Planar point-objective location problems with nonconvex constraints: A geometrical construction

The planar point-objective location problem has attracted considerable interest among Location Theory researchers. The result has been a number of papers giving properties or algorithms for particular instances of the problem. However, most of these results are only valid when the feasible region where the facility is to be located is the whole space ², which is a rather inaccurate approximation in many real world location problems.In this paper, the feasible region is allowed to be any closed, not necessarily convex, setS in ². The special structure of this nonconvex vector-optimization problem is exploited, leading to a geometrical resolution procedure when the feasible regionS can be decomposed into a finite number of (not necessarily disjoint) polyhedra.     

On the lexicographic minimax approach to location problems

When locating public facilities, the distribution of travel distances among the service recipients is an important issue. It is usually tackled with the minimax (center) solution concept. The minimax solution concept, despite the most commonly used in the public sector location models, is criticized as it does not comply with the major principles of the efficiency and equity modeling. In this paper we develop a concept of the lexicographic minimax solution (lexicographic center) being a refinement of the standard minimax approach to location problems. We show that the lexicographic minimax approach complies with both the Pareto-optimality (efficiency) principle (crucial in multiple criteria optimization) and the principle of transfers (essential for equity measures) whereas the standard minimax approach may violate both these principles. Computational algorithms are developed for the lexicographic minimax solution of discrete location problems.     

A reduction result for location problems with polyhedral barriers

In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facilities where a set of polyhedral barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the median and center objective functions). An exact algorithm and a heuristic solution procedure based on this reduction result are developed.     

A heuristic for large-sizep-median location problems with application to school location

This paper initially proposes a heuristic algorithm for thep-median problem designed for large weighted graphs. The problem is approached through the construction ofp trees whose shapes are progressively modified according to successive tests over the stability of their roots and vertices. The algorithm seems promising because: (i) on a regular PC it can handle problems of the order of 500 vertices, while the mainframe version goes indefinitely further, (ii) contrary to what normally would be expected, execution times seem to be inversely proportional top, and even for large problems, they may be reasonable, especially ifp is large relative to the number of vertices, and (iii) it produces solutions of good quality and in most of the cases studied, it outperforms the traditional heuristic of Teitz and Bart. A real application of the algorithm embedded in a methodology to evaluate the location of 85 public schools, among 389 possible vertices, in the metropolitan area of Rio de Janeiro is reported. Results confirmed the conjecture of poor location and the procedure was able to identify several micro-regions simply void of schools. The methodology is being well received by the education authorities and its extension to the whole metropolitan area is being considered.     

Dynamic location problems

A class of dynamic location problems is introduced. The relationship between a static problem and its corresponding dynamic one is studied. We concentrate on two types of dynamic problems. The first is the global optimization problem, in which one looks for the all-times optimum. The second is the steady-state problem in which one seeks to determine the steady-state behavior of the system if one exists. General approaches to these problems are discussed.Supported in part by the National Science Foundation under grants MCS-8300984, ECS-8218181 and ECS-8121741.     

A comparative study of approaches to dynamic location problems

We compare the performance of seven approximate methods for locating new capacity over time to minimize the total discounted costs of meeting growing demands at several locations. Comparisons are based on results for two industrial planning problems from India, and are given for both discrete-time and continuous-time frameworks. We also discuss strategies for combining different methods into possibly more effective hybrid approaches.     

Multicriteria Planar Ordered Median Problems

In this paper, we deal with the determination of the entire set of Pareto solutions of location problems involving Q general criteria. These criteria include median, center, or centdian objective functions as particular instances. We characterize the set of Pareto solutions of all these multicriteria problems for any polyhedral gauge. An efficient algorithm is developed for the planar case and its complexity is established. Extensions to the nonconvex case are also considered. The proposed approach is more general than previously published approaches to multicriteria location problems.The research of the third and fourth authors was partially supported by Grants BFM2001-2378, BFM2001-4028, BFM2004-0909, and HA2003-0121.     

A D.C. optimization method for single facility location problems

The single facility location problem with general attraction and repulsion functions is considered. An algorithm based on a representation of the objective function as the difference of two convex (d.c.) functions is proposed. Convergence to a global solution of the problem is proven and extensive computational experience with an implementation of the procedure is reported for up to 100,000 points. The procedure is also extended to solve conditional and limited distance location problems. We report on limited computational experiments on these extensions.This research was supported in part by the National Science Foundation Grant DDM-91-14489.     

