Production, Manufacturing and LogisticsFacility location problems in the plane based on reverse nearest neighbor queries
Introduction
Given a database, a nearest neighbor (NN) query returns the data objects that are nearer to a given query object than any other object in the database. On the other hand, in the conceptually inverse query problem, a Reverse nearest neighbor (RNN) query retrieves those objects that have a query object as their nearest neighbor. Reverse nearest neighbors queries have emerged as an important class of queries for spatial and other types of databases. The concept was first introduced by Korn et al. [19], [20]; the reader is referred to these papers for a gathering of a large number of applications in marketing and decision support systems. Also, see [30] for a survey on the current state-of-art and open geometric problems in another application area.
The RNN query itself presents several variants, ranging from monochromatic or bichromatic versions to static or dynamic versions. In the monochromatic case, all points have the same color. In the bichromatic case, the point set consists of red and blue points, and the problem turns into computing those points belonging to one of the two colors for which a query point is a bichromatic nearest neighbor. In the static version of the problem, distances between points in the set remain unchanged, whereas in the dynamic problem they may change. Some previous related work on these problems includes [6], [22], [23], [27], [29]. High-dimensional instances of RNN and BRNN (bichromatic RNN) have hardly been considered in the past, in sharp contrast with the NN problem; and it is striking to see how little research on (B)RNN has been carried out compared to the research on NN. This shows that even the planar instances of (B)RNN are still worth studying at the present time.
This paper considers the RNN query as a rule or mapping to associate points from the database to every point in a continuous space and introduces new optimization problems by using this rule. We study new geometric optimization problems in the planar static bichromatic variant, where data points belong to two categories. In particular, we will define RNN facility location problems in a two dimensional space. Some points are designated as facilities, and others as customers. In this setting, a reverse nearest neighbor query asks for the set of customers affected by the opening of a new facility at some point (query); here we will assume that all customers choose the nearest facility (Fig. 1). We point out here that we pick the name “reverse” from the data mining community and this concept is different from the “inverse” or “reverse” as used sometimes in the operational research field, where the goal is to modify the underlying space to improve the efficiency [33].
We will study optimization problems that arise when considering various optimization criteria: maximizing the number of potential customers for the new facility (MAXCOV criterion); minimizing the maximum distance to the associated clients (MINMAX criterion); and maximizing the minimum distance to the associated clients (MAXMIN criterion). The MAXCOV and MINMAX criteria deal with the location of an attractive facility (bars, discos, hospitals, schools, supermarkets, fixed wireless base stations, etc), while the MAXMIN criterion seeks the best location for a new obnoxious facility (rubbish dumps, chemical plants, etc.). Notice that these problems can be interpreted as the location of a new facility in a competitive environment. Competitive facility location addresses the problem of the placing of sites by competing market players. Typically, the expected income the new facility will generate will depend on the market share it will capture. Competitive location models have been studied in several disciplines such as geography, economics, marketing and operations research. Comprehensive surveys of competitive facility location models can be found in [14], [15], [24], [31]. A continuous analogue to the MAXCOV problem was considered in [8], [10], where the problem of placing a new facility in a location that maximizes the area of the corresponding Voronoi region is considered. Observe that the MAXCOV criterion can also be seen as a greedy step in a discrete version of the Voronoi game [2].
Finally, as already pointed out above, applications of the problems under consideration are also related to various fields that lie beyond the scope of facility location problems, for example, advanced database applications.
An outline of the paper is as follows: In Section 2 we state the optimization problems. In Section 3 we propose exact and approximate algorithms for the MAXCOV problem and we prove its 3SUM hardness. An -time algorithm for the MINMAX and the MAXMIN problems is described in Section 4. In Section 5 we also consider several variants of the problems which include the combination of criteria, the use of the and -metrics and the reverse farthest neighbor version. Finally, concluding remarks of the paper are put forward in Section 6.
Section snippets
Problem statement
In the sequel, unless otherwise stated, we will use the metric and will denote the Euclidean distance between points p and q. Let be a set of points in the plane. Given a point b in the plane, the reverse nearest neighbor set of b is defined as
For the bichromatic case, assume we have a nonempty set of n red points (clients) and a nonempty set of m blue points (facilities) such that and . Given a new
The MAXCOV problem
In this section we provide exact and approximate algorithms for the MAXCOV problem, as well as result on the hardness of the exact problem.
The MINMAX and MAXMIN problems
We are given a bichromatic set formed by a set of m blue points B (facilities) and a set of n red points R (clients), , and a constraint region .
Extensions
In this section we consider some extensions of the problems above. First, we combine the MINMAX or MAXMIN criteria with the MAXCOV criteria. Second, we solve the same problems as above under the and -metrics. Finally, we consider a different rule to associate clients to facilities, namely, the furthest neighbor rule.
Concluding remarks
Given a query blue point, the bichromatic reverse nearest neighbor problem is to find all red points for which the query point is a nearest blue neighbor under some given distance metric. Such queries repeatedly arise when designing efficient algorithms in a variety of areas. In this paper, we introduced and efficiently solved some optimization problems with a direct interpretation in the area of Competitive Facility Location. In particular, we studied three problems (MAXCOV, MINMAX, and
Acknowledgements
We are grateful to anonymous referees and Paco Gómez for many useful comments. These problems were posed and partially solved during the Second Spanish Workshop on Geometric Optimization, July 5–10, 2004, El Rocı´o, Huelva, Spain. The authors would like to thank the Ayuntamiento de Almonte for their support and the other workshop participants for helpful comments.
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2016, Expert Systems with ApplicationsCitation Excerpt :The MaxBRNN problem has many interesting real life applications in service location planning and emergency schedule such as the example in Fig. 1. MaxBRNN problem has been studied in Cabello et al. (2010), Liu, Wong, Wang, Li, and Chen (2012), Wong, Özsu, Yu, Fu, and Liu (2009), and Zhou, Wu, Li, Lee, and Hsu (2011), in which it is supposed that the service facility's capacity is unlimited. However in real cases, facilities are inevitably constrained by designed capacities, when the needs of service increase, facilities in those booming areas may be overloaded, and some facilities may run out of capacity.
- 1
Partially supported by the European Community Sixth Framework Programme under a Marie Curie Intra-European Fellowship, and by the Slovenian Research Agency, project J1-7218.
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Partially supported by project MEC MTM2006-03909.
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Partially supported by projects MCYT-FEDER-BFM2003-00368, Gen-Cat-2005SGR00692, and MCYT HU2002-0010.