Weber Location Problem Research Articles

An improved algorithm for solving the Weber location problem

An improved algorithm for solving the Weber location problem

Solving non-linear optimization problems by a trajectory approach

Abstract Accepted by: Konstantinos Nikolopoulos We propose solving non-linear optimization problems by a trajectory method. A parameter is introduced into the optimization problem. For example, a variable in the original formulation is replaced by its squared value. The parameter is the power at which the variable is raised. For a particular value of the parameter (power of 2), the optimal solution is easily obtained. The original optimization problem is defined for another value of the parameter (power of 1). As another example, the means and standard deviations of a function based on a set of variables can be calculated. We multiply the standard deviations by a factor (the parameter) between 0 and 1. Suppose that the problem is easily solvable for zero standard deviations (factor of 0). If we ‘slowly’ increase the factor, the solution moves to the desired solution for a factor of 1. A trajectory connects the easily obtained solution to the desired solution. We trace the trajectory and the solution for the optimization problem is at the end of the trajectory. The procedure is applied for solving the single facility Weber location problem, and a competitive location problem with good results.

AoI-Minimal Task Assignment and Trajectory Optimization in Multi-UAV-Assisted IoT Networks

This article investigates an unmanned-aerial-vehicle (UAV)-assisted Internet of Things system, where multiple UAVs take off from a data center (DC), collect data from multiple ground sensor nodes (SNs), distribute data for multiple users, and finally return to DC. A centralized information-sharing mechanism is designed among UAVs before data distribution, such that UAVs’ repeatedly visiting the same user can be avoided to save energy and time. A nonconvex Age-of-Information (AoI) minimization problem is formulated, where the task assignment, selection of interaction point (IPT), and UAVs’ trajectories are optimized jointly. It is decomposed into two subproblems: one is a joint task assignment and UAV trajectory optimization problem, which is solved by a multipopulation-based genetic algorithm (MPGA), as well as a hybrid algorithm combining dynamic programming (DP) with an improved <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -means clustering; and the other is an IPT selection problem, which is modeled as a Fermat–Weber location problem and is solved by the convex optimization technique. Simulation results show that the MPGA performs better than the hybrid algorithm as well as several benchmark algorithms.

Exact and approximate heuristics for the rectilinear Weber location problem with a line barrier

In this article, we propose an extension of the multi-Weber facility location problem with rectilinear-distance in the presence of passages over a non-horizontal line barrier. For the single-facility case, we develop an exact heuristic based on a divide-and-conquer approach that outperforms alternative heuristics available in literature. The multiple facilities case is solved by means of the application of an alternate-location-allocation heuristic, characterized by embedded exact and approximate procedures. For large instances, we propose a heuristic (with polynomial time complexity) which provides near-optimal solutions in a short computational time and a negligible gap. Finally, for testing purposes, we use a benchmark based on the transformation of the main problem into an equivalent p-median problem. Experimental results evidence the efficiency and validity of the proposed heuristics, which are capable of obtaining high quality solutions within acceptable computation times.

New Convergence Results for Inertial Krasnoselskii–Mann Iterations in Hilbert Spaces with Applications

We consider inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. To do these, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the inertial Krasnoselskii–Mann iteration for fixed point of nonexpansive operators in infinite dimensional real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. One of our contributions is that the convergence analysis and rate of convergence results are obtained using conditions which appear not complicated and restrictive as assumed in other previous related results in the literature. We then show that Fermat–Weber location problem and primal–dual three-operator splitting are special cases of fixed point problem of nonexpansive mapping and consequently obtain the convergence analysis of inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. Some numerical implementations are drawn from primal–dual three-operator splitting to support the theoretical analysis.

Asymmetric distance location model

In this note, we analyze location problems where the distance (time) to get to the destination by air is affected by winds. We propose two models: the asymmetric Weber location problem and the round-trip Weber location problem. The problems are analyzed and solved.

