Solving the Green Vehicle Routing Problem using Tabu Search

Transcription

1 Master Thesis in Logistics and Supply Chain Management Solving the Green Vehicle Routing Problem using Tabu Search By Aleksandra Georgieva Academic Supervisor: Sin C. Ho, Associate Professor Department of Economics and Business Economics, Aarhus University September 2015

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3 Abstract Initiatives aiming to reduce the severity of greenhouse gas emissions affect traditional logistic operations, and give rise to new variants of routing problems. This thesis studies one such problem, the green vehicle routing problem (GVRP). It is characterized by a fleet of vehicles that run on alternative fuel, have a limited driving range and face difficulties due to a limited refuelling infrastructure. It has been studied in literature only in recent years, and very scarcely. The specific features, characteristic of this problem bring new challenges to both researchers and practitioners in the field of routing optimization. Thus, providing new knowledge on solution techniques applied to the GVRP is beneficial. For that reason, this research constructed and applied a tabu search metaheuristic, using simple neighbourhood strategies that has not been implemented, in that particular configuration, to GVRP before. The objective of this research is to identify the performance of the tabu search algorithm in comparison to the solution methods used in the original study by Erdogan and Miller Hooks (2012), which introduces the GVRP. To accomplish this, the tabu search algorithm, consisting of a nearest- neighbour greedy algorithm with a random element and three local search neighbourhood operators, was coded in Visual Basic for Applications (VBA). In comparing the results to those in the benchmark research, it was identified that the algorithm considerably outperforms the Modified Clarke and Wright Algorithm and the Density Based Clustering Algorithm, used together with an improvement heuristic. For small problem instances containing 20 customers and between 2 and 10 alternative fuelling stations, tabu search found a better solution value in 31 out of 40 cases. In large problem instances, containing up to 500 customers and up to 28 stations, the algorithm showed superior performance in all cases. The highest performance gap was an improvement of solution value amounting to approximately 15%. Thus, tabu search shows superior performance to the benchmark research. This is especially relevant for real world problem instances, where due to problem size exact solution methods do not have a practical application, and therefore finding a good approximate solution technique is important for practice and applications. III

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5 Acknowledgements It is with immense gratitude and appreciation that I acknowledge the help and support of my supervisor, Associate Professor Sin C. Ho, whose guidance and valuable advice, have been of vast importance to me in the preparation of this research. I would also like to express my gratitude to my family and friends for their continuous love, support and endless motivation. V

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7 Table of Contents 1. Introduction Problem formulation Distribution Problems Vehicle Routing Problems A Green Vehicle Routing Problem: description and mathematical formulation Literature review Complexity theory Metaheuristics Classification of metaheuristics Metaheuristics important attributes in design and implementation Tabu search Design and implementation features Adaptive memory in tabu search Candidate list strategies in tabu search Elements and characteristics of the solution algorithm Creation of the initial solution Local search neighbourhood generation Tabu search algorithm Numerical results and benchmarking Scenarios of benchmark instances Comparative results Small test instances Large test instances Conclusion VII

8 7. Limitations and future research List of Figures List of Tables Reference List: APPENDIX A: Results from numerical experiments APPENDIX B: Contents of the CD- DVD VIII

9 1. Introduction Recent decades have been marked by a growing concern about the environment and the sustainability of economic growth. One of the most significant environmental problems is climate change. During the period global greenhouse gas (GHG) emissions have grown faster than in previous decades, to reach an all time- high of 49 billion tonnes of CO 2 equivalents in From those, the transport sector accounts for approximately 14% (6.7 billion tonnes) of direct GHG emissions, which is largely due to the fact that 95% of the world s transportation energy comes from petroleum- based fuels (IPCC 2014). Emissions in this sector involve the use of fossil fuels for road, rail, air and marine transportation. Increasing apprehension about the effects of anthropogenic greenhouse gases on climate, and about what some believe are the first signs of global warming, has led to many governmental, as well as inter-governmental mitigation initiatives. In the transport sector, such initiatives include the reduction of kilometres travelled by road through an increase of urban trips by walking, cycling and public transportation, and by shifting greater volumes of freight onto rail. Other initiatives focus on lowering the energy intensity of transportation (megajoule per passenger kilometre or megajoule per tonne kilometre), by use of more efficient engines and alternative, advanced vehicles, and on reducing the carbon intensity of fuels by use of alternative fuels, such as biodiesel, propane, ethanol, electricity, hydrogen, natural gas and other emerging alternative fuels (IPCC 2014). In light of the mounting concern with environmental effects of economic development, over the past two decades companies are faced with rising pressure to decrease the environmental impact of their logistic operations, a major part of which is transportation (Mckinnon et al. 2010). This is the case primarily, because freight transport holds a great potential for reduction of GHG emissions. Vehicle s carbon dioxide emissions per kilometre are considerably higher for freight transport: between 2 and 1700 grams CO 2 per tonne kilometre, than for passenger transport: between 20 and 300 grams CO 2 per passenger kilometre (IPCC 2014). Thus, organizing logistics today involves not only considerations of profitability, but also incorporation of environmental and social costs. Distribution activities hold a large potential for cost reduction, and have always been at the centre of economic development (Eilon 1977). Traditional daily logistic operations and planning involve the construction of low cost tours, from plant to customer 1

