Center location problems on tree graphs with subtree-shaped customers (original) (raw)
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Mathematical Programming, 1993
In the classicalp-center location model on a network there is a set of customers, and the primary objective is to select p service centers that will minimize the maximum distance of a customer to a closest center. Suppose that the p centers receive their supplies from an existing central depot on the network, e.g. a warehouse. Thus, a secondary objective is to locate the centers that optimize the primary objective "as close as possible" to the central depot. We consider tree networks and two p-center models. We show that the set of optimal solutions to the primary objective has a semilattice structure with respect to some natural ordering. Using this property we prove that there is a p-center solution to the primary objective that simultaneously minimizes every secondary objective function which is monotone nondecreasing in the distances of the p centers from the existing central depot. Restricting the location models to a rooted path network (real line) we prove that the above results hold for the respective classical p-median problems as well.
A finite algorithm for the continuousp-center location problem on a graph
Mathematical Programming, 1985
Let G = (V, E) be an undirected graph with positive integer edge lengths. Let m be the number of edges in E, and let d be the sum of the edge lengths. We prove that the solution value to the continuous p-center location problem is a rational pJ p:, where log p, = O(m 5 log d + m ~' log p), i = 1,2. This result is then used to construct a finite algorithm for the continuous p-center problem.
New Results on the Complexity of p-Centre Problems
SIAM Journal on Computing, 1983
An O(n log3 n) algorithm for the continuous p-center problem on a tree is presented. Following a sequence of previous algorithms, ours is the first one whose time bound in uniform in p and less than quadratic in n. We also present an O(n log2 n log log n) algorithm for a weighted discrete p-center problem.
A unifying location model on tree graphs based on submodularity properties
Discrete Applied Mathematics, 1993
Let % be the collection of nonempty subtrees of a given tree T. Each subtree is viewed as a potential facility. Let f be a real objective function defined on 9. The facility location model we consider is to select a subtree minimizing 1: This model unifies and generalizes several facility location problems discussed in the literature. We prove that the most common objective functions used in facility location theory possess the submodularity property. In particular, the ellipsoid approach provides a unified framework for polynomial solvability. when the transportation costs are linear with the distance travelled [2]. It is polynomially solvable on tree graphs and some generalizations like series-parallel graphs c2,4,91.
A Maximal Client Coverage Algorithm for the p-Center Problem
Thai Journal of Mathematics, 2012
In this work, we propose a maximal client coverage algorithm for solving the p-center problem. The algorithm is created to locate p facilities and assign clients to them in order to minimize the maximum distance between clients and the facilities. We consider both uncapacitated and capacitated cases where demands of clients and capacities of facilities are taken into account. The simulations to test the proposed algorithm are also given and compared with method given by Albareda-Sambola et al. in 2010. Optimal solutions of the test problems are found using branch and bound algorithm to compare the optimality gaps of the proposed heuristics. The proposed heuristics solutions are found to be statistically faster than the reference algorithm at the significance level α = 0.01 in both uncapacitated and capacitated cases.
A polynomial algorithm for thep-centdian problem on a tree
Networks, 1998
The most common problems studied in network location theory are the p-median and the p-center models. The p-median problem on a network is concerned with the location of p points (medians) on the network, such that the total (weighted) distance of all the nodes to their respective nearest points is minimized. The p center problem is concerned with the location of p-points (centers) on the network, such that the maximum (weighted) distance of all the nodes to their respective nearest points is minimized. To capture more real world problems, and obtain a good way to trade-off minisum (efficiency) and minimax (equity) approaches, Halpern [5, 6, 7], introduced the centdian model, where the objective is to minimize a convex combination of the objective functions of the center and the median problems. In this paper we study the p-centdian problem on tree networks, and present the first polynomial time algorithm for this problem.
A note on spanning trees for network location problems
1998
A p-facility location problems on a network N consist of locating p new facilities on N such that some function of distances among them and vertices of N is minimum. We consider a class of such problems where objective function is nondecreasing in distance. The median, the center and the centdian problems belong to this class. We prove that the optimal solution on the network and on the corresponding spanning tree are equal. Since location problems on tree network are easier to solve than on general network, we propose a descent local search heuristic that solve optimaly the problem on spanning tree in each iteration.
The complete vertex p-center problem
The Complete Vertex p-Center Problem, 2020
The vertex p-center problem consists of locating p facilities among a set of M potential sites such that the maximum distance from any demand to its closest located facility is minimized. The complete vertex p-center problem solves the p-center problem for all p from 1 to the total number of sites, resulting in a multi-objective trade-off curve between the number of facilities and the service distance required to achieve full coverage. This trade-off provides a reference to planners and decision makers, enabling them to easily visualize the consequences of choosing different coverage design criteria for the given spatial configuration of the problem. We present two fast algorithms for solving the complete p-center problem: one using the classical formulation but trimming variables while still maintaining optimality and the other converting the problem to a location set covering problem and solving for all distances in the distance matrix. We also discuss scenarios where it makes sense to solve the problem via brute-force enumeration. All methods result in significant speedups, with the set covering method reducing computation times by many orders of magnitude.
One-way and round-trip center location problems
Discrete Optimization, 2005
In the classical p-center problem there is a set V of points (customers) in some metric space, and the objective is to locate p centers (servers), minimizing the maximum distance between a customer and his respective nearest server. In this paper we consider an extension, where each customer is associated with a set of existing depots or distribution stations he can use. The service of a customer consists of the travel of a server to some permissible depot, loading of some package (e.g., a spare part) at the depot, and the delivery of the package to the customer. This model is called the customer one-way problem. In the round-trip version of the problem, the service also includes the travel from the customer to the home base of the server. In both problems the customer cost of the service is a linear function of the distance travelled by the server. The objective is to locate p servers, minimizing the maximum customer cost (weighted distance travelled by the respective server). Since the classical p-center problem is NP-hard, so are the one-way and the round-trip models we study. We present efficient constant factor approximation algorithms for these problems on general networks. Turning to special networks, we prove that the one-way problem is strongly NP-hard even on path networks. We then present polynomial time algorithms for the round-trip problem on general tree networks. We also discuss the single center case, and provide polynomial time algorithms for general networks, tree networks and planar Euclidean and rectilinear metric spaces.