Single Facility Location Problem Research Articles

Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

The aim of this paper is twofold: first, to extend the area of applications of tropical optimization by solving new constrained location problems, and second, to offer new closed-form solutions to general problems that are of interest to location analysis. We consider a constrained minimax single-facility location problem with addends on the plane with rectilinear distance. The solution commences with the representation of the problem in a standard form, and then in terms of tropical mathematics, as a constrained optimization problem. We use a transformation technique, which can act as a template to handle optimization problems in other application areas, and hence is of independent interest. To solve the constrained optimization problem, we apply methods and results of tropical optimization, which provide direct, explicit solutions. The results obtained serve to derive new solutions of the location problem, and of its special cases with reduced sets of constraints, in a closed form, ready for practical implementation and immediate computation. As illustrations, numerical solutions of example problems and their graphical representation are given. We conclude with an application of the results to optimal location of the central monitoring facility in an indoor video surveillance system in a multi-floor building environment.

An efficient algorithm for the single facility location problem with polyhedral norms and disk-shaped demand regions

The single facility location problem with demand regions seeks for a facility location minimizing the sum of the distances from n demand regions to the facility. The demand regions represent sales markets where the transportation costs are negligible. In this paper, we assume that all demand regions are disks of the same radius, and the distances are measured by a rectilinear norm, e.g. ell _1 or ell _infty . We develop an exact combinatorial algorithm running in time O(nlog ^c n) for some c dependent only on the space dimension. The algorithm is generalizable to the other polyhedral norms.

A planar single-facility competitive location and design problem under the multi-deterministic choice rule

A new customer choice rule, which may model in some cases the actual patronising behaviour of customers towards the facilities closer to reality than other existing rules, is proposed. According to the new rule, customers split their demand among the firms in the market by patronising only one facility from each firm, the one with the highest utility, and the demand is split among those facilities proportionally to their attraction. The influence of the choice rule in the location of facilities is investigated. In particular, a new continuous competitive single-facility location and design problem using this new rule is proposed. Both exact and heuristic methods are proposed to solve it. A comparison with the classical proportional (or Huff) choice rule when solving the location model reveals that both the location and the quality of the new facility to be located may be quite different depending on the patronising behaviour of customers. Most importantly, the profit that the locating chain may lose if a wrong choice is made can be quite high in some instances.

Efficient algorithms for the round-trip 1-center and 1-median problems

This paper presents improved algorithms for the round-trip single-facility location problem on a general graph, in which a set A of collection depots is given and the service distance of a customer is defined to be the distance from the server, to the customer, then to a depot, and back to the server. Each customer i is associated with a subset Ai⊆A of depots that i can potentially select from and use. When Ai=A for each customer i, the problem is unrestricted; otherwise it is restricted. For the restricted round-trip 1-center problem, we give an O(mnlg⁡n)-time algorithm. For the restricted 1-median problem, we give an O(mnlg⁡(|A|/m)+n2lg⁡n)-time algorithm. For the unrestricted 1-median problem, we give an O(mn+n2lg⁡lg⁡n)-time algorithm.

On an algebraic solution of the Rawls location problem in the plane with rectilinear metric

A minimax single-facility location problem with rectilinear metric is examined by using methods of tropical (idempotent) mathematics. This problem, known as the Rawls problem or the messenger problem, appears in the location of emergency services (hospitals, fire stations) in cities with straight rectangular streets. In terms of idempotent algebra, the problem is reduced to minimizing a functional given on the set of three-dimensional vectors by an appropriately constructed matrix and calculated by using the multiplicative conjugate transposition. The minimum of the objective function is found subject to constraints in the form of a relation that holds between components of the vectors. A new result of the spectral theory of matrices in idempotent algebra is applied, which offers a general solution to the problem of minimizing such functionals without additional constraints. Based on the result, a general solution to the problem with constraints on the feasible solution is given in terms of the tropical algebra. The solution obtained is then used to derive a complete solution to the Rawls loca-tion problem, which extends a known particular solution of the problem under consideration.

A note on the facility location problem with stochastic demands

In this research note that the single source capacitated facility location problem with general stochastic identically distributed demands is studied. The demands considered are independent and identically distributed random variables with arbitrary distribution. The unified a priori solution for the locations of facilities and for the allocation of customers to the operating facilities is found. This solution minimizes the objective function which is the sum of the fixed costs and the value of one of two different recourse functions. For each case the recourse function is given in closed form and a deterministic equivalent formulation of the model is presented. Some numerical examples are also given.

The single facility location problem with time-dependent weights and relocation cost over a continuous time horizon

In this study, we investigate the problem of locating a facility in continuous space when the weight of each existing facility is a known linear function of time. The location of the new facility can be changed once over a continuous finite time horizon. Rectilinear distance and time- and location-dependent relocation costs are considered. The objective is to determine the optimal relocation time and locations of the new facility before and after relocation to minimize the total location and relocation costs. We also propose an exact algorithm to solve the problem in a polynomial time according to our computational results.

