Residential waste management in South Africa: Optimisation of vehicle fleet size and composition

Transcription

1 Residential waste management in South Africa: Optimisation of vehicle fleet size and composition by Wouter H Bothma Submitted in partial fulfillment of the requirements for the degree of Bachelors of Industrial Engineering in the Faculty of Engineering, Built Environment and Information Technology University of Pretoria October 2010

2 Executive Summary The collection of solid waste is a highly discernible and important municipal service. Despite this fact, municipalities still make use of ineffective measures that are costly to sustain. The basic nature of our problem is that of a Capacitated Arc Routing Problem with Vehicle Site Dependencies. We present a Vehicle Decomposition Algorithm capable of determining the collection routes and the vehicle fleet size and composition for a fleet of waste collection vehicles with the objective of minimising the total cost incurred. Each vehicle class within the fleet may be constrained from traversing or servicing certain street segments due to limitations such as physical dimensions. A sensitivity analysis is performed to determine the optimal parameter values for use in the algorithm. The Vehicle Decomposition Algorithm is tested on three example problems and results are presented.

3 Contents 1 Introduction The solid waste disposal process Basic waste collection operation Problem statement Research design Research methodology Document structure Literature review: Vehicle fleet size and composition Background The Capacitated Arc Routing Problem CARP with Vehicle Site Dependencies CARP with Alternative Objective Functions Solution Approaches Solving the CARP-VSD using the VDA Phases of the solution procedure for the fleet size and mix extension of the Multi Objective CARP Tabu search Simulated Annealing search Genetic Algorithms Conclusion The Vehicle Decomposition Algorithm Terminology The use of matrices in the VDA Existing sourced components used in the VDA High level structure Step A: Create and verify vehicle class networks Step B: Estimate total work and determine initial fleet mix Step C: Partition the service network Conclusion Computational results Input data Key Parameters Sensitivity analysis i

4 4.4 Performance of the VDA Example problem one Example problem two Example problem three Conclusion Opportunities for future research 52 References 54 A Pseudo code for Algorithm 9 and Algorithm B Output from the VDA for example problem two 58 C Output from the VDA for example problem three 62 D Example problem data sets 66 ii

5 List of Figures 1.1 Typical residential neighbourhood The difference between an edge and an arc The OR process The arc sequences Diagramatic representation of the VDA Excess capacity as a function of the TRT factor and the percentage spesification The specified percentage and Target route time factor combination The vehicle class load composition for example problem one Vehicle class time distribution for example problem one The vehicle class load composition for example problem two The vehicle class time distribution for example problem two The vehicle class load composition for example problem three The vehicle class time distribution for example problem three The travel network with vehicle class limitations The first vehicle from class one The second vehicle from class one The third vehicle from class one The fourth vehicle from class one The first vehicle from class two The second vehicle from class two The third vehicle from class two iii

6 List of Tables 1.1 Edges and arcs of Figure 1.1(b) Comparison of heuristic differences VDA specific terms The vehicle preference list presented as the P ref matrix The lower triangular V matrix The vehicle preference list for the set of benchmark problems Vehicle load composition summary for example problem one Time distribution summary for example problem one Vehicle load composition summary for example problem two Time distribution summary for example problem two Vehicle load composition summary for example problem three Time distribution summary for example problem three iv

7 List of Algorithms 1 The Vehicle Decomposition Algorithm The vehicle size and mix algorithm Tabu Search algorithm Simulated Annealing algorithm High level Genetic Algorithm Complete VDA Create-Networks Generate-Vmatrix Generate-FleetInitialise Generate-Fleet Create-Partitions Pseudo code: Generate-FleetInitialise Pseudo code: Generate-Fleet Acronyms ARP CARP Arc Routing Problem Capacitated Arc Routing Problem CARP-VSD Capacitated Arc Routing Problem with Vehicle-Site Dependencies CPP FSMVRP GA GUI MCARP OR RPP RWCP SA Chinese Postman Problem Fleet Size and Mix Vehicle Routing Problem Genetic Algorithm Graphical User Interface Mixed Capacitated Arc Routing Problem Operations Research Rural Postman Problem Residential Waste Collection Problem Simulated Annealing v

8 SPP TS VDA VRP VSD Shortest Path Problem Tabu Search Vehicle Decomposition Algorithm Vehicle Routing Problem Vehicle-Site Dependencies vi

