Locating Multiple Facilities in Convex Sets with Fuzzy Data and Block Norms

ABSTRACT

In this paper we study the problem of locating multiple facilities in convex sets with fuzzy parameters. This problem asks to find the location of new facilities in the given convex sets such that the sum of weighted distances between new facilities and existing facilities is minimized. We present a linear programming model for this problem with block norms, then we use it for problems with fuzzy data. We also do this for rectilinear and infinity norms as special cases of block norms.

In this paper we study the problem of locating multiple facilities in convex sets with fuzzy parameters. This problem asks to find the location of new facilities in the given convex sets such that the sum of weighted distances between new facilities and existing facilities is minimized. We present a linear programming model for this problem with block norms, then we use it for problems with fuzzy data. We also do this for rectilinear and infinity norms as special cases of block norms.

Cite this paper

J. Fathali and A. Jamalian, "Locating Multiple Facilities in Convex Sets with Fuzzy Data and Block Norms,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 1950-1958. doi: 10.4236/am.2012.312267.

J. Fathali and A. Jamalian, "Locating Multiple Facilities in Convex Sets with Fuzzy Data and Block Norms,"

References

[1] R. E. Bellman and L. A. Zadeh, “Decision Making in a Fuzzy Environment,” Management Science, Vol. 17, No. 4, 1970, pp. 141-164. doi:10.1287/mnsc.17.4.B141

[2] Z. Drezner and H. Hamacher, “Facility Location: Applications and Theory,” Springer-Verlag, Berlin, 2002.

[3] A. Sarkar, R. Batta and R. Nagi, “Placing a Finite Size Facility with a Center Objective on a Rectangular Plane with Barriers,” European Journal of Operational Research, Vol. 179, No. 3, 2007, pp. 1160-1176. doi:10.1016/j.ejor.2005.08.029

[4] R. C. Larson and G. Sadiq, “Facility Locations with the Manhattan Metric in the Presence of Barriers to Travel,” Operations Research, Vol. 31, No. 4, 1983, pp. 652-669. doi:10.1287/opre.31.4.652

[5] J. Mu?oz-Pérez and J. J. Saame?o-Rodr??guez, “Location of an Undesirable Facility in a Polygonal Region with Forbidden Zones,” European Journal of Operational Research, Vol. 114, No. 2, 1999, pp. 372-379. doi:10.1016/S0377-2217(98)00138-6

[6] U. Bhattacharya, J. R. Rao and R. N. Tiwari, “Fuzzy Multi-Criteria Facility Location Problem,” Fuzzy Sets and Systems, Vol. 51, No. 3, 1992, pp. 277-287. doi:10.1016/0165-0114(92)90018-Y

[7] U. Bhattacharya, J. R. Rao and R. N. Tiwari, “Bi-Criteria Multi Facility Location Problem in Fuzzy Environment,” Fuzzy Sets and Systems, Vol. 56, No. 2, 1993, pp. 145-153. doi:10.1016/0165-0114(93)90139-9

[8] S. M. A. Nayeem and M. Pal, “The p-Center Problem on Fuzzy Networks and Reduction of Cost,” Iranian Journal of Fuzzy Systems, Vol. 5, No. 1, 2008, pp. 1-26.

[9] R. Francis Jr., L. F. McGinnis and J. A. White, “Facility Layout and Location: An Analytical Approach,” Prentice Hall, Upper Saddle River, 1992.

[10] G. Y. Handler and P. B. Mirchandani, “Location on Networks: Theory and Algorithms,” MIT Press, Cambridge, 1979.

[11] R. F. Love, J. G. Morris and G. O. Wesolowsky, “Facilities Location: Models and Methods,” North Holland Publishing Company, New York, 1988.

[12] P. B. Mirchandani and R. Francis, “Discrete Location Theory,” John Wiley & Sons, Hoboken, 1990.

[13] J. E. Ward and R. E. Wendell, “A New Norm for Measuring Distance Which Yields Linear Location Problems,” Operations Research, Vol. 28, No. 3, 1980, pp. 836-844. doi:10.1287/opre.28.3.836

[14] J. E. Ward and R. E. Wendell, “Using Block Norms for Location Modeling,” Operations Research, Vol. 33, No. 5, 1985, pp. 1074-1090. doi:10.1287/opre.33.5.1074

[15] H. Tanaka, T. Okuda and K. Asai, “On Fuzzy Mathematical Programming,” Journal of Cybernetics, Vol. 3, No. 4, 1973, pp. 37-46. doi:10.1080/01969727308545912

[16] H. J. Zimmermann, “Fuzzy Mathematical Programming,” Computers & Operations Research Journal, Vol. 10, No. 4, 1993, pp. 291-298. doi:10.1016/0305-0548(83)90004-7

[17] C. V. Negoita, “Fuzziness in Management,” OPSA/TIMS, Miami, 1970.

