Chapter 1. Introduction

Transcription

1

2 Chapter 1 Introduction Intermodal freight transportation describes the movement of goods in standardized loading units (e.g., containers) by at least two transportation modes (rail, maritime, and road) in a single transport chain (Bontekoning et al., 2004; Macharis and Bontekoning, 2004). The change of transportation modes is performed at specially designed terminals (also known as transshipment yards) by transferring the loading units without handling the freight itself. The route of intermodal transport is namely subdivided into the pre-, main-, and end-haulage, which denote the route segments from customer to terminal, terminal to terminal, and terminal to customer, respectively (refer to Figure 1.1). The main-haulage generally implies the longest traveling distance and is typically carried out by rail or maritime, whereas the preand end-haulage are handled by trucks (i.e., vehicles) to enable door-to-door transports. The pre- and end-haulage is also referred to as drayage. Fig. 1.1: Intermodal Freight Transportation

3 2 Chapter 1. Introduction An essence of intermodal transportation compared to the conventional unimodal transportation is the ability to combine the advantages of the distinct transportation systems. For example, road-rail intermodal transportation consolidates the flexibility of road transport for short distances and the high transportation capacity of rail for long distances. Intermodal freight transportation has received an increased attention, e.g., by support programs introduced by the European Commission s Directorate General for Mobility and Transport, to divert freight transportation from road to rail and maritime in order to reduce road congestion and environmental pollution (European Commission, 2007, 2011). However, despite of these efforts, the fraction of the overall freight transportation by rail is steadily declining, leveling off at around 10% in 2009, its lowest level since 1945 (Boysen et al., 2010; European Commission, 2007, 2011). Reasons include the lack of flexibility and reliability (e.g., train delays, variability of train sizes), as well as deficiencies in the technical operability between and within the different transportation modes (European Commission, 2007; Martínez et al., 2004). To increase the attractiveness of intermodal freight transportation, cost-effectiveness and an efficient handling of the integrated operational procedures are mandatory. In this thesis, we consider several operational planning problems that arise in intermodal freight transportation. Simply the efficient handling of the preand end-haulage will result, according to Cheung et al. (2008) and Morlok and Spasovic (1994), in a significant cost reduction. Despite of the short traveling distance, the pre- and end-haulage accounts for a large percentage (between 25% to 40%) of the total intermodal transportation costs and seriously affects the profitability of an intermodal service (Macharis and Bontekoning, 2004; Morlok and Spasovic, 1994). The operational planning problems that we address in this research are various routing problems that are encountered in the pre- and end-haulage, as well as a partitioning problem that arises at rail-rail transshipment yards.

4 1.1 Notation and Terminology Notation and Terminology We assume the reader is familiar with the general concepts of Operations Research (e.g., Domschke and Drexl (2007)), combinatorial optimization (e.g., Korte and Vygen (2006); Schrijver (2003)), graph theory (e.g., Gross and Yellen (2004)), complexity theory (e.g., Garey and Johnson (1979)), as well as with the fundamentals of linear (e.g., Bertsimas and Tsitsiklis (1997)), nonlinear (e.g., Bertsekas (1999)), and integer programming (e.g., Schrijver (1998)). In the following, we will shortly introduce the graph notation used throughout the thesis and give a brief overview of routing and partitioning problems Graph Notation The used graph notation follows standard combinatorial optimization books such as Korte and Vygen (2006) and Schrijver (2003) and is given as follows. Let G =(V,E) be an undirected graph, where V is a non-empty, finite set and E is a finite multiset of unordered pairs of elements of V. The elements of V are called vertices or nodes and the elements of E undirected edges. Each undirected edge e E is a two-element subset of V and is denoted by e =[u, v] with u, v V. A directed graph (or simply digraph) D =(V,A) consists of a set of vertices V and includes in contrast to an undirected graph a finite multiset A of ordered pairs of elements of V. The elements of A are called directed edges or arcs and are denoted by e =(u, v) A with u, v V. A mixed graph combines the elements of an undirected and a directed graph and is defined by a triple D =(V,E,A) that consists of a set of vertices V, a set of undirected edges E, and a set of directed edges A. The terminology of undirected and directed graphs directly applies to mixed graphs. Notice that throughout the thesis, the term graph either refers to an undirected, directed, or mixed graph; by an edge we mean an undirected or directed edge. We say that an edge e =[u, v] or e =(u, v) connects vertices u and v and vertices u and v are adjacent. For directed graphs, we furthermore say that

