### Abstract

For location problems in which optimal locations can be at nodes or along arcs but no finite dominating set has been identified, researchers may desire a method for dispersing p additional discrete candidate sites along the m arcs of a network. This article develops and tests minimax and maximin models for solving this continuous network location problem, which we call the added-node dispersion problem (ANDP). Adding nodes to an arc subdivides it into subarcs. The minimax model minimizes the maximum subarc length, while the maximin model maximizes the minimum subarc length. Like most worst-case objectives, the minimax and maximin objectives are plagued by poorly behaved alternate optima. Therefore, a secondary MinSumMax objective is used to select the best-dispersed alternate optima. We prove that equal spacing of added nodes along arcs is optimal to the MinSumMax objective. Using this fact we develop greedy heuristic algorithms that are simple, optimal, and efficient (O(mp)). Empirical results show how the maximum subarc, minimum subarc, and sum of longest subarcs change as the number of added nodes increases. Further empirical results show how using the ANDP to locate additional nodes can improve the solutions of another location problem. Using the p-dispersion problem as a case study, we show how much adding ANDP sites to the network vertices improves the p-dispersion objective function compared with (a) network vertices only and (b) vertices plus randomly added nodes. The ANDP can also be used by itself to disperse facilities such as stores, refueling stations, cell phone towers, or relay facilities along the arcs of a network, assuming that such facilities already exist at all nodes of the network.

Original language | English (US) |
---|---|

Pages (from-to) | 383-409 |

Number of pages | 27 |

Journal | Geographical Analysis |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2005 |

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### ASJC Scopus subject areas

- Geography, Planning and Development

### Cite this

*Geographical Analysis*,

*37*(4), 383-409. https://doi.org/10.1111/j.0016-7363.2005.03704002.x

**Dispersion of nodes added to a network.** / Kuby, Michael; Lim, Seow; Upchurch, Christopher.

Research output: Contribution to journal › Article

*Geographical Analysis*, vol. 37, no. 4, pp. 383-409. https://doi.org/10.1111/j.0016-7363.2005.03704002.x

}

TY - JOUR

T1 - Dispersion of nodes added to a network

AU - Kuby, Michael

AU - Lim, Seow

AU - Upchurch, Christopher

PY - 2005/10

Y1 - 2005/10

N2 - For location problems in which optimal locations can be at nodes or along arcs but no finite dominating set has been identified, researchers may desire a method for dispersing p additional discrete candidate sites along the m arcs of a network. This article develops and tests minimax and maximin models for solving this continuous network location problem, which we call the added-node dispersion problem (ANDP). Adding nodes to an arc subdivides it into subarcs. The minimax model minimizes the maximum subarc length, while the maximin model maximizes the minimum subarc length. Like most worst-case objectives, the minimax and maximin objectives are plagued by poorly behaved alternate optima. Therefore, a secondary MinSumMax objective is used to select the best-dispersed alternate optima. We prove that equal spacing of added nodes along arcs is optimal to the MinSumMax objective. Using this fact we develop greedy heuristic algorithms that are simple, optimal, and efficient (O(mp)). Empirical results show how the maximum subarc, minimum subarc, and sum of longest subarcs change as the number of added nodes increases. Further empirical results show how using the ANDP to locate additional nodes can improve the solutions of another location problem. Using the p-dispersion problem as a case study, we show how much adding ANDP sites to the network vertices improves the p-dispersion objective function compared with (a) network vertices only and (b) vertices plus randomly added nodes. The ANDP can also be used by itself to disperse facilities such as stores, refueling stations, cell phone towers, or relay facilities along the arcs of a network, assuming that such facilities already exist at all nodes of the network.

AB - For location problems in which optimal locations can be at nodes or along arcs but no finite dominating set has been identified, researchers may desire a method for dispersing p additional discrete candidate sites along the m arcs of a network. This article develops and tests minimax and maximin models for solving this continuous network location problem, which we call the added-node dispersion problem (ANDP). Adding nodes to an arc subdivides it into subarcs. The minimax model minimizes the maximum subarc length, while the maximin model maximizes the minimum subarc length. Like most worst-case objectives, the minimax and maximin objectives are plagued by poorly behaved alternate optima. Therefore, a secondary MinSumMax objective is used to select the best-dispersed alternate optima. We prove that equal spacing of added nodes along arcs is optimal to the MinSumMax objective. Using this fact we develop greedy heuristic algorithms that are simple, optimal, and efficient (O(mp)). Empirical results show how the maximum subarc, minimum subarc, and sum of longest subarcs change as the number of added nodes increases. Further empirical results show how using the ANDP to locate additional nodes can improve the solutions of another location problem. Using the p-dispersion problem as a case study, we show how much adding ANDP sites to the network vertices improves the p-dispersion objective function compared with (a) network vertices only and (b) vertices plus randomly added nodes. The ANDP can also be used by itself to disperse facilities such as stores, refueling stations, cell phone towers, or relay facilities along the arcs of a network, assuming that such facilities already exist at all nodes of the network.

UR - http://www.scopus.com/inward/record.url?scp=29144471179&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29144471179&partnerID=8YFLogxK

U2 - 10.1111/j.0016-7363.2005.03704002.x

DO - 10.1111/j.0016-7363.2005.03704002.x

M3 - Article

AN - SCOPUS:29144471179

VL - 37

SP - 383

EP - 409

JO - Geographical Analysis

JF - Geographical Analysis

SN - 0016-7363

IS - 4

ER -