9 Chapter 1 Introduction Waste disposal refers to the proper disposition of discarded or discharged material. For many of us the last time we think of our waste is when we take it out of our homes and leave it on the sidewalk for collection. Have you ever stopped to consider the cost of transporting your waste to an appropriate disposal facility? What about the operating costs associated with the vehicle fleet? The focus of our work will be on selecting a vehicle fleet size and composition in combination with collection routes to minimise the total cost of waste collection given specific service demands, subject to certain business constraints. 1.1 The solid waste disposal process Municipal waste collection transportation can be divided into four critical components: sectoring a service area into balanced collection days and vehicle areas; determining the location of transfer stations within a collection area; determining the vehicle fleet size and composition; and lastly developing balanced collection routes for waste collection vehicles. For the purpose of this project we assume that balanced collection vehicle routes exist and are available. The focus for the remainder of the project will be on optimising the vehicle fleet size and composition. Palmer Development Group [11] recognises waste transportation as being the most costly component of the waste management function. Waste management is transport intensive and requires various different types of vehicles to perform the function successfully. Key obstacles with regards to waste transportation in South Africa include the following: The conventional collection truck (a compactor) has a high capital outlay and, as a result, many rural municipalities make use of inexpensive and inefficient collection vehicles such as trailers. The volatile fuel price makes it difficult to determine running expenses. Most municipal waste collection vehicles are old and, as a result, are often out of order. Despite the fact that most municipalities have purchased new vehicles within the 2007 financial year, they have no vehicle rotation plan in place. Municipalities seldom have back-up vehicles to replace those that are out of order. Round balancing studies are seldom undertaken to determine optimisation of collection routes. 1

10 Viotti et al. [19] state that the collection of municipal waste can account for anything between 50 70% of the overall cost associated with any waste management system. One also needs to consider the collection process from an environmental and health viewpoint. According to Amponsah and Salhi [1] 69 96% of solid waste in developing countries consist of organic matter with a very high moisture content. Add to this the hot climate in South Africa and the result is rapid decomposition of waste that could lead to various health risks. In recent times solid waste management has enjoyed a lot of attention from both municipal and governmental parties. This is largely due to the widely acknowledged increase in solid waste production, and the increased awareness concerning environmental issues such as sustainability. According to Sbihi and Eglese [16] these factors have led local governments and agencies to devote resources to solid waste collection policy planning. Benefits of improving the waste collection process includes improved process efficiency, sustainability, reliability and in the long term, financial gain. 1.2 Basic waste collection operation Kim et al. [10] classify waste into three categories: commercial waste; residential waste and roll-on-roll-off waste. Commercial waste is stored in large bins. It involves the point-to-point collection of waste from businesses and apartment blocks in urban areas. Residential waste is stored in bags and small containers along the side of the road in residential areas. Each residence represents a demand point. It forms the largest percentage of waste collected by municipalities. Roll-on-Roll-off waste is typically found in industrial areas and other high volume locations such as construction sites. It involves the pickup, transportation and drop-off of containers. The collection vehicle can typically carry only one such container at a time and may follow one of two methods, namely, round trip or exchange trip. During a round trip the vehicle picks up a full container at the collection point, empties it at a dumping site and returns with the empty container. During an exchange trip the vehicle heads for the collection point with an empty container and exchanges it for the full container. The full container is then returned to the landfill to be emptied. For the purpose of this project the focus will be on residential waste collection, which Amponsah and Salhi [1] describe as follows: containerised refuse is collected from the sidewalk by a fleet of vehicles with limited capacities, meaning each road segment that contains refuse bins will be serviced by a collection vehicle. Each vehicle within the fleet will collect several containers of refuse until its capacity is reached. The vehicle then transports the refuse to an unloading site. In most urban areas more than one such site exists and the driver can use his discretion in selecting the most appropriate site. Furthermore, the assumption is made that every vehicle leaves the depot empty and returns empty at the end of the day. Figure 1.1(a) depicts a typical residential neighbourhood. Assume each house represents a demand point. The highlighted area in Figure 1.1(a) can be simplified to the network of Figure 1.1(b). In this figure each of the eleven vertices represent a street intersection. The service demand for each street segment is shown on the arrows connecting the vertices. When a road segment 2

11 is connected by two vertices it is assigned as either an edge or an arc. In Figure 1.1(b) the edges are illustrated with red arrows, whereas the arcs are illustrated with black arrows. In both cases the directions are indicated by the arrow heads. Figure 1.2 illustrates how edges represent segments that may be traversed and serviced in any direction, whereas arcs only in one direction. Table 1.1 shows the edges and arcs for the example in Figure 1.1(b) with their respective service demand levels. In order to service the required demand levels all the arcs and edges with a positive demand must be servived by the waste collection vehicles. A problem that requires arcs and edges to be serviced is known in literature as an Arc Routing Problem (ARP). Table 1.1: Edges and arcs of Figure 1.1(b) Arc\Edge Edge (1, 2) Edge (2, 11) Edge (11, 10) Edge (10, 9) Edge (9, 8) Edge (8, 7) Edge (7, 6) Edge (6, 4) Edge (4, 5) Edge (4, 3) Edge (3, 2) Edge (11, 7) Arc (1, 10) Demand 2kg 7kg 3kg 8kg 11kg 10kg 13kg 7kg 4kg 13kg 23kg 12kg 15kg 1.3 Problem statement Due to rapidly expanding neighbourhoods, and the ever increasing fuel price, municipalities are facing an increase in waste collection expenditure. In this project we will focus on transportation costs i.e. the cost of operating and maintaining the vehicle fleet, as well as the capital costs of acquiring additions to the vehicle fleet. Municipalities require a mathematical model that will optimise the vehicle fleet size and composition in an attempt to minimise expenditure in waste collection operations. This leads us to the following research question: How should the waste collection vehicle fleet size and composition be selected to ensure a quality and efficient service at a minimum cost? 1.4 Research design This project aims to develop a model as an add-on to an existing routing application. The model will determine the vehicle fleet size and composition that will accompany the routing 3

