Abstract We compute the robustness of Fermat–Weber points with respect to any finite gauge. We show a breakdown point of $$1/(1+\sigma )$$ 1 / ( 1 + σ ) where $$\sigma $$ σ is the asymmetry measure of the gauge. We obtain quantitative results indicating how far a corrupted Fermat–Weber point can lie from the true value in terms of the original sample and the size of the corrupted part. If the distance from the true value depends only on the original sample, then we call the gauge ‘uniformly robust.’ We show that polyhedral gauges are uniformly robust, but locally strictly convex norms are not, while in dimension 2 any uniform robust gauge is polyhedral.

The Weber problem with demand uniformly generated in discs

A swarm intelligence optimization algorithm called white shark optimizer (WSO) has been proposed and successfully applied in regard to many aspects. In this paper, the location problem of sports facilities is regarded as a multi-objective problem, and the number of residents covered by sports facilities and the Weber problem are introduced as objective functions. A multi-objective white shark optimizer (MOWSO) is proposed, and MOWSO introduced an archived mechanism to store the non-dominated solutions obtained by the algorithm. When the Pareto solutions in the archive overflow, the solutions are removed by calculating the true distance of the Pareto optimal solution. The performance of the MOWSO is verified on CEC 2020 benchmark functions, and the results show that the proposed MOWSO is better than other algorithms in the diversity and distribution of solutions. The MOWSO is applied to solve the rural sports facilities location problem, and a variety of different sports facilities location schemes are obtained. It can provide a variety of options for the location of rural sports facilities, and promote the intelligent design of sports facilities.

ABSTRACT This study proposes a novel enhancement to location allocation algorithms called the Enhanced Cell-based Algorithm (ECBA), which specifically addresses the Capacitated Multi-source Weber Problem, a variant of the Location Allocation Problem. The ECBA finds profitable cells in a continuous space to strategically locate facilities while minimising transportation costs. ECBA operates by dividing the distribution of customers into smaller cells that signify promising locations, establishes an initial facility configuration using fixed and dynamic radii and applies an Alternating Transportation Problem to determine optimal new locations. The algorithm was tested on datasets containing 50, 654 and 1060 customers, along with various facility configurations. The results showed that the algorithm outperformed previous studies in identifying strategic facility locations while adhering to capacity restrictions. The performance and effectiveness of the ECBA make it a useful tool for addressing other LAP applications with similar requirements. These contributions extend its applicability to a variety of complex location allocation challenges.

Swarm optimization approaches for the minmax regret Fermat–Weber problem with budgeted-constrained uncertain point weights

Reviewing extensions and solution methods of the planar Weber single facility location problem

Abstract In this work, new mixed integer nonlinear optimization models are proposed for two clustering problems: the unitary weighted Weber problem and the minimum sum of squares clustering. The proposed formulations are convex quadratic models with linear and second‐order cone constraints that can be efficiently solved by interior point algorithms. Their continuous relaxation is convex and differentiable. The numerical experiments show the proposed models are more efficient than some classical models for these problems known in the literature.

Several new qualitative properties of the problem of minimizing the sum of the weighted minima of the Euclidean distances of the demand points to the facilities, which is called the multi-source Weber problem and also known as the clustering problem with Euclidean norms, are obtained in this paper. Nontrivial local solutions are defined and characterized with the help of a necessary optimality condition in DC programming and the special structure of the problem in question. Since most of the existing solution algorithms for this problem just give some local solutions, our characterizations can be used to analyse and refine these algorithms.

Weber’s inverse problem in the plane is to modify the positive weights associated with n fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber’s inverse problem in the plane and generalize it to the surface of the sphere. Our study uses a subspace orthogonal to a subspace generated by two vectors X and Y associated with the given points and weights. The main achievement of our work lies in determining a vector perpendicular to the vectors X and Y, in Rn; which is used to determinate a solution of Weber’s inverse problem. In addition, lower bounds are obtained for the minimum of the Weber function, and an upper bound for the difference of the minimal of Weber’s direct and inverse problems. Examples of application at the plane and unit sphere are given.

Progress in the development of location theory and clustering methods is mainly aimed at improving the performance of algorithms. We study the continuous p-median problem with the Euclidean metric. Various algorithmic combinations of the ALA procedure embedded into more sophisticated algorithmic combinations are used. Two repeated steps of the ALA algorithm solve the simplest discrete optimization problem, and then a series of central point search problems (Weber problem) for each cluster. A parallel implementation of the ALA algorithm is used. The paper studies the issue of choosing from two strategies: solve the Weber problem until accuracy ε is achieved at each iteration, or perform only one iteration of the Weiszfeld algorithm at each ALA iteration.