10 locations, in order to save money and time in providing a competitive service level to clients. This in itself offers environmental benefits, as in minimizing operational costs, consumption of fuel is reduced and thus GHG emissions are cut down as well. However, the growth of environmental pollution and problems today calls for preservation practices of larger scale. In reducing the environmental impact of logistics companies increasingly turn to designing and building more sustainable distribution networks. This is done by switching to next generation alternative vehicles and fuels, and other green infrastructures. Regardless of how beneficial for emission reduction such an environmentally cautious logistic policy is, its implementation poses a number of challenges on traditional profit- driven operations. Routing problems, which deal with every-day distribution planning and scheduling, under such green logistic operations become more complex as they have wider objectives and more operational constraints (Lin et al. 2014). The problem of identifying a route for servicing a set of customers using a fleet of vehicles is known in literature as the vehicle routing problem (VRP) (Simchi- Levi, Chen & Bramel 2014). There are many variants of this general problem, which incorporate different operational conditions and goals. However, very few variants of VRP that incorporate sustainable transport concerns have been studied so far. Typical issues that green logistics has to manage are measurement of environmental effects of a distribution strategy, reduction of energy consumption, recycling refuse and management of waste disposal (Lin et al. 2014). These issues give rise to three major classes of green vehicle routing problems that take into account environmental concerns. The first class, pollution routing problems, deals with the minimization of polluting emissions. It aims at identifying a routing scheme that minimizes GHG emissions, while at the same time minimizing economic costs. The second class concerns vehicle routing in reverse logistics. These problems focus on the backward flows of goods or raw materials from the point of use to a point of disposal or recycling. The third class, green vehicle routing problems (GVRP), deals with the minimization of energy consumption and the recharging, or refuelling of vehicles, specifically alternative vehicles. Problems of this class address the challenges that pertain to the adoption of alternative fuel vehicles (AFV). These are limited refuelling infrastructure and limited driving range of such vehicles (Mckinnon et al. 2010). It is also the class, which is of great interest today as AFV and alternative fuels have the potential to 2

11 considerably reduce transportation related GHG emissions from corporate operations, as well as private users and help meet regulations goals. Even though GVRP, which aim at constructing low cost routes for AFV, have an application today there is very little prior research on them. The first, among a very small number of papers that address this, is the research by Erdogan and Miller-Hooks (2012). They consider the limitations on distance due to limited fuel tank capacity of AFV, and address it by allowing refuelling at alternative fuelling stations (AFS) during a tour. Their model seeks to minimize the total distance travelled by vehicles, while at the same time eliminating the risk of running out of fuel while en route. It also incorporates service time at customer locations and a limit on maximum tour duration. In solving the GVRP the authors apply two construction heuristics: Modified Clarke and Wright Savings heuristic and the Density- Based Clustering Algorithm, and a customized improvement heuristic. These approximate algorithms are appropriate, especially for real life problems that tend to be larger in size, because they find a reasonably good solution within reasonable computing time. However, there are two alternative approaches to solving this GVRP. One option is the use of exact methods, which find an optimal solution. Nevertheless, as problems get bigger in size and more complex, such algorithms become inefficient due to computational intractability (Talbi 2009). Another option is to apply a more sophisticated approximate algorithm, a metaheuristic algorithm. The significant contribution of transportation on climate change and its important role in logistic operations, make green vehicle routing problems an important area of research today. Exploration of potential reduction of the environmental effects from transportation through route planning offers valuable ideas on how companies can support greener initiatives, and still run a successful business. Alternative fuelled vehicles have a great potential of reducing GHG emissions from transport. However, there is very little research on the optimal routing of such vehicles. Therefore, this thesis will build on the research by Erdogan and Miller-Hooks, and construct a tabu search (TS) metaheuristc for solving the GVRP. The metaheuristic algorithm applied in this research consists of a specific combination of elements that has not been applied to the GVRP in prior studies. Thus, this paper will investigate whether the method applied is effective and efficient in solving the problem 3

12 at hand. This leads to the problem formulation of this study, which is outlined in the following section Problem formulation This research is going to construct and apply a tabu search metaheuristic algorithm to the GVRP. Its main purpose is discovering whether the proposed implementation of tabu search, using simple neighbourhood search strategies, is efficient and effective in solving the problem. More specifically, it aims at determining how the proposed technique performs in comparison to the methods utilized in the first study on GVRP, that by Erdogan and Miller-Hooks (2012). In order to achieve this, the research question that will be raised is: RQ: How does tabu search, using simple neighbourhood search strategies, perform for solving the GVRP in comparison to the heuristics proposed by Erdogan and Miller- Hooks (2012)? 4