12 (a) Aerial photo of a residential neighbourhood 1 2kg 3 23kg 2 13kg 15kg 7kg kg 12kg 4 7kg 3kg 7 13kg kg 8kg 9 11kg 8 (b) Simplified road network representation Figure 1.1: Typical residential neighbourhood 4

13 Edge Two-way arc One-way arc Direction of traffic Refuse containers Figure 1.2: The difference between an edge and an arc schedule for the Residential Waste Collection Problem (RWCP). A number of research contributions on the fleet composition problem is found in literature, but each model is unique to its situation. Our focus will therefore be on combining and adapting existing well known algorithms to derive algorithms that can be applied to a broad range of networks. The deliverables for this project are: 1. A research question. 2. A solution algorithm. 3. Results of the algorithm when tested on a set of benchmark problems. 1.5 Research methodology Problem Modeling Model Accessing Solving Decision Inferring Solution Figure 1.3: The OR process The Operations Research (OR) process (Figure 1.3) as adapted from Rardin [12] is followed to develop a generic model that can be used to design or redesign a municipality s vehicle 5

14 fleet composition. The reason for using OR methods is to obtain an optimal or near-optimal solution to the problem mathematically. This will improve the performance of the system. The four phases of the OR process are: Problem: A problem statement is formulated that includes all requirements and constraints. Constraints will typically include budget, demand and time. During this phase we study existing algorithms used to solve similar problems. Data regarding the travel network, service network and vehicles on hand are collected as test data to form the basis of the test case. The most important deliverable of this phase is a holistic comprehension and understanding of the theory of the algorithms on which our solution will be based. Model: The development of the model is guided by the problem definition and relative literature study to present an accurate representation of the waste collection environment. The model will consist of decision variables that will guide the solution strategy developed, constraints that limit the problem according to certain decisions and an objective function that will ensure that only good solutions are chosen for final implementation. Solution: Data gathered is used to build test cases for computational testing in order to determine whether the application is working. The mechanism for determining whether the application has passed or failed such a test is known as a test oracle and gives an indication to further refinement and improvement of the algorithm. Decision: Next we interpret the numerical results obtained in the solution phase. This phase will typically give an indication to other problems that need to undergo the same OR process. Once we are satisfied with our model the final integration of the vehicle fleet mix and composition module with the working route scheduling algorithm can commence. 1.6 Document structure In Chapter 1 we discussed the waste collection environment with specific emphasis on the vehicle fleet and defined the goal of this research project. Chapter 2 provides an overview of various literature regarding solution strategies that can be used to solve our problem. A detailed description of the development of the algorithms used to optimise the vehicle fleet size and composition is given in Chapter 3. The algorithms are then tested on a set of benchmark problems, followed by computational results in Chapter 4. Opportunities for future research are discussed in Chapter 5 and final conclusions are made. 6

15 Chapter 2 Literature review: Vehicle fleet size and composition The vehicle fleet size and composition problem involves two basic decisions: the composition of a heterogeneous vehicle fleet and the routing of this fleet. Hoff et al. [9] describe the Vehicle Routing Problem (VRP) and the Capacitated Arc Routing Problem (CARP) as belonging to a general class denoted as fleet composition and routing problems. In order to design an algorithm to optimise the vehicle fleet size and composition, we need to have a holistic understanding of the algorithms currently used to derive the routing scedule. Arc Routing Problems (ARPs) require the service and traversal of edges within a graph. This is also true for the Residential Waste Collection Problem (RWCP). Eiselt et al. [6] define the formulation for an Arc Routing Problem (ARP) as follows. Let G = (V, A) be a connected graph without loops, where V = {v 1,..., v n } is the vertex set (or node set), and A = {(v i, v j )} : v i, v j V and i j} is the arc set. Every arc set (v i, v j ) is assigned a non-negative cost, distance or time d ij ; the assumption is made that d ij = if the arc is undefined. As a practical example we refer back to Figure 1.1(b). Each undirected edge can be replaced by two directed arcs, each with the same length and demand. Note, however, that only one of the associated arcs needs to be serviced by the collection vehicle. In the RWCP each vehicle in the fleet has a limited capacity; an ARP with a capacity contraint is known in literature as a CARP. A detailed analisys of the CARP and the Fleet Size and Mix Vehicle Routing Problem (FSMVRP), and the heuristics and metaheuristics used to solve them will be dicussed in this chapter. 2.1 Background The vehicle fleet will be composed of different vehicle types, each type with unique fixed and variable costs. The objective of the problem is to minimize the total cost, which is composed of fixed utilization costs and variable traveling costs. Renaud and Boctor [14] mathematically define the FSMVRP as a graph G = (V, A),where V = {v 0,..., v n } is the vertex set and A = {(v i, v j ) : v i, v j V, i j} is the arc set. Vertex v 0 represents a depot where M different vehicle types are based. Each vertex v i V \{v 0 } corresponds to a customer and is associated with a non-negative demand q i and a service time s i. In this version of the problem, all arcs are undirected, i.e. they are edges. Every edge (v i, v j ) is associated with a non-negative cost, c ij, representing its travel cost and a non-negative time, t ij, representing its travel time. The 7

