Mathematical Programming

, Volume 102, Issue 2, pp 313–338 | Cite as

Locating tree-shaped facilities using the ordered median objective

  • J. Puerto
  • A. Tamir


In this paper we consider the location of a tree-shaped facility S on a tree network, using the ordered median function of the weighted distances to represent the total transportation cost objective. This function unifies and generalizes the most common criteria used in location modeling, e.g., median and center. If there are n demand points at the nodes of the tree T=(V,E), this function is characterized by a sequence of reals, Λ=(λ 1, . . . ,λ n ), satisfying λ 1λ 2≥...≥λ n ≥0. For a given subtree S let X(S)={x 1, . . . ,x n } be the set of weighted distances of the n demand points from S. The value of the ordered median objective at S is obtained as follows: Sort the n elements in X(S) in nonincreasing order, and then compute the scalar product of the sorted list with the sequence Λ. Two models are discussed. In the tactical model, there is an explicit bound L on the length of the subtree, and the goal is to select a subtree of size L, which minimizes the above transportation cost function. In the strategic model the length of the subtree is variable, and the objective is to minimize the sum of the transportation cost and the length of the subtree. We consider both discrete and continuous versions of the tactical and the strategic models. We note that the discrete tactical problem is NP-hard, and we solve the continuous tactical problem in polynomial time using a Linear Programming (LP) approach. We also prove submodularity properties for the strategic problem. These properties allow us to solve the discrete strategic version in strongly polynomial time. Moreover the continuous version is also solved via LP. For the special case of the k-centrum objective we obtain improved algorithmic results using a Dynamic Programming (DP) algorithm and discretization and nestedness results.