13 2. Distribution Problems A focus on distribution activities has become essential in today s competitive world. Distribution problems can be separated into three categories: strategic level, tactical level, and operational level. Efficient and effective distribution management poses a range of challenges at all three levels. While the strategic level is concerned with long term issues such as location of supply chain entities, and the tactical level is concerned with short term problems such as fleet size and planning fixed routes, the operational level is concerned with day-to-day activities such as scheduling and routing of vehicles (Eilon 1977). Daily operational costs account for a great part of a company s distributional expenses, and represent a potential source of significant logistic savings. There are a number of formal classifications of routing problems, which help in the realization of the cost savings potential of daily logistic operations. One such class of problems is the vehicle routing problem (Anbuudayasankar, Ganesh & Mohapatra 2014). This chapter will provide an introduction to vehicle routing problems and present the green vehicle routing problem, which is studied in this research. Additionally, the chapter will provide an overview of existent literature on the green vehicle routing problem. It will also examine problem complexity theory, in order to justify the implementation of approximate solution algorithms to the problem studied in this research Vehicle Routing Problems Vehicle routing problems (VRP) are central to logistics management (Anbuudayasankar, Ganesh & Mohapatra 2014). Since the influential research by Dantzig and Ramser (1959) vehicle routing problems have been the focus of extensive research attention. In the general form of these types of problems a warehouse supplies goods to a set of customers, or nodes, with certain demands using a fleet of vehicles with specific carrying capacity, where the objective is to minimize the total route cost (Simchi- Levi, Chen & Bramel 2014). One possibility for the cost function is to minimize the number of vehicles that are used. Another one is to minimize the total distance travelled by the vehicles to satisfy all customers demands, where this distance 5

14 is represented by arcs, connecting the customer nodes and the depot. From this general form of the VRP emerge many different problem classes. Bodin and Golden (1981) present a number of characteristics through which VRP can be categorized. Their framework is shown on Figure 1. On the figure, for each category of characteristics, from 1 to 12, possible options are listed. Different combinations of Figure 1: A taxonomy for classifying vehicle routing problems 1. Time to service a node/ arc Time is specified/ fixed in advance Time windows Time unspecified 2. Number of depots One depot / domocile More than one depot / domicile 3. Size of the vehicle fleet One vehicle More than one vehicle 4. Type of vehicle fleet Homogeneous fleet Heterogeneous fleet 5. Nature of demands Deterministic Stochastic 6. Location of demands At nodes On arcs Mixed 7. Underlying network Undirected Directed Mixed 8. Vehicle capacity constraints Imposed- all vehicles have the same capacity Imposed- not all vehicles have the same capacity Not imposed 9. Maximum vehicle route- times Imposed- all times are the same Imposed- not all times are the same Not imposed 9. Costs Variable or routing costs Fixed operating or vehicle acquisition costs 10. Operations Pickups only Drop- offs only Mixed 11. Objective Minimze routing costs incurred Minimize sum of fixed and variable costs Minimize number of vehicles required 12. Other (problem - dependent ) constraints Note: Adapted from Classification in vehicle routing and scheduling, by L. Bodin and B. Golden, 1981, Networks, 11(2), pp

15 these characteristics result in a vast number of possible problem settings. It should be noted that even though there are many distinguishable variants of VRP, they are often very similar in nature. Some problem settings are well known, and have received considerable research attention, while others are relatively new and lack, or have a scarce amount, of prior research on them. This is an important distinction, as the identification of algorithms to solve a given problem is simplified when there is a prior research background to support the choice of solution methodology. One variation of the VRP is the capacitated vehicle routing problem (CVRP), one of the most basic vehicle routing problems and the first VRP to be addressed in literature (Lin et al. 2014). It is characterized by a single depot, which supplies products to customers using a homogenous fleet of vehicles. The vehicles have limited capacity, and the underlying network consisting of customer locations and the depot is undirected. Only routing, or variable, costs are considered, and the objective is to construct a set of routes that start and end at the depot, and which minimize routing costs without violating the capacity constraints. In other words, to minimize total distance travelled on all routes (Simchi- Levi, Chen & Bramel 2014). Another example of a commonly studied vehicle routing problem is the VRP with time windows. In this problem the customer demands a certain quantity of goods within a specific period of time, referred to as a time window. It has the same characteristics as the capacitated vehicle routing problem, however, with the addition of a time window constraint (Simchi- Levi, Chen & Bramel 2014). For more information on VRP see Golden and Assad (1988), Fisher (1995), Crainic (1998). The CVRP and the VRP with time windows are only two variants of vehicle routing problems that can be classified using Bodin and Golden s (1981) taxonomy. Additionally, there is a great amount of variants of these two general problems, as well as of other widely studied problems. However, there are also numerous problems that can be classified and incorporate new operational considerations, and which have not or only recently became a popular topic for researchers. One such problem is the green vehicle routing problem. In the following section the formulation and main characteristics of a variant of the GVRP will be discussed. 7