5 4 Chapter 1. Introduction an arc (u, v) goes from vertex u to vertex v or leaves vertex u and enters vertex v. The vertices u and v are called the endpoints of edge e. If vertex u is an endpoint of edge e, then u is said to be incident with e (or equivalently, e is incident with u). The degree of a vertex u gives the number of edges incident with u. In case of digraphs, the out-degree and the in-degree denote the number of arcs that leave and enter vertex u, respectively. Edges e and f are called incident, if they share at least a common vertex. Two or more edges that are incident to the same pair of vertices are called multiple edges. A loop is an edge that joins a single vertex to itself. Graphs without loops and without multiple edges are called simple graphs. A complete graph is a simple undirected graph where any two distinct vertices are adjacent. If U V and both endpoints of an edge e belong to U, wesaythatu spans edge e. Given an undirected or directed graph G =(V,E), awalk is a sequence of vertices and edges W =(v 0,e 1,v 1,...,e k,v k ) such that k 1 and e i = [v i 1,v i ] E or e i =(v i 1,v i ) E for i =1,...,k. To ease the formulation, we will sometimes represent a walk only by its vertices W =(v 0,...,v k ) or by its edges W =(e 1,...,e k ). The vertices v 0 and v k are called the start vertex and end vertex of W, respectively. Furthermore, the walk W is said to connect vertices v 0 and v k and is defined between vertices v 0 and v k. If vertices v 0,...,v k are all distinct, the walk is called a path. If v 0 = v k and v 1,...,v k 1 are all distinct, we refer to a walk as a cycle. A graph that contains no cycle is referred to as an acyclic graph and a graph with at least one cycle is referred to as a cyclic graph. In case of digraphs, the terms walk, path, and cycle, are sometimes extended to directed walk, directed path, and directed cycle, respectively. Let G = (V,E) denote an undirected or directed graph. A digraph G =(V,E ) with V V, E E and where [u, v] E or (u, v) E implies u, v V is called a subgraph of G. IfE includes all edges of G that are spanned by V, G is called an induced subgraph. For each directed graph D =(V,A), an underlying undirected graph G = (V,E) is defined on the same vertex set V and contains an edge [u, v] E for each arc (u, v) A. An undirected graph is connected, if there is a path that

6 1.1 Notation and Terminology 5 connects any two distinct vertices of V. A directed graph D =(V,A) is said to be (weakly) connected, if the underlying undirected graph is connected. Moreover, D is strongly connected, if a directed path is defined on D between any two distinct vertices of V. Finally, a graph G =(V,E) is called vertex-weighted, if a vertex weight w u R is assigned to each vertex u V. Analogously, a graph G =(V,E) is referred to as egde-weighted, ifanedge weight c uv R (or c(u, v) R) is assigned to each edge [u, v] E or (u, v) E. In terms of digraphs, we also say arc-weighted graph Routing Problems Routing problems are concerned with the efficient transportation of, e.g., commodities, goods, information, and/or people. The main objective is to find optimal routes, that minimize, e.g., the traveling distance, traveling time, and/or traveling cost and which respect several side constraints. The most studied routing problem is the traveling salesman problem (TSP) which may be stated as follows (refer, e.g., to Applegate et al. (2006)): Given is a depot and a set of customers along with the traveling distance between each pair of them. The traveling salesman problem is to find a vehicle route that starts and ends at the depot, visits all the customers exactly once, and minimizes the total distance traveled. A considerable amount of literature has been published on the TSP. These studies investigate different TSP variants, as well as different solution approaches. Handbooks on the TSP are, among others, given by Cook (2012), Gutin and Punnen (2002), Lawler et al. (1985), and Reinelt (1994). Reallife applications of the TSP arise, e.g., in logistics, telecommunications, and chip design (Applegate et al., 2006). The TSP is along with most routing problems NP-hard in the strong sense (refer, e.g., to Garey and Johnson (1979)). We furthermore differentiate between symmetric and asymmetric TSPs. In the symmetric TSP, the traveling distance between two customers is identical in both directions, i.e., the distance from customer A to customer B is

7 6 Chapter 1. Introduction equal to the distance from customer B to customer A. Different traveling distances may be incurred in the asymmetric TSP. The multiple traveling salesman problem (m-tsp) is a generalization of the TSP, where multiple vehicles may visit the customers and where each customer must be visited by exactly one vehicle. All vehicles start and end at the depot. In analogy to the TSP, we search for routes that minimize the total traveling distance. A list of real-life applications, as well as a thorough literature overview are reported by Bektas (2006). The vehicle routing problem (VRP) is in turn a generalization of the m-tsp that places side constraints on the vehicles and on the customers; capacities and demands are assigned to the vehicles and to the customers, respectively. In the VRP, we search for vehicle routes with minimum total traveling distance that fulfill the demand of the customers and where each route respects the vehicle capacity. The main known results on the vehicle routing problem are, e.g., summarized by Golden et al. (2008) and Toth and Vigo (2002). By imposing time windows on the customers, we obtain the vehicle routing problem with time windows (VRPTW). In the same manner, we obtain the TSPTW and the m-tsptw. Here, a vehicle has to visit a customer within a defined time window. The time windows are further differentiated in soft and hard time windows. Soft time windows can be violated by paying appropriate penalties, while hard time windows do not allow to visit a customer after the defined time window. In the latter case, if a vehicle arrives before the beginning of a time window, the vehicle has to wait. A survey on routing problems with time windows is, e.g., given by Toth and Vigo (2002). Vehicle routing problems that include a time component are also referred to as vehicle scheduling problems. Finally, the last considered variant of the routing problem is the pickupand-delivery problem (PDP). In PDPs, commodities, goods, and/or people have to be transported between origins and destinations. Savelsbergh and Sol (1995), Berbeglia et al. (2007), Parragh et al. (2008a), and Parragh et al. (2008b) provide detailed surveys of the recent literature, as well as classification schemes. We follow Berbeglia et al. (2007) by differentiating between