16 vehicle fixed cost, the capacity and maximum travel time for vehicle type k = 1,..., M are represented by F k, Q k and T k, respectively. It is assumed that F k1 < F k2 implies Q k1 < Q k2, that vehicle types are numbered in ascending order of F k and that we can use any number of vehicles of type k. The goal of the FSMVRP is to determine a mix of vehicles as well as their routes such that: 1. Routes start and end at the depot. 2. Each customer is visited exactly once. 3. The total demand of the route does not exceed the capacity of the vehicle used. 4. Total duration of each route does not exceed the maximum allowable time T k of the vehicle type used. 5. The sum of fixed and variable costs are minimised. According to Sahoo et al. [15] the primary factor differentiating residential from commercial routes is the mandatory adherence to driving on one side of the road. Point-to-point solutions such as the FSMVRP work well for routes with specific demand points not necessarily on the same street segment, like visiting different warehouses across the city. Residential routes on the other hand, where multiple points on a single street segment require service, requires the use of arc routing solutions. The CARP is an arc routing counterpart of the VRP and Golden et al. [7] state that the problem of refuse collection can be modeled as a CARP where the goal is to spread the demand evenly among the routes. 2.2 The Capacitated Arc Routing Problem The routing of snow removal vehicles, street sweepers and waste collections vehicles are all applications of the CARP. The CARP, as first suggested by Golden and Wong [8], considers an undirected graph G = (V, E) with a set of required edges R E. Each arc (v i, v j ) incurs a cost c ij and has a non-negative demand q ij associated with it. It can be assumed that a fleet of m homogeneous vehicles of capacity W are based at a depot (located at vertex v 1 ). All arcs with q ij > 0 must be serviced, while the remaining arcs and edges of E may be traversed. The CARP is designed to minimise the total route length whilst ensuring that: 1. Each arc with positive demand is serviced by exactly one vehicle. 2. The sum of demand of those arcs serviced by each vehicle does not exceed W. 3. The total cost of the tours is minimised. An integer linear programming formulation for the undirected CARP is proposed by Eiselt and Laporte [5]. With this formulation, each edge is replaced with an arc (v i, v j ) to create a directed formulation of the CARP. The formulation is as follows. 8

17 1 if edge (v i, v j ) A is traversed from v i to v j by vehicle k, x ijk where k = {1,..., m}, i j, 0 otherwise. 1 if edge (v i, v j ) A is serviced by vehicle k while traveling from v i to v j, y ijk where k = {1,..., m}, i j, 0 otherwise. c ij The cost or distance of edge (v i, v j ). q ij The demand of edge (v i, v j ). W Capacity of the vehicles. S A given vertex set. min = m k=1 (v i,v j ) A (c ij x ijk ) (2.1) subject to (v i,v j ) A v i,v j S x ijk k=1 (v i,v j ) A x jik =0 v i V, k = 1,..., m (2.2) { m 0 if q ij = 0, (y ijk + y jik ) = 1 if q ij > 0 (v i,v j ) A (v i, v j ) A (2.3) x ijk y ijk (v i, v j ) A, k = 1,..., m (2.4) q ij y ijk W k = 1,..., m (2.5) x ijk S 1 + n 2 u s k v i S v j / S x ijk 1 wk s u s k + ws k 1 u s k, ws k {0, 1} S S V \{v 1 }; S ; k = 1,..., m (2.6) S V \{v 1 }; S ; k = 1,..., m (2.7) S V \{v 1 }; S ; k = 1,..., m (2.8) V \{v 1 }; S ; k = 1,..., m (2.9) x ijk, y ijk {0, 1} (v i, v j ) A; k = 1,..., m (2.10) The objective function (2.1) minimises the total cost incured by the k vehicles. Constraints (2.2) conserve the flow for each vehicle. Constraints (2.3) ensure that each arc with a positive demand corresponds to a service arc. Constraints (2.4) ensure that an arc is only serviced by a given vehicle if it is traversed by the same vehicle. Constraints (2.5) ensure the capacity restriction of vehicle k is adhered to. Constraints (2.6) to (2.9) constrict the solution from 9

