Efficient algorithms for center problems in cactus networks

Efficient algorithms for solving the center problems in weighted cactus networks are presented. In particular, we have proposed the following algorithms for the weighted cactus networks of size n$n$: an O(nlogn)$O(nlogn)$ time algorithm to solve the 1-center problem, and an O(nlog³n)$O\left(n{log}^{3}n\right)$ time algorithm to solve the weighted continuous 2-center problem. We have also provided improved solutions to the general p$p$-center problems in cactus networks. The developed ideas are then applied to solve the obnoxious 1-center problem in weighted cactus networks.     

Covering problems in facility location: A review

In this study, we review the covering problems in facility location. Here, besides a number of reviews on covering problems, a comprehensive review of models, solutions and applications related to the covering problem is presented after Schilling, Jayaraman, and Barkhi (1993). This survey tries to review all aspects of the covering problems by stressing the works after Schilling, Jayaraman, and Barkhi (1993). We first present the covering problems and then investigate solutions and applications. A summary and future works conclude the paper.     

Competitive facility location: the Voronoi game

We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins. We present a winning strategy for the second player, where the arena is a circle or a line segment. We permit variations where players can play more than one point at a time, and show that the first player can ensure that the second player wins by an arbitrarily small margin.     

Modeling health care facility location for moving population groups

Locating public services for nomadic population groups is a difficult challenge as the locations of the targeted populations seasonally change. In this paper, the population groups are assumed to occupy different locations according to the time of the year, i.e., winter and summer. A binary integer programming model is formulated to determine the optimal number and locations of primary health units for satisfying a seasonally varying demand. This model is successfully applied to the actual locations of 17 seasonally varying nomadic groups in the Middle East. Computational tests are performed on different versions of the model in order analyze the tradeoffs among different performance measures.     

Use of distributed data sources in facility location

Facility location decisions are usually determined by cost and coverage related factors although empirical studies show that such factors as infrastructure, labor conditions and competition also play an important role in practice. The objective of this paper is to develop a multi-objective facility location model accounting for a wide range of factors affecting decision-making. The proposed model selects potential facilities from a set of pre-defined alternative locations according to the number of customers, the number of competitors and real-estate cost criteria. However, that requires large amount of both spatial and non-spatial input data, which could be acquired from distributed data sources over the Internet. Therefore, a computational approach for processing input data and representation of modeling results is elaborated. It is capable of accessing and processing data from heterogeneous spatial and non-spatial data sources. Application of the elaborated data gathering approach and facility location model is demonstrated using an example of fast food restaurants location problem.     

Multi-dimensional dynamic facility location and fast computation at query points

We present O(nlogn) time algorithms for the minimax rectilinear facility location problem in R¹ and R². The algorithms enable, once they terminate, computing the cost of any given query point in O(logn) time. Based on these algorithms, we develop a preprocessing procedure which enables solving the following two problems: Fast computation of the cost of any query point in Rd, and fast solution for the dynamic location problem in R² (namely, in the presence of an additional facility). Finally, we show that the preprocessing always gives a bound on the optimal value, which allows us in many cases to find the optimum fast (for both the traditional and the dynamic location problems in Rd for any d).     

An integrated approach to facility location problems

An integrated analysis approach to facility location problems is described. The approach is based on integrating analytical location models and a multicriteria decision model.     

Planar geometric location problems

We present anO(n ² log³ n) algorithm for the two-center problem, in which we are given a setS ofn points in the plane and wish to find two closed disks whose union containsS so that the larger of the two radii is as small as possible. We also give anO(n ² log⁵ n) algorithm for solving the two-line-center problem, where we want to find two strips that coverS whose maximum width is as small as possible. The best previous solutions of both problems requireO(n ³) time.Pankaj Agarwal has been supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), an NSF Science and Technology Center, under Grant STC-88-09648. Micha Sharir has been supported by the Office of Naval Research under Grants N00014-89-J-3042 and N00014-90-J-1284, by the National Science Foundation under Grant CCR-89-01484, by DIMACS, and by grants from the US-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary version of this paper has appeared inProceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 449–458.     

