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Article | Open Access

2-Median Problem on a Tree with Grey Parameters

Jafar Fathali1( ), Davood Darvishi Salookolaei2
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood 3619995161, Iran
Department of Mathematics, Payame Noor University, Tehran 193954697, Iran
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Abstract

The purpose of this paper is to find solutions to the 1-median and 2-median problems on a tree network with grey parameters. Generally, the p-median problem asks for finding a set of p facilities on the vertices of a given tree such that the sum of weighted distances from vertices to the closest facility is minimized. First, the 1-median problem with grey weight of vertices and edge lengths is considered, and by showing some properties for this problem, a linear time algorithm is proposed. Then an edge deletion method with time complexity O(n2) is developed to find a solution to the 2-median problem with grey parameters. Moreover, some new concepts of networks with grey parameters are defined in this paper.

Keywords

facility locationgrey number2-median problem1-median problem

References

[1]

S. L. Hakimi, Optimum locations of switching centers and the absolute centers and medians of a graph, Oper. Res., vol. 12, no. 3, pp. 450–459, 1964.
Crossref Google Scholar
[2]

O. Kariv and S. L. Hakimi, An algorithmic approach to network location problems. I: The p-centers, SIAM J. Appl. Math., vol. 37, no. 3, pp. 513–538, 1979.
Crossref Google Scholar
[3]

A. J. Goldman, Optimal center location in simple networks, Transp. Sci., vol. 5, no. 2, pp. 212–221, 1971.
Crossref Google Scholar
[4]

P. B. Mirchandani and A. Oudjit, Localizing 2-medians on probabilistic and deterministic tree networks, Networks, vol. 10, no. 4, pp. 329–350, 1980.
Crossref Google Scholar
[5]

B. Gavish and S. Sridhar, Computing the 2-median on tree networks in O(n lg n) time, Networks, vol. 26, no. 4, pp. 305–317, 1995.
Crossref Google Scholar
[6]

A. Oudjit and M. F. Stallmann, Efficient algorithms for finding 2-medians of a tree, Networks, vol. 77, no. 3, pp. 383–402, 2021.
Crossref Google Scholar
[7]
G. Y. Handler and P. B. Mirchandani, Location on Networks : Theory and Algorithms. Cambridge, MA, USA: MIT Press, 1979.
[8]
P. B. Mirchandani and R. L. Francis, Discrete Location Theory. New York, NY, USA: Wiley, 1990.
[9]
M. S. Daskin, Network and Discrete Location : Models, Algorithms, and Applications. New York, NY, USA: Wiley, 1995.
Crossref
[10]

M. J. Canós, C. Ivorra, and V. Liern, An exact algorithm for the fuzzy p-median problem, Eur. J. Oper. Res., vol. 116, no. 1, pp. 80–86, 1999.
Crossref Google Scholar
[11]
D. Kutangila-Mayoya and J. L. Verdegay, p-median problems in a fuzzy environment, Mathw. Soft Comput., vol. 12, pp. 97–106, 2005.
[12]

F. Taleshian, J. Fathali, and N. A. Taghi-Nezhad, Fuzzy majority algorithms for the 1-median and 2-median problems on a fuzzy tree, Fuzzy Information and Engineering, vol. 10, no. 2, pp. 225–248, 2018.
Crossref Google Scholar
[13]

S. M. A. Nayeem and M. Pal, The p-center problem on fuzzy networks and reduction cost, Iranian Journal of Fuzzy Systems, vol. 5, pp. 1–26, 2008.
Google Scholar
[14]

J. Fathali and A. Jamalian, Locating multiple facilities in convex sets with fuzzy data and block norms, Appl. Math., vol. 3, no. 12, pp. 1950–1958, 2012.
Crossref Google Scholar
[15]

K. Yang, Y. K. Liu, and G. Q. Yang, Solving fuzzy p-hub center problem by genetic algorithm incorporating local search, Appl. Soft Comput., vol. 13, no. 5, pp. 2624–2632, 2013.
Crossref Google Scholar
[16]

