Locating a Single Facility in the Plane in the Presence of a Bounded Region and Different Norms

Brimberg, Jack; Kakhki, Hossein Taghizadeh; Wesolowsky, George O.

doi:10.15807/jorsj.48.135

Cited by 14 publications

(8 citation statements)

References 9 publications

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“…This problem is not new and we can find antecedents in the literature in the papers by Parlar [17], Brimberg et. al [5,6], Fathaly [13], among others, and it can be seen as a natural generalization of the classical Weber's problem (see [12]). Note that the distances between two points, depending of the region where they are located, may measured with different norms.…”

Section: Introductionmentioning

confidence: 99%

Continuous location under the effect of ‘refraction’

2016

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In this paper we address the problem of locating a new facility on a d-dimensional space when the distance measure (ℓp-or polyhedral-norms) is different at each one of the sides of a given hyperplane H. We relate this problem with the physical phenomenon of refraction, and extends it to any finite dimension space and different distances at each one of the sides of any hyperplane. An application to this problem is the location of a facility within or outside an urban area where different distance measures must be used. We provide a new second order cone programming formulation, based on the ℓp-norm representation given in [3] that allows to solve, exactly, the problem in any finite dimension space with semidefinite programming tools. We also extend the problem to the case where the hyperplane is considered as a rapid transit media (a different third norm is also considered over H) that allows the demand to travel faster through H to reach the new facility. Extensive computational experiments run in Gurobi are reported in order to show the effectiveness of the approach.Date: April 11, 2018. 2010 Mathematics Subject Classification. 90B85, 90C22, 90C30, 47A30.

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Section: Introductionmentioning

confidence: 99%

Continuous location under the effect of ‘refraction’

2016

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“…Due to the huge density difference between the inner-city and highway road networks, these approaches approximate city networks as bounded connected subspaces of R 2 and model them as vertices within a large network connected by edges representing the highways. While the results from these papers have been adopted to frame location problems in several contexts [10], the underlying metric space used therein poses major limitations in terms of the accuracy and modeling abilities when dealing with practical applications. More specifically, the assumption that the underlying space is a compact, dense subset of R k implies that metric norms like L 1 or L 2 are used to represent the distances, which often results in notable imprecisions.…”

Section: Literature Reviewmentioning

confidence: 99%

On the distance between random events on a network

2019

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In this paper, we study several statistical properties regarding the distance between events that take place on random locations along the edges of a given network. We derive analytical expressions for the arbitrary moments of such a distance, its probability density function, its cumulative distribution function, as well as their conditional counterparts for the cases in which the position of one event is known in advance. As part of this study, we implement our developments as a callable library for the Python language, to provide potential users with a computational engine able to calculate and visualize these statistics for any given network. We test our implementation on several networks of different sizes and topologies, analyze some of interesting properties we observed in our experiments, and discuss several applications for our proposed methodology. In particular, we focus our discussion on applications aimed to help with the optimal design of emergency response systems on infrastructure networks. KEYWORDSdistance distributions, emergency response systems, infrastructure networks, random events, spatial random processes INTRODUCTION AND MOTIVATIONA wide variety of problems arising from different scientific domains involve the careful study of random events that occur on scattered locations of a given network [3,11,15,24,44]. Among the many different examples, perhaps the studies that have permeated the literature the most, are the ones that involve the analysis of events that take place on infrastructure networks [2]. Of particular importance, given the major role they play in our society, are the studies aimed to optimally design emergency response systems (ERS), like police, fire, and medical services [8,25,29,40]. In this particular context, a detailed analysis of the location of criminal acts, car accidents, and medical emergencies, among others, can be enormously beneficial to maximize the coverage and efficiency of such systems.Motivated by the fact that most ERS operations require one to rapidly dispatch specialized units in response to randomly occurring critical incidents, there has been a notable interest in designing methodological tools to help analyze several random properties regarding such events [4]. Moreover, with the recent emergence of new technologies that facilitate the collection, storage, and processing of large amounts data, the need for new developments able to estimate and characterize locations, distances, and the nature of such random events has become even more prevalent in recent years [8].One of the key components pertaining to the study of random events in networks is the statistical characterization of the distances between them. Given that the events take place on random locations across a given network, studying the random variables representing these distances can bring valuable information regarding the dynamics behind these complex systems [27].A few examples of key problems in the context of ERS, whose solution approaches can benefit from these methodologies are: (a...

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“…that ℓ is vertical and p is located to the left of ℓ. It is easy to observe from equations (8) and (9) that for any α ∈ [0, π 4 ] the expression of Φ(f α , h α ) has the form b 5 are constants. Furthermore, if we progressively increase the value of α from 0 to π 4 , that expression changes whenever sets S 1 , S 2 , and S 3 change, that is, when h α crosses a demand point, f α crosses a horizontal line of G, or α = ϕ v .…”

Section: Proposition 22 There Exists An Optimal Solution Of the Ffl-mentioning

confidence: 99%

Locating a single facility and a high-speed line

Díaz-Báñez

Korman

Pérez-Lantero

et al. 2014

European Journal of Operational Research

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In this paper we study a facility location problem in the plane in which a single point (facility) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time from the clients to the facility, using the L 1 or Manhattan metric. The rapid transit line is given by a segment with any length and orientation, and is an alternative transportation line that can be used by the clients to reduce their travel time to the facility. We study the variant of the problem in which clients can enter and exit the highway at any point. We provide an O(n 3 )-time algorithm that solves this variant, where n is the number of clients. We also present a detailed characterization of the solutions, which depends on the speed given in the highway.

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Locating a Single Facility in the Plane in the Presence of a Bounded Region and Different Norms

Cited by 14 publications

References 9 publications

Continuous location under the effect of ‘refraction’

Continuous location under the effect of ‘refraction’

On the distance between random events on a network

Locating a single facility and a high-speed line

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