A Green Vehicle Routing Problem

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Abstract

A Green Vehicle Routing Problem (G-VRP) is formulated and solution techniques are developed to aid organizations with alternative fuel-powered vehicle fleets in overcoming difficulties that exist as a result of limited vehicle driving range in conjunction with limited refueling infrastructure. The G-VRP is formulated as a mixed integer linear program. Two construction heuristics, the Modified Clarke and Wright Savings heuristic and the Density-Based Clustering Algorithm, and a customized improvement technique, are developed. Results of numerical experiments show that the heuristics perform well. Moreover, problem feasibility depends on customer and station location configurations. Implications of technology adoption on operations are discussed.

Highlights

► Conceptualizes and formulates a Green Vehicle Routing Problem. ► Develops solution techniques for application on real-world problem instances. ► Tackles complexities of tracking fuel level as fuel is consumed and replenished. ► Incorporates optional visits to fueling stations to extend tour length limits. ► Studies station/customer geographic distribution impact on operational viability.

Introduction

In the United States (US), the transportation sector contributes 28% (US EPA, 2009) of national greenhouse gas (GHG) emissions. This is in large part because 97% of US transportation energy comes from petroleum-based fuels (US DOT, 2010). Efforts have been made over many decades to attract drivers away from personal automobiles and onto public transit and freight from trucks to rail. Such efforts are aimed at reducing vehicle miles traveled by road and, thus, fossil-fuel usage. Other efforts have focused on introducing cleaner fuels, e.g. ultra low sulfur diesel, and efficient engine technologies, leading to reduced emissions for the same miles traveled and greater mileage per gallon of fuel used. While each such effort has its benefits, only a multi-faceted approach can engender the needed reduction in fossil-fuel usage.

As part of such a multi-faceted approach, renewed attention is being given to efforts to exploit alternative, greener fuel sources, namely, biodiesel, electricity, ethanol, hydrogen, methanol, natural gas (liquid-LNG or compressed-CNG), and propane (US DOE, 2010). Municipalities, government agencies, nonprofit organizations and private companies are converting their fleets of trucks to include Alternative Fuel Vehicles (AFVs), either to reduce their environmental impact voluntarily or to meet new environmental regulations. This focus on truck conversion is desirable because, while medium- and heavy-duty trucks comprise only 4% of the vehicles on the roadways (US FHWA, 2008), they contribute nearly 19.2% of US transportation-based GHG emissions (US DOT, 2010). Moreover, truck traffic has had the greatest growth rate of all vehicle traffic, increasing 77% for heavy-duty trucks and 65.6% for light-duty trucks compared with only 3.3% for passenger cars between 1990 and 2006 (US DOT, 2010).

The US currently has energy policies in place with the aim of reducing fossil-fuel use so as to reduce GHG emissions, break dependency on foreign oil, increase homeland security and support renewable energy use (e.g. the Energy Policy Act, 1992, EPAct, 2005, EO 13423, 2007, Energy Independence and Security Act, 2007). These policies have led to the creation of regulations, mandates, tax incentives, etc. that motivate or require companies and agencies to use AFVs. In fact, federal agencies with a fleet of 20 motor vehicles or more are required to reduce petroleum consumption by a minimum of 2% per year through the end of fiscal year 2015 from the 2005 baseline usage. These agencies are required by executive order to increase their alternative fuel use by 10% per year relative to the previous year (EO 13423, 2007). This executive order replaced an earlier order (EO 13149, 2000) requiring a 20% reduction in petroleum use by 2005 in comparison to base year 1999. The replacement was needed, because no EPAct-covered agency could meet the reduction goal due to insufficient alternative fueling infrastructure. Federal fleets are also required to maximize use of diesel with biodiesel blends (B20) by replacing medium- and heavy-duty gasoline vehicles with diesel vehicles that can use such biodiesel blends. This requirement applies to agencies at locations where there is sufficient B20 infrastructure (current or planned). In addition, the US DOE sponsors a program called Clean Cities (US DOE, 2011a) with over 100 local coalitions to support reduction in petroleum use in the transportation sector.

Agencies consider numerous factors in the selection of a particular vehicle type, including fuel availability and geographic distribution of fueling stations in the service area, vehicle driving range, vehicle and fuel cost, fuel efficiency, and fleet maintenance costs. The lack of a national infrastructure for refueling AFVs presents a significant obstacle to alternative fuel technology adoption by companies and agencies seeking to transition from traditional gasoline-powered vehicle fleets to AFV fleets (Melaina and Bremson, 2008). In fact, approximately 98% of the fuel used in the federal government’s 138,000 AFV fleet (of which, 92.8% in 2008 are flex-fuel vehicles that can run on gasoline or ethanol-based E85 fuel) continues to be conventional gasoline as a result of a lack of opportunity for refueling using the alternative fuel for which the vehicles were designed (US DOE, 2010). Moreover, existing alternative fueling stations (AFSs) are distributed unevenly across the country and within specific regions. Additional operational challenges exist as a result of the reduced driving range of most AFVs.

Similar challenges exist for privately owned AFV fleets as noted in various reports (e.g. Chandler et al., 2000, Chandler et al., 2002, US DOE, 1997, US DOE, 2001, US DOE, 2006, ATA, 2010). FedEx, in its overseas operations, employs AFVs that run on biodiesel, liquid natural gas (LNG) or compressed natural gas (CNG). In US operations, hybrid vehicles have dominated, while LPG, biodiesel and CNG use is limited to regions with access to appropriate AFSs (Bohn, 2008).

