Vehicle routing problem with steep roads

Highlights

• We formally state the vehicle routing problem with steep roads (VRP-SR).

• We formulate the VRP-SR as an integer linear programming model (MIP).

• In our experiments including road grades in route planning can result in significant savings.

• We obtain valuable managerial insights for routing planning on steep roads:

• (1) Try to cluster customers with small altitudes differences to avoid abrupt changes.

• (2) Could be cost-efficient to split a route, even if a single route can visit all customers.

Abstract

Most routing decisions assume that routing costs are modelled as a weighted sum of total distance and time traveled by delivery vehicles. However, this assumption does not apply for logistics operations in cities with significant road grades. We study an extension to the VRP model that plans vehicle routes considering the combined impact of detailed road grade information and vehicle load-weight in fuel consumption cost. We refer to this model as the VRP with Steep Roads (VRP-SR), which is formulated as an integer linear program and solved heuristically. In simulated experiments performed for a mountainous metropolitan region in Chile, we estimate operating cost reductions up to $12.4\%$ when compared to the real cost of a plan disregarding road grade information. In addition, we obtain valuable managerial insights from our routing plans; our planned routes tend to initially use roads with relatively small grades to avoid abrupt altitude changes with a loaded vehicle. Higher altitude changes are planned after the vehicle unloads a significant fraction of its cargo. Also, we identify instances in which it is cheaper to insert an intermediate depot visit to drop off weight and split a feasible route in two subroutes to travel over mountainous areas with a lighter vehicle.

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 $q_1$ ton and $q_2$ ton, respectively. Also, assume that $q_2>q_1>0$ 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 $\alpha$ 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., $(0,2,1,0)$ in Fig. 1 and $(0,1,2,0)$ 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 $2·q_1+q_2<2·q_2+·q_1$.

If we consider the impact of road grade on the fuel consumption model, then we show in Section 2.1 that route $(0,1,2,0)$ minimizes fuel consumption if 2 is placed at a high-enough elevation $\alpha$ above SL. For instance, percentage fuel savings compared to route (0,2,1,0) are 10.6%, when $q_1=5$ ton, $q_2=8$ ton, vehicle speed is 30 km/hr and $\alpha =50$ meters (i.e., road angle 2.87°); these savings are 16.8% when $\alpha =100$ meters (5.74°) and increase as $\alpha$ 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.