[18] H. Tanaka and K. Asai, “Fuzzy Linear Programming Problems with Fuzzy Numbers,” Fuzzy Sets and Systems, Vol. 13, No. 1, 1984, pp. 1-10. doi:10.1016/0165-0114(84)90022-8

[19] H. R. Maleki, M. Tata and M. Mashinchi, “Linear Programming with Fuzzy Variables,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 21-33. doi:10.1016/S0165-0114(98)00066-9

[20] S. C. Fang and C. F. Hu, “Linear Programming with Fuzzy Coefficients in Constraints,” Computers & Mathematics with Applications, Vol. 37, No. 10, 1999, pp. 63-76. doi:10.1016/S0898-1221(99)00126-1

[21] R. Fuller and H. J. Zimmermann, “Fuzzy Reasoning for Solving Fuzzy Mathematical Programming Problems,” Fuzzy Sets and Systems, Vol. 60, No. 2, 1993, pp. 121-133. doi:10.1016/0165-0114(93)90341-E

[22] R. N. Gasimov and K. Yenilmez, “Solving Fuzzy Linear Programming Problems with Linear Membership Function,” Turkish Journal of Mathematics, Vol. 26, No. 4, 2002, pp. 375-396.

[23] C. Garsia-Aguado and J. L. Verdegay, “On the Sensitivity of Membership Functions for Fuzzy Linear Programming Problems,” Fuzzy Sets and Systems, Vol. 56, No. 1, 1993, pp. 47-49. doi:10.1016/0165-0114(93)90184-J

[24] H. R. Maleki, “Ranking Functions and Their Applications to Fuzzy Linear Programming,” Far East Journal of Mathematical Sciences, Vol. 4, No. 3, 2002, pp. 283-301.

[25] H. Mishmast Nehi, H. R. Maleki and M. Mashinchi, “Solving Fuzzy Number Linear Programming Problem by Lexicographic Ranking Function,” Italian Journal of Pure and Applied Mathematics, Vol. 15, 2004, pp. 9-20.

[26] F. Hosseinzadeh Lotfi, G. R. Jahanshahloo, F. Rezai Balf and H. Zhiani Rezai, “Finding the Minimize Summation for Location of Facility in a Convex Set with Fuzzy Data,” Applied Mathematical Sciences, Vol. 16, No. 1, 2007, pp. 749-759.

[1] R. E. Bellman and L. A. Zadeh, “Decision Making in a Fuzzy Environment,” Management Science, Vol. 17, No. 4, 1970, pp. 141-164. doi:10.1287/mnsc.17.4.B141

[2] Z. Drezner and H. Hamacher, “Facility Location: Applications and Theory,” Springer-Verlag, Berlin, 2002.

[3] A. Sarkar, R. Batta and R. Nagi, “Placing a Finite Size Facility with a Center Objective on a Rectangular Plane with Barriers,” European Journal of Operational Research, Vol. 179, No. 3, 2007, pp. 1160-1176. doi:10.1016/j.ejor.2005.08.029

[4] R. C. Larson and G. Sadiq, “Facility Locations with the Manhattan Metric in the Presence of Barriers to Travel,” Operations Research, Vol. 31, No. 4, 1983, pp. 652-669. doi:10.1287/opre.31.4.652

[5] J. Mu?oz-Pérez and J. J. Saame?o-Rodr??guez, “Location of an Undesirable Facility in a Polygonal Region with Forbidden Zones,” European Journal of Operational Research, Vol. 114, No. 2, 1999, pp. 372-379. doi:10.1016/S0377-2217(98)00138-6

[6] U. Bhattacharya, J. R. Rao and R. N. Tiwari, “Fuzzy Multi-Criteria Facility Location Problem,” Fuzzy Sets and Systems, Vol. 51, No. 3, 1992, pp. 277-287. doi:10.1016/0165-0114(92)90018-Y

[7] U. Bhattacharya, J. R. Rao and R. N. Tiwari, “Bi-Criteria Multi Facility Location Problem in Fuzzy Environment,” Fuzzy Sets and Systems, Vol. 56, No. 2, 1993, pp. 145-153. doi:10.1016/0165-0114(93)90139-9

[8] S. M. A. Nayeem and M. Pal, “The p-Center Problem on Fuzzy Networks and Reduction of Cost,” Iranian Journal of Fuzzy Systems, Vol. 5, No. 1, 2008, pp. 1-26.