18 containing any illegal subtours. The basic CARP is often too symplistic to realistically model an accurate representation of real world problems. This leads us to investigate variants of the CARP CARP with Vehicle Site Dependencies Sniezek et al. [17] describe Vehicle-Site Dependencies (VSD) as a constraint that prohibits certain vehicle classes from servicing or traversing the street because of some limitation (such as physical dimentions). To incorporate this, Sniezek et al. [17] recommend using the Capacitated Arc Routing Problem with Vehicle-Site Dependencies (CARP-VSD), which is merely a generalization of the CARP and can be solved using the Vehicle Decomposition Algorithm (VDA). The VDA solves the CARP-VSD by decomposing it into several smaller single vehicle class CARPs CARP with Alternative Objective Functions The dominating objective of routing problems is to minimise the total distance traveled. There are, however, several other real life applications where other objectives are just as important. Examples of such alternatives include minimising the total number of vehicles used, equalizing the load of the tours, or minimising the length of the longest tour. When considering the FSMVRP we can see how this version of the CARP is of particular interest. Ulusoy [18] considers a version of the CARP where a vehicle includes a fixed cost if it is used and where vehicles differ in capacity. Therefore, the objective function is to minimise the total travel cost plus the total fixed cost incurred by the use of a vehicle fleet. A heuristic is recommended which first constructs a giant tour and then splits the tour by solving a shortest path problem that takes vehicle capacities and costs into consideration. This is known in literature as the route-first-cluster-second approach. 2.3 Solution Approaches While it would be ideal to solve a problem exactly, this isn t always the case. The CARP has been proven to be N P-hard. This implies an exponential increase in the solution space as the number of customers increase, meaning it cannot be solved in reasonable computing time. This makes it necessary to use heuristics or metaheuristics to solve the CARP. Heuristics are approximate techniques that have been used to arrive at a good solution when operations researchers where confronted with N P-hard problems. Winston and Venkataramanan [21] describe heuristics as using a greedy approach to obtaining a good solution in efficient time. Heuristics incrementaly improve solutions through neighbourhood exchanges or local search techniques. The problem with this approach is that there is a high risk of finding a sub-optimal solution by settling for a local optimum instead of a global optimum. Fortunately most heuristic techniques tend to converge to a good solution with the appropriate amount of iterations. Metaheuristics, as the name implies, refers to searching beyond to find. They are based on intelligent search techniques and are a form of Artificial Intelligence. They can be based on evolutionary principles such as methods of natural selection or physical systems such 10

19 as the annealing process. Tabu search is a metaheuristic that uses short- and longterm memory to intelligently guide the search process whilst preventing the search from cycling. Metaheuristic techniques try to avoid local optima by accepting solutions that may not be an improvement or by considering several solutions simultaneously in an effort to find the global optimum. In this section we will take a look at various heuristics proposed by Sniezek et al. [17] and Ulusoy [18] to solve variants of the CARP. We will also define some popular metaheuristics such as the Tabu Search (TS), Simulated Annealing (SA), and Genetic Algorithm (GA) Solving the CARP-VSD using the VDA In this section we define the travel network, assumptions, and goals of the CARP-VSD and explain the use of the VDA to solve the CARP-VSD. Sniezek et al. [17] describe the problem of CARP-VSD with specific reference to a residential sanitation-vehicle scheduling problem, the authors successfully solved the CARP-VSD for the Philadelphian sanitation department. A routing application known as RouteSmart is used throughout. Travel Network Sniezek et al. [17] define the travel network G = (N, A) as a directed network, representing the underlying street network over which the CARP-VSD is to be solved. Each arc in G represents one side of a street segment and is directed in the direction of travel. Each set of two directed arcs are referred to as counterpart arcs. The travel network G is composed completely of counterpart arcs. In the case of a street segment permitting two-way traffic, the two counterpart arcs in G are directed in opposite directions. In the case of a one-way street segment, both counterpart arcs are directed in the same direction i.e. the direction of traffic. The following attributes are defined for each arc a(i, j) in the travel network: 11

20 W(i, j) 1 if a(i, j) is a counterpart arc of a one-way street segment. Note that counterpart arcs of a one-way street segment are denoted a(i, j) and a (i, j), as it is replicated twice in the travel network. 2 if a(i, j) is a counterpart arc of a two-way street segment. Note that both a(i, j) and a(j, i) exist in this case. 0 if a(i, j) has to be serviced one side at a time. 1 if the street segment associated with arc a(i, j) can be meandered. M(i, j) In this case both sides of the street segment is serviced with a single traversal of the street. D(i, j) Deadhead travel time on arc a(i, j). L(i, j) Length of arc a(i, j). It can be assumed that L(i, j) = L(j, i). SC(i, j) Largest vehicle class that may service arc a(i, j). It can be assumed that SC(i, j) = SC(j, i). TC(i, j) Largest vehicle class that may travel along arc a(i, j). It can be assumed that T C(i, j) = T C(j, i). S(i, j) Service time on arc a(i, j). If the arc is to be serviced as a meander (zigzag), then S(i, j) is half the service time for the street segment associated with a(i, j). Q(a(i, j)) Volume/weight of refuse to be picked up on arc a(i, j). If the arc is not to be serviced, Q(a(i, j)) = 0. Service Network Sniezek et al. [17] define the service network as G s. The arc set from the travel network G is broken up into four sets representing different service needs. These four sets are as follows. DA: Set of deadhead arcs that require no service and may be used only for deadheading. DA = {a(i, j) Q(a(i, j)) = 0}. RA: Set of arcs requiring service, and cannot be meandered, and allows two-way traffic on the street segment. RA = {a(i, j) Q(a(i, j)) > 0, M(i, j) = 0}. Note that if a(i, j) RA, service of a(i, j) must occur by traveling along a(i, j) and not its counterpart arc. MA: Set of arcs requiring service, and may be meandered, and allows only one-way travel. MA = {a(i, j) Q(a(i, j)) + Q(a (i, j)) > 0, M(i, j) = 1, W (i, j) = 1}. SE: Set of edges that represent street segments allowing two-way travel, require service, and must be serviced as a meander, SE = {e(i, j) Q(a(i, j)) + Q(a(i, j)) > 0, M(i, j) = 1, W (i, j) = 2}. The edge, e(i, j), that appears in SE represents both arc a(i, j) and arc a(j, i). The service network G s is therefore represented as a node set N, the arc sets DA, RA and MA, and the edge set SE. 12

