Robust solutions to the pollution-routing problem with demand and travel time uncertainty
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 with as the set of nodes that node 0 considered as a depot and as the set of customers. is the set of arcs and the distance from node i to node j is shown by . 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.
References (57)
- et al.
The robust vehicle routing problem with time windows
Comput. Oper. Res.
(2013) - et al.
New exact method for large asymmetric distance-constrained vehicle routing problem
Eur. J. Oper. Res.
(2013) - et al.
Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints
Eur. J. Oper. Res.
(2012) - et al.
The pollution-routing problem
Transport. Res. Part B: Methodol.
(2011) - et al.
A comparative analysis of several vehicle emission models for road freight transportation
Transport. Res. Part D: Transp. Environ.
(2011) - et al.
An adaptive large neighborhood search heuristic for the pollution-routing problem
Eur. J. Oper. Res.
(2012) - et al.
A review of recent research on green road freight transportation
Eur. J. Oper. Res.
(2014) - et al.
A selected review on the negative externalities of the freight transportation: modeling and pricing
Transport. Res. Part E: Logist. Transport. Rev.
(2015) - et al.
The vehicle routing problem: a taxonomic review
Comput. Ind. Eng.
(2009) - et al.
Solving the capacitated vehicle routing problem with environmental criteria based on real estimations in road transportation: a case study
Proc.-Soc. Behav. Sci.
(2011)
The time-dependent pollution routing problem
Transport. Res. Part B: Methodol.
The fleet size and mix pollution-routing problem
Transport. Res. Part B: Methodol.
A matheuristic approach for the pollution-routing problem
Eur. J. Oper. Res.
The vehicle routing problem: an overview of exact and approximate algorithms
Eur. J. Oper. Res.
Vehicle routing problems with time windows and stochastic travel and service times: models and algorithm
Int. J. Prod. Econ.
A review of dynamic vehicle routing problems
Eur. J. Oper. Res.
Robust possibilistic programming for socially responsible supply chain network design: a new approach
Fuzzy Sets Syst.
Mitigation of greenhouse gas emissions in vehicle routing problems with backhauling
Expert Syst. Appl.
A new truck-routing approach for reducing fuel consumption and pollutants emission
Transport. Res. Part D: Transp. Environ.
Green logistics at Eroski: a case study
Int. J. Prod. Econ.
Development of a fuel consumption optimization model for the capacitated vehicle routing problem
Comput. Oper. Res.
Design and development of a hybrid artificial bee colony algorithm for the environmental vehicle routing problem
Transport. Res. Part D: Transp. Environ.
Models and algorithms for stochastic and robust vehicle routing with deadlines
Transport. Sci.
Real-world CO2 impacts of traffic congestion
Transport. Res. Rec.: J. Transport. Res. Board
Development of a Heavy-duty Diesel Modal Emissions and Fuel Consumption Model
Robust convex optimization
Math. Oper. Res.
Robust solutions of linear programming problems contaminated with uncertain data
Math. Program.
Cited by (66)
The pollution-routing problem with speed optimization and uneven topography
2024, Computers and Operations ResearchAn integrated assignment, routing, and speed model for roadway mobility and transportation with environmental, efficiency, and service goals
2023, Transportation Research Part C: Emerging TechnologiesMixed fleet scheduling method for airport ground service vehicles under the trend of electrification
2023, Journal of Air Transport ManagementA robust multi-objective routing problem for heavy-duty electric trucks with uncertain energy consumption
2023, Computers and Industrial EngineeringA novel multi-objective green vehicle routing and scheduling model with stochastic demand, supply, and variable travel times
2022, Computers and Operations ResearchDynamic stochastic electric vehicle routing with safe reinforcement learning
2022, Transportation Research Part E: Logistics and Transportation Review