Pointing Consensus and Bearing-Based Solutions to the Fermat–Weber Location Problem

The pointing consensus problem asks each agent in a multiagent system to agree on their headings toward a common target. This paper proposes a decentralized approach to the pointing consensus problem by simultaneously solving three smaller problems: bearing-only measurement based network localization, target decision, and heading coordination. The proposed solution guarantees that all agents’ headings almost globally asymptotically target any weighted centroid of the agents’ positions. Furthermore, based on this approach, two decentralized solutions for the Fermat–Weber location problem are proposed and analyzed. Simulation results are also provided to support the analysis.

A coevolution model of population distribution and road networks

This paper proposes a coevolution model of population distribution and road networks. In the model, new roads are built with the formation of new communities, and these are connected to existing zones with higher population density with higher probability. The relative neighborhood graph and a Fermat–Weber location problem are introduced as the connection mechanism to capture the characteristics of population distribution and road networks. Moreover, two kinds of economic factors, accessibility and housing price, are considered to reflect the impacts on the coevolution between population distribution and road networks. Simulations and empirical experiments are conducted to demonstrate the interaction between population distribution and road networks. The topological attributes of the road network are characterized in the experiments using degree distribution, betweenness centrality, coverage, circuitness, and treeness. The experimental results show that the proposed model has reflect on the simulation.

A modified particle swarm optimization algorithm for location problem

In the Weber location problem which was proposed for optimal location of industrial enterprises, the aim is to find the location of a point such that the sum of weighted distance between this point and a finite number of existing points is minimized. This popular model is widely used for optimal location of equipment and in many sophisticated models of cluster analysis such as detecting homogeneous production batches made from a single production batch of raw materials. The well-known iterative Weiszfeld does not converge efficiently to the optimal solution when the solution either coincides or nearly coincides with one of the demands point which is not the optimum. We propose a modified Particle Swarm Optimization (PSO) algorithm. The velocity update of the PSO is modified to enlarge the search space and enhance the global search ability. The preliminary results of these algorithms are analyzed and compared.

Sylvester's Four-Point Problem in a Sector and its Relation to Weber Location Problem

Sylvester's Four-Point Problem in a Sector and its Relation to Weber Location Problem

A discretization result for some optimization problems in framework spaces with polyhedral obstacles and the Manhattan metric

In this work we consider the shortest path problem and the single facility Weber location problem in any real space of finite dimension where there exist different types of polyhedral obstacles or forbidden regions. These regions are polyhedral sets and the metric considered in the space is the Manhattan metric. We present a result that reduce these continuous problems into problems in a “add hoc” graph, where the original problems can be solved using elementary techniques of Graph Theory. We show that, fixed the dimension of the space, both the reduction and the resolution can be done in polynomial time.

The Weber obnoxious facility location model: A Big Arc Small Arc approach

In this paper we propose the Weber obnoxious facility location problem. As in the classic Weber location problem, the objective is to minimize the weighted sum of distances between the facility and demand points. However, the facility location is required to be at least a given distance from demand points because it is “obnoxious” to them. A practical example is locating an airport. Since in most applications the nuisance generated by the facility “travels by air”, we concentrate on the case where the required minimum distance between the facility and demand points is Euclidean. The Weber objective distance can be measured by a different norm. We develop very efficient algorithms to optimally solve the single facility problem based on geometric branch and bound and on a finite candidate set. We tested it on problems with up to 10,000 demand points using Euclidean, Manhattan, and ℓp for p=1.78 norms for the Weber objective. The largest problems were optimally solved in a few seconds of computer time. Many extensions to the basic Weber obnoxious facility location problem are proposed for future research.

Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

The Krasnoselskii---Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii---Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii---Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas---Rachford splitting method, the Fermat---Weber location problem as well as the alternating projection method by John von Neumann.

On Newton’s Method for the Fermat–Weber Location Problem

This paper considers the Fermat---Weber location problem. It is shown that, after a suitable initialization, the standard Newton method can be applied to the Fermat---Weber problem, and is globally and locally quadratically convergent. A numerical comparison with the popular Weiszfeld algorithm shows that Newton's method is significantly more efficient than the Weiszfeld scheme.

