Mathematical Problems in Engineering

Volume 2017, Article ID 6329203, 16 pages

https://doi.org/10.1155/2017/6329203

## Vehicle Routing Problems with Fuel Consumption and Stochastic Travel Speeds

School of Economics and Management, Beihang University, Beijing 100191, China

Correspondence should be addressed to Guozhu Jia; nc.ude.aaub@9018041yb

Received 30 July 2016; Revised 8 December 2016; Accepted 15 December 2016; Published 29 January 2017

Academic Editor: Guillermo Botella-Juan

Copyright © 2017 Yanling Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Conventional vehicle routing problems (VRP) always assume that the vehicle travel speed is fixed or time-dependent on arcs. However, due to the uncertainty of weather, traffic conditions, and other random factors, it is not appropriate to set travel speeds to fixed constants in advance. Consequently, we propose a mathematic model for calculating expected fuel consumption and fixed vehicle cost where average speed is assumed to obey normal distribution on each arc which is more realistic than the existing model. For small-scaled problems, we make a linear transformation and solve them by existing solver CPLEX, while, for large-scaled problems, an improved simulated annealing (ISA) algorithm is constructed. Finally, instances from real road networks of England are performed with the ISA algorithm. Computational results show that our ISA algorithm performs well in a reasonable amount of time. We also find that when taking stochastic speeds into consideration, the fuel consumption is always larger than that with fixed speed model.

#### 1. Introduction

The vehicle routing problem (VRP) is one of the most important and studied combinatorial optimization problems [1]. It has gained great attentions from many researchers, especially in distribution and logistics management fields. More than fifty years have elapsed since Dantzig and Ramser [2] firstly introduced the problem in in 1959. They described a real-world application concerning the delivery of gasoline to service stations and proposed the first mathematical programming formulation and algorithmic approach. Following this seminal paper, hundreds of models and algorithms were proposed for the optimal and approximate solution of the different versions of VRP. Then in order to be better aligned with the real-world applications, many different VRP versions have been studied. The most common version is the CVRP [3], where each vehicle has limited load capacity. The VRP with Time Window (VRPTW) [4–6] aims to find optimal route sets with minimum total travel cost while serving each customer within specified time window. Other extended versions include VRP with backhauls [7], VRP with pickups and deliveries [8], and the multidepot VRP [9].

The deterministic VRP cannot cover all the situations in reality while considering stochastic VRP components. Consequently, Stochastic VRP are developed. For example, Ritzinger et al. [10] made a detailed review of the stochastic VRP. Mehrjerdi [11] combined chance constrained programming and multiple objective programming to obtain satisfactory solutions. Taş et al. [12, 13], Ehmke et al. [14], and Laporte et al. [15] put forward different heuristic methods for VPR with stochastic travel time and time windows. Marinakis et al. [16] developed a particle swarm algorithm for VRP with stochastic demands. For more stochastic demand results, one may refer to [17–19].

Based on the NP hardness of VRP, multiple heuristic algorithms have been put forward to solve this problem. Xiao and Konak [20] present simulating annealing algorithm to solve the green vehicle routing and scheduling problem with hierarchical objectives and weighted tardiness. Kondekar et al. [21] provide a mapreduce based hybrid genetic solution for solving large-scale vehicle routing problems in dynamic network with fluctuant link travel time. Neural network is also applied to solve stochastic multiconstraint problems with different time-scales; see Zhang et al. [22] and Meyer-Bäse et al. [23].

Demir et al. [24] proposed an adaptive large neighborhood search algorithm (ALNS) to minimize the fuel consumption and the driving time with Pareto optimality. Garaix et al. [25] proposed a column generation algorithm for a dial-a-ride problem.

Due to the uncertain traffic factors, such as unexpected workload or bad weather, stochastic VRP has attracted more and more attentions. Cao et al. [26] proposed a partial Lagrange multiplier method. Ishigaki [27] considered a dynamic collection plan with stochastic demand and apply their search algorithm to an actual trash collection problem. Novel applications of the well-established problem can also be found in optical flow and vehicle systems [28–30].

In most literatures, travel speeds are assumed to be fixed or time-dependent (see, e.g., [31, 32]). However, in practice, due to the uncertainty of weather, traffic conditions, and other random factors, it is not appropriate to set travel speeds to be fixed constants in advance. The interest in stochastic VRP in this paper is motivated by both its practical relevance and its considerable difficulty: large VRP instances may be solved to optimality only in particular cases. Therefore, we study vehicle routing problems with stochastic travel speeds. Moreover, as each arc has limited speed and other random factors, the average speed of the same type of vehicles on the same arc approximately obeys normal distribution.

Figure 1 is an illustration of this VPR with stochastic average travel speed. Assume that a logistics company owns a fleet of trucks and one truck delivers goods for customers 1, 2, and 3. On the first day, the truck travels with an average speed of 25 m/s on arc. On the second day, it rains heavily when it delivers goods for customer 1. The truck has to get over the poor road conditions caused by the heavy rain; as a result, the average speed it travels on arc becomes 15 m/s. Several days later, a traffic accident occurred on arc, so it may travel with an average speed of 10 m/s. Practically, during a relatively long period of time, the average speed on arc is not fixed but with slight fluctuation. From this point of view, the average speed is a stochastic variable. Here we assume that the average speed on each arc follows normal distribution.