Abstract
A number of network design problems can be built on the following premise: given an undirected tree network, T, with node set, V, identify a single subtree, t, containing nodes, v, so that the subtree is located optimally with respect to the remaining, subset of unconnected nodes {V−v}. Distances between unconnected nodes and nodes in the subtree t can be defined on paths that are restricted to lie in the larger tree T (the restricted case), or can be defined on paths in an auxiliary complete graph G (the unrestricted case). The unrestricted case represents a class of problems that is not explicitly recognized in the literature, which is of intermediate complexity relative to the widely studied restricted case, and the general problem in which the underlying graph is general. This paper presents the Median Subtree Location Problem (MSLP), formulated as a bicriterion problem that trades off the cost of a subtree, t, against the population-weighted travel distance from the unconnected nodes to nodes on the subtree where both objectives are to be minimized. Integer programs were formulated for the travel restricted and travel unrestricted cases and were tested using linear programming and branch and bound to resolve fractions. Tradeoff curves between cost and travel burden were developed for sample networks.
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References
Aneja, Y.P. and K.P.K. Nair. (1992). “Location of a Tree Shaped Facility in a Network.” INFOR 30(4), 319–324.
Bruno, G., M. Gendreau, and G. Laporte. (2002a). Computers and Operations Research 29(1), 1–12.
Bruno, G., M. Gendreau, and G. Laporte. (2002b). “A Heuristic for the Location of a Rapid Transit Line.” Computers and Operations Research 21(1), 1–12.
Church, R.L. and J.R. Current. (1989). “Maximal Covering Tree Problems.” Working Paper Series, WPS 89–47, College of Business, Ohio State University, July.
Church, R.L. and J.R. Current. (1993). “Maximal Covering Tree Problems.” Naval Research Logistics 40, 129–142.
Cohon, J.L. (1978). Multiobjective Programming and Planning. New York: Academic Press.
Current, J.R. and D.A. Schilling. (1989). “The Covering Salesman Problem.” Transportation Science 23(3), 208–213.
Current, J.R. and D.A. Schilling. (1994). “The Median Tour and Maximal Covering Tour Problems: Formulations and Heuristics.” European Journal of Operational Research 73, 114–126.
De Jong, K. (1985). “Genetic Algorithms: A Ten-Year Perspective.” In J.J. Grefenstette (ed.), Proceedings of an International Conference on Genetic Algorithms and their Applications, Carnegie-Mellon University, Pittsburg, PA, July 24–25, 1985. Hillsdale, NJ: Lawrence Erbaum Associates.
George, J.W. (1997). “Subtree Location on Tree Networks Applied to Transportation and Telecommunications Network Design.” A Dissertation, Department of Geography and Environmental Engineering, Whiting School of Engineering, Johns Hopkins University.
George, J.W., C.S. ReVelle, and J.R. Current. (2002). “The Maximum Utilization Subtree Problem.” Annals of Operations Research 110, 133–151.
Glover, F. (1986). “Future Paths for Integer Programming and Links to Artificial Intelligence.” Computers and Operations Research 13(5), 533–549.
Glover, F. (1990). “Tabu Search: A Tutorial.” Interfaces 20(4), 74–94.
Goldman, A.J. (1971). “Optimal Center Location in Simple Networks.” Transportation Science 5, 212–221.
Hakimi, S.L., E.F. Schmeichel, and M. Labbe. (1993). “On Locating Path-or Tree-Shaped Facilities on Networks.” Networks 23, 543–555.
Hansen, P. and B. Jaumard. (1987). “Algorithms for the Maximum Satisfiability Problem.” RUTCOR Research Report 43–87, Rutgers University, New Brunswick, NJ.
Hedetniemi, S.M., E.J. Cockayne, and S.T. Hedetniemi. (1981). “Linear Algorithms for Finding the Jordan Center and Path Center of a Tree.” Transportation Science 15(2), 98–114.
Holland, J.H. (1975). Adaptation in Natural and Artificial Systems. Ann Arbor, MI: University of Michigan Press, 183 pp.