This paper is concerned with those companies or agencies that employ a fleet of vehicles to serve customers or other entities located over a wide geographical region. Such companies rely on tools to aid in forming low cost tours, so as to save money and time resulting from travel to customer locations. These routes typically begin at a depot, visit multiple customers and then return to the depot. The problem of assigning customers to vehicles and ordering the customer visits in forming these tours is known as the Vehicle Routing Problem (VRP). A variant of the VRP, the Green Vehicle Routing Problem (G-VRP), is introduced herein that accounts for the additional challenges associated with operating a fleet of AFVs.

In this paper, techniques are developed to aid an organization with an AFV fleet in overcoming difficulties that exist as a result of limited refueling infrastructure. These techniques plan for refueling and incorporate stops at AFSs so as to eliminate the risk of running out of fuel while maintaining low cost routes. The G-VRP is formulated as a mixed-integer linear program (MILP). Given a complete graph consisting of vertices representing customer locations, AFSs, and a depot, the G-VRP seeks a set of vehicle tours with minimum distance each of which starts at the depot, visits a set of customers within a pre-specified time limit, and returns to the depot without exceeding the vehicle’s driving range that depends on fuel tank capacity. Each tour may include a stop at one or more AFSs to allow the vehicle to refuel en route.

The G-VRP is illustrated on a simple example problem in Fig. 1. This example involves only one truck with a fuel tank capacity of Q = 50 gallons and fuel consumption rate of r = 0.2 gallons per mile (or 5 miles per gallon fuel efficiency (Fraer et al., 2005). Three AFSs are available in the region. The vehicle begins its tour at depot D and must visit customers C1–C6 before returning to the depot. To visit these customers, a minimum distance of 339 miles must be traversed. Travel of such a distance would consume 67.8 gallons, 17.8 more gallons of fuel than the vehicle’s tank can hold. Thus, the vehicle needs to visit at least one AFS in order to serve all customers and return to depot D. The G-VRP takes into account the vehicle’s fuel tank capacity limitation and chooses the optimal placement of AFS visits within the tour. Accounting for fuel limitations, the optimal solution to the G-VRP involves a stop at one AFS and requires the traversal of 354 miles. Thus, the tour length is 15 miles longer than the minimum tour length, where fuel tank capacity is assumed to be unlimited.

As the VRP is known to be an NP-hard problem (indicating that the computational effort required for its solution grows exponentially with increasing problem size), and the VRP is a special case of the G-VRP, the G-VRP is NP-hard. Thus, exact solution of large, real-world problem instances will be difficult to obtain. Two heuristics, the Modified Clarke and Wright Savings (MCWS) heuristic and the Density-Based Clustering Algorithm (DBCA), along with a customized improvement technique, are proposed for solution of such larger problem instances. These techniques are intended to provide decision support for a company or agency operating a fleet of AFVs for which limited fueling stations exist. These heuristics provide fast solution capability. Their steps show how the additional problem constraints can be tackled within construction and improvement heuristics. Moreover, they provide intuition for the development of more sophisticated implementations. A natural extension, for example, would be to incorporate the proposed concepts within a tabu search procedure. Numerical experiments were designed and conducted to assess heuristic performance as a function of customer location configuration, and station density and distribution. The techniques are also applied on a hypothetical problem instance meant to replicate a medical textile supplier company’s daily operations in the Washington, DC metropolitan area.

Section snippets

Background

A number of works in the literature present optimization-based approaches designed specifically for siting AFSs. The majority of these works were motivated by the Hydrogen Program that was created during the G.W. Bush administration and supported by a diverse group of governmental and private sponsors (Nicholas et al., 2004, Kuby and Lim, 2005, Kuby and Lim, 2007, Upchurch et al., 2007, Lin et al., 2008a, Lin et al., 2008b, Bapna et al., 2002). Other works focus on military applications and

Problem definition and formulation

The G-VRP is defined on an undirected, complete graph G = (V, E), where vertex set V is a combination of the customer set I = {v1, v2,  , vn}, the depot v0, and a set of s  0 AFSs, F = {vn+1, vn+2,  , vn+s}. The vertex set is V={v0}IF={v0,v1,v2,,vn+s}, |V| = n + s + 1. It is assumed that in addition to the AFSs, the depot can be used as a refueling station and all refueling stations have unlimited capacities. The set E = {(vi, vj): vi, vj ϵ V, i < j} corresponds to the edges connecting vertices of V. Each edge (vi, vj)

Solution of the G-VRP

The vehicle driving range (or fuel tank capacity) limitations and existence of a subset of vertices (the AFSs) that can, but need not be, visited, as well as the possibility of extending a vehicle’s driving range as a result of a visit to a site along the tour, introduce complications that are not present in classical VRPs or most variants thereof. Thus, heuristics designed for the classical VRP or related variants cannot be applied directly in solving the G-VRP. Not only might such heuristics

Numerical experiments

Numerical experiments were conducted to assess the quality of solutions obtained through the proposed heuristics on randomly generated small problem instances through comparison with exact solutions obtained through direct solution of the G-VRP formulation. The experiments were devised to allow consideration of the impact of customer and AFS location configuration and AFS density on the solution. A larger, more realistic G-VRP was devised using a medical textile supply company’s depot location

Concluding remarks

In this paper, the G-VRP is formulated and techniques were proposed for its solution. These techniques seek a set of vehicle tours that minimize total distance traveled to serve a set of customers while incorporating stops at AFSs in route plans so as to eliminate the risk of running out of fuel. Numerical experiments showed that these techniques perform well compared to exact solution methods and that they can be used to solve large problem instances. The ability to formulate the G-VRP, along

Acknowledgements

This effort was partially funded by the Mid-Atlantic University Transportation Center (MAUTC). This support is gratefully acknowledged, but implies no endorsement of the findings. The authors are also thankful to Dr. Rahul Nair and Ramzi Mukhar for their insight and help with implementing the developed techniques.

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