16 2.2. A Green Vehicle Routing Problem: description and mathematical formulation Green vehicle routing problems are variations of the general VRP, which have emerged in literature only in recent years (Lin et al. 2014). They have the objective of considering not only economic costs, but also environmental costs, and in doing that to reduce energy consumption from the distribution process. A variant of the green vehicle routing problem will be studied in this research, as it is described by Erdogan and Miller-Hooks (2012). The problem is defined on the undirected graph G=(V,E), where the node set V groups the set of customer nodes I={v 1,v 2,.,v n }, the single depot v 0, and the set of s alternative fuelling station (AFS) nodes F={v n+1,v n+2,.,v n+s }. Each node i ϵ V is characterized by a service time p i (hours), which is assumed to be constant for both customer nodes, where i ϵ I, and fuelling station nodes and the depot, which can also serve as a fuelling station, where i ϵ F or corresponds to the depot. All customer nodes i ϵ I have to be visited by exactly one vehicle. All AFSs nodes i ϵ F and the depot can be visited multiple times and at any time. The set E={( v i,v j ): v i,v j ϵ V, i < j) is the set of arcs that connect the nodes in V. Each arc connecting nodes v i and v j has a corresponding travel time t ij (hours), cost c ij, and distance d ij (miles). A fleet of m identical, alternative fuelled vehicles (AFV) is considered in this problem. Each vehicle has a fuel tank capacity of Q (gallons) and fuel consumption rate of r (gallons per mile). The duration of each route, which has to start and end at the depot, should not exceed T max (hours), the maximum allowed tour duration. In this model it is assumed that travel speed is constant. Also that during refuelling the tank is filled to capacity and no limit is posed on the number of visits to fuelling stations during a route. It is also assumed that each customer node can be visited by a vehicle starting and returning back at the depot, by visiting at most one fuelling station, within the pre-specified time limit on tour duration. Only routing costs are taken in consideration, and the objective is to construct a set of routes, one for each vehicle in the fleet, which start and end at the depot and minimize the distance travelled for visiting each customer exactly once, without violating the vehicle s driving range constraints and the maximum vehicle route- time constraints. 8

17 The mathematical formulation of the GVRP, as described above, is displayed on Figure 2. It is formulated as a mixed integer linear program. The objective function (1) aims to minimize the total distance travelled by the fleet of AFV to service all customers on a given day. Where x ij is a binary variable, which takes the value of one if a vehicle travels from node i to j, and zero otherwise. V = V U φ is the set of all customer nodes, the depot and the AFS nodes, together with a set s of dummy nodes φ ={ v n+s+1,v n+s+2,.,v n+s+s } for all AFS and the depot when it serves as a refuelling station. These dummy nodes correspond to the number of times stations can be visited for refuelling. This is important in order to allow for multiple visits to a given station. The number of dummy nodes for each refuelling station should be large enough not to restrict beneficial visits for refuelling, but as small as possible in order to reduce the size of the network. The first set of constraints (2) is degree constraints and ensures that each customer node is followed by exactly one customer, depot or AFS node. Constraints (3) are also degree constraints, which ensure that the fuelling stations, the depot and the associated dummy Figure 2: Mathematical formulation of the GVRP Note: From A green vehicle routing problem, by S. Erdogan and E. Miller-Hooks, 2012, Research Part E:Logistics and Transportation Review, 48(1), p

18 nodes will have at most one successor node (customer, AFS or depot). F 0 ={v 0 } U F, where F =F U φ, is the set of AFS and the depot, as well as the associated dummy nodes. The next set of constraints (4) ensures that at each node the number of arrivals is equal to the number of departures. Constraints (5) and (6) ensure that at most m vehicles are routed out of the depot and at most m vehicles return to the depot. The depot is copied in order to keep a record of arrival and departure times. Constraints (7) track the arrival time of vehicles at each node. τ j is a time variable specifying arrival time of a vehicle at node j, which has an initial value of zero when the vehicle leaves the depot. Constraints (8) give an initial value of zero to the variable tracking arrival time upon leaving the depot, and an upper bound on arrival time back at the depot, at the end of a route. Constraints (9) provide lower and upper bounds on arrival times at customer nodes and AFS, which aims to ensure that the tour duration constraint is met. Tracking of fuel level is done through constraints (10). Upon arrival at node j, if its predecessor node i is a customer node, the fuel level of the vehicle is reduced based on the distance between node j and i, and the fuel consumption rate. The variable y j specifies the remaining fuel in the vehicle s tank upon arrival at node j and is reset to Q, at each AFS or visit to the depot to recharge. Time and fuel level tracking constraints serve as sub tour elimination constraints. Constraints (11) reset the fuel level when the vehicles arrive at refuelling stations of the depot. The next set of constraints (12) ensures that all vehicles have enough remaining fuel to return to the depot, directly or by passing through a recharging station, from any customer node during a route. Lastly, constraints (13) ensure binary internality of the variable x ij. That is node j is either visited after node i or not. The succeeding section will present an overview of prior research on the green vehicle routing problem, as well as on problems that share common characteristics with it. Once the state of the art in relation to GVRP is outlined, one of the most important considerations in relation to the choice of solution methods problem complexity, will be discussed Literature review Green logistics issues give rise to many new problems in logistics operations. Sbihi and Eglese (2010) discuss the operations research perspective of green logistics and address 10