21 Vehicle classes Vehicle classes represent different types of vehicles. Each vehicle class k has a unique set of attributes that are defined by Sniezek et al. [17] as follows: Q k Number of vehicles available in class k. M k Capacity of a vehicle from class k. DT k Disposal time of vehicle from class k. This constitutes the time it takes to empty a vehicle of type k at the disposal facility. QT k Office time for a vehicle from class k. This includes the time it takes to carry out mandated functions, such as filling up with fuel and washing the vehicle. Travel network and service network for a vehicle class For every arc in G, T C(i, j) specifies the largest class k that can travel on arc a(i, j). Each vehicle class has a different set of arcs that can be traversed by a vehicle from the vehicle class. The travel network for vehicle class k, G k = (N k, A k ), is the network of arcs that can be traversed by the vehicles in vehicle class k. For each arc in G, SC(i, j) specifies the largest class k that can service arc a(i, j). The service network, G k s, for vehicle class k is created from the travel network for vehicle class k, G k t, in exactly the same way as G s is created from G, as described earlier. Arcs in A k that may be serviced by vehicle class k are placed in RA k,sa k, or SE k. Arcs in A k that cannot be serviced by class k are placed in DA k. The vehicle preference list The vehicle preference list specifies, in decreasing order of preference, a vehicle class k and the corresponding maximum number of vehicles available at that preference level. A specific class may occur twice on the list. Say for example a company owns 10 class 1 vehicles but may purchase another 4 at a capital cost to the company, typically the 10 currently owned vehicles will appear higher on the preference list than the 4 vehicles available to purchase. Other assumptions The VDA assumes every arc in RA MA and every edge in SE can be serviced by the smallest vehicle class in the preference list. A further assumption is that if a vehicle from a vehicle class of specified capacity can service/deadhead a specific street segment, then all vehicles from smaller vehicle classes having smaller capacity can also service/deadhead that street segment. The travel network G and service network G s are assumed to be connected networks. It is also assumed that all vehicles start and end their day from a single depot and make use of a single disposal facility, regardless of the class k of the vehicle. A target route time, T RT k is specified for each vehicle class k. The target route time is the target length of time for a partition being serviced by a vehicle from a specific class. 13

22 Goals and constraints of the Capacitated Arc Routing Problem with Vehicle-Site Dependencies (CARP-VSD) Each street in G s requiring service is assigned to a partition as long as G s is connected. The total nonproductive time for each travel path is minimised. The partitions interlace as little as possible. The Vehicle Decomposition Algorithm (VDA) The VDA decomposes the CARP-VSD into several smaller single vehicle class CARPs. Partitioning procedures, followed by traditional travel-path-generation techniques contained in RouteSmart are used to find an approximate minimum deadhead time travel path for the arcs and edged in each partition. Algorithm 1 describes the five steps of the VDA. Inputs to the VDA include: the travel network G; the service network G s ; the vehicle preference list; and various parameters such as the number of partitions allowed and the target route times for each. Algorithm 1 The Vehicle Decomposition Algorithm Step 0: Create and verify vehicle class networks. Step 1: Estimate the total work and determine an initial fleet mix. Step 2: Partition the service network. Step 3: Determine a travel path and balance the partitions. Step 4: Revise the estimate obtained in Step 1 and adjust fleet mix. Initially Steps 0 and 1 are carried out. terminates. The VDA then loops between Steps 2 4 until it 14

