Classical Facility Location Problem Research Articles

Optimizing the bioenergy industry infrastructure: Transportation networks and bioenergy plant locations

In the context of combating climate change and maintaining energy security, ambitious bioenergy development projects in emerging economies face considerable challenges, for example an overburdened bioenergy industry infrastructure due to the growing demand for bioenergy products. There are abundant studies on optimizing the bioenergy industry infrastructure. However, they fail to comprehensively simulate the interactions among the predominant actors of the infrastructure, especially the bioenergy plant operators in emerging economies. To fill this research gap, we develop a new dynamic agent-based model of optimized bioenergy industry infrastructure from the perspective of bioenergy plant operators. We then apply the model to Jiangsu Province of China to simulate the coordination of two types of bioenergy plants and project the optimal distribution of these plants and their corresponding transportation networks for the year of 2030. The model results suggest locating bioenergy plants closer to bioenergy feedstock source regions rather than to bioenergy products consumption sites, an answer to the classical facility location problem. A welfare analysis based on the extended model indicates that the biomass densification process aiming at mitigating the growing transport volumes incurred by the delivery of bulky bioenergy feedstock is not economically profitable in our case region. The experiences from this region further show that for emerging economies, a successful bioenergy industry infrastructure needs to take the benefits of smallholder farmers into consideration.

Dynamic Facility Location via Exponential Clocks

The dynamic facility location problem is a generalization of the classic facility location problem proposed by Eisenstat, Mathieu, and Schabanel to model the dynamics of evolving social/infrastructure networks. The generalization lies in that the distance metric between clients and facilities changes over time. This leads to a trade-off between optimizing the classic objective function and the “stability” of the solution: There is a switching cost charged every time a client changes the facility to which it is connected. While the standard linear program (LP) relaxation for the classic problem naturally extends to this problem, traditional LP-rounding techniques do not, as they are often sensitive to small changes in the metric resulting in frequent switches. We present a new LP-rounding algorithm for facility location problems, which yields the first constant approximation algorithm for the dynamic facility location problem. Our algorithm installs competing exponential clocks on the clients and facilities and connects every client by the path that repeatedly follows the smallest clock in the neighborhood. The use of exponential clocks gives rise to several properties that distinguish our approach from previous LP roundings for facility location problems. In particular, we use no clustering and we allow clients to connect through paths of arbitrary lengths . In fact, the clustering-free nature of our algorithm is crucial for applying our LP-rounding approach to the dynamic problem.

An approximation algorithm for k-facility location problem with linear penalties using local search scheme

In this paper, we consider an extension of the classical facility location problem, namely k-facility location problem with linear penalties. In contrast to the classical facility location problem, this problem opens no more than k facilities and pays a penalty cost for any non-served client. We present a local search algorithm for this problem with a similar but more technical analysis due to the extra penalty cost, compared to that in Zhang (Theoretical Computer Science 384:126–135, 2007). We show that the approximation ratio of the local search algorithm is \(2 + 1/p + \sqrt{3+ 2/p+ 1/p^2} + \epsilon \), where \(p \in {\mathbb {Z}}_+\) is a parameter of the algorithm and \(\epsilon >0\) is a positive number.

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function $$f_i(z)$$fi(z). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a ($$7.88+\epsilon $$7.88+∈)-approximation local search algorithm for this problem.

New Approximation Algorithms for the Unsplittable Capacitated Facility Location Problem

