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The Maximum Utilization Subtree Problem

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Abstract

A number of network design problems can be built on the following premise: Given a tree network, T, containing node set, V, identify a single subtree, t, containing nodes, v, so that the subtree is located optimally with respect to the remaining, unconnected nodes {Vv}. Distances between unconnected nodes and nodes in the subtree t can be defined on travel paths that are restricted to lie in the larger tree T (the travel-restricted case), or can be defined on paths in an auxiliary complete graph G (the travel-unrestricted case).

This paper presents the Maximum Utilization Subtree Problem (MUSP), a bicriterion problem that trades off the cost of a subtree, t, against the utilization of the subtree by the sum of the populations at nodes connected to the subtree, plus the distance-attenuated population that must travel to the subtree from unconnected nodes. The restricted and unrestricted cases are formulated as a two objective integer programs where the objectives are to maximize utilization of the subtree and minimize the cost of the subtree. The programs are tested using linear programming and branch and bound to resolve fractions.

The types of problems presented in this paper have been characterized in the existing literature as “structure location” or “extensive facility location” problems. This paper adds two significant contributions to the general body of location literature. First, it draws explicit attention to the travel-restricted and travel-unrestricted cases, which may also be called “limited-access” and “general-access” cases, respectively. Second, the distance-attenuated demands represent a new objective function concept that does not appear in the location literature.

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References

  1. Y.P. Aneja and K.P.K. Nair, Location of a tree shaped facility in a network, INFOR 30(4) (1992) 319-324.

    Google Scholar 

  2. R.I. Becker and Y. Perl, Finding the two-core of a tree, Discrete Applied Mathematics 11 (1985) 103-113.

    Google Scholar 

  3. T.B. Boffey, Distributed Comupting: Associated Combinatorial Problems (Blackwell, Oxford, 1992).

    Google Scholar 

  4. R.L. Church and J.R. Current, Maximal covering tree problems, Working Paper Series, WPS 89-47, College of Business, Ohio State University (July, 1989).

  5. R.L. Church and J.R. Current, The minimal cost/maximal covering forest problem on a tree, in: Fifth International Symposium on Locational Decisions, Lake Arrowhead, CA (1990).

  6. R.L. Church and J.R. Current, Maximal covering tree problems, Naval Research Logistics 40 (1993) 129-142.

    Google Scholar 

  7. J.L. Cohon, Multiobjective Programming and Planning (Academic Press, New York, 1978).

    Google Scholar 

  8. J.R. Current, Multiobjective design of transportation networks, Ph.D. dissertation, The Johns Hopkins University, Baltimore, MD (1981).

    Google Scholar 

  9. J.R. Current and H. Pirkul, The hierarchical transportation network with transshipment facilities, European Journal of Operational Research 52 (1991) 338-347.

    Google Scholar 

  10. J.R. Current and D.A. Schilling, The Covering Salesman Problem, Transportation Science 23(3) (1989) 208-213.

    Google Scholar 

  11. J.R. Current and D.A. Schilling, The Median Tour and Maximal Covering Tour Problems: Formulations and heuristics, European Journal of Operational Research 73 (1994) 114-126.

    Google Scholar 

  12. B. Gavish, Topological design of computer communications networks-The overall design problem, European Journal of Operational Research 58 (1992) 149-172.

    Google Scholar 

  13. J.W. George, Subtree location on tree networks applied to transportation and communication network design, Ph.D. dissertation, The Johns Hopkins University, Baltimore, MD (1997).

    Google Scholar 

  14. S.L. Hakimi, E.F. Schmeichel and M. Labbe, On locating path-or tree-shaped facilities on networks, Networks 23 (1993) 543-555.

    Google Scholar 

  15. S.M. Hedetniemi, E.J. Cockayne and S.T. Hedetniemi, Linear algorithms for finding the Jordan center and path center of a tree, Transportation Science 15(2) (1981) 98-114.