23 2.3.2 Phases of the solution procedure for the fleet size and mix extension of the Multi Objective CARP In this section we invesigate extending the homogeneous fleet case to the heterogeneous fleet case. This enables us to make the fleet size and composition decision variables. In order to determine the best possible vehicle fleet size and composition we need to consider the fixed cost of the vehicles simultaniously with the operational costs of the vehicles. The objective therefore is to minimise the sum of fixed and variable costs. The solution approach as developed by Ulusoy [18] for the fleet size and mix extension of the CARP can be classified as a route-first-cluster-second approach. It will be stated for the undirected case first. The process consists of four phases, namely: 1. Find a giant tour. 2. Partition the giant tour into subtours each corresponding to a feasible vehicle tour. 3. Select the least cost set of vehicle tours satisfying the demands from the set of vehicle tours generated. 4. Repeat steps one to three until some stopping criteria is satisfied. Obtaining the giant tour The giant tour on G = (V, E) is found without taking into account the fixed cost component, the capacity constraints, and any other type of constraints that might be present. The giant tour is a solution to the Chinese Postman Problem (CPP) on G when q ij > 0 for all (i, j) V. Otherwise, it is obtained by solving the Rural Postman Problem (RPP) on G. The CPP is mostly used in urban mail delivery systems. It is concerned with finding the minimum cost of servicing all streets within a network. The RPP is similar but makes use of street segments that do not require service. It seeks the minimum cost traversal by making use of the nonrequired street segments to reach all the demand points [6, 5]. All arcs on the giant tour which are not part of the required arc set V r are called no-service arcs. If an arc a(i, j) V r appears in the giant tour more than the required times, all its copies are called no-service arcs, too. The network on which the giant tour is constructed is denoted by G E. Partitioning the giant tour Once the giant tour is obtained, it is partitioned into subtours each corresponding to a feasible vehicle tour. The partitioning takes place on a transformed network, G t. The rules for constructing G t are stated as follows: First we reduce the network G E. Rule 1: If there is any chain of no-service arcs on the Euler tour, then it is reduced to an arc with initial and final nodes being those of the chain and its cost being equal to the total cost of the chain. An Euler tour of a connected, directed graph G = (V, E) is a cycle that traverses each edge of graph G exactly once, although it may visit a vertex more than once. The graph, G t, is defined by its: (i) Node set: Rule 2: Each node on G corresponds to an arc on the Euler tour except the first node. The first node on G t corresponds to the depot node (node one) of G. The arcs on the Euler tour are numbered in the same order as they appear in the tour and 15

24 are mapped to the nodes of G t in the same order with the first arc corresponding to node two on G t. If the last node of G t represents a no-service arc, then it is eliminated from the node set. (ii) Arc set: Each arc on G t corresponds to a feasible vehicle tour on G, i.e a given arc a(c, f) on G t corresponds to a vehicle tour on G which includes all the arcs represented by the nodes (c + 1),..., f. The arcs present in the arc set of G t are determinded by the following rule. Rule 3: An arc is incident from each node c into each node f with c < f, given that the corresponding vehicle tour does not violate any of the side constraints that might be present. (iii) Incidence relationships: The presence of no-service arcs on G is exploited to reduce the number of arcs on G t as determined by Rule 3. Let node i correspond to a no-service arc on G. Rule 4: No arc is incident into node i except arc (i 1, i). Rule 5: No arc is incident from node (i 1) except arc (i 1, i). These two rules are formulated so as to prevent no-service arcs from being present in the vehicle tours without any purpose, and therefore leading to inferior vehicle tours. (iv) Arc cost structure: When partitioning the Euler tour we have obtained arc sequences from the Euler tour which are to be transformed into vehicle tours. For this purpose we distinguish between two cases as illustrated in Figure 2.1. Do the arcs on the sequence form a cycle? Yes Is node 1 a member of the sequence? Yes No Then the sequence corresponds to a vehicle tour. Then connect node 1 to the node of the cycle with the least cost chain to node 1 Yes Then connect the two odd degree nodes with the least cost chain. No Is node 1 a member of the sequence? No Then connect node 1 to the two odd degree nodes using least cost chains. Figure 2.1: The arc sequences Selecting the least cost set of vehicle tours on G t Having succesfully constructed G t we are now ready to select the least cost set of vehicle tours represented on G t. This is done by solving for the shortest path from node 1 to node m, where node m corresponds to the last arc with positive service demand on the giant tour. The 16

25 shortest path can be found by applying Dijkstra s algorithm [4]. The vehicle tours correspond to the arcs of the shortest path from the least cost set of vehicle tours on G t. Stopping criterion Finding the shortest route on G t will represent the least cost set of vehicle tours for that particular G t which is obtained for a given Euler tour on G E. Obviously a different Euler tour constructed from the same G E will result in a different G t. A reasonable approach is therefore to create as many different Euler tours as G E permits. The algorithm Ulusoy [18] introduces two counters to control the number of distinct Euler tours constructed from G E, I is associated with the number of times different cycles are obtained, and J, indicates the number of distinct Euler tours constructed from the same set of cycles. Note that J N c, where N c denotes the number of cycles obtained. Ulusoy [18] proposes using Algorithm 2. To extent the problem to the directed case, simply solve a transportation Algorithm 2 The vehicle size and mix algorithm Step 0: Set i = 0 Step 1: Apply the appropriate algorithm to obtain G E. Step 2a: Construct cycles on G E covering all arcs of it. Where i i + 1 and j = 0. Step 2b: Construct a distinct Euler tour using the cycles of Step 2a. Where j j + 1. Step 3: Obtain the transformed graph G t using the Euler tour of Step 2b. Step 4: Obtain the shortest path on G t from node 1 to node m. Step 5: Determine the subtours corresponding to the arcs of the shortest path and the associated vehicle capacities. Eliminate subtours that consists of no-service arcs only. Compare the new solution with the current best solution and if it is better, declare it the current best solution. Step 6: If i = I, go to Step 7. If j = J, go to Step 2a. Else go to Step 2b. Step 7: Stop. Declare the current best solution as the approximate optimum. problem instead of the 1-matching problem when obtaining the giant tour. The rest of the algorithm applies as it is Tabu search Winston and Venkataramanan [21] describe the TS as making use of short- and long-term memory to emulate heuristic rules used by people when making day-to-day decisions. The long term memory allows searches to be conducted in the most promising neigbourhouds to 17