19 some of the problems that arise when wider environmental concerns are modelled as part of combinatorial optimization problems. However, optimization research models that incorporate such problems have gained research attention in recent times and are scarce. Lin et al. (2014) point out that GVRP research is focused on the optimization of energy consumption. Vehicle routing problems that take into consideration energy consumption, and attempt to minimize it, in direct or indirect ways, have become the subject of research attention only recently. A variation of these problems attempts to minimize the energy consumption and increase the efficiency of traditional modes of transportation. Kara, Kara and Yatis (2007) define the Energy Minimizing VRP and utilize integer linear programming. Xiao et al. (2012) extend the capacitated VRP to include a fuel consumption rate and the objective of minimizing fuel consumption. Küçükoğlu et al. (2015) address the GVRP with time windows, in which the objective is to minimize total fuel consumption and CO 2 emissions. The authors construct a fuel consumption calculation algorithm and utilize a memory structure adapted simulated annealing metaheuristic algorithm. There are a number of additional studies that consider the minimization of fuel consumption (Kuo 2010, Apaydin and Gonullu 2008, MirHassani and Mohammadyari 2014). GVRP concerning the decrease in petroleum- based fuels is another variation of green vehicle routing problems that take into consideration energy consumption, more specifically the decrease in harmful effects from it. These problems are characterised by a fleet of vehicles that run on alternative fuel. Abousleiman and Rawashdeh (2014) address some of the fundamental features of electric vehicles in their electric vehicle routing problem. These include negative path costs, battery power and energy limits and parameters that are only available at query time. Nevertheless, the authors only focus on incorporating the special features of alternative fuelled vehicles in the vehicle routing algorithm. Alternative fuelled vehicles present two additional challenges in routing compared to traditional routing problems. First, most of these vehicles have a limited driving range. Additionally, the fuelling infrastructure and network for alternative fuels is not widely available (Mckinnon et al. 2010). Thus, problems of this kind need to take into consideration the refuelling of vehicles. Conrad and Figliozzi (2011) present the first formulations of the recharging VRP, where vehicles with a limited driving range can expand it by recharging at customer locations. As a solution method a heuristic 11

20 based on iterative construction and improvement is used. The authors assume a fixed recharge time, however, they also study the impact of fleet driving range, time windows and recharge time, and construct solution bounds that are useful in predicting average tour length. However, Conrad and Figliozzi (2011) do not consider the posibility of a detour, to recharge at a refuelling station, while en route. Erdogan and Miller- Hooks (2012) are the first to take into consideration alternative fuel recharging stations in their routing model. The findings of their research are used as benchmarks in a number of papers that follow (Felipe et al. 2014; Scneider, Stegner & Goeke 2014). Scneider, Stegner & Goeke (2014) extend their model to incorporate more practical, real-world considerations, and present the electric vehicle routing problem with time windows and recharge stations. They integrate time windows, vehicle capacity limitations and variable recharge times, which are dependent on the fuel level of the vehicle upon arrival at the refuelling station. The electric VRP with time windows has the objective of minimizing the number of vehicles in the fleet, as well as the distance. As a solution method the authors utilize a hybrid metaheuristic combining variable neighbourhood search and tabu search. The results of their algorithm considerably outperform those in the benchmark study by Erdogan and Miller- Hooks. Felipe et al. (2014) study the GVRP with multiple technologies and partial recharges. They extend Erdogan and Miller- Hooks s (2012) model in an alternative way, by including partial recharges, the cost of battery amortization and multiple recharge technologies that are characterized by different recharge times and costs. The authors aim at minimizing the recharge and amortization costs, and implement constructive and a number of local search heuristics that are exploited within a simulated annealing framework. Their method outperforms the solution algorithms utilized in Erdogan and Miller- Hooks s research, and is competitive with the method utilized by Scneider, Stegner and Goeke (2014). Even though, the current base of studies that incorporate environmental concerns in vehicle routing problems is not extensive, there is a variety of researched problems that are related to the GVRP. Ichimori, Hiroaki and Nishida (1981) address the shortest path problem for a vehicle that visits only a single destination from its origin, has a limited driving range, and can visit refuelling nodes while en route. Ichimori, Hiroaki and Nishida (1983) consider determining the minimum range, such that a vehicle can reach all customer nodes, and the minimum number of recharge nodes. A number of works on military applications address issues related to limited fuel tank capacity (Mehrez, Stern 12