    Google Scholar 

  16. V.A. Hutson and Ch.S. ReVelle, Maximal direct covering tree problems, Transportation Science 23(4) (1989) 288-299.

    Google Scholar 

  17. V.A. Hutson and Ch.S. ReVelle, Indirect covering tree problems on spanning tree networks, European Journal of Operational Research 65 (1993) 20-32.

    Google Scholar 

  18. T.U. Kim, T.J. Lowe, J.E.Ward and R.L. Francis, A minimum length covering subgraph of a network, Annals of Operations Research 18 (1989) 245-260.

    Google Scholar 

  19. T.U. Kim, T.J. Lowe, J.E. Ward and R.L Francis, A minimal length covering subtree of a tree, Naval Research Logistics 37 (1990) 309-326.

    Google Scholar 

  20. R.K. Kincaid, T.J. Lowe and Th.L. Morin, The location of central structures in trees, Computers and Operations Research 15(2) (1988) 103-113.

    Google Scholar 

  21. J.B. Kruskal, Jr., On the shortest spanning subtree of a graph and the Traveling Salesman Problem, Proceedings of the American Mathematical Society 7 (1956) 48-50.

    Google Scholar 

  22. C.-H. Lee, H.-B. Ro and D.-W. Tcha, Topological design of a two-level network with ring-star configuration, Computers and Operations Research 20(6) (1993) 625-637.

    Google Scholar 

  23. Th.L.Magnanti and R.T. Wong, Network design and transportation planning: Models and algorithms, Transportation Science 18 (1984) 1-55.

    Google Scholar 

  24. J.A. Mesa and T.B. Boffey, Review of extensive facility location in networks, European Journal of Operational Research 95 (1996) 592-603.

    Google Scholar 

  25. E. Minieka, The optimal location of a path or tree in a tree network, Networks 15 (1985) 309-321.

    Google Scholar 

  26. E. Minieka and N.H. Patel, On finding the core of a tree with a specified length, Journal of Algorithms 4 (1983) 345-352.

    Google Scholar 

  27. Ch.A. Morgan and P.J. Slater, A linear algorithm for a core of a tree, Journal of Algorithms 1 (1980) 247-258.

    Google Scholar 

  28. S. Peng, A.B. Stephens and Y. Ysha, Algorithms for the fixed-charge assigning users to a sources problem, Journal of Algorithms 15 (1993) 143-159.

    Google Scholar 

  29. H. Pirkul and V. Nagarajan, Locating concentrators in centralized computer networks, Annals of Operations Research 36 (1992) 247-261.

    Google Scholar 

  30. R.C. Prim, Shortest connection matrix and some generalizations, Bell System Technical Journal (1957) 1389-1401.

  31. Ch.S. ReVelle, Facility siting and integer-friendly programming, European Journal of Operational Research 65 (1993) 147-158.

    Google Scholar 

  32. M.B. Richey, Optimal location of a path or tree on a network with cycles, Networks 20 (1990) 391-407.

    Google Scholar 

  33. P.J. Slater, Centrality of paths and vertices in a graph: Cores and pits, in: The Theory of Applications of Graphs, ed. G. Chartrand (Wiley, 1981) pp. 529-542.

  34. P.J. Slater, Locating central paths in a graph, Transportation Science 16(1) (1982) 1-18.

    Google Scholar 

  35. A. Tamir and T.J. Lowe, The generalized P-forest problem on a network, Networks 22 (1992) 217-230.

    Google Scholar 

  36. C.P. Tsouros and S.E. Kostopoulou, Multiobjective network design problem with equity constraints, Studies in Regional and Urban Planning 3 (1994) 261-277.

    Google Scholar 

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George, J.W., ReVelle, C.S. & Current, J.R. The Maximum Utilization Subtree Problem. Annals of Operations Research 110, 133–151 (2002). https://doi.org/10.1023/A:1020719718071

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