m-Median and m-Center Problems with Mutual Communication: Solvable Special Cases

Abstract

In this paper we consider the network version of the m-median problem with mutual communication (MMMC). We reformulate this problem as a graph theoretic node selection problem defined on a special graph. We give a polynomial time algorithm to solve the node selection problem when the flow graph (graph denoting the interaction between pairs of new facilities in MMMC) has a special structure. We also show that with some modification in the algorithm for MMMC, the m-center problem with mutual communication can also be solved when the flow graph has special structure.

Solving a selected class of location problems by exploiting problem structure: A decomposition approach

“…MMMC is a classical location problem that has been the focus of several papers that have appeared in the literature (see, e.g., Dearing, Francis, and Lowe [13], Picard and Ratliff [28], Erkut, Francis, and Lowe [15], Erkut, Francis, Lowe, and Tamir [16], and Chhajed and Lowe [10]). The problem can be posed in the plane or on a network.…”

“…In (20) After making appropriate changes to MAP, the decomposition approach can be applied to an instance of the above general problem, for example, the mcenter problem with mutual communication (Chhajed and Lowe [10]; Kolen [25]). …”

Solving structured multifacility location problems efficiently

“…Hooker, Garfinkel and Chen (1991) studied a large number of continuous network location problems in an effort to identify a finite set of points on the network which would contain the new facility locations in an optimal solution. They called such a set of points for a given problem a. finite domination set Problem 1. m-Median Problem (Hakimi, 1964(Hakimi, ,1965 (Goldman, 1971) (Mirchandani and Odoni, 1979) (Dearing, Francis, and Lowe, 1976;Kolen, 1986;Fernandez-Baca, 1989;Chhajed and Lowe, 1990) Problem 6. m-C enter Problem (Hakimi, 1965) (Hooker, Garfinkel and Chen, 1991). Let C be the union of all such points for all i,j and pe AinAj.…”

“…Problem 7. m-Center Problem with Mutual Communication (Dearing, Francis, and Lowe, 1976;Kolen, 1986;Chhajed and Lowe, 1990) Problem 11. Maxisum Problem (Erkut, Baptie and Hohenbalken, 1990;Tamir, 1991) This problem is the same as problem 5 except that the objective function is to be maximized.…”

Solving Structured Multifacility Location Problems Efficiently

“…Hooker, Garfinkel and Chen (1991) studied a large number of continuous network location problems in an effort to identify a finite set of points on the network which would contain the new facility locations in an optimal solution. They called such a set of points for a given problem a. finite domination set Problem 1. m-Median Problem (Hakimi, 1964(Hakimi, ,1965 (Goldman, 1971) (Mirchandani and Odoni, 1979) (Dearing, Francis, and Lowe, 1976;Kolen, 1986;Fernandez-Baca, 1989;Chhajed and Lowe, 1990) Problem 6. m-C enter Problem (Hakimi, 1965) (1979), that an FDS is the set of vertices along with the points in C.…”