21 & Ronen 1983; Mehrez & Stern 1985; Melkman, Stern & Mehrez 1986). These studies suggest extending the driving range by transferring and receiving fuel from other vehicles in the fleet. Structurally similar to GVRP is the multi depot VRP with interdepots, where intermediate depots serve as replenishment facilities (Crevier, Cordeau & Laporte 2007; Tarantilis, Zachariadis & Kiranoudis 2008). This is similar to GVRP, as intermediate depots serve to reduce the limitations from vehicle capacity on constructing a route, while recharge stations serve to reduce the limitations from vehicle fuel capacity, or driving range. Another structurally similar class of problems is the waste collection VRP. Such problems are characterized by a set of disposal facilities, a set of customers from which waste must be collected and a fleet of vehicles located at a single depot. Benjamin and Beasley (2010) study the waste collection VRP with time windows, driver rest period and multiple disposal facilities. In the formulation of their problem vehicles collect waste from customers and visit a disposal facility when necessary. The choice of solution methods to the GVRP is greatly influenced by its complexity. Thus, in the following the complexity of the green vehicle routing problem will be discussed Complexity theory Problem complexity, which is also equivalent to the complexity of the best algorithm for solving a given problem, provides guidance in the selection of solution methodologies (Talbi 2009, pp. 9-14). Problem complexity theory is concerned with decision problems, which are those problems that can be answered with a yes or a no. Two key problem complexity classes are P and NP. The class P consists of all decision problems that can be solved using a deterministic algorithm in polynomial time, while NP is the set of all decision problems that can be solved using a nondeterministic algorithm in polynomial time. Vehicle routing problems, as many other routing and scheduling problems, are NP- hard (Lenstra & Rinnooy Kan 1981). As GVRP is a variant of VRP, it follows that GVRP is also NP- hard. NP- hard problems cannot be solved in polynomial time, unless P= NP, and to be solved to optimality, they require exponential time. Thus, for real life GVRP instances, which are often large, approximate algorithms are an attractive solution 13

22 approach. Although such algorithms do not provide a guarantee on the approximation of the solution, they offer an acceptable solution in terms of quality, at acceptable computational cost. For this reason, in the current research a metaheuristic approach is proposed for solving the GVRP, as described above. Considering the limited amount of research into the GVRP, in this thesis a metaheuristic algorithm will be implemented to solve the problem. The following chapter will provide the theoretical foundations of the solution method applied in this thesis. It will give an overview to metaheuristics in general, as well as to the specific method chosen in this research, tabu search. 14

23 3. Metaheuristics This chapter will begin by presenting the theoretical background of the solution method chosen in this study, in order to address the research objectives previously outlined. This will be done by describing the main features of metaheuristics, classification criteria according to which they can be distinguished, as well as discussing two important attributes that need to be considered in the design and implementation of such an algorithm. Additionally, this chapter will provide an overview on the basic principles of tabu search, as this is metaheuristic utilized in the current research. Metaheuristics have been gaining more and more popularity over the past two decades (Talbi 2009, p. 23). The term metaheuristic was first introduced by Glover (1986) and is used to define higher level procedures, in contrast to problem specific heuristics, which attempt to explore the search space in an effective and efficient manner through combining different concepts and learning strategies. The main goal of metaheuristic algorithms is to escape from local optima by applying strategies that guide the search process. This is done by either accepting worsening moves in the search process, or by generating new initial solutions for the search, based on a more sophisticated set of rules, not just random generation (Blum & Roli 2003). Metaheuristic algorithms are approximate, and are often applied to large-size problem instances, where exact solution methods are unpractical, as they deliver satisfactory solutions within reasonable computational time (Talbi 2009, p. 23). Thus, even though metaheuristics do not guarantee finding the global optimum in a given problem, by exploring a broad range of good solutions they locate a quality solution at an acceptable cost. An attractive feature of metaheuristic algorithms is their applicability in many areas, as they are not problem specific. Such algorithms span from complex learning schemes, to basic local search procedures that implement a set of rules for moving from one solution to another and evaluate them based on a given criterion (Blum & Roli 2003). There is a wide variety of metaheuristics, which are mostly explored in an empirical fashion throughout the literature. Examples include ant colony optimization, genetic algorithms GA, guided local search, greedy randomized adaptive search GRASP, variable neighbourhood search, simulated annealing, tabu search and iterated local search (Talbi 2009, p. 24). Although, each of these utilize somewhat unique solution approaches, they 15

24 can be broadly classified based on a number of characteristics. The following section will present the most important classification schemes of metaheuristics Classification of metaheuristics One way of classifying metaheuristics is based on the origin of the algorithm. Some algorithms are inspired by natural processes, such as ant colony optimization. While others are non-nature inspired, such as tabu search (Blum & Roli 2003). For example, ant colony optimization mimics the behaviour of ants in finding the optimal path to a food source. Ants use pheromones to communicate the attractiveness of a path leading to food. The more ants pass through a given path, the stronger the pheromone trail is. Thus, the more attractive the path is. Attractive paths, on the other hand, lead to better food sources, to good solutions. Another classification is based on the memory, or search history, characteristic of algorithms. Algorithms are divided into those, which dynamically extract information throughout the search process and use it to improve the search in a given way, and those which are memory-less and do not extract and utilize information during the search. In algorithms that use a memory structure, it is common to distinguish between short and long term memory. Short term memory refers to information on recently performed moves, visited solutions or decisions taken. In contrast, long term memory is normally the accumulation of synthetic parameters about the search (Blum & Roli 2003). For example, tabu search is an algorithm that implements a memory structure. The short term memory of the algorithm keeps the recently performed moves in a list and forbids them for a given time. This is done to avoid cycling and to escape from local optima. Additionally, the algorithm can also apply long term memory, and keep information on visited solutions throughout the entire search. By recording the frequency of solution components, the algorithm can then diversify the search and visit areas of the solutions space, which have not been visited before. Metaheuristics are further divided into deterministic and stochastic. Deterministic algorithms solve a given problem by making deterministic decisions during the search. That is, using the same initial solution, will always lead to the same final solution. In contrast, stochastic algorithms implement stochastic rules throughout the search. Thus, using the same initial solution in stochastic algorithms will not lead to the same final solution (Talbi 2009, p. 25). Tabu search is an example of a deterministic algorithm that 16

