mathrm{i}\mathrm{n}\{0,z_i\}$. If $z=0$, it means that the garbage truck has received the entire volume of waste from the specific point and can proceed to the next point or return to the waste unloading point. If $z<0$ the garbage truck returns to the unloading point to empty its load before returning to the collection point to pick up any remaining waste. After this, it has two options: to proceed to the next collection point, or to make a second trip to the unloading point to fully empty its load before continuing to the next collection point. Our objective is to minimize the total expected cost over a complete cycle of visits. We denote by ${\xi }_i^j,$ the volume of waste of the category $i, i=1,2,3$, at the point $j,j=1,\dots ,8$ which is a discrete random variable. We define the vectors $\overline{z}=\left[z_1,z_2,z_3\right]$ and $\overline{{\xi }_j}$ = $\left[{\xi }_{j,1},{\xi }_{j,2},{\xi }_{j,3}\right]$. If we denote $f_j\left(z_1,z_2,z_3\right)$ for $j, j=1,\dots ,7$ the minimum expected cost from the first visit of the garbage truck to each point where bins have been placed until the end of the route, in the state $\left(z_1,z_2,z_3\right)$, then the dynamic programming equation is the following:

Case 1. If $z_1,z_2,z_3\ge 0,$ then $f_j\left(\overline{z}\right)=min\{H_j\left(\overline{z}\right),A_j\}$

Case 2. If $z=\sum _{i=1}^3{z}_i^{-}<0,$ then $f_j\left(\overline{z}\right)=2c_{j,0}+min{\{G}_j\left(\overline{z}\right),A_j\}$, where

$$H_j\left(z_1,z_2,z_3\right)=c_{j,j+1}+E{f}_{j+1}(z_1-{\xi }_{j+1,1},z_2-{\xi }_{j+1,2},z_3-{\xi }_{j+1,3})$$

(1)

$$A_j=c_{j,0}+c_{0,j+1}+E{f}_{j+1}(Q_1-{\xi }_{j+1,1},Q_2-{\xi }_{j+1,2},Q_3-{\xi }_{j+1,3})$$

(2)

$$G_j\left(z_1^{-},z_2^{-},z_3^{-}\right)=c_{j,j+1}E{f}_{j+1}\left(Q_1+z_1^{-}-{\xi }_{j+1,1},Q_2+z_2^{-}-{\xi }_{j+1,2},Q_3+z_3^{-}-{\xi }_{j+1,3}\right)$$

(3)

For collection point 8, the boundary condition is $f_8\left(z_1,z_2,z_3\right)=c_{8,0}+2c_{8,0}I\left(z<0\right)$, where $I$ is the indicator function. The minimum total expected cost during the daily route is equal to $f_0=c_{1,0}+E{f}_1(Q_1-{\xi }_{1,1},Q_2-{\xi }_{1,2},Q_3-{\xi }_{1,3})$, as the garbage truck starts its daily route from its parking point. In Equations (1)–(3), the expected values are with respect to the random vectors $\overline{{\xi }_j},j=1,\dots ,8$. The first term in Equation (1) corresponds to the energy of the garbage truck moving to the next point, while the second term corresponds to the energy of the garbage truck going to the waste unloading point. In Equation (2), the first term corresponds to the energy of the garbage truck going to the unloading point once before moving to the next point, while the second term corresponds to the energy of the garbage truck going to the unloading point twice before moving to the next point. Through the use of a similar methodology as in [6], it can be shown that the optimal garbage truck routing policy has a special form. Therefore, it is possible to develop more efficient algorithms.

4. Conclusions

A Markov decision model is proposed for the waste collection problem at the Hellenic Military Academy, where a garbage truck must collect different types of waste. The vehicle can either proceed to the next collection point or empty its load at a designated disposal site. Since the amount of waste in each bin is a random variable, the vehicle does not know in advance how much it will need to collect from each bin. The proposed mathematical model aims to determine the optimal routing policy for the vehicle. An extension of the proposed methodology could incorporate uncertainties beyond the quantity of waste collected. These factors may include variations in the vehicle’s travel costs and the availability of the driver.