Capacitated facility location problem with general setup cost

This paper presents an extension of the capacitated facility location problem (CFLP), in which the general setup cost functions and multiple facilities in one site are considered. The setup costs consist of a fixed term (site setup cost) plus a second term (facility setup costs). The facility setup cost functions are generally non-linear functions of the size of the facility in the same site. Two equivalent mixed integer linear programming (MIP) models are formulated for the problem and solved by general MIP solver. A Lagrangian heuristic algorithm (LHA) is also developed to find approximate solutions for this NP-hard problem. Extensive computational experiments are taken on randomly generated data and also well-known existing data (with some necessary modifications). The detailed results are provided and the heuristic algorithm is shown to be efficient.     

Solving the uncapacitated facility location problem using tabu search

A tabu search heuristic procedure is developed to solve the uncapacitated facility location problem. Tabu search is used to guide the solution process when evolving from one solution to another. A move is defined to be the opening or closing of a facility. The net cost change resulting from a candidate move is used to measure the attractiveness of the move. After a move is made, the net cost change of a candidate move is updated from its old value. Updating, rather than re-computing, the net cost changes substantially reduces computation time needed to solve a problem when the problem is not excessively large. Searching only a small subset of the feasible solutions that contains the optimal solution, the procedure is computationally very efficient. A computational experiment is conducted to test the performance of the procedure and computational results are reported. The procedure can easily find optimal or near optimal solutions for benchmark test problems from the literature. For randomly generated test problems, this tabu search procedure not only obtained solutions completely dominating those obtained with other heuristic methods recently published in the literature but also used substantially less computation time. Therefore, this tabu search procedure has advantage over other heuristic methods in both solution quality and computation speed.     

An LP rounding algorithm for approximating uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the facility location problem called uncapacitated facility location problem with penalties (UFLWP, for short) in which each client can be either assigned to some opened facility or rejected by paying a penalty. Existing approaches [M. Charikar, S. Khuller, D. Mount, G. Narasimhan, Algorithms for facility location problems with outliers, in: Proc. Symposium on Discrete Algorithms, 2001, p. 642] and [K. Jain, M. Mahdian, E. Markakis, A. Saberi, V. Vazirani, Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP, J. ACM 50 (2003) 795] for this variant of facility location problem are all based on primal-dual method. In this paper, we present an efficient linear programming (LP) rounding based approach to show that LP rounding techniques are equally capable of solving this variant of facility location problem. Our algorithm uses a two-phase filtering technique (generalized from Lin and Vitter's [?-approximation with minimum packing constraint violation, in: Proc. 24th Annual ACM Symp. on Theory of Computing, 1992, p. 771]) to identify those to-be-rejected clients and open facilities for the remaining ones. Our approach achieves an approximation ratio of 2+2/e (≈2.736) which is worse than the best approximation ratio of 2 achieved by the much more sophisticated dual fitting technique [K. Jain, M. Mahdian, E. Markakis, A. Saberi, V. Vazirani, Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP, J. ACM 50 (2003) 795], but better than the approximation ratio of 3 achieved by the primal-dual method [M. Charikar, S. Khuller, D. Mount, G. Narasimhan, Algorithms for facility location problems with outliers, in: Proc. Symposium on Discrete Algorithms, 2001, p. 642]. Our algorithm is simple, natural, and can be easily integrated into existing LP rounding based algorithms for facility location problem to deal with outliers.     

MIP models for connected facility location: A theoretical and computational study

This article comprises the first theoretical and computational study on mixed integer programming (MIP) models for the connected facility location problem (ConFL). ConFL combines facility location and Steiner trees: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facility opening costs and the assignment costs need to be minimized.     

Risk management in uncapacitated facility location models with random demands

In this paper we consider a location-optimization problem where the classical uncapacitated facility location model is recast in a stochastic environment with several risk factors that make demand at each customer site probabilistic and correlated with demands at the other customer sites. Our primary contribution is to introduce a new solution methodology that adopts the mean–variance approach, borrowed from the finance literature, to optimize the “Value-at-Risk” (VaR) measure in a location problem. Specifically, the objective of locating the facilities is to maximize the lower limit of future earnings based on a stated confidence level. We derive a nonlinear integer program whose solution gives the optimal locations for the p facilities under the new objective. We design a branch-and-bound algorithm that utilizes a second-order cone program (SOCP) solver as a subroutine. We also provide computational results that show excellent solution times on small to medium sized problems.     

