Robust solutions to the pollution-routing problem with demand and travel time uncertainty

https://doi.org/10.1016/j.trd.2017.01.003Get rights and content

Highlights

  • VRP with uncertain data and green routing problems are reviewed.

  • Demand and travel time uncertainty are investigated.

  • Several robust optimization formulations are presented.

  • Computational experiments are carried out on realistic instances.

Abstract

Freight transportation activities could potentially lead to detrimental effects on the natural and built environments and pose health risks. The importance of the present study is to consider demand and travel time uncertainty in green transport planning by proposing several robust optimization techniques; soft worst case, hard worst case and chance constraints. These techniques provide the most reliable solutions with very limited increase in the objective function related to fuel consumption and CO2-equivalent emissions.

Introduction

Transportation has detrimental effects on the environment such as resource depletion, land use, acidification, toxic effects on ecosystems and humans, noise and the impacts induced by Greenhouse Gas (GHG) emissions (Knörr, 2009). The emissions of CO2 are directly proportional to the amount of fuel consumed by a vehicle, which is in turn dependent on a variety of vehicle, environment and traffic-related parameters, such as vehicle speed, load and road gradient (Demir et al., 2011, Demir et al., 2014). The carbon dioxide equivalent (CO2e) measures how much global warming a given type and amount of GHG may cause, using the functionally equivalent amount or concentration of CO2 as the reference.

The Vehicle Routing Problem (VRP) is a well-known NP-hard problem which was introduced by Dantzig and Ramser (1959). Since then, VRP has been a topic of numerous studies in the literature of operations research. The traditional VRP is to find a set of vehicle routes for a set of customers with known demands. The literature on VRP and its variants are very widespread and involve many different aspects and decisions (see, for example, the latest surveys of Golden et al., 2008, Eksioglu et al., 2009, De Jaegere et al., 2014). Also, various exact (see, e.g., Baldacci et al., 2012, Almoustafa et al., 2013) and heuristics algorithms (see, e.g., Demir et al., 2012, Kramer et al., 2015) are suggested to solve such operational-level routing problems.

The traditional objective in the standard VRP is to minimize the total distance traveled by all vehicles, but this objective can be enriched through the inclusion of terms related to fuel consumption (Bektaş and Laporte, 2011). Recent developments in Green Vehicle Routing Problems (GVRPs) have heightened the importance of operations research techniques in this area. One of the successful applications in GVRP is due to Bektaş and Laporte (2011) who introduced the Pollution-Routing Problem (PRP), which is an extension of the VRP with time windows (VRPTW). In this paper, we consider a special case of the PRP where the objective function solely depends on the total fuel consumption rather than a weighted sum of fuel consumption and total driving time as in the PRP.

Demand uncertainty is one of the most common variants in non-deterministic (stochastic) VRPs. However, to the best of our knowledge, it has not been studied in the domain of GVRPs. Travel time uncertainty has been also investigated to see the impact of congested travel speeds on fuel consumption. The contributions of the paper are twofold: (i) we investigate the previous robust VRP models with stochastic demand and illustrate their weakness for the PRP, and (ii) we reformulate the PRP with several well-known robust approaches.

The remainder of this paper is organized as follows. In Section 2, a literature review on both VRPs and GVRPs is provided. In Section 3, the mathematical formulation of the PRP is presented. Section 4 introduces the proposed robust optimization approaches along with their formulations. Extensive numerical experiments and comparative analysis are provided in Section 5 to show the powerfulness of the proposed robust optimization models. Finally, Section 6 concludes the paper and introduces relevant future research directions.

Section snippets

Literature review

This section reviews the existing research literature on both VRP with uncertain data and green vehicle routing problems.

Problem description

In this section we define and formulate the corresponding mathematical model as the foundation of the proposed robust PRP. The problem on hand is defined on a complete directed graph G=(N,A) with N={0,1,2,,n} as the set of nodes that node 0 considered as a depot and N0 as the set of customers. A={(i,j):i,jN,ij} is the set of arcs and the distance from node i to node j is shown by dij. The number of homogeneous vehicles is a deterministic exogenous parameter and set of vehicles is represented

Modeling of the demand uncertainty

Stochastic optimization (SO) is one of the classical approaches to handle uncertainty, and it requires sufficient historical data to decide probability distribution of the parameters. This is not the case in many real world applications because it is hard to achieve accurate probability distribution function. According to robust optimization, a range of uncertain parameters is needed instead of finding the right probability distribution function. Interested readers are referred to Soyster, 1973

Computational experiments

This section presents the results of extensive computational experiments performed to assess the performance of our three robust approaches. We first describe the PRPLIB instances and then present the results.

Conclusions and future research directions

Vehicle routing problems have been conventionally optimized with a special focus on distance minimization or the number of vehicles reduction. With an ever growing concern for the environment, better planning algorithms were required by the logistic service providers or freight forwarders. As a result, in recent years, there have been emerging solutions provided by scientific researchers as well as practitioners to minimize fuel consumption and particularly CO2e emissions. In this paper, we

Acknowledgements

Thanks are due to the Editor and two anonymous reviewers for their useful comments and for raising interesting points for discussion.

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