A note on two source location problems

We consider Source Location ($\mathsf{SL}$) problems: given a capacitated network $G=(V,E)$, cost $c(v)$ and a demand $d(v)$ for every $v\in V$, choose a min-cost $S\subseteq V$ so that $\lambda (v,S)⩾d(v)$ holds for every $v\in V$, where $\lambda (v,S)$ is the maximum flow value from v to S. In the directed variant, we have demands $d^{\mathit{in}}(v)$ and $d^{\mathit{out}}(v)$ and we require $\lambda (S,v)⩾d^{\mathit{in}}(v)$ and $\lambda (v,S)⩾d^{\mathit{out}}(v)$. Undirected $\mathsf{SL}$ is (weakly) NP-hard on stars with $r(v)=0$ for all v except the center. But, it is known to be polynomially solvable for uniform costs and uniform demands. For general instances, both directed an undirected $\mathsf{SL}$ admit a $(\ln D+1)$-approximation algorithms, where D is the sum of the demands; up to constant this is tight, unless P = NP. We give a pseudopolynomial algorithm for undirected $\mathsf{SL}$ on trees with running time $O(|V|{\Delta }^3)$, where $\Delta ={\max }_{v\in V}d(v)$. This algorithm is used to derive a linear time algorithm for undirected $\mathsf{SL}$ with $\Delta ⩽3$. We also consider the Single Assignment Source Location ($\mathsf{SASL}$) where every $v\in V$ should be assigned to a single node $s(v)\in S$. While the undirected $\mathsf{SASL}$ is in P, we give a $(\ln |V|+1)$-approximation algorithm for the directed case, and show that this is tight, unless P = NP.     

Uncertain models for single facility location problems on networks

In practical location problems on networks, the vertex demand is usually non-deterministic. This paper employs uncertainty theory to deal with this non-deterministic factor in single facility location problems. We first propose the concepts of satisfaction degree for both vertices and the whole network, which are used to evaluate products assignment. Based on different network satisfaction degree, two models are constructed. The solution to these models is based on Hakimi’s results, and some examples are given to illustrate these models.     

A new approach to agglomeration problems

In this paper, we study a location problem with positive externalities. We define a new transferable utility game, considering there is no restriction on the transfer of benefits between firms. We prove that the core of this game is non-empty, provide an expression for it, and an axiomatic characterization. We also study several core allocations, selected by means of a certain bankruptcy problem.     

Multiple voting location problems

The facility voting location problems arise from the application of criteria derived from the voting processes concerning the location of facilities. The multiple location problems are those location problems in which the alternative solutions are sets of points. This paper extends previous results and notions on single voting location problems to the location of a set of facility points. The application of linear programming techniques to solve multiple facility voting location problems is analyzed. We propose an algorithm to solve Simpson multiple location problems from which the solution procedures for other problems are derived.     

Multiple criteria facility location problems: A survey

This paper provides a review on recent efforts and development in multi-criteria location problems in three categories including bi-objective, multi-objective and multi-attribute problems and their solution methods. Also, it provides an overview on various criteria used. While there are a few chapters or sections in different location books related to this topic, we have not seen any comprehensive review papers or book chapter that can cover it. We believe this paper can be used as a complementary and updated version.     

A memetic algorithm for multi-objective dynamic location problems

This paper describes a new multiobjective interactive memetic algorithm applied to dynamic location problems. The memetic algorithm integrates genetic procedures and local search. It is able to solve capacitated and uncapacitated multi-objective single or multi-level dynamic location problems. These problems are characterized by explicitly considering the possibility of a facility being open, closed and reopen more than once during the planning horizon. It is possible to distinguish the opening and reopening periods, assigning them different coefficient values in the objective functions. The algorithm is part of an interactive procedure that asks the decision maker to define interesting search areas by establishing limits to the objective function values or by indicating reference points. The procedure will be applied to some illustrative location problems.     

A note on the minmax regret centdian location on trees

The minmax regret optimization model of the doubly weighted centdian location on trees is considered. Assuming that both types of weights, demands and relative importance of the customers, are partially known through interval estimates, an exact algorithm of complexity O(n³logn) is derived. This bound is improved in some special cases.     

Outcomes of voting and planning in single facility location problems

For a given set of users located at the vertices of a network N, we consider the location of a single facility. Two different decision making procedures, voting among the users and minimizing total distance travelled by the users, are compared.Provided that a voting solution exists, it is shown that the two solutions sets coincide if N is a so-called cactus. For general networks, the outcomes of the two procedures are compared in terms of the cyclic structure of N and the number of users.     

Approximate formulas of average distances associated with regions and their applications to location problems

This study is concerned with the problem of measuring average distances between two points in two different coplanar regions. The objectives are: (1) to derive the approximated average distances associated with circular regions and to check their accuracy; and (2) to apply these approximated distances to location problems. Results show that the simple approximate formulas are accurate and useful. The approximated average distances can be applied to the analyses of varied kinds of movement phenomena in cities.     

Allocation search methods for a generalized class of location–allocation problems

We consider a generalized class of location–allocation problems, in which N new facilities are to be located in the plane with respect to M objects. Each object is associated with a convex cost function, specifying the expenses for serving the object from any location in the plane.     

A homological approach to representations of algebras I: The wild case

We prove that the pure global dimension of a polynomial ring over an integral domain k in a finite or countable number n?2 of commuting (non-commuting, resp.) variables is t + 1, provided $\text{|k| = ?}{}_{\mathrm{t}}$. As an application, we determine the pure global dimension of wild algebras of quiver type, also (in case k is an algebraically closed field) of the wild local and wild commutative algebras of finite k-dimension.