Optimum Locations on a Graph with Probabilistic Demands

Frank, H.

doi:10.1287/opre.14.3.409

Cited by 86 publications

(46 citation statements)

References 5 publications

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“…(1) and v 2 is a better location than v 3 . In fact, in this particular example, it turns out v 2 is the optimal location.…”

Section: Definitionmentioning

confidence: 98%

“…It is also worth noting that FRANK [3,4] has examined another type of stochastic network, one for which the node weights, h i are random variables. Under a different criterion of optimality (involving maximization of the probability that the average travel distance is below a specified value) he has shown that the "locate-facilities-at-nodes" result does not hold in this case.…”

Section: Oriented Network and Extensionsmentioning

confidence: 99%

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Locations of Medians on Stochastic Networks

Mirchandani

Odoni

1979

Transportation Science

141

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The definition of network medians is extended to the case where travel times on network links are random variables with known discrete probability distributions. Under a particular set of assumptions, it is shown that the well-known theorems of HAKIMI and of LEVY can be extended to such stochastic networks. The concepts are further extended to the case of stochastic oriented networks. A particular set of applications as well as formulations of the problem for solution using mathematical programming techniques are also discussed briefly.

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“…(1) and v 2 is a better location than v 3 . In fact, in this particular example, it turns out v 2 is the optimal location.…”

Section: Definitionmentioning

confidence: 98%

Section: Oriented Network and Extensionsmentioning

confidence: 99%

Locations of Medians on Stochastic Networks

Mirchandani

Odoni

1979

Transportation Science

141

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“…He considered the minimization of the probability of falling below a prespecified threshold. Frank [51,52] considered a model of minimizing the probability that the cost function in the Weber or minimax problems (Love et al [70]) on a network exceeds a given threshold. The threshold concept has been employed in financial circles as a form of insurance on a portfolio, either to protect the portfolio or to protect a firm's minimum profit.…”

Section: The Threshold Objectivementioning

confidence: 99%

Modeling Competitive Facility Location Problems: New Approaches and Results

Berman¹,

Drezner²,

Drezner³

et al. 2009

Decision Technologies and Applications

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The gravity model as applied to competitive facility location is described in this tutorial. The gravity model is used mainly by marketers to estimate the market share attracted by competing retail facilities. A demand area consists of potential customers who spend their discretionary buying power at competing retail facilities. We discuss the gravity model and issues related to the determination of the parameters of the models. The basic model is to find the best locations for new facilities that would attract the maximum buying power from customers in the area. Extensions to this basic model include modeling uncertainty about future market conditions, minimizing the probability that the attracted market share falls short of a certain target, demand generated by intercepting a flow of customers, market expansion and lost demand, and allocation of a given budget among one's facilities to improve their attractiveness. We describe special tools that are utilized in the solution procedures. These are the generalized Weiszfeld procedure, the big triangle small triangle global optimization approach, and the tangent line approximation.

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“…In these models, the weight of O. Berman et al 243 each nodal point and the length of each link are assumed to be constant and known a priori. Frank (1966) argued that the weight of a nodal point, e.g., the number of messages originating from a node of a communication network, may not be deterministic, but is better represented as a random variable with some probability distribution. This leads to several possible stochastic generalizations of the concepts of medians and centers: in the expected value sense ("absolute expected" median and center), in the sense of maximizing the probability of achieving some threshold ("maximum probability" median and center), or in the sense of variance minimization ("minimum variance" absolute median).…”

mentioning

confidence: 99%

“…This leads to several possible stochastic generalizations of the concepts of medians and centers: in the expected value sense ("absolute expected" median and center), in the sense of maximizing the probability of achieving some threshold ("maximum probability" median and center), or in the sense of variance minimization ("minimum variance" absolute median). These concepts, which were introduced and analyzed in Frank (1966), are discussed in Sect. 11.2 below.…”

mentioning

confidence: 99%

Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. PrefaceThis book is the final result of a number of incidents that occurred to us while discussing location issues with colleagues at conferences. Frequently we attended presentations, in which the authors quoted the well-known references that helped to make the discipline what it is today. Upon further inquiry, though, it turned out that some of these colleagues had never actually read the original papers. We then discussed among ourselves which contributions could be credited with shaping the field. And, lo and behold, we found that we, too, had neglected to read some of the papers that form the foundation of our science. Whether it was laziness or other things that got in the way, it had become clear that something had to be done. Our first thought was to collect the original contributions (once we could agree on what they were) and reprint them. When discussing this possibility with a publisher, we immediately ran into a roadblock in the form of copyright. While this appeared to have stopped our enthusiastic effort dead in its tracks, we kept on collecting and reading what we considered original contributions.This went on until we met Camille Price, who suggested that, rather than reprinting the original contributions, we should invite some of the leaders in our field and ask them to describe the original contribution, explain and interpret it, and comment on the impact that it had to the field. This was, of course, an excellent idea, and the response by our colleagues to our pertinent requests was equally enthusiastic. What you hold in your hands is the result of this effort.In other words, the purpose of this book is to provide easy access to the main contributions to location theory. The book is organized as follows. The introductory chapter provides an overview of some of the many facets of location analysis. This is followed by contributions in the main three fields of inquiry: minisum, minimax, and covering problems. The next chapters are part of an ever-growing list of nonstandard location models: models including competitive components, those that locate undesirable facilities, those with probabilistic features, and those that allow interactions between facilities. The following chapters discuss solution techniques: after a discussion of exact and heuristic techniques, we devote an entire chapter to Weiszfeld's method, and another to Lagrangean techniques. The last chapters of this book deal with the spheres of influence that the facilities generate and that attract viii customers to them, an...

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Optimum Locations on a Graph with Probabilistic Demands

Cited by 86 publications

References 5 publications

Locations of Medians on Stochastic Networks

Locations of Medians on Stochastic Networks

Modeling Competitive Facility Location Problems: New Approaches and Results

Hub Location Problems: The Location of Interacting Facilities

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