Weber Problem Research Articles

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

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

One of the classic foundational constructs of Location Science was proposed by Alfred Weber in 1909. His construct involved finding the location for a single production facility which minimized the sum of weighted distances of transporting the needed raw materials from localized sources along with the sum of weighted distances in delivering the final product to one or more markets.The first objective of this paper is to review the major advancements in this simple classic single facility location problem and its variations. One can find in the literature a very large number of algorithms to solve the standard Weber problem. Some are iterative and others are finite even for geometric Euclidean and rectilinear spaces. Moreover, some schemes are efficient (theoretically) and others are practically quite fast.The second goal of this paper is to show that many extensions of the standard Weber problem can be solved by solving a polynomial number of standard Weber problems. This unifying result implies, in particular, that all these extensions are polynomially solvable since the standard Weber problem can be solved in polynomial time. In addition, with this unifying approach we solve some important planar non-convex Euclidean location problems in polynomial time.

Second‐order cone programming models for the unitary weighted Weber problem and for the minimum sum of the squares clustering problem

AbstractIn 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.

Local solutions of the multi-source Weber problem

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.

The Inverse Weber Problem on the Plane and the Sphere

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.

Hybrid algorithms with alternative embedded local search schemes for the p-median problem

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.

Multiplant Location Involving Resource Allocation

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

Distributionally robust Weber problem with uncertain demand

Location Problems with Cutoff

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

We investigate a location-allocation-routing problem where trucks deliver goods from a central production facility to a set of warehouses with fixed locations and known demands. Due to limited capacities congestion occurs and results in queueing problems. The location of the center is determined to maximize the utilization of the given resources (measured in throughput) and the minimal number of trucks is determined to satisfy the overall demand generated by the warehouses. Main results for this integrated decision problem on strategic and tactical/operational level are: (i) The location decision is reduced to a standard Weber problem with weighted distances. (ii) The joint decision for location and fleet size is separable. (iii) The location of the center is robust against perturbations of several system parameters on the operational/tactical level. Additionally, we consider minimization of travel times as optimization target. By numerical examples we demonstrate the consequences of neglecting available information on long-term (rough) demand structure.

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

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

Fuzzy Clustering With Derivative–Free Search Algorithm for Location of Biogas Energy Systems

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

One of the classics in the field of Location Science is the book on the theory of industrial location by Weber (1909). Weber used a simple construct comprised of a 3-point triangle to describe important issues, including where raw materials for manufacturing are sourced. Virtually all of the research conducted in the last 50 years related to Weber’s construct has overlooked major elements of his work. This includes the issue of sourcing needed raw materials, which can be limited, as an integral part the location problem. This paper explores one form of raw material sourcing first described by Weber in which each raw material source is limited by a fixed capacity. We show that most instances of this location problem are non-convex as well as propose a solution procedure. We also explore a related problem where the facility itself can be of limited capacity and not all demands can be served. These two models can serve as building blocks for a greater exploration of many of the important problem facets proposed by Weber in his seminal work.

A Variational Inequality-Based Location-Allocation Algorithm for Locating Multiple Interactive Facilities

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.

A New Algorithm for the Single Source Weber Problem with Limited Distances

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

Let K=Q(N) be the Nth layer in the cyclotomic Zˆ-extension. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,…) prove results on the p-class groups CK. We enlarge “Weber's problem” to the Tate–Shafarevich groups IIIK1≃CKSp[p] and IIIK2 having same p-rank as the more easily computable torsion group, TK, of the Galois group of the maximal abelian p-ramified pro-p-extension of K; but TK is often non-trivial, which raises questions for class groups since ▪, where RK is the normalized p-adic regulator. We give a new method testing TK≠1 (Theorem 4.6, Table in Appendix A.7) and characterize the fields K1=KQ(p) with CK1≠1 (Main Theorem 1.1 affirming, for short, that CK1≠1 if and only if p totally splits in K and TK≠1); this highlights the analytical results and justifies the eight known examples. All PARI/GP programs are given for further investigations.

Research minimax and minisum Weber problems on a plane with forbidden zones

Weber problem is a well–known location problem of connected facilities. Two optimality criteria are considered. The first criterion is minimization of the total cost of connections among facilities. The second criterion is minimization of the maximum connection between facilities. This article deals with Weber problems on a plane with rectangular forbidden zones. Location inside in the forbidden zones is not allowed. The sides of forbidden zones are parallel to the coordinate axes. To measure distances rectangular metric is used. An overview of studies for these problems is provided. Some properties of the problems are described. For example, the possibilities of reduction of admissible areas in the search for an optimal solution are presented. Variants of different bounds of the objective functions are provided. Models of integer linear programming with Boolean variables are given. Computational experiments with using the branch and bound algorithms, the constructed models and IBM ILOG CPLEX package were carried out. Usage of the properties of reductions of admissible areas is promising both in solving of the problems by combinatorial methods and using the integer optimization apparatus.

Hybrid cell selection-based heuristic for capacitated multi-facility weber problem with continuous fixed costs

Location-allocation problem (LAP) has attracted much attention in facility location field. The LAP in continuous plane is well-known as Weber problem. This paper assessed this problem by considering capacity constraints and fixed costs as each facility has different setup cost and capacity limit to serve customers. Previous studies considered profitable areas by dividing continuous space into a discrete number of equal cells to identify optimal locations from a smaller set of promising locations. Unfortunately, it may lead to avoid choosing good locations because unprofitable areas are still considered while locating the facilities. Hence, this allows a significant increment in the transportation costs. Thus, this paper intelligently selected profitable area through a hybridization of enhanced Cell Selection-based Heuristic (CSBH) and Furthest Distance Rule (FDR) to minimize total transportation and fixed costs. The CSBH divides customer distribution into smaller set of promising locations and intelligently selected profitable area to increase possibility of finding better locations, while FDR aims to forbid the new locations of the facilities to be close to the previously selected locations. Numerical experiments tested on well-known benchmark datasets showed that the results of hybrid heuristic outperformed single CSBH and FDR, while producing competitive results when compared with previously published results, apart from significantly improving total transportation cost. The new hybrid heuristic is simple yet effective in solving LAP.

The evaluation of voting rules based on voting networks: Weber problem with support domain

This study focuses on improving the Weber model to obtain a more optimised solution, by which we can measure effectively the quality of the voting rules. By introducing the voter’s supporting domain and the candidate’s supported degree to depict the emotional factors of voters and the real voting network, we propose the concept of validity for a candidate. Upon the traditional Weber model, two improved models are presented and the corresponding global optimal candidate is used as the evaluation benchmark for the voting rules. The experiments show that the optimal solution of the two models has better robustness in complex voting networks, can be used as a standard to evaluate the voting rules. When the support degree of voter is the main factor, the Condorcet rule is optimal in most cases, and when the validity of the candidate is taken as the main factor, the Approval rule must be the best.

ADMM-type methods for generalized multi-facility Weber problem

<p style='text-indent:20px;'>The well-known multi-facility Weber problem (MFWP) is one of fundamental models in facility location. With the aim of enhancing the practical applicability of MFWP, this paper considers a generalized multi-facility Weber problem (GMFWP), where the gauge is used to measure distances and the locational constraints are imposed to new facilities. This paper focuses on developing efficient numerical methods based on alternating direction method of multipliers (ADMM) to solve GMFWP. Specifically, GMFWP is equivalently reformulated into a minmax problem with special structure and then some ADMM-type methods are proposed for its primal problem. Global convergence of proposed methods for GMFWP is established under mild assumptions. Preliminary numerical results are reported to verify the effectiveness of proposed methods.</p>