Double bound method for solving the \(p\)-center location problem.

*(English)*Zbl 1348.90384Summary: We give a review of existing methods for solving the absolute and vertex restricted \(p\)-center problems on networks and propose a new integer programming formulation, a tightened version of this formulation and a new method based on successive restrictions of the new formulation. A specialization of the new method with two-element restrictions obtains the optimal \(p\)-center solution by solving a series of simple structured integer programs in recognition form. This specialization is called the double bound method. A relaxation of the proposed formulation gives the tightest known lower bound in the literature (obtained earlier by S. Elloumi et al. [INFORMS J. Comput. 16, No. 1, 84–94 (2004; Zbl 1239.90103)]). A polynomial time algorithm is presented to compute this bound. New lower and upper bounds are proposed. Problems from the OR-Library and TSPLIB are solved by the proposed algorithms with up to 3038 nodes. Previous computational results were restricted to networks with at most 1817 nodes.

##### Keywords:

\(p\)-center location; multi-center location; covering location; minimax location; set covering
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\textit{H. Calik} and \textit{B. C. Tansel}, Comput. Oper. Res. 40, No. 12, 2991--2999 (2013; Zbl 1348.90384)

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##### References:

[1] | Elloumi, S.; Labbé, M.; Pochet, Y., A new formulation and resolution method for the p-center problem, INFORMS Journal on Computing, 16, 1, 84-94, (2004) · Zbl 1239.90103 |

[2] | Beasley, JE. OR-LIBRARY. June 2012. URL 〈http://people.brunel.ac.uk/ mastjjb/jeb/info.html〉. |

[3] | Reinelt, G., TSPLIB—a traveling salesman problem library, ORSA Journal on Computing, 3, 4, 376-384, (1991) · Zbl 0775.90293 |

[4] | Dearing, P.; Francis, R.; Lowe, T., Convex location problems on tree networks, Operations Research, 24, 4, 628-642, (1976) · Zbl 0341.90042 |

[5] | Hakimi, S. L., Optimum locations of switching centers and the absolute centers and medians of a graph, Operations Research, 12, 3, 450-459, (1964) · Zbl 0123.00305 |

[6] | Minieka, E., The m-center problem, SIAM Review, 12, 1, 138-139, (1970) · Zbl 0193.24204 |

[7] | Kariv, O.; Hakimi, S. L., An algorithmic approach to network location problems. part ithe p-centers, SIAM Journal on Applied Mathematics, 37, 3, 513-538, (1979) · Zbl 0432.90074 |

[8] | Hooker, J.; Garfinkel, R.; Chen, C., Finite dominating sets for network location problems, Operations Research, 39, 1, 100-118, (1991) · Zbl 0744.90049 |

[9] | Megiddo, N.; Tamir, A.; Zemel, E.; Chandrasekaran, R., An \(O(n \log^2 n)\) algorithm for the k-th longest path in a tree with applications to location problems, SIAM Journal on Computing, 10, 2, 328-337, (1981) · Zbl 0456.68071 |

[10] | Tansel, B.; Francis, R.; Lowe, T.; Chen, M., Duality and distance constraints for the nonlinear p-center problem and covering problem on a tree network, Operations Research, 30, 4, 725-744, (1982) · Zbl 0486.90037 |

[11] | Megiddo, N.; Tamir, A., New results on the complexity of p-centre problems, SIAM Journal on Computing, 12, 4, 751-758, (1983) · Zbl 0521.68037 |

[12] | Jaeger, M.; Kariv, O., Algorithms for finding p-centers on a weighted tree (for relatively small p), Networks, 15, 3, 381-389, (1985) · Zbl 0579.90024 |

[13] | Shaw, D. X., A unified limited column generation approach for facility location problems on trees, Annals of Operations Research, 87, 363-382, (1999) · Zbl 0924.90103 |

[14] | Tansel, B. C.; Francis, R. L.; Lowe, T. J., State of the art—location on networksa survey. part I: the p-center and p-Median problems, Management Science, 29, 4, 482-497, (1983) · Zbl 0513.90022 |

[15] | Tansel, B. C.; Francis, R. L.; Lowe, T. J., State of the art—location on networksa survey. part II: exploiting tree network structure, Management Science, 29, 4, 498-511, (1983) · Zbl 0513.90023 |

[16] | Handler, G. Y.; Mirchandani, P. B., Location on networkstheory and algorithms, vol. 979, (1979), MIT Press Cambridge, MA |

[17] | Daskin, M. S., Network and discrete locationmodels, algorithms, and applications, (1995), Wiley New York |

[18] | Tansel, B. C., Discrete center problems, (Eiselt, H. A.; Marianov, V., Foundations of location analysis, (2011), Springer New York), 79-106, [Chapter 5] · Zbl 1387.90122 |

[19] | Cornuéjols, G.; Nemhauser, G. L.; Wolsey, L. A., A canonical representation of simple plant location problems and its applications, SIAM Journal on Algebraic Discrete Methods, 1, 3, 261-272, (1980) · Zbl 0501.90032 |

[20] | Christofides, N.; Viola, P., The optimum location of multi-centres on a graph, Operational Research Quarterly, 145-154, (1971) · Zbl 0219.05069 |

[21] | Toregas, C.; Swain, R.; ReVelle, C.; Bergman, L., The location of emergency service facilities, Operations Research, 19, 6, 1363-1373, (1971) · Zbl 0224.90048 |

[22] | Garfinkel, R.; Neebe, A.; Rao, M., The m-center problemminimax facility location, Management Science, 23, 10, 1133-1142, (1977) · Zbl 0369.90125 |

[23] | Daskin, M. S., A new approach to solving the vertex p-center problem to optimalityalgorithm and computational results, Communications of the Operations Research Society of Japan, 45, 9, 428-436, (2000) |

[24] | Ilhan T, Pinar M. An efficient exact algorithm for the vertex p-center problem. 2001. URL 〈http://www.ie.bilkent.edu.tr/ mustafap/pubs/〉. |

[25] | khedhairi, Al-A.; Salhi, S., Enhancements to two exact algorithms for solving the vertex p-center problem, Journal of Mathematical Modelling and Algorithms, 4, 2, 129-147, (2005) · Zbl 1093.90004 |

[26] | IBM. IBM ILOG CPLEX Optimization Studio. 2013 URL 〈www.ibm.com/software/products/us/en/ibmilogcpleoptistud/〉. |

[27] | González, T. F., Clustering to minimize the maximum intercluster distance, Theoretical Computer Science, 38, 293-306, (1985) · Zbl 0567.62048 |

[28] | Floyd, R. W., Algorithm 97shortest path, Communications of ACM, 5, 6, 345-346, (1962) |

[29] | XINOX Software. JCreator—Java IDE. 2012. URL 〈http://www.jcreator.com/〉. |

[30] | Calik H. New formulations and exact solution methods for the capacitated and uncapacitated p-center location problem. PhD dissertation. Ankara, Turkey: Department of Industrial Engineering, Bilkent University; to appear. |

[31] | Francis, R. L.; Leon, F.; McGinnis, J.; White, J. A., Facility layout and locationan analytical approach, (1992), Prentice Hall Upper Saddle River, NJ |

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