Vehicle routing problem with steep roads
Introduction
Planning effective and cost-efficient urban distribution operations is essential for logistics service providers. Last-mile logistics deals with the transport of goods via a fleet of vehicles dispatched from urban distributions centers to geographically located customers within a city representing small-scale merchants and residential locations demanding relatively small volumes of product per delivery. According to Hochfelder (2017), it is the most expensive part of the supply chain network and can represent up to 28% of a product’s total transportation cost measured from its manufacturing place.
To reduce costs, logistics service providers search to consolidate deliveries into routes executing multiple customer visits. The Vehicle Routing Problem (VRP) models such a setting and plans cost-efficient routes starting and ending at a depot. routes are required to cover all customers demanding service and satisfy a set of operational constraints such as a limited vehicle capacity and route duration.
The scientific literature on the VRP is extensive (Toth and Vigo, 2014), but operating costs of a VRP model are typically modeled as a weighted sum of total distance traveled and time spent by all routes representing fuel consumption costs and driver wages, respectively. Extensions of the VRP model, such as the Pollution Routing Problem (PRP), consider a more realistic fuel consumption model that incorporate the effect of weight transported; see for example Bektaş and Laporte, 2011, Figliozzi, 2010, Kuo and Wang, 2011, Demir et al., 2014b, Goeke and Schneider, 2015, Rao et al., 2016.
Most of the available vehicle routing literature assumes that routes are executed over a flat and horizontal terrain. The work by Goeke and Schneider, 2015, Rao et al., 2016, Zhou and Lee, 2017 includes the impact of road grade information on vehicle fuel consumption, but estimates the road grade between any two customer locations using their altitude difference over distance. This simplifying assumption is inappropriate when the altitude along the route connecting two customers varies as a nonlinear function of the distance traveled. For example, two locations may be at the same altitude, but be connected through a hilly road, which differs in terms of fuel consumption with respect to traversing a flat road. Furthermore, in a real network there are typically multiple routes connecting two locations.
Consequently, a routing plan may be inefficient if it is designed without considering the combined impact in fuel consumption of (i) a detailed road grade information between customers, and (ii) varying vehicle weight along a route.
We consider a detailed urban network obtained from a Geographic Information System (GIS), such as Google Maps or Open Street Maps, which is equipped with real altitute information for each node. We estimate the cost per segment in the urban network and find minimum cost paths for each ordered pair customer nodes depending on the vehicle load-weight. To estimate costs per segment, we use a realistic fuel consumption function found in Barth and Boriboonsomsin (2009), which takes into account vehicle characteristics, distance traveled, vehicle speed, road grade and load-weight. In that study, the authors comment that it is cheaper to traverse roads downhill than uphill and account that the impact of weight transported on costs depends on road grades, but no routing decisions are discussed.
Fig. 1 shows a canonical example presenting a single vehicle instance with a depot (0) and two customer locations (1 and 2). Each location is a vertex of an equilateral triangle with side 1 km. The quantity of product demanded by customers 1 and 2 are ton and ton, respectively. Also, assume that and that the vehicle has enough capacity to transport both orders simultaneously. The depot and customer 1 are both at sea level (SL), while customer 2 is at meters above SL. Also, consider that all roads are straight and have constant road grades. When distance (or time) is minimized, then both feasible routes, i.e., in Fig. 1 and in Fig. 1(b), are optimal. If we assume a flat terrain and minimize fuel consumption, then we should first visit the heaviest demand and travel lighter most of the road reducing the ton-km traveled. In our example, we should deliver first to customer 2, who requested the heaviest load, and later to customer 1. In this case, the total ton-km transported is .
If we consider the impact of road grade on the fuel consumption model, then we show in Section 2.1 that route minimizes fuel consumption if 2 is placed at a high-enough elevation above SL. For instance, percentage fuel savings compared to route (0,2,1,0) are 10.6%, when ton, ton, vehicle speed is 30 km/hr and meters (i.e., road angle 2.87°); these savings are 16.8% when meters (5.74°) and increase as increases. Intuitively, the vehicle incurs in less fuel consumption by climbing the hill to customer 2 lighter even if it incurs additional ton-km.
The previous exercise illustrates the value of considering the effect of slopes on fuel consumption. This problem is specially relevant to urban logistics operations in cities with significant road grades, such as coastal cities located between mountains and the sea, e.g., Valparaiso, Chile (Fig. 2(a)), where some customers are located at SL and others might be at 300 meters above SL; also, it is important to cities located in valleys surrounded by mountains, such as such as Santiago, Chile (Fig. 2(b)) with elevations between 500 and 1000 meters above SL. The problem becomes even more important when vehicle load-weight varies significantly along the route traveled, as it occurs in outbound ground operations from Valparaiso’s port to the city.
Section snippets
Literature review
We now present a review of related fuel consumption and vehicle routing models available within the literature.
Problem statement
Now, we formally define the VRP-SR. We first describe a methodology to compute the cost incurred by a vehicle when it travels between two consecutive customer nodes, which depends on the travel time and distance traveled but also on the vehicle load and the detailed grade information of each road segment. Later, we formulate an Integer Linear Program (ILP) model, which takes previous cost information as input and defines cost-efficient vehicle routes.
Solution algorithm
We build a solution algorithm that obtains a good feasible solution to any VRP-SR instance. Our algorithm is based on heuristic frameworks found in the literature exploiting the VRP solution structure; we refer to Toth and Vigo (2003) for a seminal introduction on local search procedures for routing problems. It combines solution construction heuristics, intra-route and inter-route local search moves and route partitioning strategies adapted to the VRP-SR. To do this last, all potential
Computational experiments
In this section, we estimate the potential benefits of including road grades in the cost function of a VRP. We present a comparison of the results obtained from the application of our model, which explicitly considers the vehicle weight and road grades in fuel consumption, versus the Flat-VRP benchmark model, in which road grades are not considered. To do so, we conduct computationally simulated experiments with customers geographically placed in Valparaiso, Chile, which is a particularly
Conclusions
We study an extension to the VRP model that plans vehicle routes considering the combined impact of road grade information and vehicle load-weight in fuel consumption cost. Our model takes into account detailed road grade information per route segment, which is obtained from GIS data.
In our computational experiment for the city of Valparaiso, Chile, we obtained cost saving opportunities up to 12.4% for particular instances and average cost savings of 2.7%. This suggests that such a refinement
CRediT authorship contribution statement
Carlos Brunner: Software, Validation, Investigation, Formal analysis, Data curation, Visualization, Writing - original draft. Ricardo Giesen: Conceptualization, Methodology, Writing - review & editing, Supervision, Project administration, Funding acquisition. Mathias A. Klapp: Conceptualization, Methodology, Visualization, Writing - original draft, Writing - review & editing, Supervision. Luz Flórez-Calderón: Software, Validation, Investigation, Formal analysis, Data curation, Visualization,
Acknowledgements
We would like to thank the support by Fondecyt 1171049, CEDEUS, CONICYT/FONDAP 15110020, and the BRT + Centre of Excellence funded by VREF. We would also like to thank all anonymous referees for their useful feedback.
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