Author:  Tang, Huajun 
Title:  A study of network location problems 
Degree:  Ph.D. 
Year:  2009 
Subject:  Hong Kong Polytechnic University  Dissertations. Industrial location. Network analysis (Planning) 
Department:  Dept. of Logistics and Maritime Studies 
Pages:  viii, 82 p. : ill. ; 30 cm. 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2275585 
URI:  http://theses.lib.polyu.edu.hk/handle/200/5057 
Abstract:  One of the most important branches of logistics management is to investigate where to locate new facilities such as transportation hubs, air and sea ports, retail outlets, and so on. There is a wide variety of applications of facility location models. These include, but not limited to, locating a given number of ambulances to minimize the maximum response time (the time between a demand point and the nearest ambulance), and locating warehouses within a supply chain to minimize the average transportation time to market. The above two problems have different objective functions: minimax (center) for the former and minisum (median) for the latter. Facility location models differ not only in objective functions (center, median or others), but also in decision space (planar, discrete or networks), the number (singefacility or multifacility), capacity limit (uncapacitated or capacitated) and shape (isolated point or connected structure) of the facilities to locate, the nature of the inputs (static or dynamic; demands known with certainty or uncertainty), and other problem parameters. This thesis focuses on studying the network location problems as follows: Demand points with certainty are taken to be at the nodes of the network, and to be served by their own nearest facilities, which are to be located anywhere in the given network. The facilities may be isolated points, or connected structures such as paths, trees, and so on. The objective is to locate a given number of uncapacitated facilities to minimize the ordered median function (OMf) and its special cases. The cost of satisfying the demand points depends on the distances between demands and facilities, which are measured by the shortest paths through the network. The organization of this thesis is as follows: Chapter 1 introduces a taxonomy of location problems and presents the definition of the ordered median problems (OMP) proposed by Nickel and Puerto (2005), and describes two main methodologies applied in this thesis. Chapter 2 presents a literature review of the network location models, including two main clues on the history and development of the ordered median problems. Chapter 3 deals with the multifacility ordered median problem in undirected networks, in which multiple isolated facilities are to be located. Multifacility OMP in general networks are NPhard, since the pmedian and the pcenter problems are special cases. In this chapter we use a finite dominating set (FDS) to study some special instances of the OMf in networks. FDS is a finite set of points to which some optimal solutions must belong, and is very useful for solving a variety of optimization problems, which enables one to restrict one's attention to a finite set of possible solutions. We first characterize an FDS for a special convex OMP in general networks, where the convex OMP is an important class in the OMP family. The FDS result generalizes some known results in the literature. Then, based on the FDS result, we obtain a polynomial size FDS and solve the problem confined to tree networks in polynomial time, which extends some results in the literature. Chapter 4 is devoted to the multifacility OMP in directed networks, since most of the networks in the real world are directed and not symmetric (undirected networks can be viewed as symmetric directed networks). For instance, routes are usually directed in a bus traffic system. In this chapter we again apply FDS to identify some possible solutions for a multifacility OMP in a strongly connected directed network. We first prove that the OMP has an FDS in the node set, which generalizes the FDS result on the singlefacility OMP in the literature. Furthermore, we show that the OMP can be solved efficiently based on the FDS result when the number of facilities is fixed and small. However, if the number of facilities is large, it is not practical for us to obtain an optimal solution in an efficient manner, since the OMP in directed networks is NPhard. Hence, instead of finding an optimal solution, we resort to some approximation algorithms for some nearoptimal solutions. At the end of Chapter 4, we present a 6 2/3approximation algorithm for the pmedian problem in directed networks. Chapter 5 focuses on the OMP in tree networks, in which the facilities to locate are not isolated points but connected structures (e.g., paths, trees, etc). These problems are motivated by specific decision problems related to routing and network design. In this chapter we use the "nestedness property" to investigate subtree OMP in tree networks, where the nestedness property is the property that for any optimal solution x to the point OMP, there exists an optimal subtree to the corresponding subtree OMP including x. The nestedness property provides researchers with a powerful tool to develop some efficient algorithms. First, we prove the nestedness property for a special convex OMP in tree networks. This finding extends some classical results concerning the nestedness property. Second, we solve the problem in polynomial time by applying the nestedness property result. Finally, we provide one counter example to show that the nestedness property does not hold for the nonconvex case. Chapter 6, the last chapter, concludes the major findings of the thesis and suggests some directions for future research. 
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