Geometrical Solutions for Some Minimax Location Problems

Elzinga, Jack; Hearn, Donald W.

doi:10.1287/trsc.6.4.379

Cited by 309 publications

(156 citation statements)

References 6 publications

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“…Elizinga and Hearn [33,34] improved this to 0(n 2 ). Much work was done from an Operations Research perspective by viewing the problem as a minimax facility location problem, where the Euc!idean center is the point whose greatest distance to any point of the set is minirnized [41, 84, 48J.…”

Section: Constrained Euclidean Centermentioning

confidence: 94%

Geometric and computational aspects of manufacturing processes

Bose

1994

Computers & Graphics

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Section: Constrained Euclidean Centermentioning

confidence: 94%

Geometric and computational aspects of manufacturing processes

Bose

1994

Computers & Graphics

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“…Elzinga and Hearn [12] proposed an efficient geometrical-based algorithm for solving optimally the problem. Other authors attempted some enhancements to speed up the search, such as Xu et al [31] and Elshaikh et al [11] and references therein.…”

Section: Introductionmentioning

confidence: 99%

An adaptive perturbation-based heuristic: An application to the continuous p-centre problem

Elshaikh

Salhi

Brimberg

et al. 2016

Computers & Operations Research

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A self-adaptive heuristic that incorporates a variable level of perturbation, a novel local search and a learning mechanism is proposed to solve the p-centre problem in the continuous space. Empirical results, using several large TSP-Lib data sets, some with over 1300 customers with various values of p, show that our proposed heuristic is both effective and efficient. This perturbation metaheuristic compares favourably against the optimal method on small size instances. For larger instances the algorithm outperforms both a multi-start heuristic and a discrete-based optimal approach while performing well against a recent powerful VNS approach. This is a self-adaptive method that can easily be adopted to tackle other combinatorial/global optimisation problems. For benchmarking purposes, the medium size instances with 575 nodes are solved optimally for the first time, though requiring a large amount of computational time. As a by-product of this research, we also report for the first time the optimal solution of the vertex p-centre problem for these TSP-Lib data sets.

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“…A brief historical review is given by Elzinga and H earn [6]1 who deal with the case involving the Eucl idean dista nce in R" and the identity cost fun ction. Elzinga and Hearn [5] and Na ir and Chandrasekaran [23] develop additional solution procedures for the same problem in R2. A multi-facility version of the Euclidean distance probl em in R2 is considered as a convex programming probl em by Love, Wesolowsky and Kraemer [20).…”

Section: Introductionmentioning

confidence: 99%

“…Geometrical properties motiva te solutions b y Elzinga and Hearn [5] and by Fra nc is [8] to a one-fac ility rectilinear distance problem inR 2 using the identity cost fun c ti on. Wesolowsky [24] , and Dea rin g and Francis [2] solve a multi-fac ility version of the rectilinear distance probl em in R2 .…”

Section: Introductionmentioning

confidence: 99%

“…Elzinga and Hearn [5] and Na ir and Chandrasekaran [23] develop additional solution procedures for the same problem in R2. A multi-facility version of the Euclidean distance probl em in R2 is considered as a convex programming probl em by Love, Wesolowsky and Kraemer [20).Geometrical properties motiva te solutions b y Elzinga and Hearn [5] and by Fra nc is [8] to a one-fac ility rectilinear distance problem inR 2 using the identity cost fun c ti on. Wesolowsky [24] , and Dea rin g and Francis [2] solve a multi-fac ility version of the rectilinear distance probl em in R2 .…”

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confidence: 99%

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Minimax location problems with nonlinear costs

Dearing¹

1977

J. RES. NATL. BUR. STAN.

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Pre vious studies of one-fac ility minimax location problems are extended by pennitting the cost of travel to be gi ve n by any (stric tly) increa sing, continuous function of travel d ista nce. Previous solution procedures for the rec tilinear dista nce problem in the plane a nd for the proble m on a tree network are extended to these general cost fu nctions.Ke y words: Facility location; location the ory; minimax ; ne tworks. . IntroductionMinimax location problems have rece ived considerable attention in the literature as models for locating fac ilities tha t are to provide eme rge ncy or convenient servi ce to a set of ex isting facilities. In most of these problems the re is given a set of ex isting facilities whose locati ons are represe nted as points in some space, and new fac ility locations are also to be spec ified as points in that space . A dista nce fun c ti on is chosen to represe nt the travel di stance between the new a nd ex isting fac ility locations . The minimax obj ective is to locate the new fac ilities so that the max imum di stance, or a fun cti on of di stance, between th e ne w and existing fac ility locations is minimized.Differe nt problems may be spec ified by the choice of the space a nd of the dista nce fun cti on used . Most of the problems in the literature may be pl aced into one of two classes: those using a no rm-de rived di sta nce in the space R" for some n , and those on a network using network distances. Also, diffe rent proble ms may be spec ifie d by the choice of the cost-representing fun cti ons of tra vel distance. These fun c ti ons a re refe rred to genericall y as "cost" funct io ns, but they may measure cost, time or some othe r fo rm of inconve ni e nce.This pa per considers one-fac ility minimax locati on p ro blems that permit quite ge neral cost fun ctions, name ly any continuous (strictl y) increasing fun c tion of the travel di stance. In th e prev ious cons iderations of minimax location proble ms, cost fun ctions we re assumed to be linea r, or in many cases the ide ntity fun cti on.The initial formul ation and analys is a re given for a pro blem in R " using norm-derived distances; however, the results obtain ed also hold for probl e ms on a ne tw ork and are disc ussed subsequentl y. In addition, for minimax location problems using rectilinear distance, this paper extends prev ious solution procedures to these more ge neral cost fun ctions in the space R2. For the problem on a tree network , previou s solution procedures are also e xtende d to the ge neral cost fun ctions.Most of the literature on minimax location problems has appeared in the las t te n years, although one ve rsion of the problem was first formul ated in 1857 as a " minimum covering s phe re problem." A brief historical review is given by Elzinga and H earn [6]1 who deal with the case involving the Eucl idean dista nce in R" and the identity cost fun ction. Elzinga and Hearn [5] and Na ir and Chandrasekaran [23] develop additional solution procedures for the same problem in R2. A multi-...

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Geometrical Solutions for Some Minimax Location Problems

Cited by 309 publications

References 6 publications

Geometric and computational aspects of manufacturing processes

Geometric and computational aspects of manufacturing processes

An adaptive perturbation-based heuristic: An application to the continuous p-centre problem

Minimax location problems with nonlinear costs

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