26 ensure the best solution is chosen, while the short term memory prevents cycling around a local neighbourhood in the solution space by temporarily forbidding moves that would return to a solution recently visited. The TS records the best solution found thus far, and when the search is terminated, it acceps the solution as an approximate optima for the problem. Brandao and Eglese [3] succesfully made use of the TS algorithm to produce efficient solutions to the CARP.Rardin [12] lists the seven steps of the TS in Algorithm 3 Algorithm 3 Tabu Search algorithm Step 0: Initialisation Choose a starting feasible solution x (0) and an iteration limit t max. Set the current best solution (Also known as the incumbent solution) x x (0) and solution index t 0. No moves are now tabu. Step 1: Stopping If no non-tabu move x in move set β leads to a feasible neighbour of current solution x (t), or if t = t max, then stop. Incumbent solution x is an approximate optimum. Step 2: Move Choose some non-tabu feasible move x β as x t+1 Step 3: Step Update x (t+1) x (t) + x (t+1) Step 4: Incumbent Solution If the objective function value of x (t+1) is superior to that of incumbent solution x, replace x x (t+1). Step 5: Tabu List Remove from the list of forbidden moves any that have been on it for a sufficient number of iterations, and add a collection of moves that includes any returning immediately from x (t+1) to x (t). Step 6: Increment Increment t t + 1, and return to Step Simulated Annealing search The annealing process in metallurgy inspired the design of the SA algorithm. When a metal is heated to high temperatures it becomes structurally weak and unstable untill it cools down to form itself into a structurally strong configuration. The process of slowly lowering the temperature is what allows the energy exchange to reach true equilibrium in each stage until the global minimum energy level is reached [21]. The temperature at each state corresponds to an improvement in the objective function value, with the minimum energy level being the approximate optimum. SA has the valuable feature of accepting not only improving solutions, but also accepting inferior solutions with some probability. This enables the process to escape being stuck on a local minima, and helps the SA algorithm to search for the best possible solution in the solution space [13]. Rardin [12] lists the seven steps of SA in Algorithm Genetic Algorithms GAs originate from an attemp to parallel the process of biological evolution to find better and better solutions [12]. Examples of such natural evolutionary behaviour include natural 18

27 Algorithm 4 Simulated Annealing algorithm Step 0:Initialisation Choose a starting feasible solution x (0), an iteration limit t max and a relatively large initial temperature q > 0. Set the current best solution x opt x (0) and solution index t 0. Step 1: Stopping If no move x in move set β leads to a feasible neighbour of current solution x (t), or if t = t max, then stop. Current best solution x opt is an approximate optimum. Step 2: Provisional Move Choose some feasible move x β as a provisional x t+1, and compute the net objective function improvement obj for moving from x (t) to (x (t) + x (t+1) ) (Increase for a maximise and decrease for a minimise) Step 3: Acceptance If x (t+1) improves, or with probability e obj/q if obj 0, accept x (t+1) and update x (t+1) x (t) + x (t+1). Otherwise, return to Step 2. Step 4: Incumbent Solution If the objective function value of x (t+1) is superior to that of incumbent solution x opt, replace x opt x (t+1). Step 5: Temperature Reduction If a sufficient number of iterations have passed since the last temperature change, reduce temperature q. Step 6: Increment Incrument t t + 1, and return to Step 1. selection for mating and mutation for diversity. In 1975 Holland introduced the algorithm by stating that given certain conditions on the problem domain, GAs would tend to converge to an approximate optimum [21]. Good heuristic optima is derived by a GA because of the combination of individual solutions of an improving solution population. According to Rardin [12] this ensures that the current best solution will always be in the solution space. Winston and Venkataramanan [21] lists the six steps of the GA in Algorithm 5. Table 2.1 summarises the main differences between SA-, GA- and TS-algorithms. Table 2.1: Comparison of heuristic differences Heuristic Memory Nr. of solutions Search type SA No memory One Random GA No memory Population Random TS Memory-based One/population Systematic 2.4 Conclusion The RWCP was identified as being a CARP due to characteristics such as capacity constraints and multiple vehicles. After investigating heuristics previously used to sorve the CARP and 19

28 Algorithm 5 High level Genetic Algorithm Step 1: Generation Choose a population size p, crossover rate p c and generation limit t max. Set t 0. Generate a feasible polulation of initial starting solutions T (1),..., T (p). Step 2: Evaluation If t = t max, stop and report the best solution as the approximate optimum. Otherwise score each solution by a fitness fuction (Typically this will be the objective function). Step 3: Selection Probabilistically, choose from the current solutions the p c parents for the next generation (t + 1). A string with a better fitness value has a higher probability of selection than a string with a worse fitness value. Step4: Reproduction Crossover the parents at a random point in the gene string to ensure the offspring consists of a portion of each parent. Reproduce for p offspring. Step5: Mutation Randomly alter the genes of the offspring to ensure that local optima are avoided. Step6: Repear Increase t t + 1 and repeat from Step 2. some popular metaheuristics, it was established that we will make use of the VDA to solve the CARP-VSD to best model a real world representation of the RWCP. 20