25 does not apply any random rules in the search. It moves from one solution to the next based on the fitness of the solution and based on the tabu status of the moves performed, to reach the new neighbouring solution. Algorithms are also categorized into greedy and iterative. Most metaheuristics are iterative, meaning that the algorithms start with a complete solution and transform it during the search. However, there are also greedy algorithms, which are initiated with an empty solution that is expanded throughout the search (Talbi 2009, p. 25). Furthermore, algorithms are divided into such that use a single neighbourhood structure to look for a solution and such that utilize various neighbourhood structures (Blum & Roli 2003). For example, the variable neighbourhood search algorithm applies a set of neighbourhood operators to diversify the search process. Although, algorithms can formally be classified in that way, it should be noted that hybrid approaches can be created. Such hybrids comprise of the characteristics of multiple metaheuristics, as well as other optimization techniques. Last but not least, metaheuristics can be classified as population based and single solution based. Population based metaheuristics focus more on exploring the search space, that is on diversification. They manipulate and change multiple solutions throughout the search process. Single solution based search, on the other hand, are intensification oriented. That is, they have properties that intensify the search within a given region. In this class of algorithms, only a single solution is evolved (Talbi 2009, pp ). In the following, the two attributes intensification and diversification will be discussed Metaheuristics important attributes in design and implementation In designing and implementing a metaheuristic algorithm two complementary attributes need to be balanced. These are intensification and diversification (Talbi 2009, p. 24). Intensification can be broadly described as the exploitation of the best solutions found. An algorithm that is intensification oriented will quickly converge to either a local minimum or a local maximum, depending on whether the goal is to minimize or maximize the objective function. The main benefit of intensification is that it helps in quickly identifying promising regions in the solution space. However, the shortcoming 17

26 of this is that the algorithm could easily get trapped in a local minimum. Thus, although the intensification attribute of an algorithm helps in identifying peaks and valleys in the solutions space, unless there is an approach to escape from them and identify new promising regions, the algorithm could get stuck in local optima. An example of an intensification oriented metaheuristic is local search. This method starts with a given initial solution and at each iteration adopts a new, improving solution from the neighbourhood of the current one. The algorithm stops when none of the solutions in the neighbourhood have a better value of the objective function. Thus local optimum is reached. The diversification attribute of an algorithm addresses the issues related to getting trapped in local optima. Diversification represents the exploration of the search space, as opposed to exploring only promising regions (Blum & Roli 2003). This is done through applying a form of random search or other procedures that contribute to exploring more thoroughly the solution search space. For example, to introduce diversification in local search, the algorithm can be re-initialized with a different, randomly generated solution, each time it reaches a local optimum. This would increase the search area, while at the same time identify a greater number of promising solutions and possibly a better final solution. Although, the result could improve by introducing diversification as well as intensification, there is no guarantee that global optimum will be reached even if the algorithms is ran many times. However, introducing those two attributes, as well as increasing the number of iterations the algorithm passes through, will increase the probability that a global optimum is reached Tabu search Tabu search, a metaheuristic developed by Fred Glover (1986), is a deterministic, iterative and single solution based method that uses local search and memory structures to perform an intelligent search of the solution space. The TS algorithm acts like a steepest descend/ ascend local search, with the distinction that once local optimum is reached TS accepts non-improving solutions from the neighbourhood to escape from it (Talbi 2009, p. 140). To prevent cycling throughout the continuous transformation of the neighbourhood, this method manages a list of recently applied moves and forbids them, the tabu list. The moves recorded as tabu represent solution features. However, using specific solution features to forbid the consideration of a neighbouring solution 18