Polynomial Time Transportation Cost Weighted Distance Demand Point Continuous Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alstrup, S., Lauridsen, P.W., Sommerlund, P., Thorup, M.: Finding cores of limited length. In: Algorithms and Data Structures, Lecture Notes in Computer Science n. 1271, Dehne, F., Rau-Chaplin, A. Sack, J-R., Tamassia R. (eds.), Springer, 1997, pp. 45–54Google Scholar
  2. 2.
    Becker, R.I., Perl, Y.: Finding the two-core of a tree. Discrete Appl. Math. 11, 103–113 (1985)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Becker, R.I., Lari, I., Scozzari, A.: Efficient algorithms for finding the (k,l)-core of tree networks. Networks 40, 208–215 (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Boffey, B., Mesa, J.A.: A review of extensive facility location in networks. European J. Operational Res. 95, 592–600 (1996)zbMATHGoogle Scholar
  5. 5.
    Bramel, J., Simchi-Levi, D.: The Logic of Logistics: Theory, Algorithms and Applications for Logistics Management. Springer 1997, BerlinGoogle Scholar
  6. 6.
    Cho, G., Shaw, D.X.: A depth-first dynamic programming algorithm for the tree knapsack problem. INFORMS J. Comput. 9, 431–438 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Faigle, U., Kern, W.: Computational complexity of some maximum average weight problems with precedence constraints. Operations Res. 42, 688–693 (1994)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Fischetti, M., Hamacher, H.W., Jornsten, K., Maffioli, F.: Weighted k-cardinality trees: complexity and polyhedral structure. Networks 24, 11–21 (1994)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Francis, R.L., Lowe, T.J., Tamir, A.: Aggregation error bounds for a class of location models. Operations Res. 48, 294–307 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Francis, R.L., Lowe, T.J., Tamir, A.: Worst-case incremental analysis for a class of p-facility location problems. Networks 39, 139–143 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Groetschel, M., Lovasz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer-Verlag 1993, BerlinGoogle Scholar
  12. 12.
    Hakimi, S.L., Schmeichel, E.F., Labbe, M.: On locating path-or tree shaped facilities on networks. Networks 23, 543–555 (1993)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Hedetniemi, S.M., Cockaine, E.J., Hedetniemi, S.T.: Linear algorithms for finding the Jordan center and path center of a tree. Transportation Sci. 15, 98–114 (1981)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Johnson, D.S., Niemi, K.A.: On knapsack, partitions and a new dynamic technique for trees. Math. Operations Res. 8, 1–14 (1983)zbMATHGoogle Scholar
  15. 15.
    Kalcsics, J., Nickel, S., Puerto, J.: Multi-facility ordered median problems: A further analysis. Networks 41, 1–12 (2003)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kalcsics, J., Nickel, S., Puerto, J., Tamir, A.: Algorithmic results for ordered median problems defined on networks and the plane. Operations Res. Lett. 30, 149–158 (2002)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kim, T.U., Lowe, T.J., Tamir, A., Ward, J.E.: On the location of a tree-shaped facility. Networks 28, 167–175 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    McCormick, S.T.: Submodular function minimization. To appear as a chapter in Handbook on Discrete Optimization, edited by K. Aardal, G. Nemhauser and R. WeismantelGoogle Scholar
  19. 19.
    Minieka, E.: Conditional centers and medians on a graph. Networks 10, 265–272 (1980)MathSciNetGoogle Scholar
  20. 20.
    Minieka, E.: The optimal location of a path or tree in a tree network. Networks 15, 309–321 (1985)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Minieka, E., Patel, N.H.: On finding the core of a tree with a specified length. J. Algorithms 4, 345–352 (1983)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Morgan, C.A., Slater, J.P.: A linear algorithm for a core of a tree. J. Algorithms 1, 247–258 (1980)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Nickel, S., Puerto, J.: A unified approach to network location problems. Networks 34, 283–290 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Ogryczak, W., Tamir, A.: Minimizing the sum of the k largest functions in linear time. Inf. Processing Lett. 85, 117–122 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Peng, S., Lo, W.: Efficient algorithms for finding a core of a tree with specified length. J. Algorithms 20, 445–458 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Peng, S., Stephens, A.B., Yesha, Y.: Algorithms for a core and k-tree core of a tree. J. Algorithms 15, 143–159 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Queyranne, N.M.: Minimizing symmetric submodular functions. Math. Programming 82, 3–12 (1998)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Rodríguez-Chía, A.M., Nickel, S., Puerto, J., Fernández, F.R.: A flexible approach to location problems. Math. Meth. Oper. Res. 51, 69–89 (2000)CrossRefzbMATHGoogle Scholar
  29. 29.
    Shioura, A., Shigeno, M.: The tree center problems and the relationship with the bottleneck knapsack problems. Networks 29, 107–110 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Slater, P.J.: Locating central paths in a graph. Transportation Sci 16, 1–18 (1982)MathSciNetGoogle Scholar
  31. 31.
    Tamir, A.: A unifying location model on tree graphs based on submodularity properties. Discrete Appl. Math. 47, 275–283 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Tamir, A.: An O(pn 2) algorithm for the p-median and related problems on tree graphs. Oper. Res. Lett. 19, 59–64 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Tamir, A.: Fully polynomial approximation schemes for locating a tree-shaped facility: a generalization of the knapsack problem. Discrete Appl. Math. 87, 229–243 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Tamir, A.: The k-centrum multi-facility location problem. Discrete Appl. Math. 109, 292–307 (2000)MathSciNetGoogle Scholar
  35. 35.
    Tamir, A.: Sorting weighted distances with applications to objective function evaluations in single facility location problems. Oper. Res. Lett. 32, 249–257 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Tamir, A., Lowe, T.J.: The generalized p-forest problem on a tree network. Networks 22, 217–230 (1992)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Tamir, A., Puerto, J., Mesa, J.A., Rodriguez-Chia, A.M.: Conditional location of path and tree shaped facilities on trees. Technical Report, School of Mathematical Sciences, Tel-Aviv University, August 2001Google Scholar
  38. 38.
    Tamir, A., Puerto, J., Perez-Brito, D.: The centdian subtree on tree networks. Discrete Appl. Math. 118, 263–278 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Vaidya, P.M.: An algorithm for linear programming which requires O((m+n)n 2 + (m+n)1.5 nL) arithmetic operations. Math. Programming 47, 175–201 (1990)Google Scholar
  41. 41.
    Wang, B-F: Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network. J. Algorithms 34, 90–108 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Wang, B-F: Finding a two-core of a tree in linear time. SIAM J. Discrete Math. 15, 193–210 (2002)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • J. Puerto
    • 1
  • A. Tamir
    • 2
  1. 1.Facultad de MatemáticasUniversidad de Sevilla and Fundacion CentraSpain
  2. 2.School of Mathematical SciencesTel Aviv UniversityIsrael

Personalised recommendations