Stochastic facility location with general long-run costs and convex short-run costs

This paper addresses the problem of minimizing the expected cost of locating a number of single product facilities and allocating uncertain customer demand to these facilities. The total costs consist of two components: firstly linear transportation cost and secondly the costs of investing in a facility as well as maintaining and operating it. These facility costs are general and non-linear in shape and could express both changing economies of scale and diseconomies of scale. We formulate the problem as a two-stage stochastic programming model where both demand and short-run costs may be uncertain at the investment time. We use a solution method based on Lagrangean relaxation, and show computational results for a slaughterhouse location case from the Norwegian meat industry.     

The heuristic concentration-integer and its application to a class of location problems

We propose a new metaheuristic called heuristic concentration-integer (HCI). This metaheuristic is a modified version of the heuristic concentration (HC), oriented to find good solutions for a class of integer programming problems, composed by problems in which p   elements must be selected from a larger set, and each element can be selected more than once. These problems are common in location analysis. The heuristic is explained and general instructions for rewriting integer programming formulations are provided, that make the application of HCI to these problems easier. As an example, the heuristic is applied to the maximal availability location problem (MALP), and the solutions are compared to those obtained using linear programming with branch and bound (LP+B&B)$(\mathrm{LP}+\mathrm{B}\&\mathrm{B})$. For one-third of the instances of MALP, LP+B&B$\mathrm{LP}+\mathrm{B}\&\mathrm{B}$ can be allowed to run until the computer is out of memory without termination, while HCI can find good solutions to the same instances in a reasonable time. In one such case, LP-IP was allowed to run for nearly 100 times longer than HCI and HCI still found a better solution. Furthermore, HCI found the optimal solution in 33.3% of cases and had an objective value gap of less than 1% in 76% of cases. In 18% of the cases, HCI found a solution that is better than LP+B&B. Therefore, in cases where LP+B&B$\mathrm{LP}+\mathrm{B}\&\mathrm{B}$ is unreasonable due to time or memory constraints, HCI is a valuable tool.     

A simple filter-and-fan approach to the facility location problem

The design of effective neighborhood search procedures is a primary issue for the performance of local search and advanced metaheuristic algorithms. Several recent studies have focused on the development of variable depth neighborhoods that generate sequences of interrelated elementary moves to create more complex compound moves. These methods are chiefly conceived to produce an adaptive search as the number of elementary moves in a compound move may vary from one iteration to another depending on the state of the search. The objective is to achieve this goal with modest computational effort. Although ejection chain methods are currently the most advanced methods in this domain, they usually require more complex implementations. The filter-and-fan (F&F) method appears as an alternative to ejection chain methods allowing for the creation of compound moves based on an effective tree search design.     

A review of hierarchical facility location models

In this study, we review the hierarchical facility location models. Although there have been a number of review papers on hierarchical facility problems, a comprehensive treatment of models has not been provided since the mid-80s. This review fills the gap in the literature. We first classify the hierarchical facility problems according to the features of systems studied, which are based on flow pattern, service availability at each level of the hierarchy, and spatial configuration of services in addition to the objectives to locate facilities. We then investigate the applications, mixed integer programming models, and solution methods presented for the problem. With an overview of the selected works, we consolidate the main results in the literature.     

A facility location model for logistics systems including reverse flows: The case of remanufacturing activities

After a brief introduction to the basic concepts of reverse logistics, we present a two-level location problem with three types of facility to be located in a specific reverse logistics system, named a Remanufacturing Network (RMN). For this problem, we propose a 0–1 mixed integer programming model, in which we simultaneously consider “forward” and “reverse” flows and their mutual interactions. An algorithm based on Lagrangian heuristics is developed and the model is tested on data adapted from classical test problems.