In this paper, we consider the Unsplittable (hard) Capacitated Facility Location Problem (UCFLP) with uniform capacities and present new approximation algorithms for it. This problem is a generalization of the classical facility location problem where each facility can serve at most u units of demand and each client must be served by exactly one facility. This problem is motivated by its applications in many practical problems including supply chain problems of indivisible goods (Verter in Foundations of location analysis, chapter 2. International series in operations research and management science. Springer, Berlin, 2011) and the assignment problem in the content distribution networks (Bateni and Hajiaghayi in Proceedings of the nineteenth annual ACM-SIAM symposium on discrete algorithms, pp 805---814, 2009). While there are several approximation algorithms for the soft capacitated version of this problem (in which one can open multiple copies of each facility) or the splittable version (in which the demand of each client can be divided to be served by multiple open facilities), there are very few results for the UCFLP. It is known that it is NP-hard to approximate this problem within any factor without violating the capacities. So we consider bicriteria $$(\alpha ,\beta )$$(?,β)-approximations where the algorithm returns a solution whose cost is within factor $$\alpha $$? of the optimum and violates the capacity constraints within factor $$\beta $$β. Shmoys et al. (Proceedings of the twenty-ninth annual ACM symposium on theory of computing, pp 265---274, 1997) were the first to consider this problem and gave a (9, 4)-approximation. Later results imply (O(1), 2)-approximations, however, no constant factor approximation is known with capacity violation of less than 2. We present a framework for designing bicriteria approximation algorithms for this problem and show two new approximation algorithms with factors (9, 3 / 2) and (29.315, 4 / 3). These are the first algorithms with constant approximation in which the violation of capacities is below 2. The heart of our algorithm is a reduction from the UCFLP to a restricted version of the problem. One feature of this reduction is that any $$(O(1),1+{\epsilon })$$(O(1),1+∈)-approximation for the restricted version implies an $$(O(1),1+{\epsilon })$$(O(1),1+∈)-approximation for the UCFLP and we believe our techniques might be useful towards finding such approximations or perhaps $$(f({\epsilon }),1+{\epsilon })$$(f(∈),1+∈)-approximation for the UCFLP for some function f. In addition, we present a quasi-polynomial time $$(1+\epsilon ,1+\epsilon )$$(1+∈,1+∈)-approximation for the (uniform) UCFLP in Euclidean metrics, for any constant $${\epsilon }>0$$∈>0.

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

We consider the facility location problem with submodular penalties (FLPSP) and the facility location problem with linear penalties (FLPLP), two extensions of the classical facility location problem (FLP). First, we introduce a general algorithmic framework for a class of covering problems with submodular penalties, extending the recent result of Geunes et al. (Math Program 130:85---106, 2011) with linear penalties. This framework leverages existing approximation results for the original covering problems to obtain corresponding results for their counterparts with submodular penalties. Specifically, any LP-based $$\alpha $$?-approximation for the original covering problem can be leveraged to obtain an $$\left( 1-e^{-1/\alpha }\right) ^{-1}$$1-e-1/?-1-approximation algorithm for the counterpart with submodular penalties. Consequently, any LP-based approximation algorithm for the classical FLP (as a covering problem) can yield, via this framework, an approximation algorithm for the counterpart with submodular penalties. Second, by exploiting some special properties of submodular/linear penalty function, we present an LP rounding algorithm which has the currently best $$2$$2-approximation ratio over the previously best $$2.375$$2.375 by Li et al. (Theoret Comput Sci 476:109---117, 2013) for the FLPSP and another LP-rounding algorithm which has the currently best $$1.5148$$1.5148-approximation ratio over the previously best $$1.853$$1.853 by Xu and Xu (J Comb Optim 17:424---436, 2008) for the FLPLP, respectively.

Solving the Bilevel Facility Location Problem under Preferences by a Stackelberg-Evolutionary Algorithm

This research highlights the use of game theory to solve the classical problem of the uncapacitated facility location optimization model with customer order preferences through a bilevel approach. The bilevel model provided herein consists of the classical facility location problem and an optimization of the customer preferences, which are the upper and lower level problems, respectively. Also, two reformulations of the bilevel model are presented, reducing it into a mixed-integer single-level problem. An evolutionary algorithm based on the equilibrium in a Stackelberg’s game is proposed to solve the bilevel model. Numerical experimentation is performed in this study and the results are compared to benchmarks from the existing literature on the subject in order to emphasize the benefits of the proposed approach in terms of solution quality and estimation time.

Uncapacitated facility location problem with self-serving demands

In classical uncapacitated facility location problems (UFLP) the goal is to satisfy requirements of some demand points by setting up some servers, among potential facility locations, such that the total cost including service costs and fixed costs are minimized. In this paper a generalization of UFLP is considered in which some demand points, called self-serving, could be served exclusively by a new server at that point. Numerical experiments show that near optimal solutions are achieved by the proposed method.