8 1.1 Notation and Terminology 7 many-to-many, one-to-many-to-one, and one-to-one PDPs. The most frequently encountered PDPs are the ones with a one-to-one structure where each, e.g., commodity has a defined pickup and delivery location. Problems of this type arise, e.g., in courier and door-to-door transportation (refer, e.g., to Cordeau and Laporte (2003)). The PDP with unit vehicle capacity, unit supply/demand of the customers, and a one-to-one correspondence is a special case since it can be transformed into an asymmetric TSP for a single vehicle and into an asymmetric m-tsp for multiple vehicles (Savelsbergh and Sol, 1995). In problems with a one-to-many-to-one relationship, commodities are initially located at the depot and are delivered to so-called delivery customers, whereas commodities that are picked up at pickup customers are destined to the depot. Real-world applications arise, e.g., in the delivery of beverages and the collection of empty bottles (refer, e.g., to Gendreau et al. (1999)). In PDPs with a many-to-many dependency, the supply (demand) of any pickup (delivery) customer can be accommodated by any other delivery (pickup) customer. Problems of this type arise in the preand end-haulage of intermodal freight transportation and are considered in this thesis. PDPs that generalize the TSP or the VRP are further denoted as pickup-and-delivery traveling salesman problem (PDTSP) or pickup-anddelivery vehicle routing problem (PDVRP), respectively. Variants of the introduced routing problems consider, e.g., different objectives, multiple depots, heterogeneous vehicles, unit vehicle capacity and/or unit customer supply and demand. In this thesis, we exclusively address routing problems with unit capacity and unit supply/demand of the vehicles and the customers, respectively, and refer to them as full-truckload routing problems (Parragh et al., 2008b). Finally, each presented routing problem has a graph theoretical interpretation. For example, an equivalent graph formulation of the symmetric traveling salesman problem is given as follows (refer, e.g., to Korte and Vygen (2006)): Given a complete graph G =(V,E) with edge weight c(u, v) R + for every edge [u, v] E, find a cycle W with minimum weight, [u,v] W c(u, v), that contains each vertex exactly once.

9 8 Chapter 1. Introduction Partitioning Problems Graph partitioning problems (or simply partitioning problems) are, in general, concerned with the partitioning of the vertex set of an undirected or a directed graph into disjoint subsets (also known as clusters) such that the sum of the arc weights within the clusters is maximized (or equivalently, the sum of the edge weights between different clusters is minimized). Graph partitioning problems arise in a wide area of applications, including VLSI (Very Large Scale Integration) design (Alpert and Kahng, 1995), qualitative data analysis (Grötschel and Wakabayashi, 1989, 1990), computer register allocation (Kernighan, 1971), and finite element computation (Ferreira et al., 1998). The problem instances that have motivated this research are encountered at rail-rail transshipment yards. In the thesis, we consider a variant of the graph partitioning problem, namely the acyclic partitioning problem. The acyclic partitioning problem is defined on an arc- and vertex-weighted digraph that searches for a partition of the vertex set such that the sum of the vertex weights within the clusters satisfies an upper bound and such that the sum of the arc weights within the clusters is maximized. Additionally, the digraph is enforced to partition into a directed, acyclic graph, i.e., a graph that contains no directed cycle. 1.2 Outline This thesis is divided into 5 chapters. Chapter 2 (Truck Scheduling Problem) is concerned with a routing problem that arises in the pre- and end-haulage of intermodal transportation and where containers need to be transported by trucks between customers and terminals and vice versa. A problem formulation based on a full-truckload pickup-and-delivery problem with time windows is presented, as well as a 2-stage heuristic solution approach. In Chapter 3 (1-Full-Truckload Pickup-and-Delivery Traveling Salesman Problem) we address a simplification of this routing problem, namely the one-commodity full-truckload pickup-and-delivery traveling salesman problem. We formulate this problem as an integer linear and an integer nonlinear programming

10 1.2 Outline 9 problem and suggest exact solution approaches based on the classical and the generalized Benders decomposition. In Chapter 4 (Acyclic Partitioning Problem) we focus on a partitioning problem that arises, among others, at rail-rail transshipment yards. We propose different problem formulations for this acyclic partitioning problem and introduce solution approaches based on a branch-and-bound and a branch-and-price framework. Finally, Chapter 5 (Summary) summarizes the thesis and gives an outlook of further research. The core chapters (Chapters 2-4) share the same outline: We first give a detailed problem description (Problem Description), state the relevant literature (Literature Review), and introduce the used notation (Notation). Thereafter we present several problem formulations (Mathematical Formulations) and propose solution approaches (Solution Approach(es)). We assess the quality of our solution approaches in a computational study (Computational Study) and end each chapter with a summary (Summary). All solution methods that are presented in the following were implemented in Java 2 under Windows XP and were run on an Intel Pentium Core 2 Duo, 2.2 GHz PC with 4 GB system memory. We used ILOG CPLEX 12.2 Concert Technology with default settings as standard solver.