Grey Median Problem and Vertex Optimality

  • Jafar Fathali Shahrood Univercity of Technology
Keywords: facility location; grey number; median problem

Abstract

The median problem is a basic model in location theory and transportation sciences. This problem deals with locating a facility on a network, to minimize the sum of weighted distances between the facility and the vertices of the network. In this paper, the cases that weights of vertices, edge lengths or both of them are grey numbers, are considered. For all these cases, we show that the set of vertices of network contains a solution of the median problem. This property is called vertex optimality. Median problem with grey parameters and its properties are first considered in this paper.

References

M. J. Canos, C. Ivorra and V. Liern, An exact algorithm for the fuzzy p-median problem, European Journal of Operational Research, vol. 116, pp. 80-86, 1999.

M. S. Daskin, Network and Discrete Location: Models, Algorithms and Applications, Wiley, NewYork, 1995.

D. Darvishi and P. Babaei, Grey prediction in linear programming problems, International Journal of Applied Operational Research, vol. 9, pp. 11-18, 2019.

D. Darvishi and S.H. Nasseri, A dual simplex method for grey linear programming problems based on duality results, Grey Systems: Theory and Application, vol. 10, pp. 145-157, 2020.

D. Darvishi, S. Liu and J. Yi-Lin Forrest, Grey linear programming: a survey on solving approaches and applications, Grey Systems: Theory and Application, vol. 11, pp. 110-135, 2021.

S. L. Hakimi, Optimum locations of switching centers and the absolute centers and medians of a graph, Operations Research, vol. 12, pp. 450-459, 1964.

H.Y. Handler and P.B. Mirchandani, Location on networks: theory and algorithms, MIT Press, Cambridge, MA, 1979.

G.H. Huang, B.W. Baetz and G.G. Patry, A grey linear programming approach for municipal solid waste management planning under uncertainty, Civil Engineering Systems, vol. 9, pp. 319-335, 1992.

G.H. Huang, B.W. Baetz and G.G. Patry, Grey fuzzy integer programming: an application to regional waste management planning under uncertainty, Socio-Economic Planning Sciences, vol. 29, pp. 17-38, 1995.

G.H. Huang and R.D. Moore, Grey linear programming, its solving approach and its application, International Journal of Systems Science, vol. 24, pp. 159-172, 1993.

J. Fathali and A. Jamalian, Locating multiple facilities in convex sets with fuzzy data and block norms, Applied Mathematics, vol. 3, 1950-1958, 2012.

O. Kariv and S.L. Hakimi, An algorithmic approach to network location problems. I: The p-Centers, SIAM Journal on Applied Mathematics, vol. 37, pp. 513-538, 1979.

Q.X. Li, The cover solution of grey linear programming, Journal of Grey System, vol. 19, pp. 309-320, 2007.

S.F. Liu and Y. Lin, Grey Information, Theory and Practical Applications, Springer-Verlag, London, 2006.

D.K. Mayoya and J.L. Verdegay, p-median problems in fuzzy environment, Mathware and Soft Computing, vol. 12, pp. 97-106, 2005.

P.B. Mirchandani and P.L. Francis, Discrete location theory, Wiley, New York, 1990.

S.H. Nasseri and D. Darvishi, Duality results on grey linear programming problems, Journal of Grey System, vol. 30, pp. 127-142, 2018.

S.H. Nasseri, A. Yazdani and D. Darvishi, A primal simplex algorithm for solving linear programming problem with grey cost coefficients, Journal of New Researches in Mathematics, vol. 1, pp. 121-141, 2016.

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.

J.A.M. Perez, J.M.M. Vega and J.L. Verdegay, Fuzzy location problems on networks, Fuzzy Set and Systems, vol. 142, pp. 393-405, 2004.

N.A. Taghi-Nezhad, The p-median problem in fuzzy environment: proving fuzzy vertex optimality theorem and its application, Soft Computing, vol. 23, pp. 11399–11407, 2019.

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, pp. 225-248, 2018.

F. Taleshian and J. Fathali, A mathematical model for fuzzy p-median problem with fuzzy weights and variables, Advances in Operations Research, 2016, pp. 1-13.

F. Taleshian, J. Fathali and N.A. Taghi-Nezhad, Finding the absolute and vertex center of a fuzzy tree, Transportation Letters, vol. 14, pp. 591-599, 2022.

K. Yang, Y.K. Liu and G.Q. Yang, Solving fuzzy p-hub center problem by genetic algorithm incorporating local search, Applied Soft Computing, vol. 13, pp. 2624-2632, 2013.

Published
2023-01-18
How to Cite
Fathali, J. (2023). Grey Median Problem and Vertex Optimality. Statistics, Optimization & Information Computing, 11(3), 670-676. https://doi.org/10.19139/soic-2310-5070-1527
Section
Research Articles