Hub location problems: A review of models, classification, solution techniques, and applications

Hub location problem (HLP) is a relatively new extension of classical facility location problems. Hubs are facilities that work as consolidation, connecting, and switching points for flows between stipulated origins and destinations. While there are few review papers on hub location problems, the most recent one (Alumur and Kara, 2008. Network hub location problems: The state of the art. European Journal of Operational Research, 190, 1–21) considers solely studies on network-type hub location models prior to early 2007. Therefore, this paper focuses on reviewing the most recent advances in HLP from 2007 up to now. In this paper, a review of all variants of HLPs (i.e., network, continuous, and discrete HLPs) is provided. In particular, mathematical models, solution methods, main specifications, and applications of HLPs are discussed. Furthermore, some case studies illustrating real-world applications of HLPs are briefly introduced. At the end, future research directions and trends will be presented.

Using Voronoi diagrams to solve a hybrid facility location problem with attentive facilities

In this paper, we introduce a novel class of facility location problems, and propose solutions based on Voronoi diagrams. Our solutions locate a set of facilities on a two dimensional space, with respect to a set of dynamic demand. The information about these demand is gathered through modifications of the overall system, into a central decision unit. This influences our objective of minimizing the total loss function. Considering a continuous space and discrete time, facilities are assigned to meet demands in each time cycle. Two distinct approaches are proposed and thoroughly studied, followed by a case study. We call our main algorithm Reactive Agent Dynamic Voronoi Diagram Facility Spread. We also test our solutions empirically through a set of experiments. Considering n and p to be the number of demand points, and the number of facilities in hand, respectively, the time complexity of the algorithm is O(c(n2+p)logn) for a complete run of c cycles.

A unified dual-fitting approximation algorithm for the facility location problems with linear/submodular penalties

In this paper, we study two variants of the classical facility location problem, namely, the facility location problem with linear penalties (FLPLP) and the facility location problem with submodular penalties (FLPSP), respectively. We give a unified dual-fitting based approximation algorithm for these two problems, yielding approximation ratios 2 and 3 respectively.

Exact algorithms for integrated facility location and production planning problems

We consider a class of facility location problems with a time dimension, which requires assigning every customer to a supply facility in each of a finite number of periods. Each facility must meet all assigned customer demand in every period at a minimum cost via its production and inventory decisions. We provide exact branch-and-price algorithms for this class of problems and several important variants. The corresponding pricing problem takes the form of an interesting class of production planning and order selection problems. This problem class requires selecting a set of orders that maximizes profit, defined as the revenue from selected orders minus production-planning-related costs incurred in fulfilling the selected orders. We provide polynomial-time dynamic programming algorithms for this class of pricing problems, as well as for generalizations thereof. Computational testing indicates the advantage of our branch-and-price algorithm over various approaches that use commercial software packages. These tests also highlight the significant cost savings possible from integrating location with production and inventory decisions and demonstrate that the problem is rather insensitive to forecast errors associated with the demand streams. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 419-436, 2011

An approximation algorithm for the k-level capacitated facility location problem

We consider the k-level capacitated facility location problem (k-CFLP), which is a natural variant of the classical facility location problem and has applications in supply chain management. We obtain the first (combinatorial) approximation algorithm with a performance factor of $k+2+\sqrt{k^{2}+2k+5}+\varepsilon$ (?>0) for this problem.

Single facility collection depots location problem in the plane

In this paper we consider an extension of the classical facility location problem where besides n weighted customers, a set of p collection depots are also given. In this setting the service of a customer consists of the travel of a server to the customer and return back to the center via a collection depot. We have analyzed the problem and showed that the collection depots problem using the Euclidean metric can be transformed to O ( p 2 n 2 ) number of different classical facility location problems and this bound is tight. We then show the existence of small coresets for these problems. These coresets are then used to provide ( 1 + ϵ ) -factor approximation algorithms which have linear running times for fixed customer weights and ϵ.

An improved approximation algorithm for uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the classical facility location problem called uncapacitated facility location problem with penalties (UFLWP for short) in which each client is either assigned to an opened facility or rejected by paying a penalty. The UFLWP problem has been effectively used to model the facility location problem with outliers. Three constant approximation algorithms have been obtained (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001; Jain et al. in J. ACM 50(6):795–824, 2003; Xu and Xu in Inf. Process. Lett. 94(3):119–123, 2005), and the best known performance ratio is 2. The only known hardness result is a 1.463-inapproximability result inherited from the uncapacitated facility location problem (Guha and Khuller in J. Algorithms 31(1):228–248, 1999). In this paper, We present a 1.8526-approximation algorithm for the UFLWP problem. Our algorithm significantly reduces the gap between known performance ratio and the inapproximability result. Our algorithm first enhances the primal-dual method for the UFLWP problem (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001) so that outliers can be recognized more efficiently, and then applies a local search heuristic (Charikar and Guha in Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pp. 378–388, 1999) to further reduce the cost for serving those non-rejected clients. Our algorithm is simple and can be easily implemented.

