Discrete OptimizationThe bi-objective Pollution-Routing Problem
Introduction
Freight transportation lies at the forefront of logistics planning. Until now, the planning of freight transportation activities has mainly focused on ways of saving money and increasing profitability by considering internal transportation costs only, e.g., fuel cost, drivers’ wages (Crainic, 2000, Forkenbrock, 1999, Forkenbrock, 2001).
Freight transportation in the United Kingdom (UK) is responsible for 22% of the CO2 emissions from the transportation sector, amounting to 33.7 million tonnes, or 6% of the CO2 emissions in the country, of which road transport accounts for a proportion of 92% (McKinnon, 2007). The 2008 Climate Change Act commits the UK to an ambitious and legally binding 80% reduction in greenhouse gases (GHG) emissions by 2050, from a 1990 baseline. The situation in Europe is not much different. According to the TERM 2011 Report published by the European Environment Agency, transport (including international maritime) contributed 24% of the overall GHG emissions in the EU-27 countries in 2009, with road transport accounting for 17% of the total GHG emissions Vicente (2011). The transportation sector therefore has an important role to play, as one of the largest GHG contributor, in achieving reduction targets (Tight, Bristow, Pridmore, & May, 2005).
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 selection of GHGs to include in the carbon footprint is an important issue. Wright, Kemp, and Williams (2011) suggest that a significant proportion of emissions can be captured through measurement of the two most prominent anthropogenic GHGs, CO2 and CH4. The emissions of CO2 are directly proportional to the amount of fuel consumed by a vehicle. This amount is dependent on a variety of vehicle, environment and traffic-related parameters, such as vehicle speed, load and acceleration (Demir, Bektaş, & Laporte, 2011). On the other hand, the emissions of CH4 are a function of many complex aspects of combustion dynamics and of the type of emission control systems used.
Freight companies also generate significant amounts of air pollution besides GHG, including particulate matter (small particles of dust, soot, and organic matter suspended in the atmosphere), carbon monoxide (colorless, odorless, poisonous gas produced when carbon-containing fuel is not burned completely), ozone (formed when emissions of nitrogen oxides (NOx) and volatile organic compounds (VOCs) chemically react in the presence of sunlight) and hazardous air pollutants, also referred to as air toxics (chemicals emitted into the atmosphere that cause or are suspected to cause cancer or other severe health effects) (PSRC, 2010).
Freight transportation planning has many facets, particularly when viewed from the multiple levels of decision making. Arguably the most famous problem at this level is the well-known Vehicle Routing Problem (VRP), which consists of determining routes for a fleet of vehicles to satisfy the demands of a set of customers. The traditional objective in the standard VRP is to minimize a cost function which is traditionally considered to be the total distance traveled by all vehicles. Taking a more explicit look at externalities of freight transportation, and in particular vehicle routing, Bektaş and Laporte (2011) introduced the Pollution-Routing Problem (PRP) which aims at minimizing a total cost function comprising fuel and driving costs in the presence of time windows.
Most real-world problems involve multiple objectives. In the context of the PRP, two important objectives should be taken into account, namely minimization of fuel consumption and the total driving time. Fuel consumption depends on the energy required to move a vehicle from one point to another, and is proportional to the amount of emissions. As discussed in Demir, Bektaş, and Laporte (2012) for each vehicle there exists an optimal speed yielding a minimum fuel consumption. However, this speed is generally lower than the speed preferred by vehicle drivers in practice. Another important issue in road transportation is time management. In freight transport terminology, time is money and it is essential for firms to perform timely deliveries in order to establish and keep a good reputation. In practice, drivers’ schedules tend to be flexible, with different numbers of hours worked each day, subject to driving time regulations. If a saving of one hour can be achieved on a given vehicle route, this would imply reducing the corresponding driver’s costs by an hour (Fowkes & Whiteing, 2006). Reduction in time spent on a route can be achieved by traveling at higher speed, but this, in turn, increases fuel costs and emissions. Since the two objectives of minimizing fuel and time are conflicting, the problem requires the use of multi-objective optimization to allow an evaluation of the possible trade-offs.
