Waste collection inventory routing with non-stationary stochastic demands

Highlights

• Models and solves a rich stochastic waste collection inventory routing problem from practice.

• Uses a purpose-built forecasting model validated on real data for predicting container levels and estimating the forecasting error.

• Proposes a mixed integer non-linear formulation capturing dynamic probabilistic information on container overflows and route failures.

• Develops a powerful ALNS algorithm for solving benchmark and real-world instances.

• Demonstrates the strength and robustness of the approach in comparison to alternative policies on a set of waste collection instances from Geneva, Switzerland.

Abstract

We solve a rich routing problem inspired from practice, in which a heterogeneous fixed fleet is used for collecting recyclable waste from large containers over a finite planning horizon. Each container is equipped with a sensor that communicates its level at the start of the day. Given a history of observations, a forecasting model is used to estimate the expected demands and a forecasting error representing the level of uncertainty. The problem falls under the framework of the stochastic inventory routing problem and our main contribution is the modeling of the dynamic probability-based cost of container overflows and route failures over the planning horizon. We cast the problem as a mixed integer non-linear program and, to solve it, we develop an adaptive large neighborhood search algorithm that integrates a purpose-designed forecasting model, tested and validated on real data. We demonstrate the strength of our modeling approach on a set of rich inventory routing instances derived from real data coming from the canton of Geneva, Switzerland. Our approach significantly outperforms alternative deterministic policies in its ability to limit the occurrence of container overflows for the same routing cost. Finally, we show the benefit of a rolling horizon solution and derive lower and upper bounds on its cost.

Introduction

Waste collection is one of the most important logistical activities performed by any municipality, and also one of the most expensive, with collection costs alone accounting for more than 70% of waste management costs (Tavares et al., 2009). Recycling, on the other hand, can alleviate problems related to landfill capacity and pollution, and many countries have already set ambitious target levels for recycling. As part of its Circular Economy Strategy, the European Union (EU), for example, has adopted legislative proposals to set a common EU target for recycling 65% of municipal and 75% of packaging waste by 2030, limiting at the same time the use of landfills (European Commission, 2016). Given the high cost of waste management and the significant proportion of collection costs, even small improvements in the latter can lead to substantial financial savings for waste collectors, municipalities and the taxpayer.

In this context, we solve a rich recyclable waste collection problem, which can be described as follows. A heterogeneous fixed fleet is used for collecting recyclable waste from large containers. Each container holds and each vehicle collects a given waste flow, e.g. white glass, colored glass, paper, etc. Therefore, a separate problem is solved for each waste flow. As shown in Fig. 1, a tour starts and ends at the depot, and is a sequence of collections followed by disposals at the available dumps. There is a mandatory visit to a dump just before the end of a tour, i.e. a tour terminates with an empty vehicle. Dumps are recycling plants. There could be multiple dumps for the collected waste flow and they can be used when and as needed along the tour. We consider time windows for the containers, depots and dumps. For depots and dumps, time windows represent working hours, whereas for containers they can be set to avoid collections during rush hours, school times, etc. A tour is also limited by the legal duration of the working day. Accessibility restrictions apply to certain points, for example containers located in narrow streets that cannot be accessed by big collector trucks.

Containers are equipped with ultrasound sensors that communicate their waste levels via the GSM network at the start of the day. Yet, planning today’s collections while ignoring future container demands is myopic. For instance, it may be cheaper to postpone a collection if we know that the container in question fills up slowly. On the other hand, it may be worthwhile collecting an almost empty container today if we know that it will experience high demand in the coming days. Since future demands are unknown, we use historical data and forecasting techniques to estimate their expected values and the distribution of the forecasting error term over a multi-day planning horizon.

Deviations of the demand realizations from their expected values may lead to undesirable events. If a container fills up more quickly than expected and is not collected on time, it may overflow. Experience suggests that such containers continue serving demand because people place the waste beside them. Nevertheless, the municipal regulations require that overflowing containers should be collected on the same day. By the same logic, if the containers planned for collection on a given day are fuller than expected, the vehicle may run out of capacity before its scheduled dump visit, resulting in a so-called route failure. These undesirable events require recourse actions, such as the emergency collection of an overflowing container or an unplanned visit to a dump in the case of route failure. Recourse actions are expensive. Therefore, minimizing collection costs requires forward-looking plans with collection schedules that incorporate the probabilities of undesirable events.

Given the multi-day planning horizon and the uncertainty implied by the forecasting error, our problem falls under the framework of the Stochastic Inventory Routing Problem (SIRP). Markov et al. (2018) present a unified framework for rich routing problems with stochastic demands, which allows the explicit modeling of recourse actions, their probabilities and costs in a computationally tractable way. The present work builds on this unified framework by going much deeper into the specifics of our problem. We apply a forecasting model and derive the expressions for the recourse costs of container overflows and route failures. Containers additionally incur an overflow cost that the collector pays to the municipality in the form of a fine. The correct attribution of these costs to the objective function involves the calculation of conditional probabilities, which are day-dependent and dynamically affected by previous collections during the planning horizon, and which complicate the solution methodology by introducing non-linearities.

While the problem is stochastic, it is also dynamic with new container level information revealed daily. The latter is integrated in a rolling horizon fashion by solving an SIRP on each day and implementing the decisions for that day. Considering a multi-day planning horizon at each rollover thus takes advantage of future probabilistic information for making forward-looking decisions today. As a result, the collection tours executed on each day reflect the anticipation of future demands, balancing collection costs and the expected costs of undesirable events and their recourse actions. In our problem, demand is revealed in discrete time periods, i.e. with the start of each day. Conceptually, the availability of continuous demand information can be handled using a finer time discretization under the same modeling framework.

The remainder of this article is organized as follows. Section 2 positions our work with respect to the relevant VRP and IRP literature. Section 3 outlines the forecasting model and develops the mathematical formulation of our problem. Section 4 describes the solution methodology, followed by Section 5, which presents the numerical experiments. Finally, Section 6 concludes and explores future work directions.