27 may create issues. That is, a solution that has not yet been generated by the algorithm may be rejected because it was produced by a recently applied move. To avoid this restrictive feature of the tabu list, the metaheuristic uses an aspiration criterion that sets a rule for accepting solutions generated by tabu moves. When this criterion is satisfied, the tabu status of a move is overridden and the move is accepted. For a thorough review of tabu search see Glover and Laguna (1997). Tabu search has applications in a wide range of fields, such as scheduling, telecommunications, graph optimization, technology and most importantly, for the current research, routing (Glover & Laguna 1997, p.625). Its broad application in theoretical studies provide an extensive background on the methodology and implementation, as well as aid in making an inference on the efficiency and effectiveness of tabu search in optimization. In the following sections, the most important design and implementation considerations when utilizing TS will be discussed. Additionally, the different types and uses of memory structures in tabu search will be reviewed Design and implementation features In designing and implementing tabu search there are four important aspects to consider. First, the neighbourhood structure needs to be defined. An adequate neighbourhood for a given problem is characterized by the property locality (Talbi 2009, p. 89). In order for a neighbourhood structure to have strong locality, a small change in the representation of the current solution must lead to a small change in the solution value. For example, consider a problem consisting of twenty customers, for which a neighbour solution is created by switching the position of two customers. As this is a small move, it should not lead to a large change in the objective function value if the neighbourhood has a strong locality. Locality is a desirable feature, as it indicates that the solution space will be thoroughly and meaningfully searched. Another important issue in the design phase of TS is the creation of the initial solution (Talbi 2009). As tabu search is an iterative algorithm, it starts with an initial solution that is later repeatedly modified. The attributes quality and computational cost can have an impact on the performance of the metaheuristic algorithm. An initial solution of high quality, will have high computational cost, but can help the metaheuristic algorithm in 19

28 converging faster (Talbi 2009, p. 101). Thus, when deciding on the method for building an initial solution it is very important to consider the specifications of the problem at hand, and balance the two attributes. Defining the neighbourhood and creating the initial solution are design issues common for all single solution based metaheuristics. However, there are two additional aspects that are specifically associated with tabu search (Talbi 2009). These are the tabu list and the aspiration criterion. In successfully utilizing a tabu list, it is important to identify how moves will be recorded and decide on the appropriate size of the tabu list, or how long a move stays tabu tabu tenure. The range of the tabu list affects the area of the search space explored by the algorithm (Boussaïd, Lepagnot & Siarry 2013). That is why, identifying the correct dimension of the list is important. In practice, the size of the tabu list is commonly identified through preliminary testing. For example in the research by Scneider, Stegner and Goeke (2014), as well as in that by Benjamin and Beasley (2010), the size of the tabu list was determined by computational experimentation. The specification of aspiration criteria has the property of considerably improving the search process (Boussaïd, Lepagnot & Siarry 2013). These criteria provide a set of rules to override the tabu status of a move when it is beneficial to execute it, regardless of the fact that it was recently performed. Thus, such rules can signifficantly affect the quality of the tabu search algorithm and need to be carefully chosen. A frequently applied criterion is performing a tabu move, if it improves the objective function value compared to the best solution found so far in the search (Talbi 2009) Adaptive memory in tabu search A distinctive feature of TS compared to memory-less metaheuristics is the use of adaptive memory to explore the search space in a responsive manner. Tabu search can use both explicit and attributive memory. Explicit memory keeps complete solutions. Most often these are the elite solutions found throughout the search. Attributive memory is used to guide the search, and record attributes of the solution that change when moving from one solution to the next. There are four principal dimensions of memory in TS: recency, frequency, quality and influence (Glover & Laguna 1997). The use of these dimensions can be beneficial in developing a successful tabu search algorithm. In the following, each dimension will be described in turn. 20

29 The recency dimension refers to the most commonly used short term memory, which records recently performed moves in the search process (Glover & Laguna 1997). In the basic form of TS this dimension is incorporated into the algorithm by the use of tabu. As discussed earlier, a guiding principle in TS is to forbid moves, recently applied in the search, for a specific number of algorithm iterations. This is enabled by the recency information. Attributes remain tabu until a predetermined number of iterations have been executed. This number is specified by the parameter tabu tenure. Tabu tenure can be a fixed amount of iterations, or can be dynamic in which case the search process can change during the execution of the algorithm. Small tabu tenure would imply that the algorithm is more focused on finding the local optima. This is the case, because a small tenure means that a move is freed after only a few iterations, thus the probability of cycling is higher and the algorithm may not be able to escape from local optimum. In contrast, big tabu tenure helps the algorithm escape local optima and perform a more thorough search of the solution space. Considering that greater tabu tenure aids exploration and small tabu tenure intensifies the search to find local optima, it is not straight forward to determine its appropriate length for the problem at hand. That is why, it is common that tabu tenure is determined by means of experimentation, or it is changed dynamically during the search. Another dimension of the memory in TS is frequency. This dimension can be used to complement the recency memory, and to help in choosing the next solution to be visited in the search process (Glover & Laguna 1997). Frequency memory can be used to store the number of times a specific solution attribute has been revisited and use that to move away from solutions contacting such attributes. This dimension can help diversify the search, together with recency. The quality dimension is related to the ability of the TS algorithm to identify what constitutes a good solution, or the path to such a solution (Glover & Laguna 1997). That is, memory in tabu search can be used to locate elements common to good solutions, and then the inclusion of these elements in the solutions throughout the search can be encouraged. The quality dimension can be used in the same manner to penalize unattractive solutions. Both uses of this memory dimension result in a more effective and efficient TS. 21