A Computational Method for the Solution of Optimizing Facility Location Problem in the Non-Convex Set

Facility location problems arise for planning and stationing serve centers including waste disposal sites, hospitals, distribution centers, post offices, and fire stations. The single facility location problem can be modeled by a fixed point problem. Commonly the definition set of variables are requested as a convex set. In real world applications, the collection of feasible sites to locate always is non-convex set. The contribution of this paper is to apply the theorem of existence of the solution of fixed point problem to discuss the facility location problem in non-convex definition set of variables and obtain the computational result. Numerical example shows that the numerical method by tracing the homotopy pathway with predictor-corrector is an effective method to solving the facility location problem and a global optimal solution can be obtained.

Covering Part of a Planar Network

The problem of locating facilities in a feasible area covering some parts of network links within a given radius is analyzed. The feasible area can be the interior (convex hull of the nodes) of a planar network or any union of convex polygons. Both minimization and maximization of coverage are considered. The single facility location problem is solved by the global optimization approach “Big Triangle Small Triangle.” The multiple facility maximization problem is solved by a specially designed heuristic algorithm. The idea of the heuristic algorithm may prove to work well on other planar multiple facility location problems. Computational experience with problems of up to 40,000 links demonstrate the effectiveness of the single facility and multiple facilities algorithms. The largest single facility minimal cover problem is solved in about one minute and the largest single facility maximal cover problem is solved in less than 4 minutes.

Heuristics for a continuous multi-facility location problem with demand regions

We consider a continuous multi-facility location allocation problem where the demanding entities are regions in the plane instead of points. The problem can be stated as follows: given m (closed, convex) polygonal demand regions in the plane, find the locations of q facilities and allocate each region to exactly one facility so as to minimize a weighted sum of squares of the maximum Euclidean distances between the demand regions and the facilities they are assigned to.We propose mathematical programming formulations of the single and multiple facility versions of the problem considered. The single facility location problem is formulated as a second order cone programming (SOCP) problem, and hence is solvable in polynomial time. The multiple facility location problem is NP-hard in general and can be formulated as a mixed integer SOCP problem. This formulation is weak and does not even solve medium-size instances. To solve larger instances of the problem we propose three heuristics. When all the demand regions are rectangular regions with their sides parallel to the standard coordinate axes, a faster special heuristic is developed. We compare our heuristics in terms of both solution quality and computational time.

The Quintile Share Ratio in location analysis

The inequality measure “Quintile Share Ratio” (QSR or sometimes S80/S20) is the primary income inequality measure in the European Union’s set of indicators on social cohesion. An important reason for its adoption as a leading indicator is its simplicity. The Quintile Share Ratio is “The ratio of total income received by the 20% of the population with the highest income (top quintile) to that received by the 20% of the population with the lowest income (lowest quintile)”. The QSR concept is used in this paper in the context of obnoxious facility location where the inequality is in distances to the obnoxious facility. The single facility location problem minimizing the QSR is investigated. The problem is investigated for continuous uniform demand in an area such as a disk, a rectangle, and a line; when demand is generated at a finite set of demand points; and when the facility can be located anywhere on a network. Optimal solution algorithms are devised for demand originating at a finite set of demand points and at nodes of the network. Computational experiments demonstrate the effectiveness of the algorithms.

Revisiting several problems and algorithms in continuous location with $$\ell _\tau $$ ℓ τ norms

This paper addresses the general continuous single facility location problems in finite dimension spaces under possibly different $$\ell _\tau $$ l ? norms, $$\tau \ge 1$$ ? ? 1 , in the demand points. We analyze the difficulty of this family of problems and revisit convergence properties of some well-known algorithms. The ultimate goal is to provide a common approach to solve the family of continuous $$\ell _\tau $$ l ? ordered median location problems Nickel and Puerto (Facility location: a unified approach, 2005) in dimension $$d$$ d (including of course the $$\ell _\tau $$ l ? minisum or Fermat-Weber location problem for any $$\tau \ge 1$$ ? ? 1 ). We prove that this approach has a polynomial worst case complexity for monotone lambda weights and can be also applied to constrained and even non-convex problems.

Applying metaheuristic approaches on the single facility location problem with polygonal barriers

In this paper, we describe, improve, and compare the performance of four metaheuristic algorithms on the single facility location problem with barriers. The barriers are polygonal regions on a plane where neither facility location nor travelling is permitted. The objective function in this problem is non-convex justifying the application of metaheuristics and their robustness. We argue that this problem has interesting geometric properties such as size and distribution of barriers on the plane which can affect the choice of metaheuristics. Various test instances which reflect these geometric properties are designed in this paper. Through computational experiments, we discuss the effect of these properties on different variants of metaheuristics and propose the suitable metaheuristic and parameter setting for the single facility location problem with barriers.