The Fermat–Weber location problem in single integrator dynamics using only local bearing angles

This paper proposes a modified gradient control law to solve the Fermat–Weber location problem. The control law may guide an autonomous agent to the point that minimizes the weighted sum distance with regard to some stationary beacons using only bearing measurements. Based on previous results from Fermat–Weber location problem and Lyapunov stability theory, we prove that the agent will globally asymptotically reach the destination only using local bearing measurements. We also consider a scenario when the beacons are not stationary. Simulation results are provided to verify the mathematical analysis.

ALGORITHM FOR WEBER PROBLEM WITH A METRIC BASED ON THE INITIAL FARE

We introduce a non-Euclidean metric for transportation sys- tems with a dened minimum transportation cost (initial fare) and in- vestigate the continuous single-facility Weber location problem based on this metric. The proposed algorithm uses the results for solving the We- ber problem with Euclidean metric by Weiszfeld procedure as the initial point for a special local search procedure. The results of local search are then checked for optimality by calculating directional derivative of modi- ed objective functions in nite number of directions. If the local search result is not optimal then algorithm solves constrained Weber problems with Euclidean metric to obtain the nal result. An illustrative example is presented.

Robust solutions to multi-facility Weber location problem under interval and ellipsoidal uncertainty

In this paper, we consider the multi-facility Weber location problem (MFWP) with uncertain location of demand points and transportation cost parameters. Equivalent formulations of its robust counterparts for both the Euclidean and block norms and interval and ellipsoidal uncertainty sets are given as conic linear optimization problems.

Weiszfeld’s Method: Old and New Results

In 1937, the 16-years-old Hungarian mathematician Endre Weiszfeld, in a seminal paper, devised a method for solving the Fermat---Weber location problem--a problem whose origins can be traced back to the seventeenth century. Weiszfeld's method stirred up an enormous amount of research in the optimization and location communities, and is also being discussed and used till these days. In this paper, we review both the past and the ongoing research on Weiszfed's method. The existing results are presented in a self-contained and concise manner--some are derived by new and simplified techniques. We also establish two new results using modern tools of optimization. First, we establish a non-asymptotic sublinear rate of convergence of Weiszfeld's method, and second, using an exact smoothing technique, we present a modification of the method with a proven better rate of convergence.

A new mathematical model for the Weber location problem with a probabilistic polyhedral barrier

With the wide application of location theory in a variety of industries, the presence of barriers ‎merits the attention of managers and engineers.‎ In this paper, we assess the Weber location problem in the presence of a polyhedral barrier which probabilistically occurs on a given horizontal barrier route in the rectilinear space. A left triangular distribution function is used for the starting point of the barrier and therefore an expected rectilinear barrier distance function is formulated. In addition, a modification of the polyhedral barrier is presented which is equivalent to the original problem. Therefore, a mixed integer nonlinear programming model, which has a nonconvex solution space, is presented. Furthermore, by decomposing the feasible space into a finite number of convex solution spaces, an exact heuristic solution method is proposed. Then, a lower bound problem based on the forbidden region is applied. Some theorems and an example are reported.

Accelerating the convergence in the single-source and multi-source Weber problems

The modified Weiszfeld method [Y. Vardi, C.H. Zhang, A modified Weiszfeld algorithm for the Fermat–Weber location problem, Mathematical Programming 90 (2001) 559–566] is perhaps the most widely-used algorithm for the single-source Weber problem (SWP). In this paper, in order to accelerate the efficiency for solving SWP, a new numerical method, called Weiszfeld–Newton method, is developed by combining the modified Weiszfeld method with the well-known Newton method. Global convergence of the new Weiszfeld–Newton method is proved under mild assumptions. For the multi-source Weber problem (MWP), a new location-allocation heuristic, Cooper–Weiszfeld–Newton method, is presented in the spirit of Cooper algorithm [L. Cooper, Heuristic methods for location-allocation problems, SIAM Review 6 (1964) 37–53], using the new Weiszfeld–Newton method in the location phase to locate facilities and adopting the nearest center reclassification algorithm (NCRA) in the allocation phase to allocate the customers. Preliminary numerical results are reported to verify the evident effectiveness of Weiszfeld–Newton method for SWP and Cooper–Weiszfeld–Newton method for MWP.