Hua, L.K. et al. (1962). “Applications of Mathematical Models toWheat Harvesting.” Chin.Math. 2, 77–91.
Hutson, V.A. and C.S. ReVelle. (1989). “Maximal Direct Covering Tree Problems.” Transportation Science 23(4), 288–299.
Hutson, V.A. and C.S. ReVelle. (1993). “Indirect Covering Tree Problems on Spanning Tree Networks.” European Journal of Operational Research 65, 20–32.
Kim, T.U., T.J. Lowe, A. Tarmir, and J.E. Wang. (1996). “On the Location of a Tree-Shaped Facility.” Networks 28, 167–175.
Kim, T.U., T.J. Lowe, J.E. Ward, and R.L Francis. (1989). “A Minimal Length Covering Subgraph of a Network.” Annals of Operations Research 18, 245–260.
Kim, T.U., T.J. Lowe, J.E. Ward, and R.L Francis. (1990). “A Minimal Length Covering Subtree of a Tree.” Naval Research Logistics 37, 309–326.
Kincaid, R.K., T.J. Lowe, and T.L. Morin. (1988). “The Location of Central Structures in Trees.” Computers and Operations Research 15(2), 103–113.
Mesa, J.A. and T.B. Boffey. (1996). “A Review of Extensive Facility Location in Networks.” European Jornal of Operational Research 95, 592–603.
Minieka, E. (1985). “The Optimal Location of a Path or Tree in a Tree Network.” Networks 15, 309–321.
Minieka, E. and N.H. Patel. (1983). “On Finding the Core of a Tree with a Specified Length.” Journal of Algorithms 4, 345–352.
Morgan, C.A. and P.J. Slater. (1980). “A Linear Algorithm for a Core of a Tree.” Journal of Algorithms 1, 247–258.
Peng, S., A.B. Stephens, and Y. Yesha. (1993). “Algorithms for a Core and k-Tree Core of a Tree.” Journal of Algorithms 15, 143–159.
Peng, S. and W. Lo. (1996). “Efficient Algorithm for Finding a Core of a Tree with Specified Length.” Journal of Algorithms 20, 445–458.
ReVelle, C. (1993). “Facility Siting and Integer-Friendly Programming.” European Journal of Operational Research 65(2), 147–158 (Invited review).
ReVelle, C. and R. Swain. (1970). “Central Facilities Location.” Geographical Analysis 2, 30–42.
Richey, M.B. (1990). “Optimal Location of a Path or Tree on a Network with Cycles.” Networks 20, 391–407.
Slater, P.J. (1981). “Centrality of Paths and Vertices in a Graph: Cores and Pits.” In G. Chartrand (ed.), The Theory of Applications of Graphs. New York: Wiley, pp. 529–542.
Slater, P.J. (1982). “Locating Central Paths in a Graph.” Transportation Science 16(1), 1–18.
Tamir, A. (1993). “A Unifying Location Model on Tree Graphs Based on Submodularity Properties.” Discrete Applied Mathematics 47, 275–283.
Tamir, A. (1998). “Fully Polynomial Approximation Schemes for Locating a Tree-Shaped Facility: A Generalization of the Knapsack Problem.” Discrete Applied Mathematics 87, 229–243.
Tamir, A., J. Puerto, and D. Perez-Brito. (2002). “The Centdian Subtree on Tree Networks.” Discrete Applied Mathematics 118.
Tsouros, C.P. and S.E. Kostopoulou. (1994). “A Multiobjective Network Design Problem with Equity Constraints.” Studies in Regional and Urban Planning 3, 261–277.
Wang, B.F. (2002). “Finding a Two-Core of a Tree in Linear Time.” To appear in SIAM J. Discrete Mathematics.
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George, J., ReVelle, C. Bi-Objective Median Subtree Location Problems. Annals of Operations Research 122, 219–232 (2003). https://doi.org/10.1023/A:1026154708960
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DOI: https://doi.org/10.1023/A:1026154708960