J. A. M. Pérez, J. M. M. Vega, and J. L. Verdegay, Fuzzy location problems on networks, Fuzzy Sets Syst., vol. 142, no. 3, pp. 393–405, 2004.
Crossref Google Scholar
[17]

F. Taleshian and J. Fathali, A mathematical model for fuzzy p-median problem with fuzzy weights and variables, Adv. Oper. Res., vol. 2016, pp. 1–13, 2016.
Crossref Google Scholar
[18]

F. Taleshian, J. Fathali, and N. A. Taghi-Nezhad, Finding the absolute and vertex center of a fuzzy tree, Transp. Lett., vol. 14, no. 6, pp. 591–599, 2022.
Crossref Google Scholar
[19]

J. L. Deng, Control problems of grey systems, Syst. Contr. Lett., vol. 1, no. 5, pp. 288–294, 1982.
Crossref Google Scholar
[20]

G. Huang, B. W. Baetz, and G. G. Patry, A grey linear programming approach for municipal solid waste management planning under uncertainty, Civ. Eng. Syst., vol. 9, no. 4, pp. 319–335, 1992.
Crossref Google Scholar
[21]

G. H. Huang and R. D. Moore, Grey linear programming, its solving approach, and its application, Int. J. Syst. Sci., vol. 24, no. 1, pp. 159–172, 1993.
Crossref Google Scholar
[22]

G. H. Huang, B. W. Baetz, and G. G. Patry, Grey fuzzy integer programming: An application to regional waste management planning under uncertainty, Socio Econ. Plan. Sci., vol. 29, no. 1, pp. 17–38, 1995.
Crossref Google Scholar
[23]

Q. X. Li, The cover solution of grey linear programming, J. Grey Syst., vol. 19, pp. 309–320, 2007.
Google Scholar
[24]

S. H. Nasseria, A. Yazdani, and D. Darvishi, A primal simplex algorithm for solving linear programming problem with grey cost coefficients, J. New Res. Math., vol. 1, no. 4, pp. 121–141, 2016.
Google Scholar
[25]

H. Nasseri and D. Darvishi, Duality results on grey linear programming problems, J. Grey Syst., vol. 30, no. 3, pp. 127–142, 2018.
Google Scholar
[26]

D. Darvishi and P. Babaei, Grey prediction in linear programming problems, Int. J. Appl. Oper. Res., vol. 9, pp. 11–18, 2019.
Google Scholar
[27]

D. D. Salookolaei and S. H. Nasseri, A dual simplex method for grey linear programming problems based on duality results, Grey Syst. Theory Appl., vol. 10, no. 2, pp. 145–157, 2020.
Crossref Google Scholar
[28]

D. Darvishi, S. Liu, and J. Y. L. Forrest, Grey linear programming: A survey on solving approaches and applications, Grey Syst. Theory Appl., vol. 11, no. 1, pp. 110–135, 2021.
Crossref Google Scholar
[29]

J. Fathali, Grey median problem and vertex optimality, Stat., Optim. Inf. Comput., vol. 11, no. 3, pp. 670–676, 2023.
Crossref Google Scholar
[30]

D. Darvishi, J. Forrest, and S. Liu, A comparative analysis of grey ranking approaches, Grey Syst. Theory Appl., vol. 9, no. 4, pp. 472–487, 2019.
Crossref Google Scholar
[31]
S. F. Liu and J. Y. L. Forrest, Grey Information : Theory and Practical Applications. London, UK: Springer-Verlag, 2006.
Fuzzy Information and Engineering
Volume 15 Issue 4,
December 2023
Pages 362-377
DOI: 10.26599/FIE.2023.9270027
Cite this article:
Fathali J, Salookolaei DD. 2-Median Problem on a Tree with Grey Parameters. Fuzzy Information and Engineering, 2023, 15(4): 362-377. https://doi.org/10.26599/FIE.2023.9270027

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Received: 26 May 2023
Revised: 10 September 2023
Accepted: 05 November 2023
Published: 02 January 2024
© The Author(s) 2023. Published by Tsinghua University Press.

This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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