In this paper we define and analyze a class of two-dimensional location-allocation problems that can be solved with a one-dimensional dynamic programming algorithm. We define a criterion that must be satisfied in order that a problem can be classified as having a one-dimensional intrinsic property. An algorithm is developed to test any given problem to see if it possesses this property. We then show that any problem possessing the intrinsic property can be solved by means of an efficient dynamic programming algorithm developed earlier by one of the authors.

This paper considers a general form of the single facility minisum location problem (also referred to as the Fermat-Weber problem), where distances are measured by an l p norm. An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances. Global convergence of the algorithm is proven for any value of the parameter p in the closed interval [1, 2], provided an iterate does not coincide with a singular point of the iteration functions. However, for p > 2, the descent property of the algorithm and as a result, global convergence, are no longer guaranteed. These results generalize the work of Kuhn for Euclidean (p = 2) distances.

We consider the problem of locating a spherical circle with respect to existing facilities on a sphere, such that the sum of distances between the circle and the facilities is minimized or such that the maximum distance is minimized. The problem properties are analyzed, and we give solution procedures. When the circle to be located is restricted to be a great circle, some simplifications are possible. The models may be used in preliminary studies on the location of large linear facilities on the earth’s surface, such as superhighways, pipelines, and transmission lines, or in totally different contexts such as search-and-rescue missions and medical or biological studies.

We consider the problem of locating a line or a line segment in three-dimensional space, such that the sum of distances from the facility represented by the line (segment) to a given set of points is minimized. An example is planning the drilling of a mine shaft, with access to ore deposits through horizontal tunnels connecting the deposits and the shaft. Various models of the problem are developed and analyzed, and efficient solution methods are given.

We formulate and solve a location problem that determines where to position punt returners to maximize the number of punts caught. The problem is unusual within the location literature because it includes the dimension of time as well as Euclidean distance. The parameters of the model are estimated from actual punt return data. Our major finding is that the standard horizontal configuration of two punt returners results in only a small increase in the percentage of punts fielded over the case where a single returner is used. Moreover if a punter is not “directionalizing,” the vertical configuration of two returners (the punter and two returners are colinear) outperforms a horizontal configuration.

The multisource Weber problem is to locate simultaneously m facilities in the Euclidean plane to minimize the total transportation cost for satisfying the demand of n fixed users, each supplied from its closest facility. Many heuristics have been proposed for this problem, as well as a few exact algorithms. Heuristics are needed to solve quickly large problems and to provide good initial solutions for exact algorithms. We compare various heuristics, i.e., alternative location-allocation (Cooper 1964), projection (Bongartz et al. 1994), Tabu search (Brimberg and Mladenović 1996a), p-Median plus Weber (Hansen et al. 1996), Genetic search and several versions of Variable Neighbourhood search. Based on empirical tests that are reported, it is found that most traditional and some recent heuristics give poor results when the number of facilities to locate is large and that Variable Neighbourhood search gives consistently best results, on average, in moderate computing time.

We consider a given set of offshore platforms and onshore wells producing known (or estimated) amounts of oil to be connected to a port. Connections may take place directly between platforms, well sites, and the port, or may go through connection points at given locations. The configuration of the network and sizes of pipes used must be chosen to minimize construction costs. This problem is expressed as a mixed-integer program, and solved both heuristically by Tabu Search and Variable Neighborhood Search methods and exactly by a branch-and-bound method. Two new types of valid inequalities are introduced. Tests are made with data from the South Gabon oil field and randomly generated problems.