In practice, companies would like to minimize their total operating cost, including those related to fuel and time. However, costs of fuel, emissions and time might differ from one organization to another, and in some cases rather significantly. As an example, it is found that driver costs are paid as hourly wages in some countries (e.g., UK and USA) whereas they are a monthly salary in others. Fuel costs also differ between countries. Finally, carbon costs vary significantly (£60–£225 per tonne) as discussed in Bektaş and Laporte (2011). In this paper, we investigate a bi-objective vehicle routing problem in which one of the objectives is related to fuel consumption and the other to driving time. The two objectives are treated in their natural units of measurement in order to eliminate the bias resulting from the cost differences just mentioned. The benefit is that managers or users of the approach described in the paper can attach cost figures relevant to their organization and can produce tailored trade-off curves for their operations.
We propose a solution method based on an enhanced adaptive large neighborhood search (ALNS) and a specialized speed optimization algorithm described in Demir et al. (2012). The scientific contribution of this study is threefold: (i) to introduce of a bi-objective variant of the Pollution-Routing Problem, (ii) to apply and test multi-objective techniques to solve the bi-objective PRP, and (iii) to perform extensive computational experiments using four a posteriori methods evaluated by means of two performance indicators. In contrast to existing studies on the “green” VRP (for which a brief review is presented below), this paper breaks away from the literature by considering two objectives, one of them being a comprehensive emissions function incorporating the effect of load and speed. This study also contributes to the multi-objective optimization literature by presenting a comprehensive comparison of four methods on the bi-objective PRP.
The remainder of this paper is organized as follows. In Section 2 we provide a general overview of multi-objective optimization and we summarize the existing literature on multi-objective and “green” VRPs. Section 3 presents the bi-objective PRP along with a mathematical programming pformulation. Section 4 includes a brief description of the heuristic algorithm. Section 5 presents the generation of the instances and the results of extensive computational experiments, together with managerial insights. Conclusions are stated in Section 6.
Section snippets
Multi-objective optimization
Multi-objective optimization (MOO), also known as multi-objective programming, multi-criteria or multi-attribute optimization, is the process of simultaneously optimizing two or more conflicting objectives subject to a number of constraints. In this section, we consider a MOO problem of the formwhere fk: are k ⩾ 2 objective functions to be minimized simultaneously. The decision variables x = (x1, … , xn)T belong to a non-empty feasible region (set)
The bi-objective Pollution-Routing Problem
We now formally describe the bi-objective PRP. This problem is defined on a complete directed graph where is the set of nodes, 0 is a depot and and i ≠ j} is the set of arcs. The distance from node i to node j is denoted by dij. A fixed-size fleet of m vehicles, each of capacity Q, is available to serve the nodes. The set is a customer set, and each customer has a non-negative demand qi as well as a time interval [ai, bi] in which service must start;
A bi-objective adaptive large neighborhood search algorithm with speed optimization procedure
To solve the bi-objective PRP, we use an enhanced version of the ALNS algorithm introduced in Demir et al. (2012). The ALNS algorithm is used as a search engine to find a set of non-dominated solutions. Here, we only briefly describe the ALNS for reasons of space but refer the interested reader to Demir et al. (2012) for further details.
The ALNS metaheuristic is an extension of the large neighborhood search (LNS) heuristic first proposed by Shaw (1998), and based on the idea of modifying an
Computational results
This section presents the results of extensive computational experiments performed to assess the performance of the multi-objective methods using our ALNS algorithm with speed optimization procedure. We first describe the generation of the test instances, the parameters and the quality indicators used to assess the performance of the proposed methods. We then present the computational results. The parameters used in the experiments are given in Table 1.
For the parameters listed in Table 1,
Conclusions
We have studied the bi-criteria PRP in which one of the objectives is related to CO2e emissions, and the other to driving time. An enhanced adaptive large neighborhood search (ALNS) algorithm was used for the generation of non-dominated/Pareto optimal solutions. The algorithm integrates the classical ALNS with a specialized speed optimization procedure. The proposed algorithm first calls the ALNS using fixed speeds as inputs, then optimizes speed on each route. Our approach implicitly assumes
Acknowledgements
The authors gratefully acknowledge funding provided by the University of Southampton School of Management and by the Canadian Natural Sciences and Engineering Research Council under Grant 39682-10. Thanks are due to two anonymous reviewers for their useful comments and for raising interesting points for discussion.
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