Designing minimum-cost recycling collection networks with required throughput

The problem of designing minimum-cost recycling networks with required throughput is a subset of the broader class of facility location problems in which the recycler wishes to determine the optimal number and location of receiving centers as well as the correct financial incentive to be offered in order to stimulate collection of used or unrecoverable products to a degree required for regulatory reasons or otherwise. This paper presents a conceptual framework, an analytical model, and a three-stage algorithmic solution for this problem. An illustrative case study in the recycling of unrecoverable tires in southern states of Brazil is presented.

Mixed-integer nonlinear programming: Some modeling and solution issues

We examine various aspects of modeling and solution via mixed-integer nonlinear programming (MINLP). MINLP has much to offer as a powerful modeling paradigm. Recently, significant advances have been made in MINLP solution software. To fully realize the power of MINLP to solve complex business optimization problems, we need to develop knowledge and expertise concerning MINLP modeling and solution methods. Some of this can be drawn from conventional wisdom of mixed-integer linear programming (MILP) and nonlinear programming (NLP), but theoretical and practical issues exist that are specific to MINLP. This paper discusses some of these, concentrating on an aspect of a classical facility location problem that is well-known in the MILP literature, although here we consider a nonlinear objective function.

Stochastic p-robust location problems

The two most widely considered measures for optimization under uncertainty are minimizing expected cost and minimizing worst-case cost or regret. In this paper, we present a novel robustness measure that combines the two objectives by minimizing the expected cost while bounding the relative regret in each scenario. In particular, the models seek the minimum-expected-cost solution that is p-robust; i.e., whose relative regret is no more than 100p% in each scenario. We present p-robust models based on two classical facility location problems. We solve both problems using variable splitting, with the Lagrangian subproblem reducing to the multiple-choice knapsack problem. For many instances of the problems, finding a feasible solution, and even determining whether the instance is feasible, is difficult; we discuss a mechanism for determining infeasibility. We also show how the algorithm can be used as a heuristic to solve minimax regret versions of the location problems.

A Family of Location Models for Multiple‐Type Discrete Dispersion

One of the defining objectives in location science is to maximize dispersion. Facilities can be dispersed for a wide variety of purposes, including attempts to optimize competitive market advantage, disperse negative impacts, and optimize security. With one exception, all of the extant dispersion models consider only one type of facility, and ignore problems where multiple types of facilities must be located. We provide examples where multiple‐type dispersion is appropriate and based on this develop a general class of facility location problems that optimize multiple‐type dispersion. This family of models expands on the previously formulated definitions of dispersion for single types of facilities, by allowing the interactions among different types of facilities to determine the extent to which they will be spatially dispersed. We provide a set of integer‐linear programming formulations for the principal models of this class and suggest a methodology for intelligent constraint elimination. We also present results of solving a range of multiple‐type dispersion problems optimally and demonstrate that only the smallest versions of such problems can be solved in a reasonable amount of computer time using general‐purpose optimization software. We conclude that the family of multiple‐type dispersion models provides a more comprehensive, flexible, and realistic framework for locating facilities where weighted distances should be maximized, when compared with the special case of locating only a single type of facility.

A constant factor approximation algorithm for the fault-tolerant facility location problem

We consider a generalization of the classical facility location problem, where we require the solution to be fault-tolerant. In this generalization, every demand point j must be served by r j facilities instead of just one. The facilities other than the closest one are “backup” facilities for that demand, and any such facility will be used only if all closer facilities (or the links to them) fail. Hence, for any demand point, we can assign nonincreasing weights to the routing costs to farther facilities. The cost of assignment for demand j is the weighted linear combination of the assignment costs to its r j closest open facilities. We wish to minimize the sum of the cost of opening the facilities and the assignment cost of each demand j. We obtain a factor 4 approximation to this problem through the application of various rounding techniques to the linear relaxation of an integer program formulation. We further improve the approximation ratio to 3.16 using randomization and to 2.41 using greedy local-search type techniques.