[9] R. Francis Jr., L. F. McGinnis and J. A. White, “Facility Layout and Location: An Analytical Approach,” Prentice Hall, Upper Saddle River, 1992.

[10] G. Y. Handler and P. B. Mirchandani, “Location on Networks: Theory and Algorithms,” MIT Press, Cambridge, 1979.

[11] R. F. Love, J. G. Morris and G. O. Wesolowsky, “Facilities Location: Models and Methods,” North Holland Publishing Company, New York, 1988.

[12] P. B. Mirchandani and R. Francis, “Discrete Location Theory,” John Wiley & Sons, Hoboken, 1990.

[13] J. E. Ward and R. E. Wendell, “A New Norm for Measuring Distance Which Yields Linear Location Problems,” Operations Research, Vol. 28, No. 3, 1980, pp. 836-844. doi:10.1287/opre.28.3.836

[14] J. E. Ward and R. E. Wendell, “Using Block Norms for Location Modeling,” Operations Research, Vol. 33, No. 5, 1985, pp. 1074-1090. doi:10.1287/opre.33.5.1074

[15] H. Tanaka, T. Okuda and K. Asai, “On Fuzzy Mathematical Programming,” Journal of Cybernetics, Vol. 3, No. 4, 1973, pp. 37-46. doi:10.1080/01969727308545912

[16] H. J. Zimmermann, “Fuzzy Mathematical Programming,” Computers & Operations Research Journal, Vol. 10, No. 4, 1993, pp. 291-298. doi:10.1016/0305-0548(83)90004-7

[17] C. V. Negoita, “Fuzziness in Management,” OPSA/TIMS, Miami, 1970.

[18] H. Tanaka and K. Asai, “Fuzzy Linear Programming Problems with Fuzzy Numbers,” Fuzzy Sets and Systems, Vol. 13, No. 1, 1984, pp. 1-10. doi:10.1016/0165-0114(84)90022-8

[19] H. R. Maleki, M. Tata and M. Mashinchi, “Linear Programming with Fuzzy Variables,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 21-33. doi:10.1016/S0165-0114(98)00066-9

[20] S. C. Fang and C. F. Hu, “Linear Programming with Fuzzy Coefficients in Constraints,” Computers & Mathematics with Applications, Vol. 37, No. 10, 1999, pp. 63-76. doi:10.1016/S0898-1221(99)00126-1

[21] R. Fuller and H. J. Zimmermann, “Fuzzy Reasoning for Solving Fuzzy Mathematical Programming Problems,” Fuzzy Sets and Systems, Vol. 60, No. 2, 1993, pp. 121-133. doi:10.1016/0165-0114(93)90341-E

[22] R. N. Gasimov and K. Yenilmez, “Solving Fuzzy Linear Programming Problems with Linear Membership Function,” Turkish Journal of Mathematics, Vol. 26, No. 4, 2002, pp. 375-396.

[23] C. Garsia-Aguado and J. L. Verdegay, “On the Sensitivity of Membership Functions for Fuzzy Linear Programming Problems,” Fuzzy Sets and Systems, Vol. 56, No. 1, 1993, pp. 47-49. doi:10.1016/0165-0114(93)90184-J

[24] H. R. Maleki, “Ranking Functions and Their Applications to Fuzzy Linear Programming,” Far East Journal of Mathematical Sciences, Vol. 4, No. 3, 2002, pp. 283-301.

[25] H. Mishmast Nehi, H. R. Maleki and M. Mashinchi, “Solving Fuzzy Number Linear Programming Problem by Lexicographic Ranking Function,” Italian Journal of Pure and Applied Mathematics, Vol. 15, 2004, pp. 9-20.

[26] F. Hosseinzadeh Lotfi, G. R. Jahanshahloo, F. Rezai Balf and H. Zhiani Rezai, “Finding the Minimize Summation for Location of Facility in a Convex Set with Fuzzy Data,” Applied Mathematical Sciences, Vol. 16, No. 1, 2007, pp. 749-759.