A new heuristic algorithm for the planar minimum covering circle problem

The minimum covering circle problem is widely utilized in the studies of single or multiple facility location problems. It may be employed to locate the essential locations of factories, schools, fire departments, hospital establishments, and other facilities, which could be considered as a point in a plane theoretically. In this paper, a new geometrical algorithm is presented which determines the minimum covering circle of all points on a plane in four steps. The model validity was considered by studying the coordinates of points with random numbers and different distributions. In order to show the accuracy of the proposed algorithm, numerical experiments were carried out and compared with other studies in the field. The results show that the proposed algorithm extremely outperforms other examined algorithms.

Algorithms for Location Problems Based on Angular Distances

This paper describes four mathematical models for the single-facility location problems based on four special distance metrics and algorithms for solving such problems. In this study, algorithms of solving Weber problems using four distance predicting functions (DPFs) are proposed in accordance with four strategies for manipulator control. A numerical example is presented in this proposal as an analytical proof of the optimality of their results.

An Improved Genetic Algorithm for the Multi Level Uncapacitated Facility Location Problem

In this paper, an improved genetic algorithm (GA) for solving the multi-level uncapacitated facility location problem (MLUFLP) is presented. First improvement is achieved by better implementation of dynamic programming, which speeds up the running time of the overall GA implementation. Second improvement is hybridization of the genetic algorithm with the fast local search procedure designed specially for MLUFLP. The experiments were carried out on instances proposed in the literature which are modied standard single level facility location problem instances. Improved genetic algorithm reaches all known optimal and the best solutions from literature, but in much shorter time. Hybridization with local search improves several best-known solutions for large-scale MLUFLP instances, in cases when the optimal is not known. Overall running time of both proposed GA methods is signicantly shorter compared to previous GA approach.

An efficient method for single-facility location and path-connecting problems in a cell map

This article presents an efficient method for simultaneously finding both the Weber cell and optimal connective paths in a grid. As numerous barriers of arbitrary shape and weighted regions are distributed in the cell map of this research, the problem scenario is similar to working out a real-life facility location selection and path-routing problems in a geographical map. In this study, the Weber problem of finding a single-facility location from an accumulation cost table is generated by a grid wave propagation method (higher-geometry maze router). After finding the Weber point (cell), optimal connective paths with minimum total weighted cost are backtracked between the Weber location cell and the demand cells. This new computation algorithm with linear time and space complexity can be integrated as a spatial analytical function within GIS.

Integrated dynamic single-facility location and inventory planning problems

This article considers a class of dynamic single-article facility location problems in which the facility must determine order and inventory levels to meet the dynamic demands of the customers over a finite horizon. The motivating application of this class of problems is in military logistics and the decision makers in this area are not only concerned with the logistical costs of the facility but also with centering the facility among the customers in each time period in order to be able to provide other services. Both the location plan and inventory plan of the facility in the problem must be determined while considering these different metrics associated with the performance of these plans. Effective dynamic programming algorithms for this class of problem are provided for both of these metrics. These dynamic programming algorithms are utilized in order to construct the efficient frontier associated with these two metrics in polynomial time. Computational testing indicates that these algorithms can be used in planning activities for military logistics. [Supplemental materials are available for this article. Go to the publisher’s online edition of IIE Transactions for a worst-case example of constructing the efficient frontier.]

Serving regional demand in facility location

AbstractLocation modelling is employed in urban and regional planning to site facilities that provide services of some sort. Issues to be considered usually include the number of facilities to locate, where to site those facilities and how demand is to be served. Given the geographic nature of location problems, a key issue is how to represent facilities and demand in geographic space. Traditionally, spatial abstraction as discrete demand is assumed as it simplifies model formulation and reduces computational complexity. However, errors in derived solutions are likely not negligible, especially when demand varies continuously across a region. This paper discusses a single facility location problem that considers demand to be continuously distributed and allows a facility to be located anywhere in space, the continuous Weber problem. An approach for dealing with continuous demand is proposed that is integrated through geographical information system (GIS) functionality. Empirical results highlight the advantages of the developed approach and the importance of solution integration with GIS.

Nonlinear Programming Model For The Facility Location Problem In The Presence Of Arc-shaped Barrier

In this paper we consider the single facility location problem with respect to a given set of existing facilities in the presence of an arc-shaped barrier. A barrier is considered a region where neither facility location nor travelling is permitted. We present a mixed-integer nonlinear programming model for this single facility location problem. The objective of this problem is to locate this single facility such that the sum of the rectilinear distances from the facility to the demand points is minimized. Test problems are presented to illustrate the applicability of the proposed model.