Recently, a multisource, raw material allocation form of Weber's classic single‐facility location problem was rediscovered and recognized for its significance in contemporary planning and decision‐making. This variation of the Weber problem investigates the location of a production plant while permitting the selection of each required raw material source. This article reviews the Weber problem with an emphasis on its extension to incorporate multiple facilities. The only formulated multiplant Weber problem involving resource allocation remains unsolved due to its complexity. An effective approach integrating GIS processing (i.e., the Voronoi diagram and vector‐based overlay) with the classic optimization algorithm (i.e., the Weiszfeld algorithm) is developed to address raw material sourcing in the process of siting facilities. The implementation relies entirely on open‐source Python packages, making the work reproducible, replicable, and expandable. Application findings demonstrate that the utility and computational efficiency of the proposed method to tackle this challenging problem are superior to those of the most advanced commercial optimization software.

Distributionally robust Weber problem with uncertain demand

In this paper, we study a generalized version of the Weber problem of finding a point that minimizes the sum of its distances to a finite number of given points. In our setting, these distances may be cut off at a given value [Formula: see text], and we allow for the option of an empty solution at a fixed cost [Formula: see text]. We analyze under which circumstances these problems can be reduced to the simpler Weber problem, and also when we definitely have to solve the more complex problem with cutoff. We furthermore present adaptions of the algorithm of Drezner, Mehrez and Wesolowsky (1991 [The facility location problem with limited distances. Transportation Science, 25(3), 183–187, INFORMS]) to our setting, which in certain situations are able to substantially reduce computation times as demonstrated in a simulation study. The sensitivity with respect to the cutoff value is also studied, which allows us to provide an algorithm that efficiently solves the problem simultaneously for all [Formula: see text].

The Weber problem in logistic and services networks under congestion

Customized Alternating Direction Methods of Multipliers for Generalized Multi-facility Weber Problem

In the past decades, new models and algorithms have been developed for solving various types of facility location problems using different versions of the fuzzy c-means algorithm and its hybrid combinations. On the other hand, the need for renewable energy sources has become even more important than ever due to the increasing population, environmental pollution, and climate change. In this study, we proposed two revised weighted fuzzy c-means clustering-based hybrid algorithms for solving real-life biogas facility location problem. The first algorithm is Nelder-Mead simplex algorithm, and the second is center-of-gravity approach. Problem is solved as multi-facility Weber problem, and results have been analyzed.

Extensions to the Weber problem

Multi-source Weber problem (MWP) is an important model in facility location, which has wide applications in various areas such as health service management, transportation system management, urban planning, etc. The location-allocation algorithm is a well-known method for solving MWP, which consists of a location phase and an allocation phase at each iteration. In this paper, we consider more general and practical case of MWP–the constrained multi-source location problem (CMSLP), i.e., the location of multiple facilities with considering interactive transportation between facilities, locational constraints on facilities and the gauge for measuring distances. A variational inequality approach is contributed to solving the location subproblem called the constrained multi-facility location problem (CMFLP) in location phase, which leads to an efficient projection-type method. Then a new location-allocation algorithm is developed for CMSLP. Global convergence of the projection-type method as well as local convergence of new location-allocation algorithm are proved. The efficiency of proposed methods is verified by some preliminary numerical results.

The single source Weber problem with limited distances (SSWPLD) is a continuous optimization problem in location theory. The SSWPLD algorithms proposed so far are based on the enumeration of all regions of [Formula: see text] defined by a given set of n intersecting circumferences. Early algorithms require [Formula: see text] time for the enumeration, but they were recently shown to be incorrect in case of degenerate intersections, that is, when three or more circumferences pass through the same intersection point. This problem was fixed by a modified enumeration algorithm with complexity [Formula: see text], based on the construction of neighborhoods of degenerate intersection points. In this paper, it is shown that the complexity for correctly dealing with degenerate intersections can be reduced to [Formula: see text] so that existing enumeration algorithms can be fixed without increasing their [Formula: see text] time complexity, which is due to some preliminary computations unrelated to intersection degeneracy. Furthermore, a new algorithm for enumerating all regions to solve the SSWPLD is described: its worst-case time complexity is [Formula: see text]. The new algorithm also guarantees that the regions are enumerated only once.

Tate–Shafarevich groups in the cyclotomic [formula omitted]-extension and Weber's class number problem